,,,,,, A NEW CHARACTERIZATION OF CESARO-‘PERRON INTEGRALS USING PEANO DERIVATES Thesis for the Degree of Ph. ,ng‘é" W \, MICHIGAN STATE UNIVERSITY . ' - JOSEPH ANTHONY BERGIN 5’" 1972 LIBRAR Y Michigan State nivcrsity This is to certify that the . ; thesis entitled r ' \ A NEW CHARACTERIZATION OF CESARO-PERRON INTEGRALS USING PEANO DERIVATES presented by Joseph Anthony Bergin has been accepted towards fulfillment of the requirements for Ph. D. degree in Mathematics ”JIM M/W% Date WW I 0-7639 ABSTRACT A NEW CHARACTERIZATION OF CESARo-PERRON INTEGRALS USING PEANO DERIVATES BY Joseph Anthony Bergin Using the methods of O. Perron the Zn-integral is defined. A majorant of a function f on an interval [a,b] is, by definition, the n-th exact Peano derivative of a function P which satisfies -w < 5n+1P(X) 2_f(x) (X e [a,b]). where 6n+lP is the (n+1)th lower generalized derivate of P. Using a modification of a theorem of James it is shown that such majorants lead to a reasonable defini- tion of integration. Some of the useful properties of this integral follow. 1. Every n-th exact Peano derivative is Z -integrable. n-l 2. The Zn-integral is a positive linear functional defined on certain of the Lebesgue classes. 3. The indefinite Zn-integral is itself an n-th exact Peano derivative. 4. The Zo-integral is equivalent to the Perron integral and the Zn+l-integral properly generalizes the Zn-integral. Joseph Anthony Bergin 5. A non-negative function is Zn-integrable if and only if it is Lebesgue integrable. 6. The Zn-integral of a product may be computed using an integration by parts formula if certain natural restrictions are fulfilled. 7. The Zn-integral is exactly equivalent to the CnP-integral of Burkill. The need to correct and simplify Burkill's work motivated this study. This approach brings to light some interesting relationships between Cesaro derivatives and Peano derivatives. Namely: 8. A function is Cn—continuous on [a,b] if and only if it is an exact Peano derivative. 9. The (n+l)th Peano derivative of f is the Cn-derivative of the n—th Peano derivative of f. REFERENCES [l] J.C. Burkill, The Cesaro—Perron scale of integration, Proc. London Math. Soc. (2) vol. 39, (1935) pp.54l-552. [2] R.D. James, Generalized n-th primitives, Trans. Amer. Math. Soc. vol. 76 (1954) pp.149-176. \ A NEW CHARACTERIZATION OF CESARO-PERRON INTEGRALS USING PEANO DERIVATES BY Joseph Anthony Bergin A THESIS Submitted to Michigan State university in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1972 g); ’4' TO LYNDA ii ACKNOWLEDGMENTS I would like to thank Professor Jan Marik, under whose direction this thesis was prepared, for constant, even unrelenting help and encouragement. 'Without his ability and patience I am sure this would never have been finished. iii Introdu CHAPTER I. II. Peano Derivatives . . . . . . . . . . III. The Theorem of R.D. James . . . . IV. The Central Theorem . . . . . . V. The Zn-Integral . . . . . . . . . VI. Properties of the Zn-Integral . . . . VII. Integration by Parts for the Zn-Integral VIII. Relation Between the Z —Integral and the CnP—Integral . . . 9 . . . . . . . . IX. Examples . . . . . . . . . . . . . . X. An Improvement in the Definition of the Zn—Integral . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . TABLE OF CONTENTS ction . . . . . . . . . . . . Burkill's CnP-Integral . . . . . iv 16 26 28 32 51 66 71 76 85 I NTRODUC TI ON In a paper published in 1914, O. Perron, [ll], attempted to give a new characterization of the Lebesgue integral but in the process defined a new and more general integral. He apparently was only trying to simplify Lebesgue’s definition, and indeed it.was easily seen that for the class of bounded measurable functions the ideas of Perron and Lebesgue were the same. Perron realized however that the integral did not depend upon the boundedness of the integrand. A year later H. Bauer, [6], developed some of the properties of the integral of Perron and in so doing established that the new integral is more general than Lebesgue's. In fact, he showed that Perron's integral is capable of reconstructing a function from its finite derivative, while the Lebesgue integral is not capable of this without using extra assumptions. Perron's idea was very similar to what we know today as the theorem of Vitali-Carathebdory (not completed until 1918). That is, if f is Lebesgue integrable on [a,b] and G > 0 then there is a function u which is lower semi-continuous and bounded below and a function v, upper—semicontinuous and bounded above such that v g.f g_u on [a,b] and fb (u-v) < E- If we define U(x) = I: u and VTX) = I: v then U is an example of what Perron called an Oberfunktion (majorant or superfunction) for f and V is an example of an Unterfunktion (minorant or subfunction) for f on [a,b]. That is, the upper derivate of V is g_f on [a,b] and the lower derivate of U is 2_f on [a,b] (also, the upper derivate of V is bounded above and the lower derivate of U is bounded below). The idea of using these functions to define the integral comes from the formulas U(b) g_J: f g_V(b) and U(b) - V(b) < e. For full details the reader should consult the excellent account given in [4:11, p.157f]. Mbre recently the idea of using majorants and minorants to define integrals has been adapted to derivatives other than the ordinary derivatives. In 1931, J.C. Burkill, [8], applied the method to approximate derivatives and thus defined the AP integral, and in 1932-33 ([1] and [2]) he defined a new derivative, the Cesaro derivative and then developed the corresponding integral. Unfortunately there were serious errors. Still more recently R.D. James [3] and L. Gordon [9] have used the method in dealing with still other types of derivatives. Our purpose here is to define yet another integral, one which is naturally associated, according to Perron's method, with the Peano derivatives. We show that the integral has several desirable pr0perties, among which is a certain "Fundamental Theorem," a form of "continuity" for the integral and an integration by parts theorem. We shall show that the integral we give is identical to that of Burkill (thus correcting his errors) and also give some interesting relationships \ between the Peano derivatives and the Cesaro derivatives. CHAPTER I BURKILL ' S CnP - INTEGRAL Burkill ([1] and [2]) attempted to define a notion of differentiation, continuity and integration inductively. We present his definitions. Let the C P-integral denote the usual Perron O integral (see [4,II p.157f] or [5, p.201f]). Then CO-continuity is the ordinary continuity and the CO derivative is the ordinary first derivative. Suppose that the notions of Cn -continuity, differentiation -1 and the C P-integral are defined. If f is n-l Cn_1P-integrable on [a,b] then the number (Cn_1P) [b f(x)dx denotes the value of this integral. a Definition 1.1. For a positive integer n, the n—th Cesaro mean of a C P-integrable function n-l in (x,x+h) is defined to be +h _ cn -m for each x 6 [a,b], d) M(a) = O. Then Cn—minorants are defined by replacing b) and c) by * b') CnD M(x) g_f(x) for each x 6 [a,b], * c') CnD M(x) < +m for each x c [a,b]. Definition 1.5. The upper CnP-integral of f is * (CnP )[b f = inf{M(b):M is a C -majorant of a n f on [a,b]}. The lower CnP-integral of f is (C P*)J';b f = sup[m(b):m is a C -minorant of n a n f on [a,b]}- Suppose now that we have a proof of the following theorem. Theorem. Let M be defined on [a,b]. Let CnD*M 2.0 on [a,b]. Then M is non—decreasing on [a,b]. Then if f is any function and M is a Cn- majorant and m is a Cn-minorant of f on [a,b], we * see that CnD*(Mrm) 2.CnD*M - CnD m 2.0 and so M—m is non—decreasing. But M(a) = m(a) = 0, so that I _ * M(b) 2 m(b). Thus it follows that (CnP*)[: f g_(CnP )I: f. When these are equal and finite we say f is CnP-integrable and put (CnP)f: f for the common value. Moreover, if f is CnP-integrable then for each g > 0 there is a majorant M and a minorant m such that MCb) - m(b) < 6. And so 0 g M(x) - m(x) < E for each x 6 [a,b]. From this it easily follows that if f is CnP—integrable on [a,b] then it is also CnP-integrable on any sub-interval, and in particular on [a,x], a 3.x g_b. (We make the usual convention that I: f = 0 always). If we define F(x) = (CnP)f: f, then we see easily that F is the uniform limit of a sequence of majorants of f. For these reasons and others the theorem stated above is central to the development of these integrals and will be called the "Validity Theorem." The first and most fundamental problem with Burkill's paper is that his proof of the validity theorem is in error. Mbreover, because of the way in which the n—th Cesaro mean is defined an integration by parts theorem is essential to the develOpment. The induction by which Burkill defined the CnP—integral depends essentially on integration by parts to transform a CnP-integral to one of lower order. However Burkill gives only the briefest sketch of a proof of this theorem and although we believe that the details can be supplied the complete proof is very long. We will take a different approach. We first define an integral, the Zn-integral, which at first does not seem to be related to the CnP-integral. The definition does not use an induction and so is technically much simpler. The notion of derivation on which it is based is also much neater than the notion of Cn-derivatives. On the other hand we shall show that the Zn—integral is equivalent to the CnP-integral. CHAPTER II PEANO DERIVATIVES Definition 2.1. Let F(x) be defined on an interval [a,b]. Let XO 6 (a,b). Let n be a natural number. If there are constants dl,...,dn depending on xO but not on h such that k k: = o(hn) as haO (2.1) F(x +h) — F(x ) — d o o k=l then an is called the generalized derivative or Peano derivative of F at x0. This is denoted by F(n)(xo)' It is eaSIly seen that If F(n)(xo) eX1StS then so do F(k)(xo) (l g_k 3.“) and then n k ~ h # n F(xo+h) - F(xO) - k:: k: F(k)(xo) — o(h ) as haO. In particular F(l)(xo) = F(l)(xo), the ordinary derivative. ‘We also say that F(O)(Xo) = F(xo) when F is continuous. By restricting h, say h > O, we may also define one-sided generalized derivatives, denoted F(n) +(X0) etc. A function f defined and finite on an interval I will be called an n-th exact Peano derivative (e. P. d.) on I provided that there is a (continuous) function F on I such that F(n)(xo) = f(xo) for each p01nt xO In the interior 10 of I and, in case I contains its end points, that the corresponding one-sided n-th derivatives of F at these points equal f there. Similarly if we say that F has an n—th Peano derivative in I we mean that F(n) exists in the interior and the one—sided derivatives exist at the end points when these are in I. Definition 2.2. Let n be a natural number. Let F be defined in the interval [a,b]. Let x0 6 [a,b]. If n = O we assume F is continuous at x0. If n > O we assume F(n-l)(xo) eXlStS. Define en(F,xO,h) for h such that xo+ h 6 (a,b) by hn n—l k (2.2) B? 9n(F,xO,h) = F(xo+h) - kg) fi- F(k) (x0). Note that in case x0 is a or b we agree that all these generalized derivatives are "one—sided." Define AnF(xo) = F(n)(xo) = limgup 9n(F,xO,h) (2.3) 6nF(xO) = E(n)(xo) = limgnf 9n(F,xo,h). Then AnF(XO) is called the n-th upper generalized derivate of f at x0 and 6nF(XO) is called the n-th lower gener— alized derivate of f at x0. It is clear that F(n)(xo) exists if and only if AnF(xO) = 6nF(xO) and both are finite. In this case F(n)(xo) is the common value. But as AnF, énF need not be finite we can say that F(n)(xo) ll exists (in the finite or infinite sense) whenever AnF(xO) = 5nF(xo). Again the one-Sided derivates are easy modifications of (2.3) and, as above, when we speak about AnF(xO) , etc., on a closed interval we shall mean it in the one-sided sense at end points. These derivatives have been extensively studied. (For example see [10]). We would like to note that these derivatives are more general than the ordinary derivatives. For example if f(x) is the characteristic function of the rationals then x3f(x) has a second generalized derivative at 0 but no second ordinary derivative there. The same is true of the function g(x) = x3 sin x , x # O , g(O) = 0. Definition 2.3. Let F be defined in the interval [a,b]. Then F will be called l—convex iJI [a,b], if it is non-decreasing there. It will be called 2-convex :if it is convex and continuous in [a,b]. If n > 2 then (n—2) F will be called n-convex if F exists in [a,b] and is 2-convex there. We also define n—convexity in an open interval (a,b) by simply drOpping the restrictions at a and at 'b. Proposition 2.1. If f(n)(x) exists then f(n)(x) does as well, and they are equal. 12 Proof. Apply Taylor's theorem. F Proposition 2.2 Assume f(n)(x) (n 2_l) . t ex1sts on [a,b] and F(t) = I f where c 6 (a,b). c Then F(k)(x) = f(k_1)(x) (k = l,...,n+l; x 6 (a,b)). +h Proof. F(x+h) - F(x) = [X f(t)dt = x = fh f(x+s)ds = o n k [f(x) + '2 §—.— f (x) + 0(sn) ds 1: k=1 k. (k) I n hk+l n+1 hf(X) + kEl W f(k) (X) + 0(h ). [3 Notation. Let em +f(x) signify liminf en (f: X! h) I h40+ the lower right hand generalized derivate of f at x. We similarly define the symbols 6n _f(x), An +f(x), I and An _f(x). I Corollary 2.3. Under the conditions of prOposition NM _>_ 6 f(X)- 5n+2 n+1 Proof. Let M < 6n+lf(x). Then for all sufficiently small positive h, 13 I [f( ) f() En 5k f ( ) d >| Sn” Md + — — _r _— . o x s x k=1 k. (k) x ] s o s (n+1) : That is, n+1 hk hn+2 F(X+h) — F(X) - k§1 El— F(k) (X) 2 W M: thus 6n+2,+F(X) > M, and so 6n+2’+F(x) 2_6n+lf(x). If h < O and n is even then n k S J‘h Sn+l El E1- f(k) (X) ]ds > O W MdS. [h [f(x+s)-f(x)- O k: while if n is odd n k h Sn+1 I: [f(x+s)-f(x)-k:: ET fik)(X)]dS<< o THIITT Mds. Thus in either case we have I n k %§LL {F(X+h) “F(X)— Z '13—.- F(k) (X)] > MI h k=l ° for h sufficiently small and negative. Thus 6n+2’_F(X) > M whence 6n+2'_F(x) 2_5n+lf(x). The rest is obvious. B Corollary 2.4. If f is defined in [a,b] and if f(m) exists there and if 14 § 5 [X l k—l for x 6 [a,b], then f a X " F 5 F261“ f. m+k+1 +1 Proof. This follows by iteration from proposition 2.2 and corollary 2.3. D We need the next three prOpositions in the next chapter. Lemma 2.5. Let f’ be finite in (a,b) and let f’(a+) exist (finite or infinite). Then f(a+) exists. Proof. If f’(a+) is finite then f’ is bounded in some interval (a,x0) so that f is uniformly continuous there and so f(a+) exists. If f’(a+) = +m then f’ is positive in some interval (a,xo) and so, by the mean value theorem, f is monotone in (a,xO). Thus f(a+) exists. The case f’(a+) = —w is similar. U 15 Lemma 2.6. Let f be continuous in [a,b] and let f have a finite derivative on (a,b). Let f’(a+) exist. Then f“l (a) exists and equals f’(a+). Proof. Apply L'Hdpital's rule to the limit f (a+x) - f (a) x lim an D Preposition 2.7. Let n be a natural number. Let f be defined in [a,b] so that f(n) is finite in (a,b), f(n) (a+) exists (finite or infinite) and f(n)+(a) Is finite. Then (n)+ f(n)+(a) = f (a) = f(n)(a+). Proof. As f(n)+(a) lS finite, f is continuous at a and hence, in some interval [a,bl]. By lemma 2.5, f(n-l)(a+),...,f'(a+) existo By lemma 2.6, f’i-(a) = f’(a+) and since f’4 (a) = f(1)+(a) which is finite, we see that f' is continuous at a. Then lemma 2.6 applies also to f’. Thus f(z) (a+) = f(2)+(a) = f(2)+(a) (2) which is finite so that f is continuous at a. Finitely many such steps finish the proof. D CHAPTER III THE THEOREM OF R.D. JAMES In this section we prove a theorem that is very important to the deve10pment of what follows. It is merely a modification of a theorem of James [3]. Theorem 3.1. If f is defined in [a,b] and if 5nF.2 O in [a,b] then F is n-convex in (a,b). The proof of this theorem requires some additional definitions and lemmas. Let n be a natural number. Let f be defined on an interval J and let xo,xl,...,xn be distinct points of J. Then there is a unique polynomial P of degree g.n such that P(xj) = f(xj)(j = O,...,n). We write P(x) = P(xo,xl,...,xn;x) to express the . n . dependence on Xo’°°"Xn' Since P( ) IS a constant, we define _ . _ (n). Vn(f,xk) — Vn(f,xo,xl,...,xn) — P /n. . l6 17 It is easily seen that n (x-x )...(x-xk )(x-x )...(x-x ) o -1 k+l n F(X) = kEg f(Xk) (xk-x )...(x -x )(x -x )...(xk-x )' — o k k—l k k+l n since this is clearly a polynomial of degree g'n having the required prOperties. If we expand this eXpression we see that n f(xk) Vn(f;xo,...,xn) = Z) WjTE—TI' where k=0 k n w(x) = II (x-xk). k=O It is also easily verified that (X-X )P(X pooo'X :X)-(X-X )P(X pooopx 7X) P(x ,...,x :x) = O l n n O n-l O n X - X n 0 since this polynomial also has the required values at the points xo,...,xn. From this we see that Vn_l(f;Xl,...,Xn) "' Vn_l(f7Xo,...,Xn_l) V (f;x,...,x) = X — n o n n xo Another property easily seen from the definition is the fact that a permutation of the points xo,...,xn leaves P and Vn(f;xo,...,xn) unchanged. Definition 3.1. The function f is called n— convex (d) in J provided Vn(f;xo,...,xn) 2_O for all choices of n+1 distinct points Xo""'xn from J. 18 We note that "2-convex (d)" means "convex" and "l—convex (d)" means "non-decreasing." Preposition 3.2. Let f be n—convex (d) on (a,b). Let Xo""’Xn be distinct points of (a,b) and Xn-l < Xn. Then Vn_l(f;xo,...,xn_2,xn_l) S-Vn-l(f;xo’°°"Xn—2'Xn)° Proof. We see that Vn 1(f7xo,...,x -2’Xn) - Vn—l(f7xo"°°'Xn—2’Xn—l) = Vn_l(f;xo,...,xn_2,xn) — Vn_l(f;xn_l,xo,...,xn_2) = = (Xn“Xn-l)Vn(f7Xn_lIXOI0°°ixn_2:Xn) _>_ 00 C] Remark. We see from prOposition 3.2 (and the remarks above it) that Vn_l(f;xl,...,xn) is a monotone function in each of the variables Xl'°°°’Xn when f is n—convex (d). Proposition 3.3. Let n.2 2. Let f be n—convex (d) on (a,b). Then f’+ and f'-' exist and are finite in (a,b). * Proof. Let x 6 (a,b), x e (a,x). If n > 2 choose X1"°°'xn-2 such that 1- a < xl < x2 <...< Xn—2 < x . l9 F(t) = Vn_1(f;xl,...,xn_2,x,t)(X < t < b). By proposition 3.2, F is non—decreasing on (x,b) and * * F(t) 2 Vn_l(f;xl,...,x ,x,t) 2 Vn_l(f;x1,...,xn_2,x ,x) n—2 here. Thus F(x+) exists and is finite. Let n—2 g(t) = n (t—xk), h(t) = f(t)/g(t). for t e (x*.b). Let ni? f(xj) F1”) = ._ g’(x.)(X.-X)(x.—t) ' for — 3 J J x* < t < b . (Thus 9 E l and Fl 5 0 if n = 2). Then _ f(x) f(t) F(t) -— Fl(t) + g(X) (X-t) + g(t) (t-X) —Fl(t)+D-Q%%(}—{L(XZ-—l-Ixj) 20 a Proposition 3.6. Let f be continuous in (a,b) and f’I- = O in (a,b). Then f is constant. Proof. We first show that if g is continuous + in (a,b) and g' > 0 there, then g is non-decreasing. To see this let d,B be in (a,b) and d < B. Let x0 = supIt 6 [m6]: g(t) 29(001. It follows easily from the continuity of g that g(xo) 2_g(d). Moreover, we claim that x0 = B. If this is not the case then, since g"“+(xo) > 0, there is a point t in (XO,B] such that g(t) 2.g(x0) 2_g(d), 21 and this contradicts the definition of XO. Thus x0 = B so that g(B) 2Ig(a). From this one sees easily that if g is continuous and g'i'lz O in (a,b) then g is non- decreasing. The prOposition at hand is immediate from this and the corresponding statement about a function g such that g'4"g_0. D Proposition 3.7. Let f be continuous on (a,b) such that f'+ is also continuous. Then f is differentiable. Proof. Let g be such that g' = f . Then (f—g)'4- = 0 so f—g is constant. B PrOposition 3.8. Let n > 2. Let f be n-convex (d) in (a,b). Then f’ is (n—l)—convex (d) there. Proof. As f"+ is (n-l)-convex (d) (prOposition 3.5), and continuous (corollary 3.4), we see by proposition 3.7 that f’ exists, and so f’ = f’4- is (n-l)-convex (d). C Corollary 3.9. If f is n—convex (d) in (a,b) then f is n-convex in (a,b). That is: for n > 2, 22 f(n-2) if f is n-convex (d) in (a,b) then exists and is convex. Proof. This follows trivially from proposition 3.8 by iteration. D In fact n—convexity (d) is equivalent to n-convexity, but we do not need this result and so omit the proof. (A proof may be found in [7], however this paper is quite complicated. We have included the proof of corollary 3.9 for completeness and because of its simplicity. The proof is due to J. Marik.) Definition 3.2. The generalized symmetric derivative, Dan(xO), of order 2n of a function F at x0 is defined by the relation n 2k %{F(x0+h)+F(XO—h)} — k2; ?2k)1 D2kF(XO) = 0(h2n) as h 4.0, in a manner anaLDgous to the definition of the Peano derivatives. The generalized symmetric derivative, D2n+lF(xO), of order 2n+l is defined by n 2k+1 l h 2k+l l 2 { F (Xo+h) *F (XO-h) } -- k2 W D F (X0) = O (h2n+ ) =0 . as h 4'0. These derivatives are generalizations of the Peano derivatives in the sense that if Fk(xo) exists 23 then so does DkF(xo) but not necessarily conversely. We can proceed as in Chapter II to define upper and lower symmetric derivates. Thus we define ¥h(f,xo,h) for n — — 2m by 2m m—l 2k hl___. _ l _ _ -« 1;____ 2k (2m): Y2m(f,xo,h) — §[f(xo+h)+f(xO h)] kib (2k): D f(xo) 2 (provided D kf(xo) (k = O,...,m-l) exist) and, for n = 2m+l, by h2m+l l (2m+1): Y2m+1(f'x0'h) = §(f(x0+h)-f(xO-h)} m—l 2k+l h 2k+1 k=0 (2k+1): D f(Xe) (provided D2k—1f(xo) (k.= l,...,m) exist). The k-th generalized upper symmetric derivate of f at x0 is ka(xo) = limsup Yk(f,xo,h). The function F is said to satisfy conditions A2m in (a,b) if it is continuous in [a,b] and if, for l g_k g_m-l, each D2kF(x) exists and is finite in (a,b), and if lim hY (F,x ,h) = O hao 2m 0 for all x6 in (a,b)\E, where E is a countable set. 24 The conditions are defined analogously. This A2m+1 condition can be thought of as a sort of "n-th order smoothness." We say that the finite function f has a discontinuity of the first kind at c 6 (a,b) or an ordinary discontinuity, provided that f(c+), f(c—) both exist in the finite or infinite sense but that at least one of these is different from f(c). The function F is said to satisfy conditions BZm—Z in (a,b) if it is continuous in (a,b), if, for l g_k g_m—l, each D2kF(x) exists and is finite in [a,b] and if no D2kF(x) has an ordinary discontinuity in (a,b). A similar definition is given for conditions B2m-l' PrOposition 3.10. If F satisfies conditions An and Bn-2 1n (a,b) and 1f ynF(x) 2_O in (a,b) then F(X) is n—convex (d) in (a,b). Proof. This is theorem 4.2 of James [3]. D We shall see that theorem 3.1 follows from proposition 3.10. In the first place, it is easily seen that 6kF(x) g_AkF(x) g ka(x) whenever the left hand side has a meaning, so that if 6kF(x) > O on [a,b] then also ku(x) 2_O on [a,b]. To finish we need 25 only note that conditions An and Bn are automatically -2 satisfied in case 6nF(x) 2_O. The proof of this fact is in Section 8 of James' paper. Thus theorem 3.1 is proved. Proposition 3.11. Let F be defined in [a,b] and let 6nF(x) 2_O in [a,b]. Then F is n-convex in [a,b]. Proof. By theorem 3.1 we need only show that F(n—2)+ (n—2) is continuous F(n-2) exists at aa and that F at a. (Similarly for b.) But as is convex in (a,b) we know that F(n_2)(a+) exists. Thus by proposition 2.7, F(n'2)+ (n-2) F(n-2) (a) = F(n_2)+(a) = F (a+). Thus exists and is continuous in [a,b] so that F is n-convex there. C CHAPTER IV THE CENTRAL THEOREM In this section we shall state and prove the validity theorem for the Zn—integral which we define in the next section. Theorem 4.1. Let f be defined on [a,b] and f(n_l) exists suppose 6nf(x) > O on [a,b]. Then and is non-decreasing on [a,b]. Proof. We know f(n—l) must exist at each point of [a,b]. By theorem 3.11 f(n_2) exists and is 2—convex in [a,b]. MOreover f(n-Z) + exists in (n-2) ' — [a,b) and f exists in (a,b]. Applying a one—sided version of Taylor's theorem we see that (n-2)’ + _ (n—2)’ -_ f — f(n-l) + and f — f(n-l) _. exists in [a,b] and is the derivative of a Thus f (fl-l) convex function and hence is non-decreasing (and in fact continuous). C We shall need the following simple consequence of theorem 4.1. 26 27 Theorem 4.2. Let f and g be defined on an interval I such that f = g on I. Then (n) (n) f and g differ by a polynomial of degree no more than n-l. Proof. As (f-g)(n) = O we have (f-g)(n_l) is a constant by theorem 4.1. The rest is easy. D CHAPTER V THE Zn "' INTEGRAL Definition 5.1. Let n be a natural number. Let M,f be defined in [a,b]. Then M is called an n—majorant of f in [a,b] if there is a function P on [a,b] such that 1) M = F(n) on [a,b], 2) 6n+lP(x) 2 f(x) for each x C [a,b], 3) 6n+1P(X) > —w for each x e [a,b]. The function P will be called a pre—majorant. Then n—minorants are defined similarly, replacing 2),3) by 2 ) An+lP(x) g_f(x) for each x 6 [a,b], 3') An+lP(x) < +m for each x c [a,b], and then P is called a pre—minorant. Remark. For n=O we have exactly the definition of majorant for the Perron integral. Definition 5.2. Let f be defined on [a,b]. The upper ZnQintegral of f on [a,b] is (Zn*)fb f = inf{M(b) - M(a):M is an n-majorant a of f on [a,b]}. 28 29 The lower Zn-integral of f on [a,b] is (Zn*)fb f = sup{m(b) - m(a):m is a n-minorant a of f on [a,b]]. (The infflnum of an empty set is of course +w, etc.) If the upper and lower Zn-integrals are finite and equal then we write (Zn)fb f for the common value and say a that f is Zn—integrable on [a,b]. The ZO-integral is then exactly the Perron integral. Remark. If M is an n—majorant of f and m is an n-minorant and if P and p are an associated pre-majorant and pre-minorant respectively, then 6n+l(P-p) 2'0 and so (P-p)(n) = M—m ls non-decreaSing by theorem 4.1. Thus M(b) - M(a) 2 m(b) — m(a). From * _ this it follows that (Z )fb f > (Z )fb f for every n a — {1* a function f. We can now prove a simple theorem which can be taken as the motivation for this integral. Theorem 5.1. If F exists and is finite (n+1) on [a,b] then is Zn-integrable there and F(n+1) b a (Zn)j: F(n+1) = [F(n)] 30 Proof. F(n) is’at the same time an n—majorant and an n-minorant of F Thus (n+l)° * fb b (Zn ) a Fm”) g [F(n)]a and y b (Zn*)j: F(n+1) 4;[F(n)]a But as we have remarked above, * (Zn )C F(n+1) 2 (Zn*)j: F(n+l) and so b (Zn)j: Fm”) = [F(n)]a . [3 We will frequently make use of the following simple result in the later proofs. Proposition 5.2. A function f defined on [a,b] is Zn-integrable there if and only if for each 6 > 0 there is an n-majorant M and an n—minorant m such that M(b) - M(a) - (m(b) - m(a)) < 6- Proposition 5.3. Let M be an n-majorant of f on [a,b]. Let c be any constant. Then M(x) + c is an n—majorant. Proof. Let P(n) = M on [a,b]. Let Q(X) = P(X) + cxn/hi . Then Q(n)(X) = M(x) + c and 6n+lQ(x) = 6 P(x). [:1 n+1 31 PrOposition 5.4. If f g_g on [a,b] then * * (zn)j:f_\_(zn)j:g. Proof. Clearly any n-majorant of g is also an n-majorant of f. The rest is clear. D Occasionally we shall use the more classical notation fb f(t)dt or (Zn)jb f(t)dt instead of a a (Zn) j: f. CHAPTER VI PROPERTIES OF THE Zn-INTEGRAL PrOposition 6.1. A Zn-integrable function f on [a,b] is finite almost everywhere on [a,b]. Proof. Let M be an n—majorant and m an n-minorant of f on [a,b]. We may assume M(a) = m(a) = 0. Define R(x) = M(x) — m(x). Then R is non-decreasing by theorem 4.1. Let P,p be defined on [a,b] so that P(n) = M, p(n) = m. Then P-p is (n+1)—convex by theorem 3.11 and so (P-p = (P—p)(n) = R (prOposition 2.1). Suppose f(x) = +w. Then 6n+lP(x) = +m and, as An+lp(x) < +m, we have (P-p)(n+l)(x) = +m. Similarly, if f(x) = -m, then An+lp(x) = -m and since 6n+1P(x) > -m we have (P-p)(n+l)(x) = +m. But (n) ’ ___ ) (n+1) R' = (P-p) (P—p exists and is finite a.e. in [a,b] and so f is finite a.e. in [a,b]. D Proposition 6.2. If f is Zn-integrable on [a,b] and if c 6 (a,b) then f is Zn—integrable on each of [a,c] and [c,b]. Mbreover an”: f = (an):l f + (Zn)j: f. 32 33 Proof. Let g > 0 be given. Let M,m be respectively an n-majorant and an n—minorant for f on [a,b] such that [Memlg < 6. It follows immediately from the definitions that M is an n-majorant on [a,c] and on [c,b] and that m is an n-minorant on each of these. Then, by the remark following definition 5.2 we have c \ b [M—m]a‘4 O and [M—m]c 2.0. But then 0 g_[Mrm]: + [Mrm]: = [Mrm]: < €. Thus 0 g_[M+m]: g G and 0 g [M—m]: < 6 so that f is Zn—integrable on each of [a,c] and [c,b]. Moreover b _ c b C [m]a — [m]a + [m]C g_(zn)fé f + (Zn)f: f‘g b b _<_ [M]: + [M]c [ma . Since the first and last terms differ by less than 6 the formula is proved as well. D Proposition 6.3. If f(x) is Zn-integrable on [a,c] and on [c,b], a < c < b, then it is Zn~integrable on [a,b] and the usual formula holds. Proof. Let Ml , M2 be n-majorants of f on [a,c] and [c,b] respectively. By proposition 5.3 we may assume that Ml(a) = O and M2(c) = M1(c). Let M be defined by 34 M1(X) x c. lam] M(x) = M2(X) x 6 [c,b]. Then M is an n-majorant of f on [a,b]. let P(n) = M1 and Q(n) = M2. Then P(x) x 6 [a.C] R(x) = Q(X) - Q(C) + P(C) x 6 [c,b]. is a pre-majorant on [a,b] since Rm) = M and 5n+l,+R(c) = 6n+1,+P(C) _>_ f(c) and also > -m, and 6n+1,—R(c) = 5n+l,-Q(C) 2_f(c) also > -m. If m1 and m2 are n-minorants on [a,c] and on [c,b] then we construct an n—minorant m on [a,b] in an analogous way. Moreover, if [Ml-ml]: < g and b 6 [M2 m2]C < 2 then b _ c b [M—m]a — [Ml—m1]a + [Mz-mz]C < 6. Furthermore, b fb b [m]a _<_ (zn)j: f + (Zn) c f g_[M]<_:1 . Thus (Zn)f: f ex1sts and equals (Zn)I: f + (Zn)‘j’;b f. To see this C Definition 6.1. Let f be Zn-integrable in [a,b]. Then, if x 6 (a,b], we see from proposition 6.2 that f is Zn—integrable over [a,x]. We put and [3 35 (Zn)f: f = O for any function. Then (Zn)f: f is defined for each x 5 [a,b]. An indefinite Zn-integral of f is any function of the form F(X) = c + (Zn)f: f where c is constant. Proposition 6.4. Let f be Zn-integrable on [a,b]. Let F be any indefinite Zn-integral of f on [a,b]. Then F is the uniform limit of a sequence of n-majorants on [a,b] (similarly for n-minorants). Proof. Let G > 0. Let M,m be respectively an n—majorant and an n-minorant of f on [a,b] such that M(a) = m(a) = F(a) and M(b) - m(b) < e. As Mrm is non-decreasing on [a,b] (by the remark after definition 5.2), we have 0 g M(x) — m(x) < e for each x 6 [a,b]. Since M,m are respectively an n—majorant and an n—minorant on every sub—interval of [a,b] we have m(x) - m(a) _<_ (Zn) f _<_ M(x) — M(a) . pf? Thus 0 _<_ M(x) — M(a) - (zn)j‘: f = M(x) — F(X) < e. The rest is obvious. D Proposition 6.5. Let F be an indefinite Zn-integral of f on [a,b]. Let M be an n—majorant 36 of f on [a,b] such that F(a) = M(a). Then M—F is non-decreasing. Proof. Let a g_xl < x2 g_b. Then M is an n—majorant of f on [xl,x2] and so X2 F(X2) — F(xl) = (znfiX1 f g M(x2) - M(xl). Thus M(xl) - F(Xl) g M(x2) - F(xz). D Remark. Similarly, if m is an n—majorant than F—m is non-decreasing. PrOposition 6.6. Let F be an indefinite Zn-integral of a function f on [a,b]. Let M be an n—majorant of f on [a,b]. Then M—F is continuous. Proof. Let m be an n—minorant of f on [a,b]. Then Mrm is an n—th e.I%d. of some function G on [a,b]. As 5 G.2 O on [a,b] we see that G(n-l) is n+1 convex and continuous in [a,b] by theorem 3.11. G(n)- (n)+ Mbreover exists in (a,b] and G exists in [a,b). But then it follows from Taylor's theorem that (n)+ _ (n)- _ . . G — Gm)+ and G — G(n)-' Since G(n) eXists we see that g(n) = G(n) , so that G(n) is an ordinary derivative. Thus, since G(n) = G(n) is non-decreasing, we see by the intermediate value property for derivatives that it is continuous. But M-F is the uniform limit of 37 functions of the form M—m so that M—F is also continuous. D Theorem 6.7. Let F be an indefinite Zn-integral of a function f on [a,b]. Then F is an n—th e.P.d. on [a,b]. Proof. Let M be an n—majorant of f on [a,b] and let P(n) = M there. Then as M—F is continuous we see that there is a function G on [a,b] such that (n) G = MrF there. But then (P—G)(n) = P(n) - G(n) = F. D Corollary 6.8. Let F be an indefinite Zn-integral of f on [a,b]. Then F is Z —integrable. n—l Proof. This is immediate from theorem 6.7 and theorem 5.1. U According to corollary 6.8, we can form the iterated integral 5 (6.1) f: d§o fa E o n—l dgl ... fa f(gnmgn whenever f is Zn-integrable, where the innermost integral is a Zn—integral, the next is a Zn_l—integral, etc., and the outermost is a ZO—integral. We shall use the symbol 38 j“ f(§)d(§.n) a for the (n+1)—fold iterated integral in (6.1). Note also that IX f(§)d(§.n) is a continuous function of x. a Theorem 6.9. Let F(X) be an indefinite Zn—integral of a function f on [a,b]. Let G(n) = F on [a,b]. Then G(n+l)(X) = f(x) for almost every x € [a,b]. Proof. Let E > 0. Let m(o) denote Lebesgue * measure and let m (-) denote Lebesgue outer measure. Let M. be an n—majorant of f on [a,b] such that M(a) = F(a) and M(b) - F(b) < 62. Let P(n) = M. Let R(x) = M(x) - F(x). Then R is continuous (proposition 6.6) and non-decreasing (proposition 6.5) and so (DJ: R’_<_R(b> - R(a) < 62. (n) MCreover (P-G) = R on [a,b]. Let A(E) = [xz6n+lG(x) < f(x) — e}. Let B = [sz' = (P-G)(n+1) exists and is finite}. If x €A(€) then 6n+lG(x) < as P(x) — c and so n+1 6 P(x) - 6n+1G(x) > 6. (Note that 6n+lP(x) > —a> and n+1 39 6n+lG(x) < +m). If x e B H A(E) then 6n+lP(x) 6n+1G(x) = R’(x) . To see this, write P = (P-G) + G so 6n+lP-2 R + 6n+lG’ while G = P — (P—G) and so 5n+lG 2-6n+lP - R . Thus B D A(E) ls a subset of the set 3(6) = [X=R'(X) > 6}- We now see, EHMB(6)) 3-(L)IB(€)R'.: (L)I: R’ < 62' It follows that m(B(€)) < E and so m*(A(e)) < 6. So in particular, for each positive r, m*(A(r)) < r. * So letting r = —%- we see m (A(J%)) < J%'. From this 2 2 2 'k we see that m (A(O)) < E and as G is arbitrary this implies that 6n+lG'2 f a.e. in [a,b]. In the same way we can show that An+1G g f a.e. and so G(n+l) = f a.e. on [a,b]. C PrOposition 6.10. Every Zn—integrable function is measurable. Proof. Let f be Zn—integrable. Applying theorem 6.9 let G = f a.e. Then (n+1) n+1 l n m f(x) = lim (k (n+1):{G(x + k) — 23G )(x)/k m1]) k4m m=O (m for almost every x. Thus f is the limit a.e. of a sequence of continuous functions (as G is continuous) and so is measurable. D 40 Theorem 6.11. The ZO-integral is identical to the Perron integral. If f is Zn-integrable on [a,b] then f is also Z -integrable there and n+1 (zn)f: f = (zn+l)f: f. Proof. The first statement is obvious. Let f be Zn-integrable on [a,b]. Let E > 0. Let M,m be respectively an n—majorant and an n—minorant of f on [a,b] such that [Mem]: < 6. Let P = M, p(n) = m on [a,b]. (n) Let o(x) = I: P. q(x) = I: p. Then by proposition 2.2, Q(n+l) = M, q(n+l) = m and by corollary 2.3 and its analogue for upper derivates, 6n+zQ(x) 2-5n+lP(X)-2 f(x) on [a,b] and 6n+ZQ(X) > -w on [avblo Also An+2q s. 4.1900 _<_ fix) and An+2q(x) < +m on [a,b]. Thus M is an (n+1)-majorant and m is an (n+1)-minorant . . x and so f 18 Zn+l-1ntegrable and O g_[M]a — (Zn+l)I: f.g 6. Hence (ZN): f = (Zn+1)j: f. a 41 Recall also that the Perron integral is an extension of the finite Lebesgue integral. Proposition 6.12. Let f be a non-negative measurable function on [a,b]. Then (zn*)f: f = (zn*)f: f = (L)f: f. (The last integral is the finite or infinite Lebesgue integral). In particular if f 2.0 and if f is Zn—integrable then f is Lebesgue integrable and z ) f = (L) f. m: J: Proof. If (L)fb f is finite then all follows a immediately from theorem 6.11. If (L)fb f = +m let a fk = min{f,k}. Then fk is Lebesgue integrable and it is easy to see that b (zn*)f: f 2,(zn*)fé fk = (Zn)f: fk = (L)I: fk (k = 1,2,...). Then since lim(L)fb fk = +m, we also have (Z )fb f = +m. a 11* a The rest follows at once. F Proposition 6.13. Let f,g,h be functions on * * [a,b]. Suppose that (Z )fb f + (Z )Ib g has a meaning, and that f(x) + g(x) 2_h(x) whenever the left hand side has meaning. Then 42 (6.2) (2;): h _<_ (zn*)j: f + (23)]b g. a Proof. Let M1 and M2 an n-majorant of g. (If one or the other be an n—majorant of f of these does not exist then the right hand side of formula (6.2) is +m in which case the result is Obvious). Then M1 + M2 is an n~majorant for h. To see this let P(n) = M1' Q(n) = M2. Then 6n+l(P+Q) 2-5n+1P + 6n+1Q > _m and 6n+l(P+Q) 2 f + g _>_ h whenever the middle term has meaning. If f + g has no meaning then we see that 6n+1(P+Q) = +w. Namely, one of f(x), g(x) is +m so that one of 6n+1P(x), 6n+lQ(x) is +m and the other is not —m whence 5n+l(P+Q)(x) = +m. Thus 5n+l(P+Q) 2_h on [a,b] and M1 + M2 is an n—majorant. The result follows. E Remark. We of course have a similar theorem for lower integrals. Corollary 6.14. Let f and g be Zn-integrable in [a,b]. Let h(x) = f(x) + g(x) whenever the right side has meaning. Then h is Zn-integrable in [a,b] and (211)): h= (Zn)~J: f+ (zn>):g. 43 Proof. b (Zn*)f: f + (Zn*) 9 i (Zn*)fa h S * g(zn) a)? {‘6' h g (zn*)j': f + (zn*)j: g. But the terms on the extremities are equal. F Proposition 6.15. If f is Zn—integrable in [a,b] and if f = g a.e. in [a,b] then g is also zn—integrable there and (Zn)f: f = (Zn)J: g. Proof. Let h(x) = 0 whenever f(x) = g(x) and h(x) = +w otherwise. Then 0 = (L)J: h = (zn)j: h. and by corollary 6.14 (Zn)f: f = (Zn)f: (h+f). Moreover g g_h+f whenever the right side has a meaning and so * (Zn )I: g g (Zn)f: f. Similarly considering the function defined by k(x) = -h(x) we see that (Zn*)f: g 2 (Zn)f: f. D Remark. According to proposition 6.15 the Zn-integral may be defined naturally on certain of the 44 Lebesgue classes (those which contain a Zn—integrable function). Combining propositions 6.15 and 6.14 we see that the Zn-integral, considered as a functional on these classes, is additive. Again prOposition 5.4 and 6.12 can be used to show that the Zn-integral is a positive functional. Theorem 6.16. Let f and P be defined in [a,b] so that P exists and so that (n) a) 5n+1P(x) 2.f(x) a.e. in [a,b], b) 6n+1P(x) > -m in [a,b]. Then * b (2n )f: f g [1901)]a . Proof. Let h(x) = f(x) when 6n+lP(x) 2 f(x), h(x) = -m otherwise. Then P(n) is an n—majorant for h and so .. b (2n )j: h g [P(n)]a Define k(x) = 0 when 6n+lP(x) 2_f(x), k(x) = +m otherwise. Then f(x) g_h(x) + k(x) whenever the right side has meaning and so * * b (Zn,):fg(zn)):h.og[wa. From this theorem we see that we may enlarge the class of majorants to include those functions which 45 satisfy pr0perty 2) of definition 5.1, only almost everywhere. A similar result holds of course for minorants. PrOposition 6.17. Let f be Zn—integrable on [a,b]. Let c be any finite number. Then cf is Zn-integrable on [a,b] and (211)]: (cf) = c(zn)j: f. The proof of this prOposition is easy when the cases c.2 O and c < O are considered separately. We omit the details. Combining this result with our previous results we see that the Zn-integral is a positive linear functional on the set of those Lebesgue classes for which it makes sense. PrOposition 6.18. Suppose g and h are Zn-integrable functions and f is any measurable function on [a,b]. Suppose g g_f g.h on [a,b]. Then f is Zn-integrable as well. Proof. Since we have, for almost every x, that O g_f(x) — g(x) g h(x) — g(x) and since h-g is Lebesgue integrable we see that f-g is Lebesgue and hence Zn—integrable. We now simply apply linearity. C 46 Remark. In the same way we can show that if f is Zn-integrable and bounded below (or above) by a Lebesgue integrable function, then f is already Lebesgue integrable. Proposition 6.19, Dominated convergence theorem. Suppose g and h are Zn-integrable functions on [a,b]. Let {fk] be any sequence of measurable functions on [a,b] such that g g'fk g_h on [a,b]. Suppose lim fk = f a.e. in [a,b]. Then f and each f is Zn-integrable and k 113:: (znw: fk = (2.1)): f. Proof. The integrability is obvious from proposition 6.18. To see the last formula simply apply the usual Lebesgue theorem to the sequence h-f and the limit k h-f. Then apply linearity. D Proposition 6.20. Let [fk} be a sequence of Zn-integrable functions on [a,b] which converge uniformly there to a function f. Then f is Zn—integrable in [a,b] and (Zn)jb f = lim (Zn)fb fk' a a Proof. Let E > 0. Let K be chosen so that fk - 6'3 flg fk + E for all k.2 K. Then f is Zn-integrable. Moreover as -e.g f - f _g e we see that k 47 lim(Zn)f: (f—fk) = 0 by the dominated convergence theorem. Now apply linearity. U Note also that [(Zn)f: fk} converges uniformly to (Zn)f: f in [a,b]. Namely, f — fk is Lebesgue integrable and so (L)f: (f—fk) converges uniformly to zero on [a,b]. Proposition 6.21. Let f be Zn—integrable on [a,b] and M an n-majorant. Let P = M. Then (n) P(n+l) eXists a.e. on [a,b] and is Zn-integrable there. x Proof. Let F(X) — (zn)fé f. Let Q(n) — F on [a,b]. (Such a Q exists by theorem 6.7). Then (P-Q)(n) is a continuous, non—decreasing function by prOpositions 6.6 and 6.5. But then (P—Q)(n) = (P-Q)(n). ) Namely, there is a function R such that R(n = (P-Q)(n) which implies (R—(P-Q))(n) = 0. Then R and P—Q differ by a polynomial so that (P~Q)(n) exists. Since (P-Q)(n) is increasing and continuous we see that (P-Q)(n+l) exists a.e. in [a,b]. Thus P(n+1) : (P - Q + Q)(n+l) )(n+l) (P—Q + Q(n+1) (n+1) (P—Q) + f (a.e. in [a,b]). 48 . +1 . (we have applied theorem 6.9.) Since (P-Q)(n ) is Lebesgue integrable we see that P(n+l) must be Zn—integrable. C To finish this section we would like to present a theorem of Oliver [10]. We need some preliminaries. Lemma 6.22. Let f be differentiable on J = [a,b]. Let a > 0. Let \f’\ 2_a on J. Then there is an interval JO c J such that m(JO) 2.; m(J) \ a and that \f‘ 3-4 m(J) on JO. Proof. By the intermediate value prOperty for derivatives we may assume f’ > a. Let .2110. _2, If f 2 0 on (c,b), then f(x) 2,a(X—c) for x E [c,b] and we may choose JO = [b — g, b]. If f(y) 3.0 for some y G (c,b), then f(x) < g(x—c) for x e [a,c] and we may choose JO = [a,a + 41° C Lemma 6.23. Let n be a natural number. Set q = l+...+n = n(n+l)/2. Let J be a compact interval. Let a > 0 and ‘F(n)\ 2'a on J. Then there is an n interval K C J such that m(K) 2_4- m(J) and that ‘F‘ 26mm)n - 4‘q on K. 49 Proof. For n=l this is exactly lemma 6.22. Assume that n > 1 and that the assertion is true for n-l. Set p = l+...+n-1. Then we have an interval n+ No C J such that m(KO) 2.4— 1m(J) and that I n—l -p )F \ 2_am(J) 4 on KO. By lemma 6.22 we have an interval K C KO such that l m(K) 2 Z m(KO) _>_ 4‘n m(J) and that \F‘ 2.am(J)n-l-4—po % m(K ).2 —n am(J)n-l-4_p-4 -m(J) = am(J)n4—q on K. E h/ Theorem 6.24. Let F be a p-th indefinite integral of a function which is bounded below on some neighborhood of O (p > 0). Let F(p)(0) = 0. Then F(p)(0) = 0. Proof. Let a > 0, 6 > 0. Let f > —a on (-6,6) and let F be a p-th indefinite integral of f. Let g(x) = IX f. We may assume F(O) =...= F(p-1)(0) = 0. 0 Then (6.3) F(X) = o(xp) as x 4 0. Assume that D+g(0) > 0. Then there is an e. o < e < 2a, and numbers an > 0 such that g(an) > e an (n = 1,2,...), a 4 0. 50 For a < x < an + ane/2d we have g(x) = g(an) + onf > e an— (x-an)a > 1 > c an - dang/2a — 2 a E- F(p-l) Since = g we have, by lemma 6.23, an xn in (an,an + ane/Qa) such that \ l oi p_l . -r _ p )F(Xn)‘ 4_§ an€(an 20L) 4 —-(an) Q where _ 21221 = 6p r — 2 . Q p—l r 2p-o -4 Since 6/2a < l, we have xn < 2an whence P . £L , = \F(xn)\ Z-Xn p (n 1,2,...) 2 which contradicts (6.3). Thus D+g(0) g.o. We may + show similarly that D+g(0) 2_0 so that g' (0) = 0. Analogously g, (0) = 0 so that g’(0) = F(n)(0) = 0. Corollary 6.25. Suppose M(n) exists on [a,b] and is bounded below there. Then it is an ordinary derivative. Proof. According to theorem 5.1, M(n) is Z(n_l)-integrable on [a,b]. By prOposition 6.12, it is thus Lebesgue integrable. Thus M is an n-th indefinite integral of M(n)’ It follows easily from theorem 6.24 n)( that M(n)(x) = M( x) for any x 5 [a,b]. [’1 CHAPTER VII INTEGRATION BY PARTS FOR THE Zn-INTEGRAL Notation. In what follows let BV[a,b] signify the class of functions which have finite variation on the interval [a,b]. Let (R)f: fdg, for g e BV[a,b] and f continuous, be the Riemann— Stieltjes integral of f with respect to g in the sense that the infimum of the upper sums associated with f and the positive (negative) variation of g is the same as the supremum of the lower sums. Fundamental to all that follows is the next theorem. Theorem 7.1, Integration by Parts. a) Let f be Perron integrable in [a,b], F(X) = (Zo)f: f. Let G 6 BV[a,b]. Then b (2 )fb £6 + (R)fb FdG = [FG] . o a a a b) Let n 2.1. Let f be Zn-integrable on [a,b]. Let F(X) = (Zn)j: f. Let G and y be defined on [a,b] such (n—l) that y e BV[a,b] and G is an 51 52 indefinite integral of y. Then fG is Zn—integrable and (Zn)f: fG + (zn_l)f: FG’ = [FG]: The proof is quite long and involves a chain of lemmas. Later we shall show that the strong conditions put on the function G are really necessary. This proof is due to J. Marik. We assume that a) is well known but for completeness we present a proof. Proof of Theorem 7.1, a). Assume G is positive and non-decreasing. (This is sufficient by the linearity of the integrals). Let a = x0 < xl < ... < xm = b be a partition of [a,b]. Let Yk = min[F(x):xk_l g_x g_xk]. If M is a O—majorant of f on [a,b] then one easily sees that G(M—yk) is a O-majorant of fG on [xk_l,xk]. Thus Xk * Xk (z ) fG < [G(F—Y )] o IXk-l — k x k—l Thus * b Xk (20 >1: f6 3 [GFla - i3 [(521ka 53 for each such partition, and so * (z )jb fG < [GF]b - (R)Jb FdG. 0 a " a a Now replacing F and f by —F and —f we see my”: fG _>_ [GP]: - R]: FdG. [3 Lemma 7.2. Let m,¢ be functions on [a,b]. Let q be a natural number. Let m(q) be continuous on [a,b] and let w(q—l) be absolutely continuous on [a,b]. Let R be an indefinite integral of w¢(q). Set q-l . . . v = Z (—1)3 m(q‘3‘1)¢(3) + (-1)qR. i=0 Then V' = @(q)w. Proof. (q) __ (q-l) X _ (q-l) ’___ = j:cp 41- [co Ma J: cp 1) = [cp‘q‘l’w-cp(q‘2)w’ +...+(-1)q‘1cpw(q'l’]: + (-l)qf: cpw‘q’ = : V(X) - V(a). D Definition 7.1. Let n > 1 be an integer. Let M be a continuous function on [a,b]. Let Y e BV[a,b]. (n-2) be an indefinite integral of y. Let Let g K0,...,Kh be functions on [a,b] with the following properties: a) b) c) 54 (k) Kk = Mg (k) (k = 0,1,...,n-2), Kéfiz) is an indefinite integral of My, there is a number c such that Kén-Z), is an indefinite integral of c + I: Mdy (x 6 [a,b]). Then we say that Ko""’Kn have property u) with respect to M,g on [a,b]. Lemma 7.3. Let a,b,n,g,y be as above. Let (n) M be a function on [a,b] such that M is continuous on [a,b]. Let K0,...,K.n have property w with respec Then S(n) t to M,g. Set n k n n) = M( 9. Proof. We first show that (7.1) for r = 0, r=0. If it S (r+l) 5 II (D II r . . . . S(r) = Z (_1)3 (n+er-1)M(r-3)g(3)+ i=0 3 n - ( ) + Z (-1)3(r.1)K.r j=r+l 3 J ...,n-2. The relation (7.1) is obvious for is true for some r (0 g_r < n—2), then = A + B + C + D + E where r . . . . Z (_1)J(n+33r'1) M(r+l-J)g(3), i=0 2 (_l)j(n+jfr“l) M(r-j)g(j+l)' ° 3 J: 55 C = (-l)r(n_1)Mg (r+l)’ _ r+l (r+l) D I (-l) (r+1)Kr+l ' n E: Z (-1)3(§)Kj‘r+1’. j=r+2 We can re—write B and D as B = _ Z) (_ l)j(n+j— r— 2) M;(r+l-j)g (j) ’ j= —1 3 _ r+1 (r+l) D - (-l) (r+1) M'g ° Then, applying the relation (p+l) - (g) = ( qpl)' we get r A+B = Z (_l)j(n‘j-r-2) M(r+l-j)g(j) i=0 3’ and _ r+l n-l (r+l) C+D ‘ (_l) (r+l) ME! I which completes the induction and proves (7.1). If we put r = n—2 into (7.1) we see n-2 (7.2) s‘“‘2) = Z; (—1)j(j+1)M(n‘2‘j)g(j) + (-1)n i=0 +(-—1)“K‘“2’. Set n-2 . . . R = K§212) , Tp = .Zj(_1)JM(n-2-J)g(j)+ (_l)n—l R J=P (p = Ollloooon-Z). Set 2 = R (“’2’ K(n- 2) Then n—2 Z) = p=0 p so that (7.3) We also have T : P If we put m 56 11-2 - n n E (—1)3(j+1)M‘n‘2'3’g‘3’ + (-1)“‘1(n-1)R. j=0 n—2 = Z) T + (-l)n_lz. p=0 p S(n-2) .23 (_l)i+p M(n—Z—i-p)g(i+p) + (-l)n_lR. 1:0 = (—l)pM, w = g(p), q = n—l-p then we get from lemma 7.2 that (7.4) Further t Z(x) =(cl + 1: My)— 62 - f:(e3 + fa Mdy)dt Thus (7.5) T, = (_l)p M(n-l-p)g(p). t 4 + f: (--c3 + M(t)v(t) — I; Mdv)dt (c a + NIR t ’ + f M y)dt (the c.'s are constant). a 5 3 I - X; Z(X)—c5+faMy. Now from (7.3) and (7.4) we get (7.6) and, from (7. If we put m = M', q = n-l, w = g we have w n-2 S(n—l) _ Z: (_l)pM(n—1-p)g(p)+ (_1)n—1Z, I P=0 5), Z’ is an indefinite integral of M’y. -l) n—2) (q = g( 57 on [a,b] and w(q) = y a.e. so the formula S(n) = M(n)g follows at once from (7.6) and lemma 7.2. C Lemma 7.4. Let n > 1 be an integer. Let M,g,y be functions defined on I = [a,b]. Let M (n—2) (n) exist on I and let y e BV(I). Let g be an indefinite integral of y. Then there is a function S such that (7.7) Sm) =M(n)g on I. For any such function S and for any x 6 I for which g'(x) is finite we have (7'8) Qn+l(S:Xoh)=M(n)(X)g’(X) + g(X) 9n+l(M,X,h) +O(l). Note. If n > 2 then g’(x) is finite for every. x. If n=2 then g' is finite except possibly on a countable set and even here we have, for h > 0 (respectively h < 0), that formula (7.8) holds if we + replace g'(x) ‘by g' (x) (g'-(x)), the right (left) hand derivative. Proof. Let Ko’°°°'Kn have property w ‘with n respect to M,g. Let S = Z) (-l)k(£)Kk. Let x 6 I. k=0 We may assume x=0. Let P be a polynomial of degree g_n such that the function M = M-P fulfills (7.9) M(t) = o(tn) as t +0. 58 Let Rk be functions such that Rék) = Mg(k) (k = O,...,n-2), -(n-2) _ t — -(n-2) K (t) _ (0 MY, Kn n_1 (t) = f:(f: MdY)ds, n+k) and Rfi3)(0) = 0 whenever O g_j g_k-l and k : n—l or o‘g j g_n-3 and k=n. Obviously Rk(t) = o(t 2n-l (k = O,...,n—l) and Rn(t) = o(t ). If we put _ n k n - k= we have M(t)g(t) + o(tn+l). (7.10) S(t) Set Hk = Kk — Rk. It is easy to see that Ho"°°'H have property m with respect to P,g. Define V = E} (-1)k(}’:)Hk = s - é. k=O By lemma 7.3 we have V(n) = P(n)g. Let G satisfy G(n) = g. Let a = M(n)(0). Obviously P(n) = a so ) (n). that V(n = ag = aG Thus there is a polynomial Q of degree < n such that S - S = V = aG + Q or (7.11) s = Q + as + §. There is a polynomial Q1 of degree < n such that n (7.12) G(t) = Ql(t) + $7 g(O) + o(tn). It follows from (7.9)—(7.12) that tn n S(t) = Q2(t) + ET-ag(0) + o(t ), 59 where Q2 = Q + dQl is a polynomial of degree < n. Thus S(n)(0) = dg(0). If g'(0) is finite we have g(t) = 9(0) + 0(t) and it follows from (7.9) and (7.10) that ). (7.13) S(t) = g(O)M(t) + o(tn+l)= g(O) (figgéT'9n+1(M“3't)+°(tn+l Further, tn n+1 o n+1 G(t) = Q1(t) + Ej'9(0) + 7;;377'9 (O) + o(t ). so that from (7.11) and (7.13) tn tn+1 ’ + o(tn+l) which completes the proof. D Theorem 7.5. Let n be an integer, n > 1. Let f,G,Y be functions on [a,b] such that f is Zn—integrable on [a,b], Y e BV[a,b] and G(n_2) is an indefinite integral of Y. Then for any indefinite Zn-integral F of f we have . b (2 )fb (FG + fG) = [FG] n a a Proof. Assume first that G,G’+ and G'-' are positive functions. (If n > 2 there is no need to consider G’+ and G'-' separately as 6’ exists in this case.) Let U be an n-majorant of f such that U(a) = F(a). By lemma 7.4 there is a function S such that 60 S = P G = UG. (n) (n) Moreover, by formula (7.8) and the note following lemma 7.4 6 S(x) 2_min[U(x)G’4—(x)+G(x)6n+lP(x),U(x)G’-(X)-t n+1 -+ G(x)6n+1P(x)]. Then 6n+ls(x) > —m for every x 6 [a,b] and 6n+lS(x) 2.F(x)G'(x) + G(x)f(x) whenever G'(x) exists (that is, except for a countable set when n=2, and everywhere if n > 2). Applying theorem 6.16 we see that * , b (2n )j: (FG + fG) g [UG]a and so (zni'yfi:1 (FG’ + fG) _<_ [FG]: . A similar consideration for n—minorants shows that (Zn*): (196’ + fG) 2 [PG]: and so the theorem is proved for this case. Returning to the general case we see easily that G'i- is bounded on [a,b) and G'- is bounded on (a,b]. Namely, if n > 2 then G’ is continuous and if n=2 then G is an indefinite integral of a function of bounded variation. Thus we can find a linear 61 function G2 on [a,b] such that G2 > O on [a,b], G; > 0 on [a,b] and that the function G1 = G + G2 satisfies the conditions of the special case considered above. Let Y2 ‘ 2 and Y1 = y + y2. Then Yj E BV[a,b] and Gj - is an indefinite integral of y. (j = 1,2). Then we may apply the first part of J the proof to each Gj’ so , b . z )fb FG. + £6. = FG. = 1,2 , ( n a ( J J) [ 31a (3 ) from which the assertion follows at once by linearity. C) The analogues of theorems 7.4 and 7.5 for the case n=l are slightly different. Lemma 7.6. Let M, g be functions on [0,1]. Let M ‘be ZO-integrable, let g E BV[O,1] and let var(g,[0,h]) = 0(h). Let P be an indefinite ZO—integral of M and let S be an indefinite ZO-integral of Mg. Assume P'(0) is finite. Set T(h) .2. _ h2 )2 (g(t) 9(0))dt. l = limsup h"l var(g,[0,h]). h40 62 Then l < w and -X g_6lg(0) g_liminf T(h), h4O (7.14) limsup T(h) g_Alg(o) g_1. h40 and :13. 2 (7.15) S(h) S(O) + hg(0)P’(0) + + o(h2). Proof. The inequalities 419(0) 3.1 < +m are obvious. Let B > Alg(0). Then for sufficiently small h ‘we have g(h) — g(O) < hB and 'T(h) g_3%-£: tht = B so that h limsup T(h) g Alg(0). h4O This proves (7.14). If M s 1 then P’(o) = 1 and 92(P,O,h) a o, 2 SM - 8(0) = J: g = 119(0) + J:<<3(t)-g(0))dt = hg(0) + 1’2- T(h) so that (7.15) holds in this case. If g E 1 then 2 o. S(h) - 3(0) = P(h) — p(0) = hp’(o) + gr-92(PJ),h) so that (7.15) holds in this case also. If we hold either HI T(h) M or g fixed in formula (7.15) it is easy to see that the formula is linear with respect to the other so that we may assume P'(0) = g(O) = 0. Let K(x) = P(x) — P(O). Then K(t) = o(t), g(t) = 0(t) and since var(g,[0,h]) = 0(h) we have I: Kdg = o(h2) so that (T(h>P’(0) + 9(0) 92(P.o. 11)) + 63 h s(h) - 3(0) = )2 Mg = K(h)g(h) - f6 Kdg = o(hz) which completes the proof. F Theorem 7.7. Let f,G be functions on [a,b]. Let f be Zl—integrable on [a,b]. Let G be continuous and in BV[a,b]. Suppose var(G,[x-h,x+h]) = 0(h) for each x 6 (a,b). Let F be an indefinite Zl—integral of f. Then a b (21)];b (FG + fG) = [FG] . a a Proof. If G is constant the conclusion is obvious so applying a simple linearity argument we may assume that G > 0. Let G > 0. Let M be a l-majorant of f on [a,b] such that M(a) = F(a) and MJb) < F(b) + e. Let S be an indefinite ZO-integral of MG and let P' = M on [a,b]. Using theorem 7.1 a) and the well known mean value theorem for the Riemann— Stieltjes integral we see that S' = MG on [a,b]. Moreover, from lemma 7.6 we have that 628(x) > —m for every x and that 623(X) = G'(X)M(X) + G(X)62P(X) whenever G'(x) exists. Thus 628(x) 2 G’(X)M(X) + G(x)f(x) 2 .2 G’(X)F(x) + G(x)f(x) - €|G'(x)\ a.e. in [a,b]. 64 Thus .6 fb \G’| + (21*)fb (FG’ + fG) a a = (21*)jb(-61G’\ + FG’ + fG) g a _<_ [s’]: = [M6123 [Fe]: + as]: It follows that * i (Zl )fb (FG + fG) 3 [FG] a We can show similarly that (z )j'b (FG’ + fG) > [F6] 1* a — so that the theorem is proved. C Proof of Theorem 7.1, b). Let n=1, and suppose that F,f,G and y satisfy the conditions of the theorem. Then F,f and G satisfy the conditions of theorem 7.7, so that (z ) I (FG’ + fG) = [FG]b l a a Moreover, by corollary 6.8, F is ZO-integrable and y E BV[a,b] so that FY is ZO—integrable and (20)): FY = (20)): FG , by proposition 6.15, which in turn is (Zl)fb FG' ‘by a theorem 6.11. Thus b- (b ' - 1b ' f" lFG]a —(zl) (FG + fG) — (20) FG + (21) £6 a a a as was to be shown. Let n > 1. Let F,f,G and y satisfy the conditions of the theorem. Then by theorem 6.7 there is a function P such that P(n) = F. Moreover (G o) (fl-2) is the indefinite integral of a function in BV[a,b] so, by lemma 7.4, there is a function S such that S(n) = P But then, by theorem 5.1, FG is Z -integrable. n—l Thus, since F,f,G,y satisfy theorem 7.5, we see that [F6]: (211)]: (FG’ + fG) = (211)}: fG + (Zn)J: FG’ = (Zn): fG + (Zn-1)): FG’. E CHAPTER VIII RELATION BETWEEN THE Zn-INTEGRAL AND THE CnP-INTEGRAL Theorem 8.1. The Zn-integral is identical to the CnP—integral (n = O,l,2,...). The proof is by induction. For n=O it is certainly true as both the ZO-integral and the COP- integral are precisely the Perron integral. We assume that, for 0 g_k g_n-l, the Ck k-integral. We need some lemmas P-integral is identical to the Z (which will also be useful later). Lemma 8.2. If P(n) is finite in [a,b] then for every x 6 [a,b] and for every h such that x+h c [a,b] we have n h k: P(k)(x) + HT-Cn(P(n),x,h). —1 (8.1) F(X+h) = P(X) + k=l Proof. According to the definition and the induction assumption hn 1 +h n—l 66 67 Applying theorem 7.1 and theorem 5.1 we see that this is n—l x+h l ]X (n_1);'[P(n_l)(§)(x+h—§) l +h n—2 + (nLETT'(Zn_2)f: (x+h-é) P(n_1)(s>dg n—l [:1 h l x+h n—2 = — 73:I7'P(n_1)(X)‘+ 73:377-(zn_2)jx (x+h—5) P(n_l)(§)d§ = .= (repeating the first step) hn—l hn-2 = ' (n—1): P(n-1)(X) ‘ (n-2): P(n-2)(X) "°°+ 1 +h + 5:- (20)]: PH) (€)d§ n-l hk = _ "T p k (x) + P(x+h)- P(x) as was to be shown. k=1 k. ( ) Lemma 8.3. Suppose P(n) is finite in [a,b]. Then (a) CnD*P(n) 6n+lP' * (b) (CnD P(n) = An+lp and (c) CnDP(n) = P(n+l) prov1ded one Side eXists. Proof. We prove (a)((b) is similar and (c) follows from (a) and (b)). According to lemma 8.2, 68 en+l(P'X'h) — n hk n n F(X+h) — kEO E7111.) (X) :0 c M(P( ),x h) - Ll- P(n) (x) «n+1/(n+1) ' _ n+1/ (n+1) = Cn(PL)IXIh) ‘ P(n) (X) h/(n+1) Now simply take the lower limit of both sides to see (a). C Lemma 8.4. Let M ‘be defined on [a,b]. Then M is Cn—continuous on [a,b] if and only if there is a function G on [a,b] such that G(n) = M. That is, only n-th exact Peano derivatives are Cn—continuous on an interval. Proof. Suppose G = M. Then according to (n) lemma 8.2, n-l k G(x+h) -kZO;-1-G(k)(x) lim Cn (M, x, h) = lim h40 h40 h k/k! G(n)(x) = M(x) so that M. is Cn—continuous. Conversely suppose M is Cn-continuous on [a,b]. Then it is Cn_1P-integrab1e and thus Z -integrable by n-l the induction hypothesis. Let gn-1(X) = (Zn_l)fx M. a 69 By corollary 6.8, gn_1 is Zn_2-integrable and so we may define successively gk(x) = (zk)f: gk+l (o g.k g_n—2). Put G = g0. Then applying theorem 7.1 n—2 times we have “1‘1"? (2 )f‘+h(x+h-§)n‘1M(§)d§ = (n- ). n x n—l hk = G(x+h) - G(x) — 23 ng(x). k=1 ' Then, as M is Cn-continuous, we see that G(n)(x) = M(x) for each x. (Also G(k)(X) = gk(x) (k = O,...,n-l).) C Thus from lemmas 8.3 and 8.4 we see that the class of (n+l)th e.P.d.'s is exactly the same as the class of exact Cn—derivatives. We now finish the proof of theorem 8.1 by showing that the CnP—integral is identical to the Zn-integral. Let f be any function defined on [a,b]. Let M be a Cn-majorant of f on [a,b]. Then M is Cn— continuous so, by lemma 8.4, there is a function P such that P(n) = M and, by lemma 8.3, 6n+lP(x) = CnD*P(n)(x) = CnD*M(x) so that -w < 6n+lP(x) 2 f(x). Thus M is an n—majorant 70 as well and so * b (2,1)): f 3 [ma for each such M. It follows that (zn*)f: f g (cnp*)f: f. If we consider Cn-minorants we see that (2“): f 2 (CnP*)_[: f as well, so that the Zn-integral extends the CnP-integral. 0n the other hand, if M is an n-majorant of f on [a,b] then there is a function P such that P(n) = M so that M is Cn-continuous. Moreover CnD*M(x) = 5n+lP(x) by lemma 8.3 and so M is a Cn-majorant of f. But then ‘1' (C P )fb f < [M1b for each such M n a — a. and so (cnp*)j: f _<_ (25)]b f. a Similarly considering n-minorants we see that (CnP*)J: f 2 (Zn*)~[: f. Thus we see that if f is integrable in the sense of either Zn or CnP then it is also integrable in the other sense and (CnP): f = (zn)j: f. C) CHAPTER IX EXAMPLES In this section we would like to give a few examples to illustrate the extent of generalization of the Zn—integral and also the necessity of the strong assumptions in the integration by parts theorem. Example 9.1. Let n 2.1. Let g(x) = xn+lsin x“n (x # O), g(O) = 0. Then g(t) = o(tn) as t 4 0 so that g'(0) = 9(2) (0) =...= gm) (0) = 0. Thus, as g is certainly n—times differentiable away from x=0, we see that g(n)(x) exists for every x. Thus, by theorem 5.1, g(n) is Zn_l-integrable in [-l,1]. But if n > 1, g(n) is not Zn -integrable there. Since -2 _2-integrable, its indefinite integral. would then be Zn- if g(n) were Zn g(n—l)’ 3-integrable, etc. Continuing this we see that an n-l times iterated integral of g(n) would be continuous (since the last integral is a Perron integral) and also equal to 9' plus a polynomial. But this is impossible as 71 72 ’ -n cos x—n + (n+l)xn sin x—n, x # 0 g (X) = 0 x = O which is not continuous. It also follows from this that g(n) is not an (n—l)th e.P.d. of any function, for if it were it would be Zn_2—integrab1e. Thus the Zn-integral is a proper generalization of the Zn_l-integral and, for any n there is an n-th generalized derivative on L—l,1] which is not Zn integrable on L—l,l]. —2 Example 9.2. Let n be a natural number. There is a function G ‘which is n times differentiable in a neighborhood of zero, and a function f which is an n—th exact Peano derivative and yet fG is not Zm—integrable for any m in any neighborhood of 0. Remark. Such a function f is of course Zn_l—integrable. Thus it is essential to assume more then the n-fold differentiability of G to obtain an integration by parts theorem for the Zn_l—integral. We will show that.G and F may be taken to be n2+3n-2 1 x cos n+3 x x # O, G(x) = 73 cos x # 0 n +3n-l n+3 f(x) = 0 x = 0. To establish this we shall need some lemmas. Lemma 9.1. Let 1 i M(x) = -E-exp(—E) where x > 0, r real, x x s>0, i2=-—l. Let F(x) = fl M(t)dt (x > 0). Then x F(x) = O(——l——fi +-constant r-s-l ° x Proof. If r = l+s this is trivial, so assume otherwise. We integrate F by parts to see F(X) = Il—Eit'i' $14.1 exp“? dt Xt t t —A+A——-l-—-X-i-+B fl 1 (loot - o l r-s-l e p s l x r-s exp s ’ x x X t where AO,A1,Bl are constants. Repeating this for the last term we see that l i l i F(X) = AO+Al r-s—l exp —§-+ A2 r-Zs-l exp —§-+...+ x x x x i 1 + A exp —— + B exp —— dt 74 and if we choose n such that r-ns < O we see that each term but the last is l O(*r-s—l)° D x Lemma 9.2. If r < ns and if 1 1 H x = d5 () (X1) then 1 Eras—:5) + W) H(x) = 0( where P is a polynomial of degree < n. Thus n H(x) = o(x ) + P(x), and so H(n)(0) exists and is zero. Proof. This is a trivial induction using lemma 9.1. E Lemma 9.3. We get exactly the same results as lemmas 9.1 and 9.2 if we replace M by one of a) -3L ex -:i r p S X X 1 . 1 b) 'IF Sin-j; X X r S 75 Proof. a) is obvious and b) and c) follow from a) and the above and well known trigonometric relations. D We now show that the functions f and G of example 9.2 have the desired properties. Let r = n2 + 3n—l, s = n+3. Then r - ns < O and so by lemma 9.2 (for cosine), f is an n-th exact Peano derivative in any neighborhood of O. MOreover it is easily seen that G(k)(0) = O (“'1)(x) = 0(x2) so that (k = l,...,n—l) and G G(n)(0) exists. Thus G(n)(x) exists for all x. But (fG) (X) = and this function cannot be Zm—integrable for any m. Namely, fG has constant sign in [0,1] and in [-l,0] so if it were Zm-integrable there it would be Lebesgue integrable there. But it is not Lebesgue integrable in any interval of the form [O,d] or [—d,0]. Remark. Even though G(n) exists, G(n_l) is not of finite variation. In fact it cannot be, according to theorem 7.1. CHAPTER X AN IMPROVEMENT IN THE DEFINITION OF THE Zn-INTEGRAL In this section we would like to show that it is possible to relax somewhat the requirement 3) in definition 5.1, the definition of an n-majorant. Lemma 10.1. Let do > al 2.a2 2.°-~.2 an 4 O. L t B - 1 > B > B > > B 4 O and B < Bn_l e o" 1 2 n n 2 ° Then for each k = 1,2,..., there is a function Fk on [0,1] such that 1) FR has a non-negative continuous (k+l)th derivative on (O,l], 2) Fk(0) = pk '* kwho) = o. Fk(k)— (O) =...= F (60) = a0! 3) Fk(x) 2,6: an/zk+l n = 1,2,...; k = 1,2,...). (X E [Bn'Bn—l]; Proof. Let m ibe a continuous, non—decreasing function on [0,1] with a continuous derivative in (0,1] such that m(x) = an for Bn/2 < x < 8n. Let 76 77 5 Fk(X) = [x dil I l E k-l O O d§2 ... [O o(gk)d§k [X o(§)d(§,k—1), a k—fold indefinite O integral of 6. Then for x E [Bn’Bn-11’ we have n l n ffin B F = = m) .%— fin/z 2 k—l B -a B n n n k k+l The rest is obvious. D Lemma 10.2. Let k‘Z 1 be an integer. Let M be a finite function on [-l,l] such that M(k)(0) is finite. Let E > 0 be given. Then there is a function F with the prOperties 0) F(O) = F’(0) =...= F(k)(0) = 0, (k) 1) F is continuous and non—decreasing on [—l,l], 2) [F(k)]11 < 6. 3) 6k+l(M+F)(O) 20. Proof. Let £ be a polynomial of degree g_k (k) such that 1(0) = M(O), £’(0) = M’(0),...,z (0) = M(k)(0). 78 Let Q = M—z. Then Q(t) = o(tk). Let 6n = 3"In (n = O,l,2,...). Define, for n = 1,2,..., on = sup[JQ':':-:Ll ° 2k+1 3 t E [O'Bn_1]}o Then cn is a sequence decreasing to zero. Let k dn = cn . 3 . Then for t e [Bn'Bn-lJ' k k k t ‘Q (t) ‘ < :13...— S Can-1 = 3135!}. — 2k+1 2)E+1 2k+1 ' f - _ Let dno < 2 and define an (n — 1,2,...) by dno n g.n0 an = (1n n > nO . Now define F on [0,1] as in lemma 10.1 for this choice of k, {an}. {5n}. Then for some 8 > O, (k)+- 6 (l)<§ this construction in [-l,0] and thus define F on F(x) 2_‘Q(x)| in [0.3] and F Repeat all of [—1,1] so that F(O) = F’(0) =...= F(k)(0) = 0, so that F has a continuous, non-decreasing k-th derivative in [-l,l], F(k)+(-l) > — g and F(x) 2_\Q(x)[ on [~B,O] (k odd) or F(X) g —‘Q(x)‘ on [-B.O] (k even), for some Then 6k+1(M+F)(O) = 5k+l(Q+F)(O) 2.0. The rest is obvious. E] > O. 79 Corollary 10.3. A simple argument shows that we may replace [-l,l] with any compact interval [a,b], and instead of O we may take any point c in the interior. Moreover, if we allow for one-sided derivatives we may take c to be any point in [a,b]. D Theorem 10.4. Let M ‘be a finite function on [a,b]. Let k be a natural number. Let 6 > 0. Let T be a countable set in [a,b]. Let M be (k) finite for points in T. Then there is a function F satisfying 1) F(k) is continuous and non-decreasing in [a,b], (k) ]b 2) [F a_ O (x E T). Proof. Let T = [an]: . Then, by lemma 10.2, for each n = 1,2,..., there is a function Fn such that F(k) n a) is continuous and non-decreasing, b) [1951”]: < 3% . c) 6k+1(M+Fn)(an) 2_O. We may also suppose without loss of generality that _ . _ _ (k) _ _ Fn(a) — Fn(a) —...— Fn (a) —— O (n — 1,2,...), or else we simply add to En a polynomial of degree g.k and this leaves a),b) and c) unaffected. Then 80 \ng)(x)\ < €1- (n = l,2,...;x 6 [a,b]), 2 and so if we define co f(x) = Z Frgk)(x) n=1 we see that f is continuous and non—decreasing on [a,b], and [f]: < 6- Let F(x) = [x f(§)d(§.k-1). a a k-fold indefinite integral of f. Then it is easily seen by the uniform convergence of the series defining f that F(X) = ‘2: ijrEk)(§)d(§,k-l) = )3 Fn(x). n=1 0 n=1 Moreover F(k) = f and so 1) and 2) are Obvious. To see 3) we let x = an e T. Then 0k+l(M+F)(X) = 6k+l(M+Fn+ Z Fm)(x) _>_ m#n .2 6k+1(M+Fn)(x) 2_0. The first inequality follows from the fact that 23 F(k) = ('2: Fm)(k) mfin m m n is non-decreasing, whence 6k+l(}3 Fm) 20. E3 mfin 81 Theorem 10.5. Let f,M. be finite functions on [a,b]. Let T be a countable set in [a,b]. Assume M(k) is finite on [a,b] and that ~00 < 6k+lM(x) _>_ f(x) (x A T). Then * b (2k )1: f _<_[Mm1a . Proof. Let e > 0. According to theorem 10.4 there is a function F on [a,b] such that [M(k) 1: S. [(M+F) (k) 1: S. [M(k) ]: + 6 and 6k+l(M+F)(t) > -w everywhere and 6k+l(M+F) 2_f except on T. Then applying theorem 6.16 we see that (zk*) f: f _<_ [(M+F) (10]: _<_ [MW 1: + e. As 6 is arbitrary the result is immediate. D Applying theorem 6.16 to this case we easily obtain Theorem 10.6. Let f,P be finite functions on [a,b]. Let T be a countable set in [a,b]. Assume P(k) is finite on [a,b] and that 82 a) 6k+1P(x) > —m (x E [a,b]‘\T), b) 6k+1P 2.f a.e. on [a,b]. Then * b (Zk)]:f_<_[P(k)]a. C) This result shows that if we take the following definition for n-majorant and the corresponding one for n-minorant, then the Zn—integral is left unaffected. Definition 10.1. Let M and f be defined in [a,b]. Then M is called an n—majorant of f in [a,b] if there is a function P on [a,b] such that l) M = P(n) on [a,b], 2) 5n+lP 2_f a.e. on [a,b], 3) 6n+1P(x) > —m for all but at most countably many x 6 [a,b]. To finish we illustrate how such a definition may be useful. Theorem 10.7. Suppose f is Zn—integrable on [a.fi] for each 6 < b and suppose lim (Z )IX f xab— n a exists and is finite. Then f is Zn—integrable on [a,b] and 83 Proof. Let 6 > 0. Let a = bO < bl <...< bk 41b. Let Ml,ml be respectively an n-majorant and an n-minorant for f on [a,bl] such that Ml(a) = m1(a) = O and l [Ml—m1]a < 6/2- Suppose Mk—l and mk—l have been chosen. Let Mk’mk be respectively an n-majorant and an n—minorant of f on [bk-1'bk] such that Mk(bk—1) = Mk-l(bk-l)’ mk(bk-l) = mk—l(bk—l) and b [Mk - kab:_l < 6/2k . Let M(x) = Mk(x) when x e [bk_l,bk], m(x) = mk(x) when x e [bk_l,bk]. bk Then [M—m]a < E for every k. It is easy to see that M and m are respectively an n-majorant and an n-minorant on each interval [a,B] (B < b). Thus there is a function P such that P n)(x) = M(x) (x 6 [a,b)), and ( -m < 0k+lP(x) 2 f(x) (x 6 [a,b)). We shall now define M(b) and P(b) in such a way that P(n)(b) = M(b). Let F(X) = f: f (x < b), F(b) = F(b-). Then it is easily seen that MrF is bounded and non-decreasing in [a,b) so that M(b—) = P(n)(b—) exists and is finite. We may prove analogously that m(b-) is finite. Define M(b) = M(b—), m(b) = m(b—). Mbreover 84 P = M is bounded in some interval (a,b) so it is (n) an ordinary derivative there by corollary 6.25. Thus P(n) = P(n) and, since P(n)(b-) exists and is finite we must also have that p‘k) (b-) (k = 0,1,...,n) exist and are finite. Now we define P(b) = P(b—). (k) k) It follows from L'Hopital's rule that P -(b) = P( (b-) (k = 0,...,n), so that P(n)(b) = M(b). Thus M is 7 an n—majorant of f on [a,b] using definition 10.1. Similarly m is an n-minorant of f on [a,b] and [Mem]: < 6. Thus f is Zn—integrable on [a,b] and -e < m(b) - M(b) 3 (2,1)]: f - F(b—) _<_ _<_ M(b) - m(b) < 6. so that (Zn)]: f 2 11m (Zn)J: f. F] x+h- BIBLIOGRAPHY [1] [2] [3] [4] BIBLIOGRAPHY J.C. Burkill, The Cesaro—Perron integral, Proc. London Math. Soc. (2) vol. 34 (1932) pp.314-322. J.C. Burkill, The Cesaro-Perron scale of integration, Proc. London Math. Soc. (2) vol. 39 (1935) pp.541-552. R.D. James, Generalized n-th primitives, Trans. Amer. Math. Soc. vol. 76 (1954) pp.149-176. I.P. Natanson, Theory of Functions of a Real Variable, Frederick Ungar Publishing Co. NY, vol. I (1955), vol. II 1960. S. Saks, Theory of the integral, Warsaw, 1937. H. Bauer, Der Perronsche Integralbegriff und seine Beziehung zum Lebesgueschen, Monatshefte Math. Phys. vol. 26 (1915) pp.153-198. P.S. Bullen, A criterion for n-convexity, Pacific J. Math. vol. 36 (1971) pp.81—98. J.C. Burkill, The approximately continuous Perron integral, Mat. Zeitschrift, vol. 34 (1931) pp.270—278. L. Gordon, Perron's integral for derivatives in Lr, Studia Math., vol. 28 (1967) PP.295—316. W.H. Oliver, The exact Peano derivative, Trans. Amer. Math. Soc. vol. 76 (1954) pp.444-456. 0. Perron, Ueber den Integralbegriff, S.-B. Heidelberg Akad. Wiss., vol. 16 (1914). 85 Fm——-—_ — __li,___. .,,,11_,_,_ ..- ...“, ”.-..-. ... .._ . use .4 .. . ,1, .. --.. .‘,...a.-._-( n5.;_cu~~-~--yu~ ’ " - W“ " JIM A" . ”'Cll"\'ll[)[l[[1]]![j[l[l[)j[l]ifl)[