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I .1: ......I1...»... .0. . . :2 y .i. ’4‘ f. ‘\ ‘ This is to certify that the thesis entitled AN INTRODUCTION TO THE HISTORY OF FOURIER SERIES AND THE THEORY OF INTEGRATION presented by Dean Webb has been accepted towards fulfillment of the requirements for Ph .D . Mathemat ics degree in M 441tkem/g/ .7 Major professor Date June 1, 1971 0-7639 VC‘IRIE? Consideratior fix} suggests many 1 :5 analysis. Exampl atablishing the ex‘.’ renesentatlon by Fn’ t'13: a function be r Stuéy of thee :: functions has pr: _ _— ——___ —— .—_ — alilution of analvsi 4.:1, ““9““? i8 particzl . o! I. . fit“ I? 0f fhmctions ABSTRACT AN INTRODUCTION TO THE HISTORY OF FOURIER SERIES AND THE THEORY OF INTEGRATION BY Dean Webb Consideration of the Fourier series corresponding to a function fo) suggests many interesting questions which relate to basic concepts of analysis. Examples are the generality of the integrals admitted in establishing the existence of the Fourier coefficients, uniqueness of representation by Fourier series, and general forms of convergence sudh that a function be represented by its corresponding Fourier series. Study of these and other aspects of Fourier series representation of functions has profoundly influenced the nature and direction of the evolution of analysis and, in turn, has been influenced by it. This influence is particularly apparent in the history of development of the theory of functions of a real variable. This paper is an exposition of the history of relationships between the development of successively more general conceptions of definite integral from Cauchy to Lebesgue and the concomitant evolution of»a theory of Fourier series. The unifying theme of the paper is the study Of Fourier series representation of functions and in particular, the 8march for general sufficient conditions for such representation. The study begins with the work of the first mathematicians to con- 81der the problem of trigonometric series representation of functions, d'Aletnbert, Euler, D. Bernoulli and Lagrange. Fourier's contributions are: described and I conjecture a relationship between Fourier's work and Larry's definition 11:: for convergence are presented, as is gzve: by Riemann in ‘ gazim of necessary mic series. The theories 53313:, Sorel and Le. liqueness of represv I I 0.. ," Mafia: ed Cantor's c: :zegration is descr: L‘s conception of de: sztzicient condition a. 'I o. presented as well ‘43“? 0f the Rieé ‘94:... “we convergence 2 tifunction f e L [ larleson's assertion 12-41:: or come?" th the hiSt :ces ‘n‘hiCh motivate i’eztlfied, and both .Webb -Cauchy's definition of definite integral. Dirichlet's sufficient condi- tion for convergence of Fourier series and modern conception of function are presented, as is the more general definition of definite integral given by Riemann in his Habilitationsschrift, which is devoted to investi- gation of necessary and sufficient conditions for convergence of trigono- metric series. The theories of measure created by Stolz, Cantor, Harnack, Peano. Jordan, Borel and Lebesgue are studied; efforts to solve the problem of uniqueness of representation by trigonometric series is shown to have initiated Cantor's creation of his theory of sets. Lebesgue's theory of integration is described. as are the first applications by Lebesgue of his conception of definite integral to study of Fourier series. Fatou's sufficient condition for convergence almost everywhere of Fourier series is presented as well as the work of Riesz and Fischer which led to the discovery of the Riesz~Fischer theorem. Finally, Lusin's conjecture re- garding convergence almost everywhere of the Fourier series corresponding to a function f e L2[O. 2n] is studied, and the paper concludes with Carleson's assertion of the validity of Lusin's very general sufficient condition for convergence of Fourier series. Both the historical sequence of events and the initiating influ- ences which motivated the work of contributing mathematicians are identified, and both influences and events are expressed in the words of the men who helped create the theory. The latter is accomplished by excerpting passages from the original memoirs. Several insist ‘rv‘tich the proble: beezsuccessivelv ge: sziztion, deficienci- :r;‘:.':ed to the tree ism secondary to existent controver- ::::ions of a real ‘ Webb Several insights derive from such study. These include the manner in which the problem of Fourier series representation of functions has been successively generalized in order to render it capable of partial solution, deficiencies in the work of many of the mathematicians who con- tributed to the creation of the theory (deficiencies which are, of course, clearly secondary to the accomplishments of these men), and finally, the persistent controversy which accompanied the evolution of the theory of functions of a real variable. AN INTRODUCTION TO THE HISTORY OF FOURIER SERIES AND THE THEORY OF INTEGRATION by fiwasfi- Deannwébb A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1971 To Marilyn A series c 35 c are constan‘ fzction Whose do: the integrals -) His. series An Introduction to the History of Fourier Series and the Theory of Integration Preface A series of the form c + n§1(an cosnx + businnx) where the an,bfi, and c are constants is called a trigonometric series. If f(x) is a function whose domain of definition is the interval -n §_x §_n and if the integrals n l) a - l-}f(x) cosnx n - O 1 2 n ."4 9 999°°° and 'fl 1 n 2) bn - :‘J f(x) sinnx, n - 1,2,3,... 1T exist, then the Fourier series corresponding to f(x) is the trigonometric series 3) a0 + Z (a cosnx + b sinnx). -- n-l n n 2 The constants an and bn are the Fourier coefficients of f(x). This definition requires only that the integrals l) and 2) exist in order that a Fourier series correspond to f(x); there is no requirement that the series 3) converge in any sense. Indeed it is not apparent that the series 3) should converge in the interval [-n,n] nor if it does converge at a point x that its sum should be f(x), i.e., that it should represent f at x. Since the Fourier series corresponding to f(x) does not, in general, represent f(x), we will use the symbol "N" to denote the cor- respondence between f and its Fourier series, f(x) N a0 + Z (ancosnx + businnx). -3' n-l Considers: fix) suggests car: of analysis. Exa: un'.1 ‘ { 25.... song the 4 fits of converge: 3v.- 49 ..s correspond a::' continuity, 6.: series in interva ii a trig: tier. \ “5 5W f(x, .‘v‘l ‘ kuocic with pa ii Consideration of the Fourier series corresponding to a function f(x) suggests many interesting problems which relate to basic concepts of analysis. Examples are the generality of the integrals admitted in establishing the existence of the Fourier coefficients 1) and 2), general forms of convergence of the series 3) such that a function be represented by its corresponding Fourier series, uniqueness of series representation, and continuity, differentiability and integrability prOperties of Fourier series in intervals of convergence. If a trigonometric series converges (in the usual sense) for all x, then its sum f(x) has the property f(x + 2n) - f(x), that is, f(x) is periodic with period 2n. Periodic functions are encountered in the study of a great variety of physical problems which feature periodic phenomena such as vibration or wave motion. As is known, every periodic function satisfying certain general conditions can be represented by a trigonometric series1 and most trigonometric series encountered in applied problems are Fourier series. Thus Fourier series are of both theoretical and applied interest. The pure and applied aspects of study of Fourier series have moti- vated the work of several generations of mathematicians whose efforts in this and related areas have led to the creation of a theory of Fourier series. The creation of this theory has profoundly influenced the nature and direction of the evolution of analysis and, in turn, been influenced by it. These influences are particularly apparent in the deve10pment of a theory of functions of a real variable. 1See Appendix A for one form of this assertion. 1m thee?)Y 5381573” dur years to Com generally AC of Dirichlet or that the ' appeared in E series; or ‘3 velopnents C? in his after. trigOnOfletri' Indeed, its util‘ tributing factor This paper is :ageous relation: c; A C4 . «unite late :5 a theory of F questiOn of Four be seen, much of response to delta "Per begins wit :4 + 5‘10“ : d ' Ale" 2": 4 search for s the theorems 0 f in writing iii [The theory of Fourier series] has been a source of new ideas for analysts during the last two centuries, and is likely to be so in years to come.... It is not accidental that the notion of function generally accepted now was first formulated in the celebrated memoir of Dirichlet (1837) dealing with the convergence of Fourier series; or that the definition of Riemann's integral in its general form appeared in Riemann's Habilitationsschrift devoted to trigonometric series; or that the theory of sets, one of the most important de- ve10pments of nineteenth—century mathematics, was created by Cantor in his attempts to solve the problems of the sets of uniqueness for trigonometric series. In more recent times, the integral of Lebesgue was-developed in close connection with the theory of Fourier series.... Indeed, its utility in the study of Fourier series was an important con- tributing factor in the acceptance of Lebesgue's definition of integral. This paper is an exposition of the history of such mutually advan- tageous relationships between the successively more general conceptions of definite integral from Cauchy to Lebesgue and the concomitant evolution of a theory of Fourier series. The unifying theme of the paper is the question of Fourier series representation of functions since, as will be seen, much of this part of the theory of integration developed in response to demands placed upon it by such series representations. The paper begins with the work of the first mathematicians to consider this question, d'Alembert, Euler, D. Bernoulli, and Lagrange, and following the search for sufficient conditions for such representation, ends with the theorems of Carleson and Hunt. In writing the paper, I have attempted to identify both the histori- cal sequence of events and the initiating influences which motivated the work of contributing mathematicians, and to express both influences and events in the words of the men who helped create the theory. Therefore I have frequently excerpted passages from the original memoirs. 2A. Zygmund, Trigonometric Series, Volume I, xi. I have as.C ti have used 5: I. .9. been given 5 Eh purpose cmtext within «1 fight assume we treatment of the a: Etccurageme iv I have assumed the reader has a knowledge of undergraduate analysis, and have used standard notations and terminology; proofs of theorems have not been given since they can be found in the references cited. My purpose in writing this paper has been to identify an historical context within which a study of the theory of functions of a real variable might assume greater meaning. Many texts provide a rigorous mathematical treatment of the theory, but, of necessity, little insight into its origins. My hope is that this paper will complement such texts. I should like to thank Charles Wells, John Wagner, John Kinney, John Masterson, Clifford Weil, Francis Hildebrand, Albert Froderberg, Neil Gray and particularly Gottfried Adam and Arthur Kimmel for assistance and encouragement in the course of this work. O . .:.¢Ce ‘ . . . marl On th, Contribut: Lazr‘m'3e ' Fourier CauChy ' Dirichlet R1213?“ Heine! Ca‘ Cantor; H Peano and Borel frepter 3 On tL Lebesg e TrigOROZ“ ue Ries Lusin an kilcgue . - ipendices A . . B . C . D . eférencee . CONTENTS Preface . . . . . . . . . . . . . Chapter 1 0n the Cauchy and Riemann Integrals. . . . . . . Contributions of d'Alembert, Lagrange . . . . . . . . . Fourier . . . . . . . . . Cauchy . . . . . . . . . . Dirichlet . . . . . . . . R1 emu O O O O O O C O O Euler, D. Bernoulli Chapter 2 On the Creation of Theory of Measure . . . Heine, Cantor, Hankel and Smith . . . . . . . Cantor, Harnack and Stolz Peano and Jordan . . . . . Bore 1 I O O O O O O O O 0 Chapter 3 On the Lebesgue Integral . . . . . . . . . . Lebesgue Measure and Integral Trigonometric Series . . . The Riesz-Fischer Theorem Lusin and Carleson . . . . Epilogue . . . . . . . . . . . . Appendices A . . . . . . . . . . . . B . . . . . . . . . . . . C . . . . . . . . . . . . D O O O O O O O O O O O 0 References . . . . . . . . . . . and 11 26 37 43 55 55 71 78 94 104 104 142 154 169 184 187 191 193 198 203 One of the titration of 5m the action 0i a | :ial differentia? ‘/ B'ZEI‘E a is a C0: is to find a 303 .‘r ‘ .4 string 9 tna' {-4. gash deterr‘kine UmSider‘ "a“ “hose en, enupomt! is deformed f. ‘N. t;- Qe (x’y) ( 5.‘ ‘:\3:1 t? 8(x) § Chapter 1 0n the Cauchy and Riemann Integrals Contributions of d'Alembert, Euler, D. Bernoulli and Lagrange One of the origins of the theory of Fourier series was study of the vibration of stretched strings. Given certain simplifying assumptions the motion of a stretched string is described by a solution of the par- tial differential equation 32F 2 étz 3x 1) where a is a constant.1 This equation has many solutions and the problem is to find a solution F(x,t) of l) which describes the motion of a particu- lar string, that is, which satisfies given initial and boundary conditions which determine the motion of the string and thus determine the solution F(x,t) uniquely.l Consider, for example, a stretched string of unit length in the (x,y) plane whose equilibrium position is on the xraxis, 0 5.x‘§_1, y - 0, and whose endpoints are fixed in this position through time. Assume the string is deformed from its equilibrium position at time t - 0 by a force acting in the (x,y) plane to an initial position 0 5_x.5_1, y - f(x). Let an initial velocity g(x) in the y-direction be imparted to the string at time t - 0. Then the string responds to the forces acting upon it by vibrating about its equilibrium position. If F(x,t) is the solution of 1) which describes the motion of this string then F(x,t) satisfies the boundary conditions F(0,t) - F(l,t) - O and the initial conditions F(x,0) - f(x) and 1A derivation of this equation is given in D. Widder's Advanced Calculus, Prentice-Hall, 1947, 344. -%§(x,0) - g(x). This equation describes the motion of the string in the sense that the form of the displaced string is given by the curve y = F(x,to) at a fixed instant in time to Z 0. The reader will verify that the functions Fn(x,t) = sin nnxcosnnat, where n is a positive integer, are solutions of l) which satisfy the boundary conditions Fn(0,t) =‘Fn(l,t) = 0 and initial conditions Fn(x,0) - fn(x) = sin nnx and ggn(x,0) - gn(x) = 0. Moreover, if the series 2) F(x,t) = nzlan Fn(x,t) = “Elan sin nflxcosnnat, where the an are:constants, and the series obtained from it by differentiating term-by-term are convergent, and if term-by-term differentiation is justified, then 2) is a solution of l) satisfying(§he boundary conditions F(0,t)= F(l,t) 8 0 and initial conditions F(x,0) - f(x) = nil Thus F(x,t) describes the motion of a string stretched between the points (0,0) an sin nnx and gékaO) I g(x) 8 0. and (1,0), fixed at these points, displaced in the form f(x) 8 #:lan sin nnx, 0 i x S 1, and at rest at the instant of release t = 0. The function F(x,t) is the most general solution of this particular case of the vibrating string provided it can be made to satisfy arbitrarily given initial conditions. This is the criterion for generality of solution. A solution, therefore, satisfying a particular initial condition F(x,0) = s(x) where of physical necessity s(x) is continuous on 0‘: X‘i 1 and, in the context of this case of the vibrating string g—E(x,0) - g(x) - 0, is subsumed in the general solution. Again, a solution is obtained for each initial position of the stretChed string; the general solution encompasses all of these solutions 3":ttnecusiy SinCE :ziitisns. 0i C0“? gaeral solution of ' 2:;uestion. If F(x,t) is; LI) defined on [0,1 Sasine series on 5:11:45 since k(x) C 531:5. Thus the at 73:5:ng string lea cmed by a 32:43 Cord 0f 1 "' instant of risii uncth 3‘éE-‘JEDCES to be be C0ntinu simultaneously since by definition it satisfies arbitrarily prescribed initial (D conditions. Of course whether or not F(x,t) - nglan sin nnx cos nnat is the general solution of this particular case of the vibrating string remains open to question. If F(x,t) is the general solution, then an arbitrary continuous function k(x) defined on [0,1] and such that k(O) = kgl) - 0 must admit representation by a sine series on [0,1], that is, k(x) a nzlan sin nnx for 0 S x S 1. This follows since k(x) can be interpreted as an initial position of the stretched string. Thus the attempt to find the general solution for this case of the vibrating string leads to the very interesting question of trigonometric series representation of an arbitrary function. Historically, the controversy engendered by the question of such representation helped initiate and was resolved by the creation of the theory of Fourier series. 2 D'Alembert published a solution of the equation %:§-- a2-2-%- (He assumed 8x a = 1 but I follow the usual convention.) in his 1747 memoir "Research on curves formed by a stretched vibrating cord."2 He gave his solution for a vibrating cord of length 2, fixed at the points (0,0) and (2,0) and at rest at the instant of release, in the fonnyy- f(at + x) - f(at - x) where f denotes a periodic function with period 22. D'Alembert remarked, "There are many. . . consequences to be drawn from the general solution we have just given," a rather ironic understatement in light of the historical sequence initiated by publication of his solution general. In his analysis d'Alembert took the initial position of the stretched cord to be continuous or regular, that is, a curve whose form could be expressed by a single definite function of the independent variable. Such a 2 D'Alembert, "Recherches sur la courbe que forme un corde tendue mise en vibration," 214. cure SZOOd in CC: etc-2t to be compo: m a single 13‘ll 31' P f:. their express: Euler argue .1; .. the stretched : - n3 - 1.. A :rcs, publis..e~. at: considerable maicated to CL :fzet draws very 1 1 g eeeee ' iiii WIS of the : feed on this que Euler eXpr fifty = f(x + at ii.1+at)+f(; - the which "be e Represent the ’5 13 rEgular , E C: :Qical: its 's'E-d . f°~ the 4 curve stood in contrast to discontinuous or irregular curves which were under- stood to be composed of portions of continuous curves and thus, not conforming to a single law throughout their course, to require several different functions for their expression. Euler argued against d'Alembert's restriction of the initial position of the stretched cord to a continuous curve. In his "On the vibration of 3 published in 1748, he wrote "Mr. d'Alembert was the first to attack cords," with considerable success the examination of this problem. . . and he has communicated to our Academy a very good solution of it. But as. . . one often draws very considerable profit from the comparison of several different solutions of the same problem I do not hesitate to propose the one I have found on this question." Euler expressed his solution of the differential equation 1) in the form y - f(x + at) + f(x - at) where for every t, f(at) + f(-at) - 0 and f(z + at) + f(£ - at) = 0, and concluded from these equations that every curve which "be situated alternatively above and below [the axis] is proper to represent the nature of the. . . function f. . . ." Thus, "a. . . curve, be it regular, contained in a certain equation, or be it irregular or mechanical, its arbitrary [ordinate] will furnish the functions which we need for the solution of the problem." He then gave the equation f(x) =asin%+esin;}§+ysin3—zé+ . . . as a special case of the "general solution" in which the function f(x) "is a continuous curve whose parts be bound in virtue of the law of continuity so that its nature can be understood by an equation." 3Euler, "Sur la vibration des cordes," 69. its initial veloci :3t'1is rule [of r sees by this how t': a: is determined '3 Rd 30 give it viii :izial figure of I are the greatest 1 efficiently gener 33¢. Euler mainta J'fl‘ L ”‘M‘t. whose titular curve, did 1:“ °i the cord 1 D'Alembert < treated in the . q u‘h ttly Similar 1 ”its to me Euler recognized that the vibrations of the stretched cord subsequent to time t a O are completely determined by the initial form of the cord and the initial velocities of its points. ". . .if a single vibration conforms to this rule [of regularity], all the following must observe it also. One sees by this how the state of following vibrations depends on the preceding " Furthermore "one can before letting the and is determined by them. . . cord go give it whatever figure one wishes" and therefore, "so that the initial figure of the cord can be [given] arbitrarily, the solution must have the greatest extent possible." By obtaining a solution asserted to be sufficiently general to comprehend such initial positions of the stretched cord, Euler maintained that his solution was more general than that of d'Alembert, whose solution, by assuming the initial form of the cord to be a regular curve, did not encompass, for example, the case in which the initial form of the cord is polygonal. D'Alembert rejoined Euler in a paper published in 1750.4 "Mr. Euler has treated in the Memoirs of 1748 the problem of vibrating cords by a method entirely similar to mine as to the essential part of the problem and only, it seems to me, a little longer." He cautioned his readers that "it does not suffice to transport the initial curve alternatively above and below the axis; it is necessary in addition that the curve satisfy the conditions that I have expressed in my memoir. . . . In any other case the problem will not be capable of resolution, at least by my method, and I do not know if it will not surpass the force of continuous analysis. One cannot, it seems to me, express y analytically in a more general manner than by supposing it a 4D'Alembert, "Addition au memoire sur la courbe que forme un corde tendfie, mise en vibration," 355. fzttien of x and rile: only for t: :a be enclosed in Me ispossible t 83161 Bern ralication of his W. l "Vfl ‘ 5 Ziven by the function of x and t. But in this supposition one finds the solution of the problem only for the cases where the different figures of the vibrating cord can be enclosed in a single and same equation. In all other cases it seems to me impossible to give to y a general form." Daniel Bernoulli became thhird party to this controversy in 1753 by publication of his "Reflections and enlightenments on the new vibrations of cords given by the Memoirs of the Academy of 1747 and 1748."5 Bernoulli took exception to d'Alembert and Euler's reliance upon a strictly mathematical approach to the problem. He contended such reliance demonstrates that "to listen to abstract analysis without any synthetic examination of the proposed question is more likely to surprise rather than enlighten us. It seems to me that one need only give attention to the nature of simple vibrations of cords in order to foresee without any calculations all that these two great geometricians found by the most difficult and abstract analysis. . . ." Bernoulli apprehended the problem in physical terms. Basing his arguments upon Taylor's 22 Methodo Incrementorum, he asserted the vibrating cord, as a sonorous body whose vibrations consist of a fundamental and its overtones, admits expression mathematically as the sum of terms corresponding to the fundamental and its harmonics. This led him to conclude that any initial position of the stretched cord admits representation in the form y = asin-1% + Bsin §%§_+ ysin'§%§-+ . . . . Bernoulli wrote "Here is therefore an infinity of curves found without any calculation and our equation is the same as that of Mr. Euler. . . It is 5 , Bernoulli, "Réflexions et eclaircissemens sur les nouvelles vibrations des cordes exposées dans les Mémoires de l'Académie de 1747 at 1748," 147. me that HI- Eule gazeral and that h as still other Ct Bernoulli's faction admits re :3 6 " .-..et. Hr. Berni :te solution of Ta‘ ::r:' is susceptiblé ...2'x 1‘ . . , bet gaze-rel that it e25 Ll the curves ex Estimated, have 2:2: curves. If Eli-IS also negal 5571‘s ‘5 a X; and it [1,. true that Mr. Euler does not treat this infinite multitude [of curves] as general and that he gives it. . . only for particular cases, but. . . if there are still other curves, I do not understand in what sense one can admit them." Bernoulli's contention that every curve, and hence an arbitrary function admits representation by a sine series was immediately disputed by Euler.6 "Mr. Bernoulli. . . sustains against Mr. d'Alembert and myself that the solution of Taylor is sufficient to explain all the movements of which a cord is susceptible. . . . Any argument "that the equation y - osinlfi + 2. thin-2%E + . . , because of the infinity of undetermined coefficients, is so general that it embraces all possible curves" must fail,Euler insisted, for "all the curves expressed by this equation, no matter how the number of terms be augmented, have certain characteristics which distinguish them from all other curves. If one takes the abscissa x negative, then the ordinate becomes also negative and equal to that which corresponds to the positive abscissa x; and in the same way, the ordinate which corresponds to the abscissa x'+ 2 is negative and equal to that which corresponds to the abscissa x." Therefore, asserted Euler, a sine series, being odd and periodic, cannot repre- sent a function which does not possess both of these properties, and in parti- cular, cannot represent an algebraic function. Thus Euler held Bernoulli's solution to be more restricted than the solution of d'Alembert. We should observe that neither Euler nor his contemporaries could concede the possibility of trigonometric series representation of non-periodic continuous functions even in an interval. Indeed such a notion would have been dismissed as a violation of the concept of continuous function. Coeval Opinion held that 6Euler, "Remarques sur lea mémoires précedens de M. Bernoulli," 196. the definition of everra‘nere in its band an interva. Sections were eq- titcu'astance whict tic: of a non-per: Lagrange e Researches on t”: :5 the first t‘nr :gh- Julli' s soli ‘5‘.- "“Sa“3 that tzlase all the I -- that O: 6. k 31> .. \' Lia? “at is \- ~i“-‘ -‘t‘ ‘v' flan. 1: r. ~;:- ‘ - lult- i:_i .lcsute . l} :‘\t "‘1iicn J Y :1 if. B: “s the definition of a continuous function in an interval implied its definition everywhere in its domain, i.e., that a continuous function could be extended beyond an interval of definition in only one way. Thus, if two continuous functions were equal on an interval they were held to be equal everywhere, a circumstance which made the impossibility of a trigonometric series representa- tion of a non-periodic function appear obvious. Lagrange entered the controversy in 1759 with publication of his "Researches on the nature and propagation of sound."7 In reviewing the methods of the first three protagonists, Lagrange reiterated Euler's objection to Bernoulli's solution, a criticism with which he concurred: "It would be necessary that the equation [y - asin'fl% + Bsintz%§-+ ysin §%§ + . . . .] enclose all the figures that one can give to a stretched cord, that is, all the possible curves. This cannot be because of certain properties which seem to distinguish cords comprised in this equation from all the other curves that one can imagine; . . .in other wordS, in augmenting or diminishing the abscissa of an arbitrary multiple of the axis, the value of the ordinate y does not change." Having dispatched Bernoulli's solution, Lagrange rejected the generality of that of d'Alembert, again following Euler's lead. "The construction of Mr. Euler is evidently much more general than that of Mr. d'Alembert, for the latter always supposed that the generating curve [i.e., the curve corresponding to the initial position of the stretched cord] be regular and susceptible to enclosure in a continuous equation. . . [and] . . .believed that such con- struction became insufficient whenever the generating curve did not follow the law of continuity. . . ." 7 Lagrange, "Recherches sur la nature et la propagation du son," 39. Lagrange a of iezonstration. itfetred by appli illegitimate in a‘ tit. It follows '3? its very nature. fro: the integrati Lagrange p. if the vibrating c 35 cord, each in . s. . ~e 9*“ all criticis: 5 Of a “eighth a.‘ . ‘3 5K ‘ "‘“ELQ n as the n “mediate stEP 1“(x I: a , Stung of u . u i‘\ e at: that the in; ’:“1°ular int it‘és‘itin. bus of Su— than the 7:“..G~ .-‘3En grals “‘3 t 9 Lagrange accepted Euler's solution as general but objected to its manner of demonstration. " . . .it seems undeniable that the consequences that are inferred by application of the rules of differential and integral calculus are illegitimate in all cases where this law [of continuity] is not assumed to hold. It follows from this that the construction of Mr. Euler is applicable by its very nature only to continuous curves since it is deduced immediately from the integration of the given differential equation. . . ." Lagrange proffered a demonstration of Euler's solution of the problem of the vibrating cord "in which one considers the movements of the points of the cord, each in particular, . . .to arrive at a conclusion which be sheltered from all criticism." His argument consists of finding the solution for the case of a weightless cord composed of a finite number of particles and then obtaining the solution of the continuous cord as the limit of the first solution as the number of particules is increased without bound. In an intermediate step in his analysis, Lagrange expressed this limit in the form F(x,t) = 2! 2(sin nns sin nnx cos nnat) f(s)ds for a string of unit length, whose initial position is given by f(x), and such that the initial velocity of each of its points is zero. This form is of particular interest since it is only necessary to interchange in it the operations of summation and integration, that is, to write a sum of integrals rather than the integral of a sum, and let t = 0 to obtain a sine series representation of f(x) in which the coefficients are determined as definite integrals. Lagrange undoubtedly recognized the relationship between this form and that of Bernoulli but did not anticipate Fourier's conclusions with respect as: importantly, iiéaat adnit the frztion which cc station. Seton-c7 frdtesm: "the series," and only particles conposir‘ :te smation to l Slat criticism wh rtitrary (perha; Sexes Mid expres 10 to such expansions. There are several reasons for this. First, and perhaps most importantly, Lagrange was constrained by his concept of function. He did not admit the possibility of such an expansion for any other than a periodic function which could be given analytically, i.e., expressed by a single equation. Second, Lagrange understood the integration symbol to denote a finite sum: "the integral sign I is used to express the sum of all these series," and only after summing the series did Lagrange let the number of particles composing the cord tend to infinity. Indeed, had Lagrange considered the summation to be an integral, his demonstration would have been open to the same criticism which he directed toward that of Euler, i.e., integration of the arbitrary (perhaps discontinuous) function f(x). Finally, having summed the series and expressed the limit of the sum in Euler's functional form of solution, Lagrange held his demonstration to be complete. "Here then is the theory of this great geometrician [Euler] placed beyond the reach of all criticism, being established on clear and direct principles which do not depend in any way on the law of continuity required by Mr. d'Alembert. Again, here is how it happens that the same formula serves to support and demonstrate the theory of Mr. Bernoulli on the mixture of isochronic vibrations when the number of mobile bodies is finite and reveals to us the insufficiency of [Bernoulli's theory] when the number of bodies becomes infinite." Lagrange "had formed in advance in his mind a definite conception of the path to be taken"8 and in his strict adherence to this conception and consequent disregard of any alternative furnishes "an instructive example of the ease with which an author 8Riemann, Mathematische Werke, 220. tat fail to draw another direction Lagrange v Ftrier‘ s conclus . ’a-H'lfa n . . mow-cal of the itereé is the an; 5:. P Etersburg AC2 £3193 represent at Axl] . n A I‘— A 11 can fail to draw an almost obvious conclusion if his attention is fixed in another direction."9 Lagrange was not the only mathematician to approach,but not attain. Fourier's conclusions. In discussing the problem of representation of the reciprocal of the distance between two planets by a cosine series 3) ¢(6) = A + B cos 6 +'C cos 26 + . . . , where 6 is the angle between the radii, Euler, in a memoir presented to the St. Petersburg Academy in 1777,10 asserted that if a function ¢(6) admits a series representation of the form 3), then A = l [" o(9)de B = -2- ]" o(e) cos ode n 0 ’ n 0 a result obtained by multiplying the series by cos n6 and integrating term by term. Euler only used this argument to determine the coefficients of a series representation whose existence was verified by other means, however, and per- haps for this reason his paper had no effect upon the question of trigonometric series representation of an arbitrary function. Years passed and the controvery regarding the possibility of series representation remained without conclusion. It was left to Fourier to carry Bernoulli's contention against the arguments and authority of d'Alembert, Euler, and Lagrange. Fourier Fourier was led to consider trigonometric series representation of functions in the course of his attempt to create a mathematical theory of the 9Burkhardt, "Entwicklungen nach oscillirenden Functionen und Inte- gration der Differentialgleichungen der mathematischen Physik," 32. 10Euler, "Disquisito ulterior super seriebus secundum multipla cuiusdam anguli progredientibus," 114. conduction of an leads tt insider a rec those base A j, the x-axis anc‘ each of its pc_ maintained at 1 the SOUICe A if longitudinal di towards the CO( 12 conduction of heat. An example of the type of problem studied by Fourier which leads to the question of such series representation is the following. Consider a rectangular plate whose sides B and C are of infinite length and whose base A is of length n. Place the base A on the interval (— %3 g) on the x-axis and heat it in such a way that unit temperature is maintained at each of its points. Assume that each point of the sides B and C is maintained at zero temperature. Then "heat will pass continually from the source A into the solid BAC, and will be propagated there in the longitudinal direction, which is infinite, and at the same time will turn towards the cool masses B and C, which will absorb a great part of it. The temperatures of the solid BAC will be raised gradually but will not be able to surpass nor even attain a maximum of temperature, which is different for different points of the mass. It is required to determine the final and constant state to which the variable state continually approaches."11 Fourier showed that the steady state temperature T(x,y) at a point (x,y) of the plate must satisfy the partial differential equation 4) _+_=Oo and the boundary and initial conditions T(- %3 y) = T(%3 y) = 0 and T(x,0) = 1 where, of course, xs(- 33 g). He obtained as an intermediate step in his analysis the form 5y 5) T(x,y) = a e-y cos x + b e—3y cos 3x + c e— cos 5x + . . . . 11Fourier, The Analytical Theory of Heat, 5164. "I: is evident t' the condition Ti; I $121“. is express: sizject to the f; ltacos The coeffi 5? E1Bans of this Thus it be icosine series r the interval (.3. 2 ’ 3631'. heated in SL1 mm)“ f (X) . th at a COSiDe 58rie 9.; “l 32:31 prob lems an}. onl‘ie pr filigree on DeQem’: if!) NIH 13 "It is evident that the function. . . T(x,y) satisfies [equation 4] and the condition Tthg3y) - 0. A third condition remains to be fulfilled, which is expressed thus, T(x,0) - l. . . Equation 5) must therefore be subject to the following condition: 1 - a cos xi+ b cos 3x + c cos 5x‘+ . . . . The coefficients a,b,c,... whose number is infinite are determined by means of this equation." Thus it became important to Fourier to determine the coefficients of a cosine series representation of the function which is identically one on the interval (~§3§). Similarly, had the base A of the rectangular plate been heated in such a way that the temperature at a point x was given by the function f(x), then it would have been required to represent f(x) in the form of a cosine series. This begins to explain why Fourier was interested in trigonometric series representation of.”arbitrary" functions. "The funda- mental problems of the theory of heat cannot be solved without reducing to this form [development in a series of sines and cosines of multiple arcs] the functions which represent the initial state of the temperatures." Fourier presented the first of his memoirs to the French Academy of Science on December 21, 1807. His assertion that an "arbitrary" function f(x) is represented on the interval (-2,2) by a series a f(x) - _Q_+ z (an cos 9-295- nnx 2 n-l + bn sin 2 ) where N 2 2. 1 nnx l nnx an - l J-z f(x) cos 0 dx and bn - 2 J_£f(x) sin.-I-dx :as set with “St" tatiozsschrift . "7' :2. there is stil Lagrange that he this controversy ,' imitated to R‘. arthires without ': it‘ibt withdrawn by Perhaps as 1.; tottpetition f0 ‘- 431% the maths .2 ESE-:31? the res 'a I .:l' ' H Fourier Sme' inglttly extended I ..EE 8 Of the Comp 12‘ arbOux 14 was met with disbelief. Riemann, in the historical section of his Habili- tationsschrift, wrote that Fourier's contention "was so unexpected by Mr. Lagrange that he contradicted it in the most decisive manner. It is said that there is still a document in the archives of the Paris Academy regarding this controversy," the statement of existence of such a document having been communicated to Riemann by Dirichlet. Fourier's memoir was deposited in the archives without being published and, according to Darboux, was "without doubt withdrawn by Fourier in 1810."12 Perhaps as a consequence of Fourier's work, however, the Academy set the competition for the grand prix de mathematiques for 1812 with the question: "To give the mathematical theory of the laws of the prOpagation of heat and to compare the results of this theory to exact experiments." On September 28, 1811, Fourier submitted to the Academy a work which essentially included and slightly extended his original memoir. Fourier's paper was referred to the Judges of the competition, Lagrange, Laplace, Malus, Hafiy and Legendre, and while they awarded him the prize, they were critical of the generality and rigor of his analysis. "This piece. . . contains the true differential equa- tions of the transmission of heat. . . . The newness of the subject together with its importance has led the jury to crown this work, observing however that the manner in which the author arrives at his equations is not exempt from difficulty, and that his analysis in integrating his equations leaves something to be desired relative to generality and even with respect to rigor." 2Darboux, Oeuvres d3 Fourier, vii. F"! Fourier' S taieay without the first part C cf Beat, publish taiezy after :12: ;.’i::ed in its 0 be gublished in is Hark "Fourie Pf‘iirity in an in Fourier's t: 5110f it jUStifi szlutions to app] Fiel’mena as a t“ fel'tile sou: 15 Fourier's manuscript was again deposited in the archives of the Academy without being published. Resentful of this treatment, he incorporated the first part of this memoir almost without change in his Analytical Theory of Heat, published in 1822, and, having become Perpetual Secretary of the Academy after the death of Delambre, caused this part of the memoir to be printed in its original form in the Mémoires in 1824, and the second part to be published in the Mémoires in 1826. In light of the interest excited by his work "Fourier desired without doubt to thus establish his rights of priority in an incontestable manner. . . ."13 Fourier's methods and results have continued to receive criticism, not all of it justified. It is true that Fourier was a physicist interested in solutions to applied problems and in his search for such solutions utilized natural phenomena as a guide to mathematical theory. "Profound study of nature is the most fertile source of mathematical discoveries. Not only has this study, in offering a determinate object to investigation, the advantage of ex- cluding vague questions and calculations without issue; it is besides a sure method of forming analysis itself, and of discovering the elements which it concerns us to know, and which natural science ought always to preserve. . . ."14 Faithful to this conception throughout the course of his work, Fourier re- iterated his position in a summary statement toward the end of his treatise. "The integrals which we have obtained are not only general expressions which satisfy the differential equations; they represent in the most distinct l 3Darboux, op, cit., viii. 14Fourier,'gp. cit., Preliminary Discourse. manner the net chief conditio: results of inve 'n’ten this cond; equation of the of it, in the 5 surface makes k. In holdin " . I amnion to do: GI solution of ' flung determine 1: aCOSx+b r) u Ln '1 lb :d‘-4 ‘ i, . ~3€ complete st:- for: 16 manner the natural effect which is the object of the problem. This is the chief condition which we have always had in view, and without which the results of investigation would appear to us to be only useless transformations. When this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms."15 In holding to this view, Fourier occasionally allowed his physical intuition to dominate his mathematical analysis. His discussion of uniqueness of solution of the problem described above is an example of such domination. Having determined the coefficients a, b, c,. . . of the equation 1 a a cos x'+ b cos 3x + c cos 5x +. . . to be, respectively, a -'%3 b = gg' c = 3%, d =-%%,. . . for values of x such that xe(- g3 g9, Fourier asserted "the complete solution of the problem which we have proposed" to be of the form 6) T(x.Y) = g-ey cos x - 3%“ e“3y cos 3x +-§% e-Sy cos 5x - . . . Fourier then argued, by means of physical considerations, that the final temperature distribution is unique; therefore "since the final state which must be determined is unique, it follows that the proposed problem admits no other solution than that which results from equation 6. Another form 'may be given to this result, but the solution can be neither extended nor restricted without rendering it inexact."16 Thus Fourier essentially assumed uniqueness of solution. 15Fourier, op, cit., 5428. 16Fourier, 93. cit., §204. This does SC'L'ad nathemati . I cemestrate his .Dthical or gec: Baler or Lagran: MVEYSEnce of s notions in his (3 ”magenta of ( Vergem . . I of the terms (14... establish the finch we arrive g itjr ‘ a each mo re an cw. - tit)? ‘fliillf1 I) de 0f the Se: 17 This does not imply, however, that Fourier was disinterested in a sound mathematical development of his theory. Indeed, his efforts to demonstrate his propositions rigorously, while sometimes frustrated by his physical or geometrical intuition, were often as successful as those of Euler or Lagrange. For example, Fourier held the modern conception of convergence of series and this, of course, before Cauchy formalized such notions in his Cours.d'analyse. He wrote, with respect to the question of convergence of (Fourier) series, "it is. . . easy to prove they are con— vergent. .. . This does not result solely from the fact that the values of the terms diminish continually, for this condition is not sufficient to establish the convergence of a series. It is necessary that the values at which we arrive on increasing continually the number of terms should approadh more and more a fixed limit, and should differ from it only by a quantity which becomes less than any given magnitude; this limit is the value of the series."17 Again, when discussing the series 5-= sin x --%-sin 2x +-% sin 3x - . N which was given without restrictions by Euler in his paper"Subsiduim Calculi SinuumJLpublished in 1754, Fourier remarked "This infinite series, which is always convergent, has the value g-so long as the arc x is greater than 0 and less than n. But it is not equal to-g-if the arc exceeds w; it has on the contrary values very different from-g. . . This series has been known for a long time but the analysis which served to discover it did not indicate why the result ceases to hold when the variable exceeds fl. 17Fourier, op, cit., 5228. The method whi. and the origin is subject mus: undertake. These rem "Sawing conve that the Fourie APartial answe series, FOUrieA 18 The method which we are about to employ must therefore be examined attentively and the origin of the limitation to which each of the trigonometrical series is subject must be sought,"18 an examination which Fourier then proceded to undertake. These remarks are by way of prelude to a more important question regarding convergence: how adequately did Fourier demonstrate his assertion that the Fourier series corresponding to a function converges to the function. A partial answer is suggested by Fourier's work with particular Fourier series. Fourier initiated his discussion of the convergence of the series y = sin x - %-sin 2x +-% sin 3x - %-sin 4x + . . . by considering the sum of the first ulterms of the series, m even. He expressed this finite sum 8m in the form x cos mx +-— x ( 2) S =—- dXO 2 x 2 cos-5 Seeking the limit of this integral for increasing m, Fourier repeatedly integrated by parts to obtain "a series in which the powers of (m +‘%9 cos (mx +-§) enter into the denominators," Thusllig I xsr-dx = 0, from which 2 cos-E Fourier concluded y = lim Sm ='%u Having shown this, Fourier completed his argument by demonstrating the necessity of the interval of convergence noted above. Fourier argued in this manner with respect to several explicit series. His work is of particular interest and importance since it is this type of 18Fourier, 22, cit., 5184. argzment, i.e., t the series and th finity, with whic of convergence of This remark established one f T "arbitrary" functi f(xi l9 argument, i.e., to express by an integral the sum of the first m terms of the series and then to seek the limit of this integral as m tends to in- finity, with which Lejeune-Dirichlet obtained the first rigorous theory of convergence of Fourier series. Thus Fourier anticipated Dirichlet. This remark is general. For consider the manner in which Fourier established one form of his assertion regarding the representation of an "arbitrary" function f(x) by its corresponding Fourier series, to wit, 1 1=+oo +33" 21 f(x) = E 2 I: [f(a) cos ;— (a- x)] do, ie-m — 2 where the interval of convergence is the set of all x such that ii x€(- 2’ §). Fourier first interchanged the order of integration and summation and, to simplify the work, letI- 2n and denoted a - x by r. He thus obtained the form 1 +oo 1-+oo f(x) 37f... [f(o) 2 cos 1 r] do. 13—00 i=+j i-+j Fourier next wrote the finite sum 2 cos i r in the form 2 cos i r = sin r i=-j 1--j cos j r + sin j r-i:zgg-;. He then multiplied the second member of this equation by f(a) expressed the product as an integral from -n to +w with a as the variable of integration, and sought the form of the limit as j + m. In so doing he obtained the expression 4f(x )fo "Man a 21rf(x) which concludes the proof.19 19Fourier's argument is given in greater detail in Appendix B. Fourier case in which greater than I the order of i fix) such that Fourier's reas that Dirichlet respect to trig and conclusions the criticism d 20 Fourier also considered the form of the integral when x a in and the case in which the limits of integration encompass an interval of length greater than 2n. Of course Fourier's work is flawed because he interchanged the order of integration and summation and did not restrict the function f(x) such that the integrals exist. It is true, however, that the form of Fourier's reasoning was essentially correct and that it was by this means that Dirichlet later rigorously established Fourier's contentions with respect to trigonometric series representations. Thus Fourier's methods and conclusions have received less than justice with respect to much of the criticism directed toward them and concomitantly to Fourier himself. It was with these considerations in mind that Darboux, in his OEuvres fig Fourier, commented that "Lejeune-Dirichlet. . .followed pre- cisely the way which was indicated by Fourier, but brought into the work an extreme precision which is necessary to such an important question. This is not to deny the real and considerable progress brought by the memoirs of Dirichlet to a subject where the efforts of Poisson and Cauchy had not been crowned with complete success. But it seems just to remark that Fourier, with his very profound sense of the questions of natural philosophy. . .had indicated and even gone over, though with uncertain steps, the path along which one should go to find the first exact demon- stration of these fundamental results. . . ." Part of Fourier's success can be attributed to his more general conception of function. In this respect he represents a distinct break from the traditions of his eighteenth century colleagues. When describing the functions which admit trigonometric series representation, Fourier wrote "above all, it must be remarked that the function f(x). . .is entirely stirrer; and not fix) represents ej arbitrary. An in 3'“ eQual number c- or negative or 281 D3” 13"; they 3 fire general than am his geometri equivalem to any anique) depender he takes refere“( it has 3° eXistir Eben limits . tat the CUI‘ve h X: ‘ D .- 5-- ‘e ‘ «I: ’ ' 1n anothe I 353:" : :3? e .. xanple, tha 34?; ed lines H 21 arbitrary and not subject to a continuous law. . . .In general the function f(x) represents a succession of values or ordinates each of which is arbitrary. An infinity of values being given to the abscissa x, there are an equal number of ordinates f(x). All have actual values, either positive or negative or zero. We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as if it were a single quantity."20 This description is more general than Fourier intended. It is clear that Fourier, consistent with his geometric methodology, considered the notion of function to be equivalent to any relationship between the independent variable x and the (unique) dependent variable f(x) which could be given graphically.21 Thus he makes reference to a function "such that the ordinate which represents it has no existing value except when the abscissa is included between two given limits a and b, all the other ordinates being supposed zero; so that the curve has no form or trace except above the interval x = a to x = b, and coincides with the x-axis in all other parts of its course." Again, in another context, Fourier wrote of a function "subject to no condition, and the line whose ordinate it represents may have any form; for example, that of a contour formed of a series of straight lines and curved lines." Thus the functions considered by Fourier were piecewise continuous and had at most finitely many discontinuities in the modern sense. 20Fourier,_p. cit., 5417. 21See Riemann, Mathematische Werke, 218. In addition analysis, Fourier an interval of de. raised the three is NIX l 22 In addition to subjecting discontinuous functions to the methods of analysis, Fourier demonstrated that a function could be extended beyond an interval of definition in more than one way. For example Fourier ob- tained the three analytic expressions x l 1 2 sin x - 2 sin 2x + 3 sin 3x - . . . , §~ _ .2— Sin x - _2'— Sin 3x + _"'2— Sin 5x - o o o 3 n 2 2 3 fl 5 n and 32{-=%-£cosx-—§—c033x-..., " 3 n and remarked that "these three values of-g- ought not to be considered as equal with refernce to all possible values of x; the three preceding develop- ments have a common value only when the variable x is included between 0 1T and-f. The construction of the values of these three series, and the comparison of the lines whose ordinates are expressed by them, render sensible the alternate coincidence and divergence of the values of these functions."22 In general, Fourier wrote, "It is remarkable that we can express by con- vergent series. . .the ordinates of lines. . .which are not subject to a continuous law. We see by this that we must admit into analysis functions which have equal values whenever the variable receives any values whatever included between two given limits, even though on substituting in these two functions. . .a number included in another interval, the results of the two substitutions are not the same. The functions which enjoy this preperty are represented by different lines which coincide only in a definite portion "23 of their course. . . Thus Fourier's conception of the notion of 22Fourier,‘gp, cit., 5225. 23Fourier, _p, cit., 5230. function was con It is of i discontinuous fu pending trigonom argued geometric enter into the e are the values 0 Mr) sin n x dx required . 3: any curve whe he Same part of integral t8 m I "33*1 ~ 0 n, 01' ”h 1 \I‘a“ 23 function was considerably more general than, say, that of d'Alembert. It is of interest to consider how Fourier viewed the integrals of discontinuous function when determining the coefficients of the corres- ponding trigonometric series expansions. As might be expected, Fourier argued geometrically: "We see that the coefficients a, b, c, . . . which enter into the equation %-¢(x) - a sin x + b sin 2x + c sin 3x + . . . are the values of definite integrals expressed by the general term f¢(x) sin n x dx, n being the number of the term whose coefficient is required. . . .if the function ¢(x) be represented by the variable ordinate of any curve whatever whose abscissa extends from x - O to x - n, and if on the same part of the axis the knowntrigonometric curve, whose ordinate is y = sin n x be constructed, it is easy to represent the value of any integral term. We must suppose that for each abscissa x to which corresponds one value of ¢(x) and one value of sin n x, we multiply the latter by the first and at the same point of the axis raise an ordinate equal to the pro- duct ¢(x) sin n x. By this continuous operation a third curve is formed whose ordinates are those of the trigonometric curve, reduced in proportion to the ordinates of the arbitrary curve represented by ¢(x). This done, the area of the reduced curve taken from x - O to x - n gives the exact value of the coefficient of sin nx, and whatever the given curve may be which corresponds to ¢(x), whether we can assign to it an analytical equation, or whether it depends on no regular law, it is evident that it always serves to reduce. . .the trigonometric curve, so that the area of the reduced curve has, in all possible cases, a definite value which is the value of the coefficient of sin fix in the development of the function."24 24Fourier, 22, cit., 5220. In another place for: of the curve The values of the included between evident that all 1' :4... ‘ «sure of the boc; ". ' "2: aaltrary form. This concept darned, of COUISE unzeption in defe 24 In another place Fourier remarked "Whatever be the function ¢(x), or the form of the curve which it represents, the integral has a definite value. . . The values of these integrals are analogous to that of the whole area. . included between the curve and the axis in a given interval. . . . It is evident that all these quantities have assignable values, whether the figure of the bodies be regular, or whether we give to them an entirely arbitrary form."25 This conception represents a break from the Eulerian tradition, derived, of course, from.Newton, of defining a definite integral in terms of its primitive. Fourier may have chosen to abandon the traditional conception in deference to his geometrical viewpoint; it is more likely that be conceived of the definite integral as an area for reasons given below. In any case it should be remarked that, while unsatisfactory from the modern standpoint, the notion of the definite integral as an area, which was itself an essentially undefined concept, was adequate for Fourier's purpose. That is, given the prevailing intuitive conception of area, Fourier could integrate the functions he had in mind. Fourier recognized that he had made a significant contribution. He was aware of and recognized the relationship of his work to the researches of d'Alembert, Euler, Daniel Bernoulli, and Lagrange. "If we apply these principles to the problem of the motion of vibrating strings, we can solve the difficulties which first appeared in the researches of Daniel Bernoulli. The solution given by this geometrician assumes that any function whatever may always be developed in a series of sines or cosines of multiple arcs 25Fourier,'_p. cit., 5229. Son the most com: consists in actua determined coeffi had contributed t ration of functic analysis. "It h.- in a series of s monetric eerie 05 the variable the Eunction be: it IEInains inco ‘31 80nometrie s have insisted 25 Now the most complete of all the proofs of this proposition is that which consists in actually resolving a given function into such a series with determined coefficients,"26 a work which Fourier had accomplished. He had contributed to the solution of a long-standing problem in the represen- tation of functions in such a way as to have important implications for analysis. "It had always been regarded as manifestly impossible to express in a series of sines of multiple arcs, or at least in a convergent trig- onometric series, a function which has no existing values unless the values of the variable are included between certain limits, all the other values of the function being zero. But this point of analysis is fully cleared up and it remains incontestable that [such functions] are exactly expressed by trigonometric series which are convergent or by definite integrals. We have insisted on this consequence from the origin of our researches up to the present time, since we are not concerned here with an abstract and isolated problem, but with a primary consideration intimately connected "27 It is little wonder with the most useful and extensive applications. that Darboux could refer to Fourier's Analytical Theory of Heat as a "handsome work that one can place with justice beside the most perfect of all scientific writings." 26Fourier,‘9_p. cit., 5230. 27Fourier,‘gp. cit., 5428. Before Cau definite integra knowledge of the especially by Ne had been ob taine tenuous ! howeve r *mltiHS process Integratio. Leibnitz. Newto- ; he" as the Qpe Station as a "Ca stem-em 0f cal cu i «E v tee; a1 QalCul u 26 Cauchy Before Cauchy there was no rigorous definition of the concept of definite integral. Integration had been known from the time of Archimedes; knowledge of the integral caluclus had been strengthened and extended, especially by Newton and Leibnitz, and results of fundamental importance had been obtained by its use. The basic concept had remained vague and tenuous, however, due to inadequate statements of definition of limits and limiting processes.28 Integration theory had followed complementary courses from Newton and Leibnitz. Newton regarded integration as "the inverse method of fluxions," i.e., as the operation inverse to differentiation; Leibnitz conceived inte- gration as a "calculus summatorius," or limiting summation. Both men were familiar with the relationship between the two concepts, the fundamental theorem of calculus. While both concepts were known, the view of the integral calculus as the inverse of differentiation became predominant in the course of the formal development. Euler, for example, in his Institutiones calculi integralis, published in 1768, defined integral calculus as the method of finding the relationship between functions given the relationship between their differentials, and only used the concept of a limiting summation for the approximate evaluation of integrals. Of course, one of the interpretations of the definite integral was the arithmetic expression of the geometric concept of area. Thus, as remarked above, the concept of area was essentially undefined. moreover, in the 28See Bell, The Development g£_Mathematics,for these aspects of the history. geoaetric-intui definitions, an converse was 31 fined by means work. With FOL added or subtra Cauchy w: and formalism. Cf d'M-e‘abert, metaphysics of definition of continuity, de Cauchy revind of the integra 27 geometric-intuitive morass which prevailed in the absence of precise definitions, and in the confusion between definition and interpretation, the converse was also held to be true. That is, the definite integral was de- fined by means of the concept of area. We have seen this in Fourier's work. With Fourier, mathematicians "merely said which areas had to be added or subtracted in order to obtain the integral f:f(x)dx."29 Cauchy wished to extricate analysis from the quagmires of intuitionism and formalism. To this end he successfully carried out the implied program of d'Alembert, who in 1754 had stated that "The theory of limits is the true metaphysics of the differential calculus." Thus, Cauchy first stated a definition of the concept of a limit and then defined the notions of continuity, derivative, and integral in terms of the limit concept. Cauchy revived the concept of limiting summation as the fundamental notion of the integral calculus. He defined the definite integral for an explicit class of functions, and demonstrated the existence of the integral for members of this class. He proved the fundamental theorem of calculus; he was first to create a theogy of integration, and indeed, a theory of functions of a real variable. Cauchy expressed his purpose in the introduction to his Qgg£g_ gfanalyse, published in 1821. "I have sought to give to the methods of analysis all the rigor which is demanded in geometry, in such a way as to never have recourse to reasons drawn from the generality [i.e., formalism] of algebra. Reasons of this type, although rather commonly admitted, above all in the passage from converging series to diverging series, and from real 29Lebesgue, "Sur le développement de la notion d'intégrale," 149. gtantitles to 1'2 me, as induction relationship to sziences. One 0 algebraic formul fore-alas hold or: Mtities which Values. and in f Iuses I shall d r"Ladefinite Vali aPPliCation of t 28 quantities to imaginary expressions, can only be considered, it seems to me, as inductions which sometimes suggest the truth, but which bear little relationship to the exact itude which is so prized by the mathematical sciences. One ought even to observe that they tend to . . . attribute to algebraic formulas an indefinite validity, while in reality most of these formulas hold only under certain conditions, and for certain values of the quantities which they contain. In determining these conditions and these values, and in fixing in a precise manner the meaning of the notations which I use, I shall dispel all incertitude." By attributing to formulas an "indefinite validity" Cauchy meant, for example, an uncritical (formal) application of the binomial theorem to obtain -1-(1-2)‘1-1+2+4+8+16+.. Cauchy was almost apologetic. in breaking with the micawberian tradition and attempted to justify his work. "It is true that in order to remain constantly faithful to these principles I have been forced to admit several propositions which will perhaps appear a little hard on first glance. For example, I state in Chapter 6 that a divergent series has no sum . . . . But those who read my book will recognize, I hope, that pro- positions of this nature, which bring forth the happy necessity of placing more precision in theories and of bringing useful restrictions to assertions which are too general, turn to the profit of analysis and furnish several subjects of research which are not without importance."3o Cauchy's Course of Analysis, which he had given at the Royal Polytechnical School, and had been encouraged to publish by Laplace and 30Cauchy, Cours d'analyse, iv. Pzisson, included i ”let f(x} be a fun value of x interme aiiits a unique ar berseen these 1i: increment u, the which will depenc been stated, the dams function 3&35‘311 these 1 j dezreases indef; Entinuous with ill-2.1“. “1 tely Sula] This def: I}; his Stateme 4:5: Cauchy ha: 29 Poisson, included this definition of the concept of continuous function. "Let f(x) be a function of the variable x and let us suppose that for each value of x intermediate between two given limits, the function always admits a unique and finite value. If, starting from a value of x included between these limits, one attributes to the variable x an infinitely small increment a, the function itself will receive the increment f(x + a) - f(x), which will depend on the new variable a and the value of x. This having been stated, the function f will be, between the two given limits, a con- tinuous function of the variable x if, for each value of x intermediate between these limits, the numerical value of the difference f(x + a) - f(x) decreases indefinitely with a. In other terms, the function f(x) will be continuous with respect to x between the limits given if, between these limits, an infinitely small increment of the variable always produces an infinitely small increment of the function itself."31 This definition is interesting in a number of aspects. First, Cauchy probably did not intend his concept of function to be as general as his.statement might be interpreted to imply. That is, it is unlikely that Cauchy had in mind a function like the Dirichlet function, 0 if 0 s x s 1 and x is irrational f(X) - . 1 if 0 s x s l and x is rational even though it is admissible in terms of his definition of (single-valued) function. Second, Cauchy defined the continuity of a function on an interval rather than at a point. Discontinuity of a function was similarly defined. ". . .when a function ceases to be continuous in a neighborhood of a 31Cauchy,‘gp, cit., 34. particular value of 2:: that there is f H 0 is a solutior that Cauchy's defix required that the We tum to teinical School 5 5:27: a definition «its. Cauchv r 30 particular value of the variable x, then one says that it is discontinuous, and that there is for this particular value a solution of continuity. Thus, x - 0 is a solution of continuity for the function i3" Finally, in order that Cauchy's definition of continuity of a function be meaningful, it is required that the concept of the infinitely small be defined. We turn to Cauchy's Resume gf_the Lessons Given §£_the Royal gglyf such a definition. Indeed, it is here that Cauchy developed his theory of limits. Cauchy remarked in the preface to this work "My principal goal has been to reconcile the rigor which has been my guiding principle in my Course g£_Analysis with the simplicity which follows from the consideration of infinitely small quantities." The first lesson of the test was devoted to "des variables, de leurs limites et des quantities infiniment petites." Cauchy wrote ". . . a variable quantity is that which successively receives several different values. . . , a constant quantity, on the other hand, is any quantity which receives a fixed and determined value. When the values attributed successively to the same variable approach a fixed value, so as to . . . differ from it by as little as one would wish, then the latter is called the limit of all the former. Thus, for example, the perimeter of a circle is the limit toward which the perimeters of regular inscribed polygons converge as the number of their sides is indefinitely increased."32 Cauchy then 1 sin a and (l + a); as a tends to zero. He discussed the limits of concluded the lesson by defining the concept of the infinitely small. sur lg'calcul infinitesimal, l3. . .._-—"— I' ,.._._.._u. When the success .L b ..at is, in suC‘n amber, then this- Avariable of thi In a subse function as he ha :0 define in the function over a c :53“? was aware 1“ Singled it ox ElCulus it has Existence of int 31 "When the successive values of the same variable decrease indefinitely, that is, in such a manner as to go below [i.e., be less than] any given number, then this variable becomes what one calls infinitely small . . . A variable of this type has zero as its limit."33 In a subsequent lesson Cauchy defined the concept of continuous function as he had in his Egggg’d'analyse. Given this he was prepared to define in the 25th lesson, the definite integral of a continuous function over a closed (finite) interval as the limit of a set of sums. Cauchy was aware of the importance of his work in integration theory and had singled it out for special attention in his preface. "In integral calculus it has seemed necessary to me to demonstrate in general the existence of integrals or primitive functions before develOping their diverse prOperties. In order to accomplish this it has first been necessary to establish the notion of integrals taken between given limits or definite integrals . . ." Cauchy's definition of definite integral is as follows. "Let us suppose that the function y = f(x) is continuous with respect to the variable x between two finite limits x = x0 and x s X. We will designate by x1, x2, . . ., xn_1 new values of x interposed between these limits, which always increase or decrease from the first limit to the second. These values are used to divide the difference X - xo into elements xl - x0, x2 - ’3’ ' ' ° ’ X - xn—l which will be of the same sign. This stated, let us multiply each element 33Cauchy,‘_p. cit., 16. 7:" the value to say the el‘ azd finally :1 s = ( be the sum Of depend on 1) t] have been diVi‘ the node of di'v‘ anerical value large, then the influence on thé Cauchy t} the difference b is Stall when th a, . «c then cont in UEC 32 by the value of f(x) corresponding to the origin of the element, that is to say the element x1 — xo by f(xo), the element x2 - x1 by f(xl), . . . , and finally the element X - xn-l by f(xn_1), and let be the sum of the products thus obtained. The quantity S will evidently depend on 1) the number n of elements in which the difference X - xo will have been divided, and 2) the values of these elements and,in consequence, the mode of division used. Now it is important to remark that if the numerical values of the elements become very small and the number n very large, then the mode of division will no longer have any but a very small influence on the value of S." Cauchy then proceeded to demonstrate this assertion, that is, that the difference between sums S and 8' corresponding to partitions P and P' is small when the norms of P and P' are small. It was in this demonstraf tion that Cauchy appealed to the hypothesis of the (uniform) continuity 0f f(x)- He then continued, "Let us consider simultaneously of two modes of division of the difference X - x0, in each of which the elements of the difference have very small numerical values. We will compare these two modes to a third chosen in such a way that . . . all the values of x interposed in the first two modes between the limits x0 and X be used in the third, and will find that the value of S is charged very little by passing from the first or second mode to the third . . . . Thus, when the elements of the difference x - x6 become infinitely small, the mode of division has on S only a negligible influence, and if one decreases indefinitely the numer- ical value of these elements while augmenting their number, the value of S -.-;;1 finish by 1 by attaining a < function f(x) ar iis lizit is wt Given thi prsperties" of t acted, the funda tion of definite 59-2231)! integrate. sebintervals in x “fiery 0f integ r; L. "5 had In mind. :1 (I) :ated different] IRPOSes. It rermine Vail. i- is 33 will finish by being sensibly constant, or, in other terms, it will finish by attaining a certain limit which will depend uniquely on the form of the function f(x) and the extreme values x0 and X attributed to the variable x. This limit is what one calls a definite integral."34 Given this definition, Cauchy then went on to develop the "diverse prOperties" of the integral in succeeding lessons, including, as has been noted, the fundamental theorem of calculus. He also generalized the defini- tion of definite integral to unbounded functions. In similar fashion, Cauchy integrated piecewise continuous functions by integrating over those subintervals in which there was no point of discontinuity. Thus, Cauchy's theory of integration was sufficiently general to integrate the functions he had in mind, i.e., continuous and piecewise continuous functions.35 Stated differently, Cauchy's theory of integration was adequate for his purposes. It remains to attempt to determine why Cauchy abandoned the pre- vailing conception of integration as the inverse of differentation. This is an important question, it seems to me, since Cauchy's definition of definite integral as a limit of sums decisively influenced the course of integration theory in its subsequent deve10pment. Some insight into this matter is furnished by Cauchy's "General observations and additions," published in 1823 as an addendum to an earlier memoir on the integration of differential equations. ". . . we will consider each definite integral, 34Cauchy, 22, cit., 122. 35See Hawkins, Lebesqués Theory o£_Integratign, 12. taken between twc 5:511 values of t correspond to the lizits. When one stews easily that the two limits of the I sign is fin later. and more i Catarally led by “35331. taken bet But” it seems to 1 be adapted by pre «sea, EVen those Plated Under the ”an: age ‘Df' iil“£i\rs I3 real functions E”? b' l-at 1011 into twc. "Er y Equiv siVE or e‘len disc ‘I k‘it 34 taken between two limits, as being nothing else than the sum of the infinitely small values of the differential eXpression placed under the sign I which correspond to the diverse values of the variable contained within the given limits. When one adapts this manner of viewing definite integrals, one shows easily that such an integral has a unique and finite value, whenever the two limits of the variable are finite quantities, and the function under the I sign is finite and continuous in the interval between the limits."36 Later, and more importantly, Cauchy wrote in a pg§£.scriptum "One is naturally led by the theory of quadrature to consider each definite in- tegral, taken between two real limits as being [the limit of a set of sums]. But, it seems to me that this manner of viewing a definite integral should be adapted by preference, as we have done, because it suits equally all cases, even those in which one cannot generally pass from the function placed under thesign f to the primitive. It has, in addition, the ad- vantageof always furnishing real values for the integrals which correspond to real functions. Finally, it permits us to separate each imaginary equation into two real equations. All this would not take place if one were to consider the definite integral, taken between two real limits, as necessarily equivalent to the difference of the extreme values of a primi- tive or even discontinuous function, or if one caused the variable of one limit to pass to another by a series of imaginary values. In these last two cases, one would often obtain imaginary values for the integral, like those given by Mr. Poisson."37 36Cauchy, 'bbservations générales et additions" S71. 37Cauchy,‘gp cit., 590. Recall that i :Ei‘ix) on [a,b] if atiecrem of Riemann :egtfle on [a,b] a: '3 f(t)dt+'C. This a {(1 -2€:1f(x) is Cauchy J ENE-tithe 0n [_1 ’ 35 Recall that if f(x) is defined on [a,b] then F(x) is a primitive of f(x) on [a,b] if and only if F'(x) - f(x) for every xe(a,b). Also recall a theorem of Riemann's theory of integration of the form; if f(x) is in— tegrgble on [a,b] and if f(x) has a primitive on [a,b], then a primitive is f(t)dt+C. This given, consider the function a 1 if - l §_x'§.0 f(x) - 21£0 - 11M (x+k) and ¢(x—0) - 11m, (x—k). k+0 k+0 k>0 k>0 44Dirichlet, 22, cit., 168. function, continuitj and f(x)s in order ' he worked Di for conve the quest COUld be Sider the the umbe hold, Th fired. In 0f coutin fl‘mcthOn arbitrary betweQn a {Ogether t to S, 1‘18 that going bac the integ fUnCtion . tendeci. t} function tegrabuh ShOUld be 41 function, Dirichlet required a condition to ensure the piecewise continuity and hence integrability of f(x) and the products f(x) cos mx and f(x)sin mx. Thus Dirichlet stated his hypothesis of continuity in order to assure the existence of the definite integrals with which he worked. Dirichlet wished to obtain more general sufficient conditions for convergence, i.e., to relax the restrictions on f(x). He believed the question of convergence of Fourier series of more general functions could be reduced to the above "most simple case." "It remains to con- sider the case where the assumptions that we have made with respect to the number of solutions of continuity and maxima and minima cease to hold. These singular cases can be reduced to the ones we have consid- ered. In order that the integral (7) have a meaning when the solutions of continuity are infinite in number it is only necessary that the function . . . ¢(x) be such that if one designates by a and b two arbitrary quantities included between -N and n, one can always place between a and b other quantities r and s which are sufficiently close together that the function remains continuous in the interval from r to s. The necessity of this restriction is easily seen by consider- ing that the different terms of the series are definite integrals and going back to the fundamental notion of integral. Thus one sees that the integral of a function signifies nothing except insomuch as the function satisfies the preceding stated condition." Dirichlet con- tended, therefore, that the only requirement to be satisfied by the function is that it be integrable, and a sufficient condition for in- tegrability is that the set of points of discontinuity of the function should be nowhere dense. Dirid fulfill this stant c when constant d w‘: thm defined however, it 4 integrals wh: Dirichlet id. integrable. tion theory! function was “Ch Patholoé terest’n‘bS tL Dirick that I have J restrictiens demonstrate ‘ that One can: Dental princ; QOther note axle ”Opel-t Diric' tinned to e X Mt only by N BOU f 42 Dirichlet continued, "An example of a function which does not fulfill this condition is [the function] ¢(x) equal to a fixed con- stant c when the variable x is a rational number and equal to another constant d when the variable is an irrational number. The function thus defined has a finite and determined value for every value of x; however, it cannot be substituted in the series, for the different integrals which enter into the series lose all meaning . . . ." Thus Dirichlet identified a function so discontinuous as to fail to be integrable. Dirichlet's example had little immediate impact on integra- tion theory, however, for the concept of an everywhere discontinuous function was not yet taken seriously. Indeed, this was "an age in which such pathological functions appeared to be completely devoid of in- terest,"45 there being no necessity to integrate such functions. Dirichlet concluded his memoir by remarking "the restriction that I have just made and that of not becoming infinite are the only restrictions to which the function ¢(x) is subject . . . . But to demonstrate this, in order that the work be done with all the clarity that one could desire, requires several details related to the funda— mental principles of infinitesimal analysis which will be presented in another note and in which I will also pursue some other rather remark— able prOperties of [Fourier] series." Dirichlet never published the prOposed subsequent note. He con- tinued to exert an influence upon the ensuing development, however, not only by the implications for the theory of trigonometric series aSBourbaki, Elements gfhistoire des mathématigues, 247. expres but al excite his st thtin; tion of necessa tionssc series, Cmillet during Dedekin was pre Univers tended origina matter. 0f the differe1 ‘-~“‘ 43 expressed in his 1829 memoir, and again in a memoir published in 1837,46 but also in his capacity as a teacher. In particular, Dirichlet excited an interest in this theory in Riemann while the latter was his student in Berlin, an interest which was expressed in Riemann's thtingen inaurgural dissertation, to which we now turn. Riemann Dirichlet determined sufficient conditions for the representa- tion of a function by a Fourier series. Riemann sought to establish necessary conditions for such representation in 1854 in his Habilita- tionsschrift "On the representation of a function by a trigonometric series."47 As will be seen, Riemann's work was in some respects in- complete, and perhaps for the reason the memoir remained unpublished during his lifetime. It was printed in 1867 on the authority of Dedekind. Dedekind wrote in an introductory footnote, "This treatise was presented in 1854 by the author to obtain appointment at the University of thtingen ... Although the author seems not to have in- tended to publish it, the present edition of this treatise in its original form seems to be justified by the high interest of the subject matter, and by the form of treatment of the most important principles of the infinitesimal calculus ..." Riemann viewed his paper as consisting of "two essentially different parts. The first part is a history of the investigation ... 46Dirichlet,"Ueber die Darstellung ganz will-khrlicher Functionen durch Sinus - und Cosinusreihen ," 152. 47Riemann, Mathematische Werke, 213. regar which this conce capti. inpos funct Whom ] iStO: 44 of arbitrary (graphically given) functions and their representation by trigonometric series. In its composition I was permitted to utilize a few hints of the famous mathematician to whom is owed the first pro- found work on this subject. In the second part I will present research regarding the representability of a function by trigonometric series which includes those cases which up to now have not been solved. [In this respect] it was necessary to introduce a short discussion of the concept of definite integral ... Thus Riemann's more general con- ception of definite integral was derived in a context of necessity imposed by the problem of trigonometric series representation of functions. The "famous mathematician" referred to was Dirichlet, to whom Riemann had turned for information in the preparation of the historical section of his Habilitationsschrift. In this first part, Riemann reviewed the development of the notion of trigonometric series representation of functions from d'Alembert to Dirichlet. He concluded his survey of the history with a statement of Dirichlet's sufficient conditions for such representation: the function must be integrable, have only finitely many maxima and minima, and "where the function suddenly changes values assumes the average of the mutual limit valuesf' That is, if xb is a point of discontinuity of f(x), then f(xo) -%'(f(xo + 0) +-f(xo-O)). Riemann remarked "whether and when a function which does not fulfill these three conditions can be represented by a trigonometrical series re- mains undecided by these researches."48 Given the success of Dirichlet's work with respect to functions en- countered in applied problems, and which therefore had been of primary “aRiemann, gp,cit., 223. inn tiox its the tOe by D Diri grEat to It prOpc ”hick defir 45 interest, Riemann attempted to justify his own more general investiga- tion of trigonometric series representation of functions in terms of its relationship to pure mathematics. "For all our ignorance concern— ing the manner in which the forces and states of matter vary with time and place in the infinitely small, we are able to hold as certain that the functions to which the researches of Dirichlet do not apply fail to express physical processes. Nevertheless, those cases not considered by Dirichlet seem to merit attention for two reasons. First, as Dirichlet himself remarked at the end of his treatise, this subject matter stands in the closest relationship with the principles of the infinitesimal calculus and it may serve to bring to these principles a greater clarity and certitude. In this respect the study of this sub- ject matter has an immediate interest. Second, the application of Fourier series is not restricted to physical researches; it is also applied with success in a field of pure mathematics, the theory of numbers, and here precisely those functions whose representation by trigonometrical series was not examined by Dirichlet seem to be of the greatest importance. Toward the end of his treatise Dirichlet promised to return to these cases but this promise has remained unfilled ..."49 As a preliminary to his work with trigonometric series, Riemann proposed a more general concept of definite integral. "The uncertainty which still exists in a few fundamental points of the theory of the definite integral forces us to make some introductory gtatements regard— ing this concept ... Thus, what do we understand by f(x)dx ?" a Riemann answered this question as follows 49Riemann, 22,cit., 224. Let 8,}: ,1 xl-a by él achter Bru the sun 5 a will depenc the sum ... then this \ contrary. t defined."50 Give Riemann tun "Let us She. tapt; in Vh. function 110' Rien. integrabili- the 5m S c. 1 that is. th interval. 13‘ by DR. The 50 R1! 46 Let 8.x1,x2,...,xn_l,b be a partition of the interval [a,b] and'denote xl-a by 61 , x2-x1 by 62,..., b-xn_1 by 6n and by en"einen positiven achter Bruch," i.e., a given position fraction. Then "the value of the am will depend on the choice of the interval 6 and the magnitude 6. If the sum ... approaches a limit A when the 6's become infinitely small, b then this value is called the definite integral f(x)dx. If, on the a contrary, the sum does not have this prOperty then the integral is not defined."50 Given this definition of the concept of definite integral, Riemann turned to the unresolved question of integrability of Dirichlet. "Let us shed light secondly as to the extent of validity of this con— cept: in which cases does a function admit and in which cases does a function not admit integration?" Riemann established necessary and sufficient conditions for integrability by arguing in this way. "Let us ... first suppose that the sum S converges if all 6's become infinitely small. We will designate the greatest divergence of the function between a and x1, that is, the difference between its maximal and minimal values in this interval, by D1; between x1 and x2 by D2,..., and between xn_1 and b by Dn' Then GlDl + 62D2+...+6nDn 5oRienann,g_g.cit., 225. 47 must become infinitely small with the magnitudes 6. In addition, we assume that as long as all 6's remain smaller than d, the greatest value this sum may attain is A; therefore A is a function of d which decreases with d and becomes infinitely small with this value. Now let the total length of the intervals in which the variation [of the function] is greater than 0 be 3. Then the sum 5 D1+<52D2+...+5nDn is 1 greater than or equal to 08. Consequently as j_61D1+62D2+...+5nDn'§.A, and hence a j_%-. Now-% can, if 0 is given, be made arbitrarily small by a suitable choice of the magnitude d; the same is true for s, and therefore we obtain the following: "In order for the sums S to converge as the 6's become infinitely small, it is required, in addition to the finiteness of the function, that the total length of the intervals in which the amplitudes are greater than a, no matter what 0 may be, can be made arbitrarily small by a suitable choice of the magnitude d." Riemann also held the converse to be true. "If the function f(x) is always finite and if with infinitely decreasing magnitudes 6, the total length s of all the intervals in which the amplitude of f(x) is greater than the given magnitude 0 becomes infinitely small, then the sums S converge if all magnitudes 6 become infinitely small. For those intervals in which the variations of f(x) are greater than 0 contribute less to the sum 61D1+62D2+...+6nDnthan s multiplied by the greatest variation in the function between a and b, which is finite; the other intervals contribute less than 6(b-a). Obviously we can 48 assume 0 at first arbitrarily small, and then define the length of the interval so that s becomes arbitrarily small, by which the sum 61D1+62D2+...+6nDn is given any chosen smallness, and consequently the value of the sum can be enclosed in arbitrarily chosen narrow limits. "Thus we have found conditions which are necessary and sufficient for the sums (Eta converge with infinitely decreasing magnitudes 5, and therefore, for the existence of the integral of a function f(x) from a to b."51 In modern terms, a function f(x) is Riemann integrable on [a,b] if and only if the sum n X 51D1 1-1 tends to zero as the norm of the partition P tends to zero, where the 61 denote the lengths of the subintervals of [a,b] determined by P, and each Di designates the oscillation of f(x) on the respective sub- interval of length 61. We should observe that Riemann asserted necessary and sufficient conditions for integrability of a function on an interval without re- quiring that the function satisfy a condition with respect to continuity. Indeed, Riemann immediately gave an example of a discontinuous function not integrable in the sense of Cauchy but which admits a Riemann integral. "Having examined the ... definite integral in general, i.e., without special assumptions as to the nature of the function to be in— tegrated, we should apply this investigation in special cases ... and, 51Riemann,lgg.c t., 227. and If~ Poi: 49 at first, to functions which are infinitely often discontinuous between any two limits however close together. Since these functions have not been considered anywhere it will be good to start with a specific example." Riemann's example is the function f(X)=-Z L121. n=l n2 where (nx) denotes the positive or negative difference between nx and the nearest integer, or is zero if nx is the midpoint of consecutive integers. This function converges for every x, and is discontinuous for every x of the form x s %E-where m is an integer relatively prime to Zn. This follows since, as Riemann wrote, if x - %;" then 1 1 1 2 f(x+0) = f(x) --—— (1 +—+__.+”.) ‘ f(x) __1r__ 2 9 25 2 2n 16n 1 1 1 ,2 and f(x-0) = f(x) +--(1 +—+-—-+...) = f(x) +-— . 2 9 25 2 2n 16n If x is not of this form, then f(x+0) - f(x-O) - f(x), i.e., x is a point of continuity of f. Thus the points of discontinuity of f form a dense subset of the points on the real line. The function f is Riemann integrable on any finite interval [a,b], however, since the 2 oscillation of f at any point of discontinuity is E—2 and there are 2 8n only finitely many values n such that "2 >6. Thus, in the 8n interval [a,b] there are only finitely many points at which the oscilla- tion of f is greater than 6, from which it follows that for a partition P of [a,b] of sufficiently small norm, the sum 2 61Di will be arbitrarily small. Riemann's example is obviously not Cauchy 50 integrable; thus the class of Riemann integrable functions is more ex- tensive than the class of Cauchy integrable functions, and encompasses the latter as a proper subset. Notice, however, that the Dirichlet function is not Riemann integrable. Having completed his definition and discussion of a more general conception of definite integral, Riemann returned to the objective of his Habilitationsschrift, trigonometric series representation of func- tions. In introducing his "Research on the representability of a function by a trigonometrical series without special assumptions regard- ing the nature of the function," Riemann stated his intention in this way. "Up to now works on this subject have had the purpose of donon- strating the validity of Fourier series representation of functions which arise in applied work. Thus the proof could begin for an entirely arbitrarily assumed function and later, the course of the function could be subjected to . . . restrictions, provided these would not impair the first purpose, i.e., to demonstrate representability by means of Fourier series. The former works demonstrate that if a function has this and that pr0perty then it is describable by the series of Fourier. We will consider the converse question; if a function is represented by a trigonometrical series, then what follows out of this as to its course, that is, to changes in its values corresponding to changes in its argument?"52 There are several aspects of this work which are of particular interest.53 First, Riemann investigated trigonometric series 52Riemann,‘gp_.cit., 230. 53See Hobson, The Theory of Functions g£“g_Real Variable and the Theory'2£_Fourier Series, vol. 2, 5420-426 for statements and proofs of Riemann's theorems. 51 representation of functions in general rather than restricting his attention to the special case of Fourier series. That is, in consid- ering the representation of a function f(x) in the form f(x) = a0 + alcos x + b1 sin x + a2cos 2x + bzsin 2x+..., Riemann made no a priori assumption regarding the coefficients an and bn; they might or might not be the Fourier coefficients of f. Riemann was first to study the general case, and demonstrated the distinction between the two types of series by giving several examples of trigono- metric series which are not Fourier series. Riemann utilized in a number of his proofs a new method of summing a trigonometric series. Thus, Riemann considered the sum of the series 7) A+A 0 1+A 2+ .0. a where Ao =-§Q ,A1 a a cos x + blsin x, ... , and lim A.u = O, in the form n“ 1 lim lim A0 + A1(_§_____in h)2 + A2 (£19159 +...+An(°-3-‘—-—-“h uh)2 h+0 n+0° This repeated limit is equal to the sum obtained at any point at which the series 7) converges in the usual manner, and may exist at points at which the series 7) fails to converge in the conventional sense. Riemann introduced, therefore, a more general conception of convergence of a trigonometric series. Riemann was successful in obtaining necessary and sufficient 52 conditions such that a function be represented by a trigonometric series. In particular, Riemann proved that a function f(x) of period 2w is represented (in the sense above) by a series of the form {I} a 0 §-'+ 4:1 (ancos nx + bnsin nx) where lima.n = lim bn = 0 if and only if “90:: n+0!) 1) there exists a continuous function F(x) such that lim F§x+o+§) + F(x-o-Q) - F(x+oe§) - gx -o+B) a f( ) a.s+o 40:8 x and 2) for arbitrary constants b and c, C 1&9 “2 F(x)cos u(x—a)A(x) dx = 0 u b where k(x) is a function such that A’(x) exists in (b,c) and vanishes at b and c, and Xv(x) has only a finite number of maxima and minima. Of course this trigonometric series need not be a Fourier series. Finally, Riemann ended his treatise by exhibiting a number of remarkable examples of functions with special properties. One of these is the function f(x) defined by f(x) - d(xvcos 31:) where 0 I51. , value f(x) - 31;- for x - -;’- to x > 3%; , and generally the value f(x) ' (3%)“ for X - 6%)“ to x > (_%_)n+1. This function, which is dis- continuous only at the points x - 6%)" and at these points jumps by 0%)“, does not have an infinite number of points at which the jumps surpass a fixed finite magnitude as do the examples we have just con- sidered. To be sure, the number of points at which the jumps are greater than a is finite; with decreasing 0 however their numbers in— crease constantly and without limit. The total length of the intervals in which jumps greater than 0 occur obviously can be made arbitrarily small for every 0 since these have to surround only the finitely many points of discontinuity." These examples suggested classes of points on the line. "If a v.I.X.ro...o§1e.7..4-11‘I,s\\«dc.1l . , ,. e: T, 68 set of points on a line has a certain prOperty [i.e., at each point of the set the jump of the function is greater than a] then I will say that these points fill up the line segment if in this segment, no interval, no matter how small, can be given which does not contain at least one point from this set; on the other hand, this set of points does not fill the line segment, but that the points are scattered in it, if between any two arbitrarily close points on the line segment, there can be "11 Thus the set given an interval which contains no point of this set. of points of discontinuity of a function at which the jump is greater than 0 either "fills up" or is "scattered" on the line providing it is dense or nowhere dense, respectively, on an interval. Hankel believed that the length 3 of the intervals in which the oscillation of a linear discontinuous function f(x) is greater than 0 can be made arbitrarily small if and only if the points of discontinuity of f are nowhere dense. This belief led Hankel to classify linear discontinuous functions into two types: 1) those that are totally discontinuous, i.e., whose points of discontinuity at which the jumps greater than 0 are dense in an interval, and 2) functions which are pointwise discontinuous, that is, are such that for each U>O, the set of points at which the jump of the function is greater than 0 is nowhere dense. Finally, Hankel concluded that this classification separated the integrable and non-integrable functions. Thus, Hankel believed that a function is Riemann integrable if and only if it is a simply continuous or pointwise discontinuous function. 11Hankel,,gg. cit., 87. 69 Hankel's conclusion is only partially correct as will be shown by an example below. Nevertheless, his work was a significant contri- bution to an emerging theory of the mathematically discontinuous. Originally published as part of the Gratulationsprogramm of Tubinger University on March 6, 1870, Hankel's paper was published again in the Mathematische Annalen in 1882. The editor of the Annals remarked in a footnote, " ...the work of Herman Hankel...has up to now been difficult to find due to its place of publication; since this paper is mentioned in almost all modern investigations of the concept of functions...we have made the transcription [of the Gratulationsprogramm] literal, and thus have considered as negligible any incorrect points which the work may contain." In 1875 an English mathematician, H. J. S. Smith, published an example of a linear discontinuous function which is not Riemann in- tegrable. He gave his example as follows: "Let m be any given integer greater than 2...now...divide the interval from O to 1 into m equal parts, exempting the last segment from any further division; let us divide each of the remaining m-l segments by m2, exempting the last segment of each segment; let us again divide each of the remaining (ur1)(m2-l) segments by m3, exempting the last segment of each segment; and so on continually. After k-l operations we shall have 2 2 - N - l + (m—l) + (m-l)(m -l) + ... + (m—l)(m -l) ... (m exempted segments, of which the sum will be 1 l 1 1-(1-t-n') (1-—2') (l-Tl) m m 70 This sum, when k is increased without limit, approximates to the finite limit 1 - E ( %-) where E ( %-) is the Eulerian product 3 (l - lk)’ and is certainly different from zero."12 1 m Smith went on to show that the points of division Q are nowhere dense on [0,1] and yet "a function having finite discontinuities at the points Q would be incapable of integration." Thus Smith gave the first example of a nowhere dense set with positive outer content. Smith also expressed Riemann's integrability conditions in a form which strongly suggests the notion of "length" of the set of points of discontinuity of an integrable function. "Let a be any given quantity, however small; if, in every [partition] of norm d, the sum [of lengths] of the segments, of which the ordinate differences surpass 0, diminishes without limit, as d diminishes without limit, the function admits of integration; and vice versa, if the function admits of integration, the sum [of lengths] of these segments diminishes without limit with d."13 The ordinate difference of a segment had been previously defined to be the difference between the greatest and least ordinates of f(x) on the segment. Smith had undertaken this work in order to further discuss Riemann's theorem stating necessary and sufficient conditions for integrability of a function "partly because, in one particular at least, Riemann's demonstration is wanting in formal accuracy, and partly because the theorem ... appears ... to have been made the basis of erroneous inferences." Smith considered the sufficiency part of 12Smith,"0n the integration of Discontinuous FunctionsJ'l48. 13Smith, 22, cit., 142. . 2v ....‘ . 71 Riemann's proof to be incomplete; the latter remark refers of course to Hankel's work. Thus the way was prepared to lead to the definition of a general- ized conception of length, that is, a notion which would "measure" general point sets such as the set of points of discontinuity of a function, and reduce to the familiar concept of length when used to "measure" an interval. Cantor, Harnack and Stolz The first definitions of the measure of arbitrary point sets were given by Cantor, Harnack and Stolz. These measures were defined as a limiting form of a finite covering of the given set by elementary figures whose measures were known. The adherence to finite covers was undoubtedly due to the relationship between such covers and the summa- tion, given a partition P of the interval of integration of a Riemann integrable function f, of the lengths of the (finitely many) intervals of P in which the oscillation of f is greater than 0 > O. Cantor, in the fifth installment of his "On infinite linear point sets," published in 1884, defined the content of a bounded set P of points in n-dimensional space Gn' "If there is given a [bounded] point set P of Gn’ then form around each point p of the ... set P + P’ [the addition . symbol denotes set union and P’ designates the set of limit points oflfl an n-dimensional ball with center p and radius a, which with all its interior and boundary points will be designated by K(p.p). "The ... full balls which are obtained by letting p run through 72 all points of P + P’ have one least common multiple [set union] 2K“): p) 3 P which point set is designated according to circumstances by H (p,P in Gn)’ or more simply by H(p,P) or H(p). "Now this pointset H(p) of Gn always consists of a finite number of regions, since P is assumed bounded, each of which is an n—dimensional continUum with its boundary. Consequently the n-times integral dxldxz...dxn , taken over all regions of H(p), has a specific value which depends upon p; we call this value F(D)... "... we define the content or volume of the set P [denoted I(P) after Inhalt] to be the limit value lim F(o)... and obtain therefore p+0 I (P) - lim F (0)."14 p+0 Cantor observed an additive property of the set function I(P); if P and Q are (bounded) sets which are "completely separated" then I(P+Q) - I(P) + 1(0). CantOr remarked further with respect to his definition of content 14Cantor,"Ueber unendliche, lineare Punktmannichfaltigkertenj'474. 73 in a letter addressed to the editor ofIAg£2.Mathematica in November, 1883. "I wish to eXplicitly state that this ... volume or extent of an arbitrary set P in a ... space Gn of n-dimensions is absolutely dependent upon Gn ..." Thus "a square each of whose sides is equal to one, has its extent equal to zero when it is considered as a con- stituent part of a space of three dimensions, but it has extent equal to one when it is regarded as part of a plane of two dimensions. This general notion of volume or extent is indispensable to me in my re- searches on the dimensions of continuous sets ..." Harnack, in 1883, in his paper "On the content of point sets" wrote, "I am going to develop in a series of explanations and theorems the general definition which I have given for discrete point sets within a closed linear interval. These theorems partially supplement the theorems which Mr. Cantor has in the meantime published on the same subject on the basis of a somewhat different definition ... "A point set within a linear interval of finite length is called discrete (with content zero) if all points [of the set] can be included in a finite number of intervals whose sum [of lengths] can be made arbitrarily small even if thereby the number of intervals may grow be— yond any limit. The latter always occurs if one has to do with an infinite number of points." Harnack defined the content of an arbitrary subset of an interval of length i in the following way: "If the point set is not everywhere dense in the entire interval (thereby the limit would be the magnitude 1) 74 then one should fix a length ... é-and construct the intervals [whose lengths] are greater than or equal to-%, and which do not contain a point of the set in their interior. If such an interval exists and one has [deleted] it, then one should exclude from the residue parts of the interval those [whose lengths] are greater than or equal to-% and contains no point of the set in their interior. One recognizes generally that there is always only a finite number of intervals [whose lengths] are :_%-and contain no point of the given set in their interior whenever n is any positive integer. The total length of the intervals excluded in this fashion which are-i-fi'may be [designated] by N. Then, except for the finite number of points which coincide perhaps with the endpoints of two adjoining excluded intervals and are isolated points, the points of the set lie in the interior or as endpoints of a finite number of intervals whose total length is 2 - N... One needs only to arbitrarily diminish [the lengths of] the intervals which are taken out... The limit value of z-N for n = w is the limit for which one is looking. The point set is consequently discrete if lim N = £."15 Harnack remarked that the union of a finite number of discrete sets is discrete, and continued "to the contrary the theorem is no .1onger valid if one has an infinite sequence of discrete sets, as is ‘taught by the example of the sequence of rational numbers from O to l, vfllich can be composed by the [infinite sequence of discrete sets] 13 1234 P3 ' {4’4}' P4 ' {5'5'5'5}'°°° ° Harnack considered a countable as Opposed to a finite cover in a'Péissage which is remarkable in its anticipation of Borel. "... in a Certain sense any point set which can be counted has the prOperty that \ 15Harnack,"Ueber den Inhalt von Punktmengen,"24l. 75 that all its points may be included in intervals whose sum [of lengths] is arbitrarily small. Thus, for example, although they are everywhere dense on the unit interval, one is able to surround all the rational numbers between 0 and l with intervals whose sum [of lengths] is arbi- trarily small. For, if one has a countable point set a1,a2,...then one surrounds the points with intervals of lengths cl, 22,...and choses these magnitudes in such a way that el+ez+...is less than an arbitrarily small magnitude 6. If one carries out this process with the above mentioned rational numbers...then...the points not covered by these intervals...constitute a point set whose content is greater than 1 - 6."16 Harnack did not attempt to develop this observation; the notion of an everywhere dense set with arbitrarily small content must have seemed paradoxical. Indeed he believed to have obtained a "remarkable paradox" by consideration of a countable covering of intervals. "... if from an...interval of length a, one deletes an infinite number of subintervals a1,a2,... whose sum of lengths b is less than a, then there will remain infinitely many subintervals whose sum of lengths is never greater than b-a; it may, however, be even smaller."17 Harnack con- ceived of the latter occurrence in the case in which the endpoints of the intervals and their limit points do not form a discrete set. The difficulty in such a case is that content is not additive. In his "On a limit value corresponding to an infinite point set," dated July, 1883, Stolz defined the measure of a point set in the following way: Let x’ denote an arbitrary set of points in a finite 16Harnack, 22, cit., 243. it. 7Harnack,.9_p_. c 244. 76 interval (a,b), and consider a sequence of partitions t1,t2,t3,... of (a,b) such that the norm lltnll of tn satisfies the inequality Iltn||<5n where lim an-o. "Now if one carries out within the interval (a,b) a n-m system of infinitely many such partitions t t2,t3,..., and if for each 1» partition one adds the lengths of those intervals containing points of the given set x’, then one obtains a sequence of sums 81,32,83,... such that 513821831... . Thus there exists a finite limit :3: Sn-L where L39. The...number L is independent of the considered system of partitions tm so that to each point set x’ in the interval (a,b) there corresponds a unique limit Lip which is called the interval limit [that is, content of the given set]."18 Later in his paper Stolz brought his work into relationship with Harnack's. "The point sets for which the limit value L-O coincide with the sets which have been called discrete by Mr. Harnack. Accord- ing to his definition the points of a discrete set may be included in a finite number of intervals whose sum S is smaller than c. Consequent- ly LO and as being unextended if £=O. The first attempt to classify point sets in such a way for the purpose of the theory of functions can be found with Mr. Hankel. Mr. Harnack calls the point set linear if Z>O and discrete if £20...deviating from the meaning of linear point set ac— cording to Mr. Cantor.... Mr. du Bois-Reymond has called the point set integrable if £=O...." These remarks give evidence of the influ- ence of the new concept of measure upon the theory of functions. The papers of Cantor, Harnack and Stolz were an important first step toward the creation of a theory of measure. The notion of content failed to satisfy one of the first exPectations of a measure, however, that of additivity. Thus a generalized conception of length might be expected to possess the property that the measure of the union of dis- joint sets be the sum of the measures of the sets. This expectation was fulfilled for finite collections of disjoint sets several years later in the definitions of measure proposed by Peano and Jordan. 20Pasch, "Ueber einige Punkte der Functionentheorieg'l42. 78 Peano and Jordan Peano was led to develop a theory of measure by his attempt to find a simple condition for Riemann integrability of a function. At the time he published his paper "On the integrability of functions," in 1883, the statement of existence of the Riemann integral was the following: a function f is Riemann integrable on [a,b] if and only if for arbitrary positive 0 and 6, the content (in the sense of Stolz or Harnack) of the set of points in [a,b] at which the oscillation of f is greater than 0 is less than 6. Peano sought to express the integra- bility of f in terms of the measurability of the ordinate set of f on [a,b], that is, the set \u/ {(x0,y)l O§y§f(x0)} if f is non negative. xos[a,b] "The existence of the integral of functions of a single variable is not always demonstrated with the rigor and simplicity desirable in such questions. The method of reasoning of principal writers by re- course to geometric considerations is not satisfactory. The analytical demonstrations are generally long and complicated and conditions are introduced which are too restrictive or partly useless. In the present study I prOpose to demonstrate the existence of the integral by intro- ducing a very simple condition of integrability. The reasoning will be analytical but can be interpreted geometrically in any of its parts."21 Peano began by stating Riemann's definition of integral: let f(x) be a function defined on [a,b] and be bounded above and below by 21Peano,"Sulla integrabilita delle funzionir 439. 79 A and B respectively. Partition the interval [a,b] into subintervals h1*“2 the sum u-hlyl+h2y2+...+hnyn, as h is "infinitely diminished," tends ,...hn and let y8 = f(x) where x is an arbitrary element of hs' Then if to a limit S, then the function f is integrable on [a,b] and converse- b ly, and its integral f(x)dx on [a,b] is S. 8 Peano continued in this way. Let p8 and g8 be supremum (limiti superiors) and infimum (limiti inferiore), reapectively, of the yS on the subinterval h . Let P = 2h p and Q = 2h 3 ; then s s s s s A(b-a) 3_P 3_u 3_Q 3 B(b-a). Hence the numbers P corresponding to all partitions of [a,b] admit a greatest lower bound M, and the numbers Q corresponding to all parti- tions of [a,b] admit a least upper bound N. Furthermore, Peano showed P :_M :_N Z,Q. Finally, if f(x) is an integrable function on [a,b], then M a N = 3. Thus "if the function f(x) is integrable then the quantities M and N are equal and their common value is equal to the value of the integral."22 22Peano, 22, cit., 441. It is of interest to note that Darboux had asserted, without proof, the necessity and sufficiency of this con- dition for the existence of the Riemann integral in 1875. Darboux's purpose was to investigate several of Hankel's prepositions which had laeen criticized by Gilbert, Schwarz, Klein and others. "I have imposed lipon myself the duty of going back to several [of Hankel‘s assertions] £1nd...to express them in a form such that they be sheltered from all czriticism..." In particular, Darboux was interested in identifying a CLlass of continuous functions which are not differentiable for in- finitely many values of the independent variable. Thus Darboux did ‘1CIt develop the consequences of his definition of the Riemann integral w31th respect to the concept of area. See Darboux,"Memoire sus les fCilnctions discontinues," 72. 80 Peano also proved the converse of this theorem in the form: if for an arbitrary positive 6 there exists a partition of [a,b] with the prOperty that the corresponding numbers P and Q are such that P-Qver, into equalities if the sets are measurable."26 26Jordan,‘gp, cit., 76-78. 87 Jordan gave his definition of definite integral in the section of the same title. "Let f(x,y,...) be a bounded function in the in- terior of a domain E supposed measurable. "Decompose E into elementary measurable domains e1, e2,... . Designate by M and m the maximum and minimum of the function f in E; and by MR and 111k its maximum and minimum in ek, and form the sums S = E Mkek , s = Z mkek ." Of course the symbol e represents the extent of the "measurable do- k 'main" ek in each of these summations. Jordan showed that the sum S and 8, corresponding to given decompositions of E tend toward m pfi§g§_as the diametersof the elements of the decompositions tend to zero. Designating these limits by T and t, reSpectively, Jordan wrote "This fixed number T=lim S is called the upper integral (integrale par exces) of the function f(x,y,...) in the interior of E. "... the sums s tend toward their maximum t, which will be the lower integral (integrale par defant)of f(x,y,...). "One evidently has Tit. If Tat, the function will be integrable and Tat will be its integral, and will be represented by the notation 27 SEf(x,y,...) 3' Jordan extended this definition of integrability to functions clefined on a nonmeasurable set. "We have assumed until now that the Clomain E is measurable. We are now able to suppress this restriction... consider the limit of a sequence of measurable domains E1,E2,...,En,... lrsaach of which is interior to the following and to E]28 and whose extents c(haverge toward a limit which, by definition, is the interior extent of \ 27Jordan’ .220 Cito, 81-84. 28 Jordan had imposed this condition in the recedin ara ra h and ‘ obviouslv intended it to hold here- p g p g p 88 E. Then the integral, upper or lower, taken in En tends toward a limit" for the difference between the integrals taken in En and En+p is f — f = f _<_D A(Efip) -A(En) in A (E) - A(Eg where |f| §_D. "We consider this limit of the integrals taken in En as representing the value of the integral in E." Thus, in modern notation Jordan then turned to the theorem which was the primary object of his paper. "If a function f(x,y,...) of n variables is integrable in a domain E of measurable extent, then the calculation of the multiple integral I - SE f(x,y,...) is reduced to the calculation of n suc- cessive simple integrals." Jordan proved this theorem in the following way. "For greatest simplicity we will suppose n=2 in the demonstration. The set E will be represented geometrically by a set of points (x,y) situated in a plane. "The values of y to which correspond points of E form a bounded set F. Let one of them be n; the values of x which, associated with n give points of E, form a bounded set Gn' We are not able to affirm that Gn has a measurable length, nor that the function f(x,n) is 89 integrable there; but this function being bounded, always has determined in the interior of Gn its upper and lower integrals. These are func- tions of n, that we designate by J(n) and j(n), and which are bounded in the domain F. We are able therefore to determine in the interior of F: (1) the upper integral of J(n) which we designate by K, and (2) the lower integral of j(n) which we designate by k." Observing that K.: k, Jordan succeeded in showing, using the notation of extent, "that the integral K is at most equal to the upper double integral SEf(x,y). "One shows, by a similar reasoning, that the integral k is at least equal to the lower double integral." Thus, if we denote the upper and lower integrals of f(x,y) on E by'S£f(x,y) and §Ef(x,y). respectively, Jordan demonstrated the inequalities ‘s'E£(x.y) _>. K _>_ k _>. _S,Efn, cover the center n+1 O by an interval AiBi of length 2n 1 "the sum [of lengths] of all the 13 segments, infinite in number, A B1, situated on the segment AB or on 1 its extension, is less than the length 2 of AB; therefore there exists on AB a nondenumarable infinity of points belonging to none of these segments. Let m be one of these points which does not coincide with any of the points 0 for iin, and let P be the circle of center w i passing through the points P and Q; I say that this circle I has the A required property, that is to say that the series £(;:§-)mn is n "36 Thus Borel determined a sense uniformly convergent on this circle. in which f(x) can be analytically continued across T. In a note at the end of his paper, Borel commented further on the existence of a nondenumerable infinity of points not belonging to the union of the intervals AiBi° intervals given on a line whose sum [of lengths] is less than the length "...if one has an infinity of partial of a given interval, then there exists at least one point of the in- terval contained in none of the partial intervals. It is clear that, if there is such a point, then there exists a nondenumerable infinity of them, for, if there is a denumerable infinity, then one is able to enclose them in intervals whose sum [of lengths] is as small as one wants, and chosen in such a manner that, in adjoining these intervals 36Borel, 22, cit., 25-26. 96 to those which are already given, one has a sum less than the length of the interval; it suffices therefore [to demonstrate the existence of] a point on the line belonging to none of these intervals."37 Borel succeeded in demonstrating the existence of such a point by means of what has since become known as the Heine-Borel theorem. Thus Borel was led to consider the union of a countable col- lection of intervals whose measure is defined to be the sum of lengths of the constituent intervals. This measure was apprOpriate for Borel's purpose in that it could distinguish between the measure of a countably dense set and the measure of its closure, a property not enjoyed by the measure Of Peano Jordan. Borel was probably influenced in his statement of definition of measure by Cantor and Harnack. Cantor, in the fourth installment of his "Ueber unendliche, lineare Punktmann- ichfaltigkeiten," had characterized each Open set of real numbers as a countable union of disjoint open intervals; an aspect of Harnack's work in this respect has previously been described. Borel develOped the measure theoretic implications of his 1895 paper in his treatise Lessons 235113 Theory 2_f_ Functions, published in 1898. "We now define...a notion which will be very useful to us, the notion of a measurable set. "All of the sets we consider are formed of points included between 0 and 1. When a set is formed of all the points included in a denumerable infinity of intervals which are disjoint and have a total length 3, we will say that the set has measure a. When two sets have 37Borel,‘gp, c t., 51. 97 no common points and their measures are a and s’ then their union has measure s+s’. Moreover, it is Of little importance in the definition of measure of a set, or that of the union of two sets, if we neglect . . . the denumerable infinity of endpoints of intervals. "More generally, if we have a denumerable infinity of sets which are mutually disjoint and have measures respectively sl,s2,...,sn,.... then their union has measure 3 + s l 2 "...if a set E has measure 8, and contains all the points of a +OOO+8 +0.. C 11 set E’ whose measure is s’, then the set E-E‘ formed of points of E which do not belong to B! will be said to have measure s-s’...."38 Borel thus imagined sets formed by countable unions and set theoretic difference, and their associated measure given by infinite series and arithmetic difference. A nonempty class of sets closed under the set Operations of difference and countable union is called a o-ring. The ring B of Borel measurable sets is, therefore, a o-ring. The o-ring B of Borel measurable sets contains all Open sets, closed sets and countable sets; sets of particular importance in analysis. None of these sets is. in general. measurable in the sense of Peano and Jordan. Thus the O-ring of Borel measurable sets B might be expected to be more useful in analysis than the ring of Peano-Jordan measurable sets R. The o-ring B does exhibit a deficiency: it is not complete. That is, it is not true that if E is a Borel measurable set with Borel measure zero, and if F is any subset of E, than F is a member of B. This assertion is proved by means Of a cardinality argument described 38Borel, Lecons sur'lgiTheorie des Fonctions, 46. 98 below; the importance of the prOperty of completeness of the o-ring of measurable sets will become evident in the next chapter. Borel undoubtedly had the Peano-Jordan conception of measure in mind when he wrote "the sets whose measures one can define by the pre- ceding definitions are called by us measurable, without necessarily implying by this that it is not possible to give a definition of the measure of other sets, but such a definition would be useless to us, it could even hinder us if it did not leave to measure the funda- mental prOperties that we have attributed to it in the definitions that we have given." Borel stated the "essential properties" of a measure to be the following: "the measure of the union of a denumerable infinity of sets is equal to the sum Of their measures; the measure of the differ- ence of two sets is equal to the difference of their measures; measure is never negative; each set whose measure is not zero is nondenumerable. It is above all this last property that we will use."39 The "essential prOperties" of a measure were referred to again in a footnote. "The procedure that we have used comes back to this: we have recognized that a definition of measure can be useful only if it has certain fundamental properties; we have posed a priori these prOperties and it is these which have served us in defining the class Of sets that we regard as measurable....ln all cases, it proceeds from the same fundamental idea: define the new elements which one intro- duces with the aid Of their essential properties, that is to say, of 39Borel, 92. cit., 48. 99 those which are strictly indispensible for the reasonings which must follow." Thus Borel attempted a postulational approach to the theory of measure. This was a significant contribution in itself for it made explicit important prOperties a.measure might be expected to possess. Borel did not endeavor to apply his concept of measure to the theory of integration. There are two reasons for this. First, the problems studied by Borel in the theory of functions were presumably unrelated to integration theory. Second, the theory of measure pro- posed by Borel is in a sense not as general as the concepts of measure defined by Harnack and Stolz, and Peano and Jordan. Thus Borel's work was ostensibly not as apprOpriate to the theory of integration. Borel wrote, "One will compare fruitfully the definitions that we are going to give with the more general definitions given by Mr. Jordan in his Course'gf,Analysis. The problem that we are studying here is besides, completely different from the one resolved by Mr. JordanJ' Lebesgue later attributed this remark to Borel with respect to Borel measurable sets: "By renouncing the definition of measure for an arbitrary set one founds a less general theory; that is to say, it applies to fewer cases, but more precisely in the cases in which it is applied."40 The sense in which Borel's theory of measure is "less general" is that the cardinal number of Borel measurable sets is less than the cardinal number Of measurable sets in the theories of Cantor, Harnack and Stolz, and Peano and Jordan. Any bounded set Of real numbers is 40Lebesgue, Notice sur les travaux scientifigues gg_M, Henri Lebesgue , 33. lOO measurable in the sense of Cantor, Harmack and Stolz; thus the cardin- ality of the collection of measurable sets in this theory of measure is 2c where c is the cardinality of the continuum. The Cantor ternary set is measurable in the sense of Peano and Jordan, with extent zero. It follows that all subsets of this set have extent zero and there are, therefore, at least 2c Peano-Jordan measurable sets. (There are also at most 2c sets Of real numbers measurable in the sense of Peano- Jordan. This does not mean,of course, that every subset of real numbers is Peano-Jordan measurable). There are, however, only c Borel measur- able sets. This assertion follows from the propositions that there are c Open sets, that the o-ring of Borel measurable sets is generated by the collection Of Open sets, that is, the Borel measurable sets are the elements of the smallest o-ring Of sets containing the open sets, and, if M is a class of sets and the cardinality of M is less than or equal to c, then the cardinality of the o-ring generated by M is also less than or equal to c.41 Thus there are "fewer" Borel measurable sets than sets with content or extent. It was probably with this in mind that Borel appended the follow- ing statement to his definition of measure. "It is expressly under— stood that we will be speaking of measure only with respect to sets that we have called measurable. "However, if a set E contains all the elements of a measurable set E1, of measure a, we will say that the measure of E is greater than a without inquiring whether E is measurable or not. Inversely, if E1 contains all the elements of E, we will say that the measurswof‘E is 418ee Halmos, Measure Theory, 26. 101 less than c. The words greater than and less than do not, moreover, exclude equality." This statement contributed to the subsequent history of measure theory; in particular it became one of the issues of con- troversy in the polemic between Borel and Lebesgue. Arthur Schoenflies promulgated a critique of Borel's theory of measure in his treatise The DevelOpment of the Theory p_i: £22.53; _S_e_t_:_s_. Schoenflies' work is of interest in two respects: first, it was pub- lished in 1900 and hence after Borel had published his theory of measure but before Lebesgue had published his work on this subject, and second, Schoenflies was at the time a well-known and respected mathematician, and his treatise was a standard reference, being frequently cited in the literature. Indeed, Schoenflies' text was an outgrowth of a re- port on the tOpic "curves and point sets," commissioned two years before by the German Mathematical Association.42 The work might be expected to reflect a conservative point of view'and therefore to illus- trate the skepticism with which contributions to the emerging theory of functions of a real variable continued to be received. Schoenflies began his discussion of "the content of point sets" by remarking on the existence of three essentially different theories of measure. "The consideration of the content of point sets con- stitutes a subject from which various controversies have emanated. On one hand, we utilize results that could appear to be paradoxical; on the other hand, the definition of content, like every mathematical definition, has above all a certain subjective character, and the con- sequences that proceed from it vary, if it has been chosen in accordance 42Schoenflies, Die Entwickelung der Lehre von den Punktmannig- faltigkerten, iii. 102 with the purpose to be accomplished." Schoenflies reviewed the conceptions of Hankel, Cantor and Harnack, and Peano and Jordan, and continued, "An essentially different position has been taken recently by E. Borel. Borel does not add the limit points to the point set P, and dismisses the requirement that a finite number of regions contain all the points of the set. He imagines that every point of P is surrounded by an arbitrary domain, and considers the areas of these and their limits respectively. A consequence of this definition is that all countable point sets have content zero...." Schoenflies was critical of Borel's form of definition of measure. "...Borel has taken [additivity] as the basis of his definition of con- tent. He considers this as the essential prOperty Of the concept of content....If the continuum C on a line is divided into two point sets P and P1, then it must follow that 1(a) = J(P) + J(P1). To this Borel adds the second requirement that this equation must hold for any countable collection of point sets....Now [this] second require- ment of Borel has of course by no means the same character as the first. It has above all only the character of a postulate; the question of ex- tending a prOperty of finite sums to the sum of infinitely many terms cannot be settled by assertion but requires investigation...."l'3 Borel's postulates describe the properties he wishes the concept of measure to possess. But where, Schoenflies seems to ask, are Borel's 43Schoenflies, 22, cit., 93. 103 proofs of existence and uniqueness of a measure with these prOperties? Of course these objections are sound; proof of Borel's assertions came only with Lebesgue. Schoenflies ended by rejecting the theories of both Borel and Peano and Jordan. "Since in applications it is always only a question of the outer content, I will from now on call it the content of T, designated by J(T)." Thus Schoenflies conceived the measure of Cantor and Harnack to be adequate for the theory of integration. Not all mathematicians dismissed Borel's conception of measure. Even as Schoenflies was writing his monograph, Lebesgue was develOping his theory of measure as a completion of that of Borel. "[Borel's] definition was to inaugurate a new era in analysis: in connection with the contemporaneous work of Baire, it formed the point of departure of a whole series Of researches of a topological nature on the classi- fication of sets Of points; above all, it went to serve as a basis for the extension of the notion of integral, realized by Lebesgue in m. the first years of the 20th century.‘ This is the extension to which we turn in the next chapter. 44Bourbaki,gp, cit., 250. 104 Chapter 3 On the Lebesgue Integral Lebesgue measure and integral Lebesgue undertook his study of measure and integral in an at- tempt to free classical results in analysis from restrictive continuity hypotheses. Thus, the Riemann integral solved the problems of determin- ing the primitive of a continuous derivative and the length of an arc with a continuously turning tangent line in the forms [ f’(x)dx = f(x) + c {b and <1+>2)1’2 dx, Ja respectively. Volterra's example Of a bounded derivative which is not Riemann integrable, and Scheeffer's example of a continuous increasing function whose derivative is unbounded in any interval1 had demonstrated, however, the failure of these classical forms in the absence of the continuity requirement necessary for the existence of the Riemann inte- gral. Lebesgue's investigation of these and other classical theorems led to his generalization of the concept of integral. The succession of ideas which resulted in Lebesgue's definition of integral, and which formed the basis of his thesis, can be found in a sequence of papers published by Lebesgue between June, 1899 and April, 1901. In the first Of these, "On several non ruled surfaces applicable on the plane," Lebesgue prOposed "to seek to determine if there exist 1These examples are given in Appendix C. 105 surfaces applicable on the plane other than developable surfaces."2 Bonnet had proved that a surface is applicable on the plane if and only if it is develOpable,3 a classical theorem whose generalization requires a less restrictive hypothesis than that of a continuously turning tangent plane. Lebesgue gave a procedure for identifying non develOpable sur- faces applicable on the plane, basing his procedure on the exemplar of a crumpled handkerchief or sheet of paper. He concluded, "These examples demonstrate that the question of finding all the surfaces applicable on the plane is not completely resolved by the theorem of Ossian Bonnet." Montel wrote, with respect to Lebesgue's violation of the con- ventional bounds of classical differential geometry, "This observation [of non develOpable surfaces applicable on the plane], in conjunction with the construction of polyhedrons by means of cutout cardboard, was the origin of the great discovery to which his name remains attached, this integral of Lebesgue, which for the study of discontinuous func— tions is the principal algorithm created since the series of Fourier."4 Lebesgue,"Sur quelques surfaces non régleés applicables sur 1e planf'1503. 3Roughly speaking, surfaces which can be continuously deformed into each other in such a way that length Of every arc on either of the surfaces is preserved, are called applicable. A ruled surface is a surface which can be generated by the continuous motion of a line in space. The instantaneous positions of the line are called generators of the ruled surface. A develOpable surface is a ruled surface with the prOperty that it has the same tangent plane at all points of a given generator. 4Montel,"Notice nécrologique sur M. Henri Lebesgue:'l98. 106 In his second research note, "On the definition of the area of a surface," Lebesgue stated, "the problem of measure of plane surfaces" in the following way: "To make correspond to each surface a number called its area, in such a way that two equal surfaces have equal areas, and that the surface formed by the union of a finite or infinite number of [non overlapping] surfaces, has area the sum of the areas of the composing surfaces."5 Thus Lebesgue had adopted Borel's conception Of a countably additive measure. Lebesgue's third and fourth notes are a continuation of his investigations of the concept of surface area. The fifth note in this sequence, "On a generalization of the definite integral," includes statements Of definition of Lebesgue measure and integral. Lebesgue began by reviewing prOperties of the Riemann integral with respect to existence of a primitive. "In the case of continuous functions there is an identity between the notions of integral and primitive function. Riemann defined the integral of certain dis- continuous functions, but not all derived functions are integrable in the sense of Riemann. The problem of the research of primitive functions is therefore not resolved by [Riemann] integration, and one can seek a definition of integral which includes the Riemann integral as a par- ticular case, and which permits the resolution of the problem of primitive functions."6 After reviewing Riemann's definition of integral, Lebesgue stated 5Lebesgue,"Sur la definition de l'aire d'une surfaceJ'870. 6Lebesgue,"Sur une generalisation de 1'intégrale definie:'1025. 107 his own definition: "Let y be a function [bounded below and above respectively by] m and M. We give m = m0 < 1111 < m2 < "°< mp_1 p y = m when x is a member of a set E0; mi!1 < y _<_'m.i when x is a member of a set E1. "We will define the measures >‘0, 41 of these sets below. Con- sider one or the other Of the two sums . A o 0 1 1 ' mo 0 I Z A O m1-1 1 1 if, when the maximum difference between two consecutive mi tend toward zero, these sums tend toward the same limit independently of the choice of the mi, then this limit will be by definition the integral of y, which will be called integrable."7 Lebesgue immediately defined his conception of the measure Of a set. "Consider a set of points of (a,b); one can in infinitely many ways enclose these points in a denumerable infinity of intervals; the greatest lower bound of the sum of lengths of these intervals is the measure of the set. A set E is said to be measurable if its measure plus the one of the set of points not contained in E gives the measure of (a,b)." Lebesgue continued by remarking that the countable union or intersection of measurable sets is measurable, and measure is countably additive on disjoint sequences of measurable sets. "It is natural to consider...functions such that the sets which figure in the definition of the integral are measurable. One finds that 7Lebesgue, 22, cit., 1026. 108 if a function bounded in absolute value is such that, for every A and B, the set of values of x for which A_A_(E). "Mr. Jordan calls measurable the sets whose interior and exterior extents are equal; these sets which wecall measurable (J) are therefore measur- able in the sense we have adopted...." Lebesgue completed his discussion of the Lebesgue measure of sets on the real line by demonstrating that the measure of the set "of points common to all the measurable sets E E 00- which are such that each con- 1’ 2’ tains all those that follow it, is the infimum of the sequence m (E1), m(Ez),...," Thus, if E13 E23 ..., then m (£51131) - 111m m(Ei). Lebesgue then extended his definition of measure to point sets in "a space of several dimensions," in particular to sets of points in the plane. It is Of interest to attempt to understand how Lebesgue was led to his definition of interior measure. He commented in this respect in his memoir "On the develOpment of the concept of integral," published in 1927. Lebesgue remarked that the measure in (E) of a set E formed of an infinity of disjoint intervals is defined to be the sum of the lengths of the intervals. In the general case this "leads us to proceed as follows. Enclose E in a finite or denumerable infinity number of inter- vals and let 2 2 1, 2,--- be the length of these intervals. We evidently 'wish to have m(E) ; 21 + £2 + 000. "If we seek the greatest lower bound of the second member for all possible systems Of intervals that cover E, then this bound is an upper bound of m(E). For this reason we represent it by m(E), and we have 1) m(E) _<_ m(E). "If C is the set of points of the interval (a,b) that do not 117 belong to E, we have similarly m(C) _<_ m(C). "Now we manifestly wish to have m(E) + m(C) B m ([a,b]) = b-a 3 and therefore we must have 2) m(E) 3_(b-a) — m(C). "The inequalities l) and 2) give upper and lower bounds for m(E)....When the upper and lower bounds for m(E) are equal, m(E) is defined, and we then say that E is measurable."12 In the second chapter of his thesis, Integrals, Lebesgue utilized his theory of measure to create a more general theory Of integration. His first definition of integral was expressed in a geometric form, i.e., the integral of a function as the measure of its set of ordinates. "From the geometric point Of view the problem of integration can be eXpressed as follows: "Being given a curve C by its equation y-f(x) (where f is a continuous positive function...), find the area of the domain bounded by an arc of C, a segment of the x-axis, and two parallels to the y-axis whose abscissas are a and b where a < b. "This area is called the definite integral Of f taken between b the limits a and b and is represented by f(x)dx." a Lebesgue reviewed the geometric definition of integral given by 12Lebesgue,"Sur 1e développement do 1a notion d'integrale," 153-154 0 118 Peano and Jordan. "In order that the function f be integrable it is necessary and sufficient that [the set of ordinates] E be measurable (J); the measure of E is the integral." If the function f is Of arbitrary sign, that is, not necessarily positive, then the set E is the union of the set E1 of positive ordinates and the set E2 of negative ordinates. "The intégrale122£.défaut is the interior extent Of E1 minus the exterior extent of E2; the intégrale‘pgg giggg'is the exterior extent of E1 minus the interior extent of E2. If E is measurable (J), in which case E1 and E2 are also, then the function is integrable, with integral A(El)-A(E2). "These results immediately suggest the following generalization: if the set E is measurable, in which case E1 and E2 are also, we will call the definite integral of f, taken between a and b, the quantity "The corresponding functions f are called summable." It may bear repeating that Lebesgue later called such functions "measurable." In order to avoid confusion (since the term "summable" is a member of the technical vocabulary as will be seen below), I will con- form to Lebesgue's later usage by referring to such functions as "measurable functions." Lebesgue continued, "Relative to nondmeasurable functions, if such exist, I will define the inferior and superior integrals as equal to mi(E1)~me(E2) me(E1)-mi(E2) . 119 "These two numbers are included between the intégrale‘225_défaut and the intégrale 233; fly” Thus if a function is Riemann integrable it is Lebesgue integrable, and since the Dirichlet function is integrable in the sense of Lebesgue, Lebesque's definition of integral is a generalization of that of Riemann. Lebesgue next turned his attention to an analytic definition of integral. His first result in this direction was an analytic character— ization of measurable functions. Arguing by means Of the definition Of measurable function, the relationship between Borel measurable and Lebesgue measurable sets identified above, and the fact that the inter- section Of a planar Borel measurable set by a line parallel to a coordin- ate axis is a linear Borel measurable set, Lebesgue succeeded in estab- lishing this analytic form of measurability: a bounded function f(x) is measurable if and only if for arbitrary real numbers a and b, a > b, the set {xIa>f(x)>b} is measurable. Lebesgue immediately utilized this characterization of measura- bility in his analytic definition of integral. As a prelude to his definition he wrote, "Let f(x) be a continuous increasing function de— fined between G and B (o n and the measurability of the limit inferior of the sequence {en}. The "great importance" of this prOposition will become apparent below. Lebesgue stated a definition of integral for a bounded measur- able function defined only for the points of a (measurable) set E. "Let AB be a segment containing E and define a function ¢ to be equal to f for the points of E and equal to zero for the points of CA (E). The integral of f taken in E is, by definition, the integral of ¢ taken in AB." Thus Lebesgue could integrate functions over more general sets than intervals. In particular, "If E is the union of E1,E2,..., all these sets being measurable and [mutually disjoint], and if the function f is measurable in E, then one has f(x)dx = Z f(x)dx ." E E1 Lebesgue extended his definition of integral to unbounded functions in the following way. "The geometric method which was so useful to us at the beginning of this chapter, being based on the concept of the measure of a bounded set, applies only to bounded 16 functions. To the contrary, the analytic method [of definition] can be applied almost without modification to [unbounded] functions. "A function is called measurable if, for arbitrary a and b, Lebesgue observed in a footnote, however, that "there is no difficulty in stating the problem of the mea f bounded or not." sure 0 points for all sets, 125 the set of values of x for which a < f(x) < b is measurable. "Let f(x) be a summable function. Choose the numbers -o-b, the set {x|a>f(x)>b} is measurable (B). Let C be the Cantor set on [0,1] and E be a subset of C which is not Borel measurable. Define f(x) - 0 for every xs[0,l] and o if xs[0,l] - c g(x) - 1 if xe C-E -1 if XEEO Then f is Borel measurable, f = g a.e, and yet g is not Borel measurable. Thus, in this circumstance, the measurability of a function may be lost by changing its values on a set of measure zero. Lebesgue's measure is complete and therefore if f is an integrable function and if f - g a.e. then g is integrable; furthermore 131 This property of completeness is also eXpressed with respect to the measurability of the limit a.e. of a sequence of measurable functions. Define, for example, the sequence {fn} of Borel measurable functions by .1 if xe [0,1] - C n (-1)n if xeC. Then lriimfn - g a.e. where g is the function defined above. Thus, if Lebesgue's bounded convergence theorem were expressed in this more general form, i.e., lémfn - f a.e., for Borel measurable functions, then an additional hypothesis would be required to ensure the measura- bility of the limit function f. Of course the limit a.e. of a sequence of Lebesgue measurable functions is Lebesgue measurable. The validity of Lebesgue's bounded convergence in this more general case may help to explain why a measure which is complete is preferred to one which is not. Lebesgue gave the Cours Peccot on integration theory at the College of France during the academic year 1902-1903. His lectures were collected and and published in 1904 as one of the Borel monographs on the theory of functions under the title, "Lessons on integration and research on primitive functions." Lebesgue devoted six of the seven chapters of his text to the history of the development of the concept of integral. "A complete history could not be given in twenty lessons; thus leaving aside many important results . . . [and] numerous defini- tions which have been successively prOposed for the integral of real valued functions of one real variable, I have retained only those that 132 in my Opinion are indespensible to know in order to understand well all the transformations the problem of integration has undergone and in order to comprehend the relationships that exist between the notion of area, so simple in appearance, and certain analytical definitions of the integral with very complicated aspects."20 Again, Lebesgue attempted to justify his work. "One may ask, it is true, if there is any interest in considering such complications and if it would not be better to limit oneself to the study of functions which only necessitate simple definitions. This has hardly any ad- vantage . . . [for] as one will see in these lessons, if one wishes to limit onself to consideration of these simple functions, it will be necessary to renounce the resolution of many problems with simple statements which have been asked for a long time. It is for the resolution of these problems, and not through love for complications, that I have introduced in this book [my] definition of the integral . . . ." Lebesgue expostulated with those who objected to the study of discontinuous functions. "Those who read me carefully, while regretting perhaps that things are not simple, will agree with me, I think, that this definition is necessary and natural. I dare to say that it is in a certain sense simpler than that of Riemann, as easy to comprehend as his, and that only habits of mind which have been acquired earlier can make it seem more complicated. It is more simple because it places in evidence the most important prOperties of the integral . . . ." Lebesgue noted that his researches on primitive functions and 20Lebesgue, Lecons sur l'intégration.g£.l_ recherche des MW. v. 133 rectification of curves were given in his course as applications of his definition of integral. "To these two applications I wished to join another which is very important: the study of the trigonometric development of functions, but in my course I have given this subject such scant attention that I have decided not to reproduce it here." Lebesgue had pursued research on trigonometric series representation of Lebesgue integrable functions during his Cours Peccot tenure, the results of which were published as separate papers. Having described the theories of integration of Cauchy and Riemann, Lebesgue began his investigation of the "problem of integration" by stating conditions to be satisfied by an integral "if one wants that there be some analogy between [it] and the integral of continuous functions." Lebesgue prOposed "to attach to each bounded function f(x), defined in a finite interval (a,b), a finite number, positive, negative, or zero, be(x)dx, that we call the integral of f(x) in (a,b) and which satizfies the following conditions: 1. For arbitrary, a, b, h, one has b b+h f(x)dx - f(x-h)dx. a a+h 2. For arbitrary a, b, c, one has b c a f(x)dx + f(x)dx + f(x)dx - 0. a b c L _ 134 b b b 3. (f(x) + ¢(x))dx . f(x)dx + ¢(x) dx. a a a 4. If fzp and b>a then b f(x)dx _>_ 0. a 5. One has ldx a l O 6. If ft5x) tends increasingly toward f(x), the integral of fn(x) tends toward the integral of f(x)."21 "In enunciating the six conditions of the problem of integration, we define the integral. This definition belongs to the class Of defini- tions that one can call descriptive; in these definitions one states the characteristic prOperties of the object that one wants to define. In constructive definitions, one states which Operations it is necessary to complete in order to obtain the Object that is defined. "When one states a constructive definition it must be shown that the indicated Operations are possible; a descriptive definition is also subject to certain conditions: it is necessary that the con- ditions stated be compatible [i.e., non contradictory] . . . It is also necessary to study the possibly ambiguous nature of the Objects that one wants to define. Suppose, for example, that one has demonstrated the impossibility of the existence of two different classes of Objects satisfying the conditions indicated and that, in addition, one has 21Lebesgue,‘gp, cit., 98. 135 demonstrated the compatibility of these conditions by choosing a class of Objects satisfying them. This class of objects is then uniquely defined, so that the constructive definition which has served to effect the choice is exactly equivalent to the descriptive definition. "We seek a constructive definition equivalent to the descriptive definition of the integral." Lebesgue stated in this respect the primary purpose of his work. ". . . one can say that the investigations reported in this treatise have as their principal goal the discovery of a constructive definition equivalent to the descriptive definition of primitive functions."22 Lebesgue succeeded in demonstrating, for the class of Lebesgue integrable functions, that there is exactly one constructive definition of integral which satisfies the descriptive definition, and that is the definition of the Lebesgue integral. Thus "the reasonings employed show that the problem of integration is possible and in only one way, if it is posed for summable functions." Lebesgue resolved a question which had remained Open in his thesis by proving this theorem: ". . . the indefinite integral Of a summable function admits this function as its derivative except at the points of a set Of measure zero." That is, if f(x) is integrable on [a,b], then x f - f(x) a.e 22Lebesgue, 22, cit., 100. 136 This remarkable theorem is only one of many stated and proved by Lebesgue with respect to "the research of primitive functions," which is of course the focal point of the Lecons.23 23It is necessary to mention the work Of W. H. Young in his attempt to generalize the Riemann integral as the natural generalization of the definition given by Darboux. In a paper titled "0n the General theory Of Integration" published in 1904, Young recalled Darboux's definition Of Riemann integral, and in particular, that the interval of Of integration is partitioned into a finite number Of subintervals. "The progress of the modern theory of sets of points . . . due, as is well known, chiefly to G. Cantor, though taking its origin in Riemann's paper 'Ueber die Darstellbarkeit . . .' naturally leads us to put the question how far these definitions can be generalized. This theory has in fact taught us on the one hand that many of the theorems hitherto stated for finite numbers are true with or without modification for a countably infinite number, and on the other hand that closed sets of points possess many Of the prOperties of intervals. "What would be the effect on the Riemann and Darboux definitions, if in those definitions, the word 'finite' were replaced by 'countably infinite,‘ and the word 'interval' by set of points? A further question suggests itself: are we at liberty to replace the segment (a,b) itself by a closed set of points, and so define integration with respect to any closed set of points?" In the course of his investigations of these problems, Young created a definition of measure and integral equivalent to that of Lebesgue. The accomplishments of the two men, however, are reflected in their intentions. Young wished to generalize the definition Of integral and succeeded in this. Lebesgue wished to resolve fundamental problems in analysis and created his theory Of measure and integral as a means to this end. Lebesgue's attempts to reinstate the classical forms relating integral and derivative, for example, led to his bounded convergence theorem and its consequent, integration of any bounded 137 It is of interest for our purpose to call attention to Lebesgue's publication Of a convergence theorem Of which the bounded convergence theorem is a special case. In "On the method of Mr. Goursat for the resolution Of the equation of Fredholm," printed in January, 1908, Lebesgue stated and proved this theorem: "A convergent sequence of summable functions f is integrable term by term when there exists 1 a summable function F such that, whatever be i and the variable x, 24 lfil 5. IF Of Lebesgue measure, if {fn} is a sequence of summable functions and In modern terms, and taking advantage of the completeness if h is a summable function such that ltimfn - f a.e. and Ifn]§_h for every n, then f is summable and - n n n This prOposition is called Lebesgue's dominated convergence theorem. It is to be expected that Lebesgue's work was rejected by some members of the mathematical establishment. Denjoy, in a memoir "Henri Lebesgue, the scholar, the teacher, the man," published in 1957, derivative. Young did not discover these results because he was con- cerned with the question of definition rather than application of his integral. See Pesin, Classical‘gng.Modern Integration Theories for a description Of Young's work and its relationship with that of Lebesgue. 24Lebesgue,"Sur la méthode de M. Goursat pour la résolution de l'equation de Fredholm,"12. 138 observed that Lebesgue "presented in his thesis . . . to mathematical service a tool Of extraordinary power." Yet "The acceptance Of the works of Lebesgue by the masters Of the time was rather reserved. Many feared to see installed a teratology of functions. Darboux, whom one might have thought favorable because of his memoir of 1875 on dis- continuous functions, was hostile to him. Boussinesq was supposed to have said, 'But a function has every interest in having a derivative!‘ He was speaking of the interest of he who uses it. Only Picard defended the research of Lebesgue and appreciated its qualities."25 Lebesgue reflected on these and other aspects of the objections which had been raised against the study of the functions of real variables in the Introduction of his "Notice on the scientific works Of Henri Lebesgue," published in 1922. "In order to demonstrate the state of mind at the time when I began my research, I will indicate certain re- sistance which I encountered; all those who have taken up the same type of studies have met analogous resistance. I can do this without hesita- tion, for it has never been anything but conflicts in ideas, and I have always found the greatest personal goodwill in the case Of those very peOple to whom my works were the least agreeable.26 "In 1899 I remitted to Mr. Picard a note on non ruled surfaces applicable on the plane; Hermite wished for a moment to Oppose its insertion in the Comptes Rendus of the Academy; Mr. Picard had to de- fend my note. One knows how much, however, Hermite was filled with 25Denjoy,"Henri Lebesgue, ES Savant, Le Professeur, L'Homme,"15. 26Lebesgue, Notice sur les travaux scientifiques de M, Henri Lebesgue, 13. 139 goodwill and praise, but this was near the time he was writing to StieltjEs, 'I turn away with fright and horror from this lamentable plague of functions which have no derivatives,‘ and he wished to see excluded from the domain of mathematics all research in which these horrifying functions intervene." Lebesgue observed that he became, for many mathematicians, the man of functions without derivatives, a charge which Lebesgue denied. Nevertheless, "as the horror manifested by Hermite was felt by nearly all, as soon as I tried to take part in a mathematical conversation there was always an analyst to tell me 'this cannot interest you; we are talking about functions with derivatives," and a geometer to repeat in his language 'We are taking up surfaces having a tangent plane.'" Lebesgue expressed doubt that Darboux ever entirely pardoned his memoir on applicable surfaces and noted that "for a long time he was hardly interested in my memoir on integration . . . .It is said that in 1875 Darboux was somewhat criticized for having allowed himself to study such questions; whether because of these remonstrances, or whether because Of the beauty and importance of the problems he took up after that, Darboux made no Other incursion into the domain of non analytical functions." Lebesgue believed that Borel was the first "to think that my work would have practical utility in some way. He did, in any case, think it before I did. I saw myself still hesitant before deciding to present as a doctoral thesis the memoir where I took up nearly all the research that I have since developed . . . .A little later, in 1903, I insisted on the necessity of these studies in the preface of my Lessons 140 22 Integration. In an analysis of the book, Mr. Picard, while encourag- ing me as he has always done . . ., allowed some uneasyness to show through on the subject of possible exagerations of the tendency that I represented." Given wideSpread resistance to the study of functions of real variables, Lebesgue asked if these Objections might have merit. He responded to this question by Observing that most Of the prior works on real functions, except for those relating to trigonometric series, were without comprehension in the sense that one could attribute to them a coherent body Of theory. Many statements were negative; positive statements were sought after but seldom achieved. ". . .if one searched to generalize [a prOperty or definition], then one would too Often end up with a notion certainly new but serving nothing other than being defined. . . .If one sought for the most general functions possessing a certain prOperty or to which is applied a certain definition, then one would end up with a class of functions variable with the prOperty or definition envisaged, and which by consequent, could not naturally be put into any research; such had been the case for the class of functions integrable in the sense of Riemann." Lebesgue argued that since the work was essentially an explora- tion of a disordered mass of functions, properties, and definitions, without knowledge of interest or application, and no criteria with which to judge such questions, mathematicians could be led to think that researches with respect tO real functions could be suspended until the necessity of such researches became more apparent. Yet, "in spite 141 of the indifference and sometimes the opposition manifested with re- gard to the theory Of functions of real variables, . . .it happened, as in the past with trigonometric series, that one encountered func- tions whose analyticity did not have to be assumed. It was thus for example in the study of solutions of differential equations by the method of Cauchy-Lipschitz, or by that of successive approximations by Mr. Picard. . . .Sometimes certain of the data or solutions could be or even necessarily were discontinuous functions, . . .at other times, as in the questions studied by Mr. Borel, the solution is con- tinuous but non analytic. Thus one could become familiar with the idea that a discontinuity or a singularity is not necessarily a monstrosity."27 The reviews Lebesgue's work received in the Jahrbuch fibggngig Fortschritte der Mathematik.were mixed. Haussner wrote a three line review of Lebesgue's memoir "Sur une generalisation de l'integrale définie" as follows: "The author gives a generalization of the definite integral which encompases Riemann's definition as a special case and simultaneously permits the solution of the problem of primitive func- tions." On the other hand, Lebesgue's "Intégrale,longeur aire" received a very complete and unbiased review from Kowalewski. Again, Stackel, reviewer Of Lecons _8_l_l_£ l'intégration _et _13 recherche £13 fonctions rimitives, was moved to write "The definition of Lebesgue [integral] . . . corresponds to a need which Obviously cannot be CORCGSted- However, if one goes so far to say, as does the author, that in a certain sense it is more simple than Riemann's theory, and is just as easy to comprehend, since only certain methods of thought which had been assumed previously 27Lebesgue, op, cit., 16. 142 let them appear complicated, then all these things are debatable to the reviewer. At any rate it will have to be expressed in a set theoretic context and will hardly be amenable to elementary lectures, for which purpose however the definition of Riemann is very suitable." Of course Lebesgue was not without support. Picard's advocacy has been noted. Lebesgue received his first university appointment in Rennes in 1902; he was chosen to give the Cours Peccot in 1902-1903. Indeed, in his inaugural lecture at the Collége de France, Lebesgue remarked how "the great authority Of Camille Jordan gave to the new school a valuable encouragement which amply compensated for the few reproofs it had to suffer." General acceptance of Lebesgue's ideas was another matter, however, and was achieved only as Lebesgue and others exploited the remarkable prOperties of his integral. One of the areas in which Lebesgue's conceptions were first applied was the study of trigonometric series. "Of all branches of analysis in which the use of an integral more powerful than Riemann's offered a rich reward, none was so promis- ing as the theory of trigonometrical series,"28 and it was to the in- vestigation Of such series that Lebesgue turned. Trigonometric Series The theory of trigonometric series representation of functions had continued to develOp in the period in which there was created a theory of measure and a more general theory of integration. The reader will recall the memoirs Of Heine and Cantor in which the uniqueness of representation of trigonometric series of functions was 28Burkill, "Henri Lebesgue," 58. 143 demonstrated for a particular class of functions, after the uniqueness of such representation had been called into question by the discovery of the relationship between uniform convergence and term by term in- tegration of infinite series. The propositions of Heine and Cantor reinstated the uniqueness of representation of trigonometric series under certain conditions but did not speak to the question Of the form Of coefficients of such representations. Ascoli considered this aspect Of the problem in his memoir "On trigonometrical series," dated April, 1872. "In a treatise on trigonometrical series...Mr. Heine has verified the following theorem: "A function which is in general continuous but not necessarily finite can be developed in at most one way in a trigonometrical series of the form 3) Z(a sinnx + b cosnx) 0 n n if the series is required to converge uniformly in general. "Shortly after that Mr. Cantor...demonstrated how a function given by a trigonometrical series which is convergent in general for every value of x cannot be represented by another series of the same form. From this it seems to follow that the preconditions which have been made in Heine's theorem on the continuity of the function and on the type Of convergence of the series are unnecessary. "Also it seems to me that if a periodically repeating function in the interval [0,2n] which is continuous in general...is representable by a trigonometric series of the form 3), then the develOpment is not only unique but must be the development of Fourier."29 29Ascoli, "Uber trigonometrische Reihen," 231. 144 Thus, Ascoli asserted, the coefficients of the unique trigonometric series representation of a function continuous except at a finite number Of points are the Fourier coefficients. Dini published a paper, "0n the series of Fourier," also dated April, 1872, which extended Ascoli's result to functions whose points of discontinuity have at most a finite number of limit points.30 These propositions were generalized in turn by du Bois Reymond. In his "Proof that the coefficients Of the trigonometric series f(x) 8 1T pm 1 Z (a cos x + b sin x) have the values a ='- f a do a - p=0 p p p p ().p 0 2n +w n -n .1 [f(a)cospodo, bp 8 iff(o)sin podo, whenever these integrals are 1T -1T 1T finite and determinate," published in 1875, du Bois Reymond argued that the coefficients of the trigonometric series corresponding to a Riemann integrable function are the coefficients of Fourier. Du Bois Reymond noted that "the main theorems of the theory of trigonometric series are called into question by introduction of the concept Of uniform conver— gence, these being the theorems "if a trigonometrical development Of the form f(x) - pg“ (apcospx+bpsinpx) is given then there is no second p-0 of the same form," and "the coefficients of the developments can only be expressed [in the form of the Fourier coefficients]. "Both these theorems were proven by term by term integration Of the series f(x) and both were suddenly left without meaning."31 3ODini, "Sopra la serie di Fourier," 161. 31duBois Reymond, "Beweis, dass die Coefficienten der 145 In his section "History of the further evolution of the study of trigonometrical series. The first theorem is restored," duBois Reymond Observed that the work Of Heine and Cantor "not only restored the theorem Of the unambiguity of trigonometric representations...but also gave to it a general applicability of which one had not thought before these events took place." Turning to the purpose Of his paper in the section "On the second main theorem. The author announces that he is able to restore it," du Bois Reymond wrote, "With regard to the first main theorem of the theory of trigonometric series, we can consider the research as being closed. There seems to be.no publication on the subject of the second theorem, that under certain necessary conditions on f(x) the coefficients have the form discovered by Fourier, so nothing has been clarified here. "I have known for some time that the second theorem can also be restored if one assumes f(x) to be continuous except at special points. trigonometrischen Reihe =00 f(x) - Z (a cospx + b sinpx) p-0 p 9 die Werthe ‘ TI' 1T II = .1;— _ l. g _l_ f(o)sinpodo a0 2" f(o)da,ap fl f(o)c08pado , bp fl haben, jedesmal wenn diese Integrale endlich und bestimmt sind,"121. 146 "This condition on f(x) is more restrictive, however, than integrability in general, which suffices for the existence of Fourier coefficients. The task which challenged my mathematical curiosity most was to find if the coefficients Of the series of f(x) have the form Of Fourier whenever f(x) is integrable.... "Now I believe I am able to solve the problem and in a most general fashion."32 Du Bois Reymond's proof of his theorem is extremely long and complicated.33 Indeed the length and difficulty of the proof were such thatchLBois Reymond was moved to comment in this respect in a concluding statement. "If the primeval saga is a juxtaposing of enjoyment without effort in paradise and the hard work of the just after the fall, then our science shows a similar anthesis. After the first analytical epoch, which ended approximately with Fourier and Poisson, and in which there ‘were many new discoveries of formulas and theorems but little concern with their precise formulation and range of validity... we, having eaten from the tree Of cognizance [of uniform convergence], have had to struggle with profound difficulties... in order to gain again these results for science."34 Some years later, in 1881, in a paper titled "On the integration of trigonometric series,"du.Bois Reymond Observed, "Heine's theorem shows that a Fourier series which converges [uniformly in general] to a function continuous at all but a finite or suitably grouped infinite set Of points 3au Bois Reymond,‘22, cit., 123. 33See Gibson, "On the History of the Fourier Series," 163, for a partial synopsis. 3duBois Reymond, 22, cit., 160. 147 admits integration in its members, but nothing can be said on the basis of Heine's theorem of the term by term integration of a trigonometrical series, assuming only that the sum is integrable."35 Du 3°13 Reymond was able to demonstrate that a trigonometric series which converges to a Riemann integrable function can be integrated term by term. Thus the Significance Of uniform convergence for such integration of a trigono- metric series was shown tO be less important than initially assumed. Of course the theorems of Argela and Osgood are generalizations of this proposition of du Bois Reymond. In the second section of the same volume of the Abhandlungen‘dgr Bayerischen Academie in whichchJBois Reymond proved his theorem that if a trigonometric series converges in (-n,w) to a Riemann integrable function f(x), then the series is the Fourier series of f(x), du 3018 Reymond published another long and arduous paper in whigh, by consider- ing particular forms of f(o) in the Dirichlet integralJ f(o)§i§hgda, he demonstrated the existence of a continuous function 0whose qurier series does not converge at a particular point, and more generally, a function continuous in (-w,w) whose Fourier series does not converge at the points of an everywhere dense set.36 Thus the question of Fourier series representation at every point of every continuous function was decided in the negative. Jordan obtained a simplified sufficient condition for convergence in his memoir "On the series of Fourier," published in 1881. The sufficient 3duBois Reymond, "Ueber die Integration der trigonometrischen Reihe," 260. Ban Bois Reymond,"Untersuchungen fiber die Convergenze und Divergenz der Fourierschen Darstellungsformeln,"72. 148 conditions Of Dirichlet wrote Jordan "depend upon the two following prOpositions: b 1) lim f(x) sinpx - O if<0n monotonically increasing sequence of real numbers. Therefore limbn exists 11% as a finite real number'or-ir. This limit is defined to be the limit inferior of the sequence {a } and is denoted lim inf a . If liminfan - a and b is n n+¢ n n+w any limit point Of the sequence {a&}then agb, that is, the limit inferior Of {afilis the least limit point of the sequence. This may help to explain the interest in Fatou's lemma. SlFatou, 22. cit., 379. 157 b 4) f(x) + K(x,y)f(y)dy = u(X) 8. where the functions K(x,y) and u(x) are given and f(x) is unknown, had introduced the concept of a complete orthogonal system.of functions {¢n}, and had, in the course of solution of the integral equation 4) sought the "generalized Fourier coefficients" of the unknown function f(x) with respect to {on}, that is, b ‘6‘... ' I cn - f(x)¢n(x)dx.52 f a Riesz commented in this regard in the paper in which he published his form Of the Riesz—Fischer theorem, "On orthogonal systems of func- tions," dated March 11, 1907. "Mr. Hilbert has introduced a general 'method for the resolution of certain functional equations of the type... of Fredholm. This method consists of relating the resolution of these functional equations to the resolution of an infinite system Of linear equations in an infinite number Of unknowns. Mr. Hilbert makes the con- nection between these two problems by using an orthogonal system of functions; the coefficients, like the unknowns of the latter equations, [are] integrals Obtained from the given functions and the unknown functions «of the problem, in a manner analogous to the coefficients of Fourier, *with the aid of an orthogonal system of functions. 52The definition of a complete orthogonal system of functions and .a description of Hilbert's method of solution of the linear integral equation 4) is given in Appendix D. 158 "For the method of Mr. Hilbert, the following question is of great importance: "Being given an orthogonal system of functions [on] a determined interval, attribute to each function of the system a real number. Then under what conditions will there exist a function such that for each function of the system [of orthogonal functions], the integral of the product Of this fmction and the function in question, taken on the in- terval, will be equal to the given number?" That is, if {¢n} is an orthogonal sequence of functions defined on [a,b] and if {cu} is a given sequence Of real numbers, than under what conditions does there exist a function f(x) such that b cn - f(x)¢n(x)dx for n - l,2,...? a Riesz continued, "For the class of summable functions, bounded or not, but whose square is summable, the theorem that I am going to give completely resolves the question."53 After Observing that "an orthogonal system of functions of which ..54 none has integral zero, must be finite or denumerable, Riesz stated 53Riesz, "Sur les systemes orthogonaux de fonctions," 616. 5"As a generalization of a theorem Of Schmidt, Riesz had proved this assertion for bounded functions in his memoir "Sur les ensembles de fonctions," published in November, 1906. Riesz Observed that the theorem "can be extended without difficulty to all square summable functions," i.e. , whether or not the functions are bounded. 159 his theorem. "Let ¢1(x), ¢2(x),... be a ... system of functions, de- fined on an interval [a,b], orthogonal two by two, bounded or not, sum- mable and square summable.... Attribute to each function ¢1(x) of the system a number a1. Then the convergence Of 2a: is a necessary and sufficient condition for the existence of a function f(x) such that one has b i-j f(x)¢i(x)dx a a1 5.. a . for each function ¢i(x) and each number a1." Riesz Observed, "The necessity of the given condition follows im- mediately from the well known inequality Of Bessel, given for continuous functions, but which remains true for arbitrary functions, summable and square summable." The necessity of this condition is also seen by appeal to Fatou's form of Parseval's equality (which is, of course, a strengthened form of Bessel's inequality in the presence of a complete orthonormal sequence). As Riesz stated, therefore, what must be shown is the suf- ficiency of the given condition. That is, if {ah} is a square summable sequence, then does there exist a square summable function f such that for every n f a = f ¢n ? Riesz was successful in demonstrating the existence of such a function f. In the proof of existence of this function Riesz utilized in an essential way the convergence properties of the Lebesgue integral; 160 properties which fail for the Riemann integral. Thus a Riesz-Fischer theorem.is not possible in the context Of Riemann's theory Of integration. Fischer demonstrated an equivalent form Of this theorem.in his note "0n convergence in mean," published April 27, 1907. He wrote "On the 11th Of March, Mr. Riesz presented to the Academy a note on ortho- gonal systems Of functions.... I had arrived at the same result and had demonstrated it at a conference Of the mathematical society in Brflnn on ‘March 5th. Thus my independence is evident, but the priority of publica- tion belongs to Mr. Riesz."55 As a prelude to the statement and proof of his form of the Riesz- Fischer theorem, Fischer introduced the notion Of convergence in mean. Let Q be the set Of real [valued] functions of a real variable x such that f and f2 are summable...on a finite interval (a,b). Then "a sequence f1,f2,... of functions belonging to Q is said to converge in mean if b lim (f —f )zdx=0 m n ° m’n-Hao a [The sequence] converges in mean toward a function f of 9 if b 2 lim (f-f ) dx - 0; new n a 'we will write then the 'equivalence' limfn mf. This does not imply the n 55Fischer, "Sur la convergence en moyenne," 1023. 161 existence Of a limit in the ordinary sense of the word." That is, the sequence fn need not converge a.e. to f. Indeed, consider the following sequence of functions. For each natural number n define n functions. n n n f1 ’ f2 '°"' fn on the interval (0,1] by 1 if 121-( 2 1’2 which is a generalization of the finite dimensional Euclidian distance n 2 1 2 d(X.y) -( 2 (xi-Y1) ) I . 1-1 If f and g are in L2[a,b], then the product fg is sumable. Thus the inner product ([f],[g]) of [f] and [g] is defined by 58See Berberian, Introduction 52Hilbert Space for an elementary account of the theory. 165 b ([f],[g]) - f8. a which is a generalization of the finite dimension Euclidean inner product n (x,y) . z X y e 1_1 i 1 Finally, the existence of an inner product makes possible the definition of the norm of [f], b 2 1/2 llIfJII-RTTTHYTT-(f . a which is a generalization of the norm of a vector in a finite dimensional Euclidean space. Now if [f] and [g] are two equivalence classes of functions of L2[a,b], then the distance between [f] and [g] is M [f] - [g] H . Thus a sequence of equivalence classes of functions {[fnl} converges to an equivalence class of functions [f] if and only if lim [Ilf] - [full] - 0, that is, convergence in the space of equivalence clgsses of functions is convergence in mean. In this context the Riesz-Fischer theorem asserts that if {[fnl} is a sequence of equivalence classes of functions of L2[a,b], then a necessary and sufficient condition that there exists on feL2[a,b] such that {[fnl} converges to [f] is that lim II [fa] - [fm] [I - O. This theorem, which is analogous to thenCauchy convergence theorem, shows that any Cauchy sequence {[fnl} converges, i.e., the space Of equivalence classes of functions of Lzla,b] is complete. It might be observed in this respect that the form of the Riesz-Fischer theorem proved by Riesz implies the form given by Fischer since Hilbert space is 166 complete in the inner product norm. Thus the two forms are equivalent which explains the name of the theorem. Fischer emphasized the necessity Of Lebesgue's conceptions for the validity Of the Riesz-Fischer theorem in a note "Applications of a theorem on convergence in mean" published on May 27, 1907. "I shall prove that use of the notions Of Mr. Lebesgue is necessary for our subject. Let H be the set of continuous functions, and n1, n2, ... [be an ortho- normal sequence of continuous functions]. Then there exists in 059 a L‘un—____—_‘.a.- . 1 function X which is essentially different from all the functions of n. Suppose Then the series of continuous terms alfll + azwz + ... converges in mean without converging in mean toward any continuous function; therefore for H, the theorem fails in general."60 Riesz remarked with respect to geometric aspects of the Fourier coefficients Of square summable functions on June 24, 1907, in a memoir titled "On a type of analytic geometry of systems of summable functions." He wrote "In a lecture given in GSttingen to the mathematical society, February 26th of this year, I set forth the results of my research on 59Recall that in his previous Cgmptes Rendus note Fischer had defined 0 to be the set of all real valued functions of a real variable x such that f and f2 are summable on a finite interval (a,b). 60 Fischer, "Applications d'un theorem sur la convergence en moyenne," 1150. 167 systems of summable functions. Afterwards I communicated the principal of these results in two notes published in Cgmptes Rendus.... "The goal of my research was to investigate the method of co- ordinates applied to the study of systems of summable functions. To whom goes the credit for having introduced the notion of coordinates into the theory of summable functions? It would be difficult to say. What is certain is that after the fundamental results relative to r! Fourier series...the idea of representing a function by its Fourier constants ought to have become very familiar. In this fashion, one ~-« arrived at representing the set of summable functions by a subset of the space of a denumerable infinity of dimensions. What is this subset? Until today no one could really say. "Now, for a more special class, the system of square summable functions, the solution of the prOblem no longer carries with it so many difficulties. For this class there exists a more intimate bond between the function and its Fourier series.... For this class Of functions one can define a notion of distance and can found upon this notion a geometric theory Of systems of functions, a theory which resembles synthetic geometry. On the other hand, the notion Of distance can also be defined in a simple manner for a subset of points of our space, the set of points whose sum of squares of coordinates converge. Now thanks to the theorem on the integration of the product of two functions represented by their Fourier constants, the bond between these two notions of distance is very intimate; it permits one to make a correspondence between the synthetic geometry of functions and an analytic geometry. This parallelism of the two theories becomes complete only through my theorem of existence which 168 assures that each point playing a role in this analytic geometry can be regarded as the image of a square summable function. Then, the whole geometry of our subset Of points, a geometry which can be developed without difficulty, can be translated into a theory of systems of square summable functions.... "...it is the analytical theory I had in mind. On the contrary, in his two notes...Mr. Fischer developed, in a very elegant manner, the synthetic theory. . . ."61 It remains to make explicit the relationship between the Riesz- Fischer theorem and the theory of Fourier series. This special case of the Riesz-Fischer theorem is given as follows: if {an} and. {bn} are sequences Of real numbers such that 1 2 w 2 2 1904-1151 (an+bn) <00 then there exists a function f e L2 [0,2H] such that 5) [260 + g (a cosan+ b sinnx) n-l n n is the Fourier series corresponding to f, and conversely. Thus the Riesz-Fischer theoremimplies a very general solution to the prOblem of existence of a function f such that a given sequence of con- stants {ch} is the set of Fourier coefficients of f. It should be noted, however, that the Riesz-Fischer theorem does not speak to the questions of pointwise convergence or representation of a function by its corres- ponding Fourier series. The sequence of partial sums of 5) converges in 61Riesz, "Sur une espece de Geometric analytique des systems de fonctions sommables," 1409. 169 mean; this does not imply convergence of the series at a point xs [0,2n], nor in case of convergence, convergence to f(x). It is natural to attempt to determine conditions such that the Fourier series corresponding to a square summable function represents the function. In 1915 Lusin conjectured that if a function is square summable then it is represented almost everywhere by its corresponding Fourier series. This sufficient condition for Fourier series representation remained open until 1966 when Carleson succeeded in affirming its validity. Lusin and Carleson The first sufficient condition for representation almost everywhere of a function feL2[0,2n] by its corresponding Fourier series was given ’"criteria for convergence by Fatou. "One can seek," Fatou remarked, [Of trigonometric series], or, supposing that convergence takes place, seek the prOperties of the functions thus defined. These prOblems, which appear difficult, have been little studied." With respect to the former, criteria for convergence, Fatou Obtained the following proposition as a consequence Of Parseval's equality. "...1et an and bn be the Fourier coefficients of f(x); if the series 2n(a: + bi) is convergent, then f(x) [is represented by its corresponding] Fourier series, except perhaps for a set of values Of x of measure zero; pratically this proposition does not seem very useful."62 After examining several examples Of trigonometric series which diverge at certain points, Fatou proposed this problem. "Here is a —__ 62Fatou, op, cit., 379. 170 question...which appears interesting to me and for which I have not been able to find a solution: consider a trigonometric series whose coef- ficients tend toward zero; we have seen that [such a trigonometric series] can have points of divergence in every interval, but the set of points for which we can demonstrate divergence... is always of measure zero. ,Can one give an example of a trigonometric series, whose coefficients tend toward zero, and which is divergent for all values for a set of non— ‘eero measure of values of the argument?"63 Three years later, in 1909, Weyl obtained a generalization of Fatou's proposition regarding sufficient conditions for convergence almost everywhere of a trigonometric series. Weyl considered the con— vergence Of the series c1¢1(x) + c2¢2(x) + c3¢3(x) + ... where the orthonormal functions ¢1(x),¢2(x),... are defined on the in- terval 0 §_x‘§.1. He succeeded in demonstrating, by a method due to Jerosch and developed by Weyl, that if the functions ¢n(x) are such that I¢n(x)l< M for every n and x£(0,l), then the trigonometric series 2 °n¢n(x) n-l converges almost everywhere on (0,1) if an 1 2c: .3 is convergent.64 63Fatou, 22, cit., 398. 64Weyl, "fiber die Konvergenz von Reihen, die nach Orthogonalfunktionen fortschreiten," 241. 171 In the axverse direction, Lusin gave an affirmative answer to Fatou's question regarding divergence of a trigonometric series on a set of positive measure by publishing in 1911 the first example of a trigonometric series whose coefficients tend to zero, and which di— verges almost everywhere on [0,211].65 Hobson contributed a generalization Of Weyl's theorem in 1912. He demonstrated the following prOposition. "If ¢1(x),¢2(x), ... [is] a sequence of [orthonormal] functions, and if the series lk ci + chg + ... + nkc: + ... converges for some value of k that is greater than zero, then the series c1¢1(x) + c2¢2(x) + ... + cn¢n(x) + ... converges at all points of the interval for which the [orthonormal] functions are defined, with at most the exception Of a set of points of ... measure ... zero."66 Hobson Observed that "the particular case Of [this] theorem which arises when k has the value 1/2 was established by Weyl... Weyl also established the theorem for the case k - 1/3, on the assumption that the functions ¢n(x) are less in absolute value than some fixed positive number, for all ... values of n and x; this last restriction has been shown... to be unnecessary." In a paper published in 1913 Plancherel wrote, "By modifying in its details the method given by Mr. Hobson... I have Obtained the follow— ing theorem, of which the theorems of Weyl and Hobson are corollaries: If the functions ¢n(x) (n - 1,2,3,...) form an [orthonormal] system of functions in the interval (a,b) ..., and if, moreover, the real constants 6SLusin, "Uber eine Potenzreihe," 386. 6Hobson, "0n the convergence Of series of orthogonal functions," 307. 172 cu are such that 2c: (logn)3 converges, then the series ch¢n(x) converges 1 1 67 almost everywhere in the interval (a,b)." Hardy contributed to the theory by proving in a paper also pub- lished in 1913 that the convergence of Z c:(log,n)2 is a sufficient n-l condition for convergence almost everywhere of the corresponding trig- onometric series.68 Hardy's was the most general result known when Lusin wrote his thesis, Integral ang_Trigonometric Series, which was published in 1915. This work contains Lusin's famous conjecture: the Fourier series of a square summable function converges almost everywhere. Lusin inferred the truth of his supposition from two lines of evidence. The first is related to the concept of the series conjugate to a given trigonometric series. The conjugate of the trigonometric series so a 6) 0 + Z (ancosnx + bnsinnx) 2 n=1 is defined to be the series on a 0 , 7) 2 + nil (-bncosnx + ansinnx). We might Observe in justifying the name "conjugate series" that these are the real and imaginary parts, respectively, of the series an n n§1(a“ - ibn)z on the circle [2] = 1. 67Plancherel, "Sur la convergence des series de fonctions ortho- gonales," 540. 68H81‘d "O ' H Y» n the summability of Fourier a series. 365- 173 If 6) is the Fourier series of a function feL2[0,2n], then 2 (a2 n n-l a square summable function f such that 7) is its Fourier series. The + bi) < co; therefore by the Riesz-Fischer theorem there exists function f is called the conjugate of f. Lusin showed that if f is square summable then chan be expressed in terms of f independently of the Fourier series 6) and 7). He demonstrated that in this case, f is defined almost everywhere by the integral 9 n _ l f(x+t)-f(x-t) 69 8) '1'; t dt. 2 tent?) 0 The integral in 8) can be expressed in the form n 9) fQH't);f(X't)dt, 0 for the integrals 8) and 9) exist or fail to exist simultaneously since ___1.__.. .1. 2 tan (12;) t is a continuous function on [0,"]. Lusin was fascinated by 9) and showed that it exists because of the mutual effect of the positive and negative quantities in the integral. Indeed, he considered this interference to be of fundamental importance in the convergence of Fourier series. "It is necessary to examine the interference of the positive and negative magnitudes of the expression x+o - x-o o 69Lusin, Integral and Trigonometric Series, 200 ff;(l951 edition). 174 as the true source Of the convergence Of Fourier-Lebesgue series. All investigations which have been carried out up to this time concerning convergence of Fourier series have been based on an examination of the absolute value Of only one or another expression. It is necessary therefore to consider these investigations as approximate and not actual- ly entering into the [ cause of] convergence Of Fourier series. Un— fortunately, the fact of the existence of a limit value of the integral 9) is deeply hidden in the Riesz-Fischer theory...."70 521 Lusin demonstrated the existence of 9) in the course of his proof of the following theorem: the Fourier series of a square summable function f(x) is convergent a.e. in [-n,n] if and only if almost every- where in [-n,n] 1T 10) I113 .f-(x+t) -?(x-tl t cosntdt - 0. O Lusin held these considerations to be a first argument for the validity of his conjecture. For "having noticed that the integral in equation 10)... differs from the integral 9) only by the factor cosnt, which acquires positive and negative values which are uniformly dis- tributed over the interval [0,2w] when n converges to +w, we are led to expect that the Fourier-Lebesgue series of any square summable func- tion f(x) is always a series which converges almost everywhere in the interval [0,2n]. All the results which have been obtained up to this time in the theory of trigonometric series confirm the probability of this hypothetical proposition."71 70Lusin, Op. cit., 218. 71Lusin, Op. cit., 219. 175 A second line of reasoning which led Lusin to his conjecture is related to the collection of papers cited above, i.e., the papers of Fatou, Weyl, Hobson, Plancherel and Hardy. Lusin argued in the follow- ing way. "It is possible, in general, to study the trigonometric series a oo 11) -%-+ z (a cosnxib sinnx) n=l n n from two points of view. First, [it is possible] to study the question of convergence or divergence of 11) in direct dependence on the numerical character of the coefficients an, bn (n-l,2,3,...). It is, for example, possible to study the convergence or divergence [of 11) as it depends] on the character of the magnitude of an, bn with increasing n. Classical analysis Often applies this point of view. Second, it is possible in the study of these questions to completely exclude the numerical character of the coefficients of the trigonometric series, expressing these co— efficients directly through the function; for example, determining them according to the formulas Of Fourier.... From this point of view the questions of convergence or divergence of the trigonometric series or other analogous questions are no longer related to the properties of the coefficients, but to the properties of the function f(x) itself.... This point of view is primarily the point Of view of the theory of functions. "Both of these points of view almost coincide when we limit our— selves to the examination of square summable functions. In this case the natural necessary and sufficient numerical characteristic of the co- efficients is the convergence Of the series 2 (a2 + b2). n-l n n human-.0011 176 Earlier we were at the second or the function theoretic point of view in the study of trigonometric series, and we saw a great probability that the Fourier-Lebesgue series Of every square summable function f(x) converges almost everywhere in the interval [0,2w]. The probability of the indicated hypothetical prOposition becomes clear with an examination of this question from the first point of view." Lusin then reviewed the sufficient conditions for convergence almost everywhere of Fourier series cited above, the tests of Fatou through Hardy. He continued, "Examining this table of tests for con- vergence, we see that their general type is the convergence of the series °° 2 2 12) Z W(n) (a +1: ) n n n=l where W(n) is a positive increasing function. We shall call such a test for convergence Weyl's test since Weyl first attracted attention to tests for convergence of this type. The function W(n) is a positive non- decreasing function such that the convergence of the series 12) implies the convergence almost everywhere of the corresponding trigonometric series on a -Q-+ Z (a cosnx+b sinnx). 2 n=1 n n We shall call such a function Weyl's function. "The more slowly Weyl's function W(n) increases, the more extensive the class Of trigonometric series which converge on the basis of this test, and consequently the greater the generality of this test of con- vergence. Hence the problem of convergence of the trigonometric series of Fourier—Lebesgue of a square summable function f(x) leads to the task 177 of seeking out the least increasing functions of Weyl. "...It is easy to see that if each increasing function ... is a Weyl function, then the Fourier series of each square summable function converges almost everywhere."72 Lusin was thus led to a conjecture whose validity remained unde- cided for over fifty years after it was first expressed. New ground was broken with respect to study of the divergence of Fourier series with Kolmogorov's publication in 1922 Of the first "example of a summable function whose Fourier series diverges almost everywhere...."73 Kolmogorov noted that the function constructed "is not square summable and I know nothing of the magnitude of the coef- ficients of its Fourier series." It might be Observed that the Fourier series of the function constructed by Kolmogorov diverges almost every- where because the sequence of partial sums of the Fourier series is almost everywhere unbounded. In a paper published in 1923, Menchov stated and proved the following generalization of Hardy's test for con- vergence almost everywhere of Fourier series: "If the functions ¢n(x). (n = 1,2,...) form an [orthonormal] system of functions in the interval (a,b), and if the series 2 a2(log n)2 n n=l converges, then the series oo 2 an¢n(X) n=1 72Lusin, op, cit., 227ff. 73Kolmogorov, "Une série de Fourier—Lebesgue divergente presque partout," 324. 178 74 (The reader will converges almost everywhere in the interval (a,b)." recall that Hardy proved this theorem for the case of trigonometric series). It may be of interest to note that Rademacher had stated this generalization Of Hardy's theorem without proof in a paper published in 1822.75 Menchov's work, however, was independent of that of Rademacher. The principal result of Menchov's paper was original and striking. Having Observed that the notation we» = o [(logn>21 means "the order of increase of W(n) is less than that of (logn)2," Menchov stated and proved the following theorem: "If W(n) is an arbi- trary positive function satisfying the condition W(n) - 0 [(logn)2], then there exists an [orthonormal] system of functions ¢n(x),n - 1,2,3,..., and a sequence Of real constants an such that the series 2 an¢n(X) n=1 diverges everywhere in (0,1), whereas the series 2 a: W(n) n=1 converges."76 Thus Menchov's first theorem above is the best possible for arbitrary systems of orthonormal functions. 4Menchov, "Sur les series de fonctions orthogonales," 82. 75Rademacher, "Einige Satze uber Reihen von allgemeinen Orthogonal funktionen," 112. 76Menchov, op, cit., 89. 179 Menchov's result did not negate the possibility of replacing the Weyl function (logn)2 by a function W(n) satisfying the condition W(n) - 0[(logn)2] for the case of trigonometric series, and this Opportunity was exploited by Kolmogorov and Seliverstov in a paper published in E225. These men demonstrated that in the case Of trigonometric series, the factor (logn)2 in Weyl's test for convergence can be replaced by the factor (logn)1+6 where 6 > 0.77 This was accomplished by proving the theorem: "If the series °° 2 2 13) uglrm) (an + bn) and m 14) Z 33%;3-where I(n) < T(n+l) n=1 converge, then the series as Z (a cosnx + b sinnx) n n n=1 converges almost everywhere," i.e., the convergence of co 2 2 2 nEll-(105m) (an + bn) can be replaced by the convergence of both Of the series 13) and 14) where in the latter series, I(n) < T(n+1). In the same year, 1925, Plessner Obtained a generalization of the result of Kolmogorov-Seliverstov. By utilizing the methods of Jerosch 77Kolmogorov and Seliverstov, "Sur la convergence des series de Fourier," 303. 180 and Weyl, as had Kolmogorov and Seliverstov, Plessner proved that if so n§1(1ogn)(a: + bi) is convergent, then oo 2 (a cosnx + b sinnx) n n n-l is convergent almost everywhere.78 This result remained the most general ‘in-nit.” ...-m is" .I , known for over forty years. Turning again to the study of divergence of Fourier series, Kol- mogorov published in 1926 an example of a summable function whose Fourier series diverges everywhere.79 Even more significant for our purpose is the assertion, due to Kolmogorov, and published in 1927 in a paper authored by Kolmogorov and Menchov, that there exists a function feL2[0,2n] with the prOperty that the terms of its Fourier series can be rearranged to form a series which diverges almost everywhere.80 Kolmogorov stated this proposition without proof. Assuming the validity Of Kolmogorov's theorem, Ulyanov published a generalization of it in 1958. To understand Ulyanov's generalization requires that we de- fine the concept of an LP space. Let 1 §.p §_w; then Lp is the space of all measurable functions f such that If]p is summable. We have already encountered the two special cases p - l and p - 2; these are 8Plessner, "Uber Konvergenz von trigonometrischen Reihen," 16. 79Kolmogorov, "Une serie de Fourier-Lebesgue divergente partout," 1327. 80Kolmogorov and Menchov, "Sur la convergence des series de fonctions orthogonales," 433. 181 the spaces of summable and square summable functions, respectively. We might observe that the use of the term "space" is meaningful for, as in the case of L2, it is possible to define a norm on the elements of Lp such that Lp is a normed linear (or vector) space.81 Given the definition of an Lp space, Ulyanov's generalization of Kolmogorov's theorem can be stated as follows: if p > 2 then there exists a function feLp such that the terms of the Fourier series of f can be rearranged to form a series which is divergent almost everywhere.82 The first construction of such a Fourier series was given in 1960 in a paper by Zahorski. Zahorski sketched a means of determining the coefficients of the Fourier series of a function feL2 and indicated how to permute the terms of the series in order that the rearranged series diverges almost everywhere.83 Zahorski thus verified in print Kolmogorov's proposition and Ulyanov's generalization of it. In the converse direction, and "[lending] some support to the con- jecture of N. N. Lusin," Garsia proved the following two theorems in a paper published in 1963. "The Fourier series of every function in L2(-w,n) can be so rearranged as to converge almost everywhere," and, "If successively and independently for each k we permute at random the terms of the Fourier series Of f whose indices are comprised between mk and mk+1 among themselves, then with probability one, the resulting rearranged series will converge almost everywhere,‘ where f is a function 81See Royden, Real Analysis, 93. 82Ulyanov, "On unconditional convergence and summability," 828. 83Zahorski, "Une série de Fourier permutée d'une fonction de classe L divergent presque partout," 501. 182 in L2(-n,n) and {mk} is a sequence Of integers such that S (x,f) + f(x). “’1. Having noted that these theorems are valid for arbitrary orthonormal expansions, Garsia Observed that "the construction of divergent ortho- normal expansions is elaborate.... The corresponding form of [carsia's second theorem] may explain this difficulty, since convergence almost everywhere appears to be more the rule than the exception."84 Such was the situation when Carleson published his paper in 1966. Carleson stated his purpose in this way. "In the present paper we shall introduce a new method to estimate partial sums of Fourier series. This will give quite precise results and will in particular enable us to solve the long Open problem concerning convergence almost everywhere for functions in L2."85 In particular, Carleson proved that if feL2[0,2n]then the sequence of partial sums of the Fourier series of f converges almost everywhere. Kahane reviewed Carleson's paper in Mathematical Reviews:"The spectacular discovery contained in this article is the validity Of Lusin's hypothesis.... The coherence [of results Of previous workers] and [their] great difficulty... had made specialists think that they were probably the best possible; from which followed a skepticism, justifiable until a few months ago, with respect to proposed demonstrations of Lusin's hypothesis. 4Garsia, "Existence of almost everywhere convergent rearrangements for Fourier series of L2 functions," 623. 5Carleson, "On convergence and growth of partial sums of Fourier series," 135. ‘T—‘.' '_‘;'m- ‘ . {alerts-y'— 183 "[Carlson's] proof is very delicate and demands admiration.... The techniques used...are refined but classical (maximal Hilbert trans— forms, harmonic functions in a half plane, convolutions, and Young's inequalities on Fourier transforms). "The article Of the author is very difficult to read. It would be desirable to find either a more rapid demonstration by another method or several general theorems suggested by the author's method from which the theorems on convergence of Fourier series would follow...." Kahane Observed that Carleson's results "...are without doubt not the best possible; one can conjecture that...[the sequence of partial sums] sn(x) is convergent a.e. when feLp, p > 1." Hunt succeeded in demonstrating this generalization Of Carleson's theorem in a paper published in 1968. By utilizing essentially Carleson's proof but modifying certain of Carle- son's definitions and constructions, Hunt showed that if feLp[0,2n] where l < p < w, then the Fourier series corresponding to f converges almost everywhere.86 Thus the theorem of Carleson and its generalization by Hunt provide a very general answer to the question Of sufficient conditions for rep- resentation almost everywhere of a function by its corresponding Fourier series. 7' 86Hunt, "On the convergence Of Fourier series," 235. 184 Epilogue Lebesgue remarked that " ... many branches of our science have died just at the time when general results seemed to guarantee them a new activity. I cite as examples the theory of forms and elliptic functions--so completely ignored since Weierstrass presented the gen— eral theorems about them. General theories reply to the questions asked of them. Unfortunately, they reply too easily, without requiring Of us any effort, and since they give us the solution of problems before we have studied them, they weaken our curiosity and deny us the intimate 86 We need not share knowledge which would have led to new problems." Lebesgue's concern regarding continued study of Fourier series. Carle- son's result is very general, yet many questions with respect to Fourier series representation of functions remain Open. This has been and is one of the great problems Of analysis. Investigation of it can be ex- pected to continue and I predict the pursuance of such investigation will continue to contribute to the development of analysis. The question of Fourier series representation of functions has been successively generalized. Such generalization, by making the problem more rather than less comprehensive, renders it capable Of partial solu- tion. Thus the question of representing a generalized Fourier series ganon, which may be convergent or divergent, by a function f derived from the series, i.e., which is related to the series by the equations b 6Lebesgue, "Humbert et Jordan, Roberval et Ramus," 192. 185 and agrees in value with the series at almost all points of (a,b) at which the series converges, becomes possible only with the introduction of a more general conception of integral and convergence of series. Saks, describing the "regularity and harmony, unhoped for by the older methods, concerning, for instance, the existence of a limit, a deriva- tive, or a tangent, remarked that . . . many branches of analysis. . . have lost none of their elegance where they have been inspired by methods of the theory Of real functions. On the contrary, we have learned to admire in the arguments not only cleverness of calculation, but also the generality which, by an apparent abstraction, often enables us to grasp the real nature of the problem."87 Indeed, the generalization of a problem in order to grasp its "real nature" and therefore to make it amenable to solution is characteristic of the evolution of the theory. It is evident from reading the almost literal excerpts that there were many ambiguities and omissions in the statements of definitions and assertions of results in the original papers. Borel's definition of measure, for example, is in my Opinion almost incomprehensive in its indefiniteness. Proofs were often wanting in accuracy; even Lebesgue slipped into occasional error in his Lecons §g£.1flnt§gration g£_l§_ Recherche des Fonctigns Primitives. Such logical deficiencies are clearly secondary, however, to the very real accomplishments of the men who contributed to the creation of the theory: the explication of new ideas, lines of investigation, methods of attack, and fruitful generaliza- tions.88 87Saks, 22, cit., x. 88See Kline, "Logic versus Pedagogy," 264. 186 Finally, mention should be made of the controversy which accom- panied the creation of the theory of functions Of a real variable. There was polemic between Bernoulli and d'Alembert and Euler; Fourier was criticized by Lagrange. Cauchy's formulation of the notion of limit was criticized by Cournot and ignored by Poisson.89 While I have no evidence to support such a contention, I suspect Riemann did not publish his Habilitationschrift in part because of the criticism he knew would follow. Cantor was criticized by Kronecker, Borel by Schoenflies, and Lebesgue by Hermite. In all Of this there is the question of the role Of authority in the determination Of what is admissable in mathematical science. It is apparent that mathematics was not the Open and impartial discipline described by the stereotype. Indeed, evidence suggests that this is true of science in general even today;90 hOpefully, the possibility of such controversy no longer exists in the science of mathematics. 89Boyer, The Histogy gf_the Calculus and its Conceptual Develog- ment, 283. 90See de Grazia, The Velikovsky Affair; The Warfare of Science and Scientism. ..— 187 Appendix A Let f be a real valued function of a single real variable such that f is periodic with period 2n, i.e., f(x + 2n) - f(x); and f’ and f” exist and are continuous. First we will show that f can be repre- sented by a Fourier series on [-",“]. Let T! 1 an a ;- f(t)cosntdt for n - 0,1,2,..., and -n 11' 1 bn - ?- f(t)sinntdt for n - 1,2,... . Then -F {N 1 la | = - f(t)cosntdt n n J-n 1 r" 1 ... f(t)d(iflli) n n 1.1: n n 1 sinnt sinnt l) 8 fl f(t) n - f (t) n dt -n -n by integration by parts. Evaluate l) to obtain r1! 1 . -'| f (t)sinntdt nn J-n in = .1_ I f’(t)d -cosnt 1111' J-" 188 n 11’ _ lg' , cosnt ,, cosnt _nfl -f (t) n + f (t)—-r-‘-—-dt 7! -n l , . _<_ 2 f (1T) + f (-11) +21IM n n where M = max {If”(t)| n §_t §_n}. Thus there exists a constant M1 such that M1 Ianl £7 for n= 1.2.... e n n Similarly, there exists a constant M such that 2 M2 lbnl_i-§ for n = 1,2,... . 1111' Therefore the series a 0 2) 2 + (ancosnx + bnsinnx) ntoB n 1 converges uniformly and absolutely on [-n,n] to a (continuous) function, say g(x). That is, a g(x) - 39-+ Z (ancosnx + bnsinnx) for every xe[-n,n]. n=1 189 Now let n , 1 an = ;- g(x)cosnxdx for n - 0,1,2,... , —n and 11' , 1 bn a ;' g(x)sinnxdx for n = 1,2,... . -w Since the series 2) converges uniformly, n 1 a0 °° a; -';- {§-+ mil (amcosmx + bmcosmx)}cosnxdx n w n w a0 m m b - - cosnxdx + Z - cosmxcosnxdx + 49- sinmxcosnxdx . 2n m=l n n Using the facts " o if m + n cosmx cosnxdx = if m - n -n and TI sinmx cosnxdx = 0 , -11 it follows that an a an for every n. Similarly, bn = bn. 190 Thus f and g have exactly the same Fourier coefficients. It is easy to show that the function which is identically zero is the only continuous function with the prOperty that all its Fourier coefficients are zero. Thus f = g, and a 3) f(x) = 39'+ nél (ancosnx + bnsinnx) for all xe[-n,n]. Now since f(x) has the property that f(x + 2n) - f(x) for every real number x, f is represented for every real number by the Fourier series 3). 191, Appendix B The difficult part of the proof is to evaluate the integral lim I: f(u) sin jr riglE—£-da. An analytic analog of Fourier's geometric j+m -cos r argument might be as follows. Express the integral in the form sin r N-x “_xf(x+r) Qin J r l-cos £>d r recalling, of course, the relationship r =<¥ - x. Now let u - jr. Then the integral is written sin2 New" f(x-+19) (sin 1.) (%)(-—-L)du. 30'1"") l-cosg- 3 5 Appealing to ghe fzmiliar equalities sin y = y - §T-+-§T-- . . . and cos y = l - §T’+'%T" . . . , we obtain the integral in the form u u3 u5 1 — - —— + -——- - . . j(1r-x) u (1" j 3313 5315 , f(x-l-) (sin u) —- du. (-1r-x) j 2 2 4 6 i) _9_ _ .11.. + _2_ - 1 2:32 4') 63j Now consider the limit Of this integral as j-+ w. The limit is of the form f(x)]dlsin u) -- du. 1L 2 Hence, completing the evaluation, we write +nsin u 2f(x )fdo du.= 4f(x)f+w81+ u du = 2wf(x). Thus, Fourier concluded i-+on n f(x) = -- Z I” [f(o) cos i (a- x)] do. "in—m It is of interest to observe that Fourier's proof is quite different from the argument given here. Fourier argued geometrically rather than analytically. 192 He Obtained the integral 2f(x)fg°einj r) 35:11: .E 2 by asserting that the area under the curve whose abscissas are a and ordinates are f(u) (sin j r) lgigsgf is zero "except for certain intervals infinitely small, namely, when the ordinate IgEEEEF- becomes infinite. This will take place if r - o - x is zero; and in the interval in which a differs infinitely little from x, the value of f(u) coincides with f(x). Hence the integral , sin r ff(o)(31n j r)I:E;;—;-do becomes 2f(x)f;(sin.j r)£é-dr. . . ." r 7 It is enjoyable to attempt to visualize Fourier's assertion; such an exercise will also begin to reveal remarkable depth of insight and geometric intuition which Fourier almost invariably brought to bear upon such problems. 193 Appendix C Volterra's1 example of a bounded derivative which is not Riemann integrable may be described as follows: In analogy with the construc- tion of the Cantor ternary set, construct a nowhere dense perfect subset E of [0,1] of positive Lebesgue measure,2 i.e., such that the sum of lengths of the Open intervals (a,b) of [0,1] whose union is the complement of E is less than 1. Recall that the function f defined by xzsini'if O N, 1 1 1 a < a +(2n+l)n < a + 2;; <'b. Therefore for n > N,{a +(2n+l)w } and {a +*§%;J are sequences contained in (a,b) which converge to a , a) - l and h’ (a +-l—', a) a -l. and are such that h’(a + 2nn ___1.__.. (2n+1)n By the intermediate value prOperty of derivatives, for every n > N there exists an xe (a +<§;:%3;-, a.+- -E%;-) such that h’(x,a) - 0. Thus there are countably many zeros of h7(x,a) in the interva1(a,'§§2']. Let Y denote a fixed zero of h’ (x,a) in this interval. Now define g(x) by h(x,a) if a < x: Y g(x) = h(Y,a) if Y : x<__ a + b-Y h(a+b-x,a) - -f(x,b) if atb-Y:x0 and y€[0,l] such that Iy-xl2>1’2 has no meaning is constructed in the following way. "Let wl, w2,...be the set of all rational numbers. With the se- quence w1,w2,...associate a sequence Of positive quantities c1, c2,... so that f(x) = r21 cr(x-wr)l/3 is uniformly convergent for all values of x from x0 to x1....The function 197 f(x) is then continuous and increasing with x...it follows that the curve between any two points (xo,yo) and (x1,y2) has a determined length. The integral x l 1) (1 + f’(x)2)1/2dx xo by which the length usually is expressed is in this case completely mean— .Wfl ingless, since...at all points x = wr a differential quotient exists and has the value +00."3 That is, for any xs(x0,x1) the oscillation at f’(x) [ is +00 and therefore the integral 1) does not exist. Scheeffer, "Allgemeine Untersuchungen fiber Rectification der Curven," 66. 198 Appendix D It should be remarked at the outset that Hilbert worked with Riemann integrals and therefore the integral symbol J is understood to denote an integral in the Riemann sense. We will begin by attempting to develOp some intuition regarding how a linear integral equation b l) f(s) + K(s,t) f (t)dt = u(s), A!“ n a where the functions K and u are given and f is unknown, can be reduced to a system of infinitely many linear equations in infinitely many un- knowns and why this might be of assistance in obtaining a solution f(x) of the integral equation. Let the interval of integration [a,b] be partitioned into n subintervals of equal length by the points x0,x1,...xn. Then xi = a+'$£;2:21'where i = 0,1,2 f(xi) by f1, K(xi,xj) by K ,...,n. Designate the functional values ij and u(xi) by ui. Then the integral equation 1) assumes the form n 2) fi + X K f = u i=0 ij j 1 ’ 1 = 0.1.2.--..n. that is, a system of n+1 linear equations in the n+1 unknowns f1. Written out, these equations are of the form 199 (1 + Koo>fo + K01f1 + K02f2 +. . . + K nfn II C‘. 0 0 x f + (1+K )f + K f _ 10 0 11 1 12 2 + . . . + Klnfn ul KZOfO + K21f1 + (1+K22)f2 + . . . + Kann . u2 K f + n0 0 Knlfl + Kn2f2 + . . . +(1+Knn)fn = un . For a given n, let f0,f1,...,fn be a solution of 2) and consider the set of points P = {(xi’fi) I Ojiin}. Now if K and u are sufficiently well behaved, then as n tends to infinity, the systems of linear equations 2) tend to the integral equation 1), and the sets of points Pn tend to a solution function f(x) of the integral equation. This may help to explain why Hilbert reduced the equation 1) to a system of linear equations. More specifically, Hilbert argued as follows. He first introduced the concept of a complete orthogonal sequence of functions {¢n}. Such a sequence of functions, each of which is defined and continuous on [a,b], is orthogonal, that is, b ¢.¢ = 0 if i + j . a and complete, i.e., if s(x) is any continuous function defined on [a,b] and if 200 5n = s(x)¢n(x)dx is defined to be the generalized nth Fourier coefficient of s(x) with respect to {¢n}, then {¢n} is complete if and only if This is, of course, a more general form of Parseval's equality. Hilbert did not give this definition of completeness of an orthogonal set but he did demonstrate the equivalence of the above condition and his defini— tion. We note that an orthogonal sequence of functions {¢n} is said to be orthonormal if and only if for every n, ¢n=1- The sequence 1 , cosx, sinx, ... , cosnx, sinnx,... me/T VT.”— is a complete orthonormal sequence of functions defined on [~n,n]; this example may help to eXplain the origin of the conception of complete orthonormal sequence and generalized Fourier coefficients. Hilbert then reduced the linear integral equation 1) to a system 201 of infinitely many linear equations in infinitely many unknowns 3) x. + 32° a. x = b , 1 = 1,2,3,..., where aij and bi are the generalized Fourier coefficients of K and u, reapectively, in reference to a complete orthogonal sequence of functions {¢n}o That is, b b aij = K(s,t)¢i(s)¢j(t)dsdt , a a a "double" Fourier series, and b bi = u(s)¢i(s)ds . a This enabled Hilbert to view the problem of obtaining a solution of 1) as the search for a solution (x1,x2,x3,...) of 3) where x1 is the i th generalized Fourier coefficient of the unknown function f with respect to {¢n}. Hilbert showed that the bi are square summable, that is, E b: < w. Therefore, by a theorem previously demonstrated by Hilbert, thire are two possibilities for 3): there exists a unique square summable solution or no unique solution exists. We will consider only the first of these two possibilities. If (01, 02,...) is the unique square summable solution of 3) then Hilbert proved, 202 where b Ki(s) = K(s,t)¢i(t)dt , Ja converges uniformly on [a,b] to a (continuous) function, u(s). Finally, Hilbert showed if f(x) is defined by f(x) 8 u(x) -a(x) then b f(x)¢i(x)dx = oi a and hence, since a1 is the i th generalized Fourier coefficient of the solution function of l) with respect to {¢n}, f(x) is a solution of the integral equation. Conversely, the conditions satisfied by (a1,a2,...) must be satisfied by the generalized Fourier coefficients with respect to {¢n} of any continuous solution of 1). Thus, Hilbert concluded, if 3) has a unique solution then there exists a unique continuous solution of 1). This argument is given in Hilbert's Grundzfige §i223_Allgemeinen Theorie ESE linearen Integralgleichungen, 174 ff., a collection of six papers of which the first five appeared in the Gfittingen Nachrichten in 1904-1906; the collected works were published in 1912. 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