0N PROPERTIES OF THE APPROXIMATE PEANO DERIVATIVE ' Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY BRUCE SCOTT BABCOCK 1973 ' I :1 w ' u.. -“ 5 Lush/1v" L“ Michige- a State , University it“ ‘ . *' "r5 ewe‘ ' 4T .— This is to certify that the . “1““ thesislzm :I if 0N PROPERTIES OF THE AEPROXJMATE -. :. PEANO DERIVATIVE .l presented by Bruce Scott Babcock has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics CW '8, WM fl V Major professor Date May 23, 1973 0-7639 MCI em AN STATE UNIVflRSITV LIBRARIES 1293 00651 8405 IL “W "i4 199‘s? ABSTRACT 0N PROPERTIES OF THE APPROXIMATE PEANO DERIVATIVE By Bruce Scott Babcock The ordinary derivative has been studied extensively and many properties of it have been discovered. Although an ordinary derivative need not be continuous, it does possess certain properties worth investigating. The following four properties, defined here for an arbitrary function 9, have been shown to hold for an ordinary derivative: 1. g is in the first class of Baire. 2. g has the Darboux property. 3. g has the Denjoy property. 4. g has the Zahorski property. C. E. Neil has recently introduced a new property which he calls property Z. He has shown that property Z is stronger than the Zahorski property in the class of functions having the Darboux property. In addition he has shown that an ordinary derivative has property 2- H. w. Oliver showed more generally that if a function f has a kth Peano derivative fk then fk has properties 1, 2, and 3 listed above. C. E. Weil showed, furthermore, that fk also has property 4 and property Z. '-_l Bruce Scott Babcock A. P. Calderon and A. Zygmund have generalized the kth Peano derivative by means of the kth Lp derivative, where 0 < p 5 w. M. J. Evans has recently shown that if a function h f has a kt Lp derivative fk p’ where 1 f p f m, then f k,p has the four properties listed above. h The notion of kt Lp differentiation, where O < p 5 w, is contained in one that is more general. It is called kth approximate Peano differentiation. M. J. Evans was the first to investigate this type of differentiation and has further shown that if a function f has a kth approximate Peano deriv- ative f(k) then f(k) has property 1 given above. In this paper we first examine the concepts discussed above. We then proceed to prove our main result that when a function f has a kth approximate Peano derivative f(k) and if _ fm _ a f(k) is bounded above or below on an interval then f(k) the ordinary kth derivative. From our main result the proper- ties 2 and 3 given above are then easily shown to hold for a kth approximate Peano derivative from known theorems. As our final result we prove that a kth approximate Peano derivative th . has Property 4 by verifying that a k approx1mate Peano derivative satisfies the stronger property Z. ON PROPERTIES OF THE APPROXIMATE PEANO DERIVATIVE By Bruce Scott Babcock A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1973 To JoAnn and Mom and Dad ACKNOWLEDGEMENTS I would like to take this opportunity to express my deepest appreciation to my major Professor C. E. Neil for his helpful guidance, for the interest he has dis- played, and especially his patience in the preparation of this thesis. I am also deeply appreciative to my wife, JoAnn, for her encouragement, moral support, and also her faith in me. iii Chapter 1. Chapter 11. Chapter III. Chapter IV. Chapter V. BIBLIOGRAPHY. TABLE OF CONTENTS Introduction and Definitions. Examples. A Preliminary Result. The Major Theorem Property Z for Approximate Peano Derivatives . . . iv . 12 . 18 . 29 . 47 59 CHAPTER I INTRODUCTION AND DEFINITIONS All of the functions in this paper are assumed to be real-valued, measurable functions defined on a nondegenerate closed interval I = [a,b], unless specified otherwise. Also whenever we write x+h (x E I) our assumption is that x+h E I. Furthermore, the notation E-lim will denote that the limit is computed only for those valuzs of y in E. Finally, if E is a measurable set then we denote the measure of E by either m(E) or IE]. Since the turn of the century the ordinary derivative has been studied extensively and many properties of it have been discovered. Although an ordinary derivative need not be continuous, it does possess certain properties worth investigating. The simplest of these properties is that every ordinary derivative is a function of Baire class one. Definition 1.1. A function f, defined on I, is said to be a function of Baire class one on I if there is a SEQUence {fn}:=1 of continuous functions, each defined on I, such that AwmfnU)= HxL for each x in I. Perhaps the best known property for an ordinary derivative is the Darboux property or intermediate value property. Definition 1.2. A function f, defined on I, is said to have the Darboux property if whenever x1 and x2 are distinct points of I and y is a number between f(x1) and f(x2), there is an x3 between x1 and x2 such that f(x3) = y. In 1916, A. Denjoy [2] proved that an ordinary derivative has what we shall call the Denjoy property. Definition 1.3. A function f, defined on I, is said to have the Denjoy property on I if for every open interval (a,8), f‘1((a,8)) either is empty or has positive measure. The Denjoy property was further strengthened by Z. Zahorski in 1950 [12]. We call this property the Zahorski property and define it in the following manner. Definition 1.4. A sequence of closed intervals {In}:_1 is said to converge to a point x if x is not in the union of the In's and if every neighborhood of x contains all but a finite number of the intervals In‘ Definition 1.5. A function f, defined on I, is said to have the Zahorski property on I if for every open interval (a,6), x in f'1((o,8)) and {In}:=1, a sequence of closed subintervals of I converging to x with m(f'1((a.e)) n1") = 0 for every n, implies m(I ) lim n n-em - . dist.(x,In) where dist.(x,In) = ianIx-yl : y e In}. Zahorski showed that an ordinary derivative also possesses the above property. Recently C. E. Neil [11] has introduced a new property, property Z, which is stronger than the Zahorski property. Definition 1.6. A function f, defined on I, has property Z on I if for each x E I, each 6 > 0 and each seQuence {In}:=1 of closed subintervals of I converging to x such that for each n, f(y) 3 f(X) on In or f(y) S TIX) on In, m{y E In : If(y)-f(x)l 3 5} lim = O. ”T“ mun)+dim.U,%) In addition Neil has shown that an ordinary derivative has the stronger property Z. These five properties, however, do not classify derivatives. Indeed, they are possessed by more general types of derivatives. Suppose a function f, defined on I, possesses an ordinary derivative f’(x) at a point x E I. Then f(x+h)-f(x) lim h-O h or equivalently, as h-O. This last equation motivates a more general first order derivative in terms of the Lm-norm. In order to understand this definition, recall that if g is a function defined on an interval J then [[9 Hm’J is defined by = 9(t)l H9 Hm,J essesgpl where ess sup|g(t)l = inf{M : m{t E J : |g(t)| > M} = O}. t E J Definition 1.7. A function f, defined on I, is said to have a first LCD derivative at a point x E I if there exists a number f1 0°(x) such that if g = f-f-tf1,m(x> then IIg “w,(0,h) = 0(h)a where (0,h) = [0,h] if h > O and (0,h) = [h,O] if h < O. The number f1,m(x) is called the first Lco derivative of f at x. Replacing the Lm-norm by the Lp-norm where o < p < m, suitably normalized so that the function iden- tically 1 has Lp-norm 1, gives the following definition. Definition 1.8. A function f, defined on I, is said to have a first Lp derivative at a point x e I, 0 < p < m, if there exists a number f (x) such that l,p (x)|pdt ]1/P = 0(h) [%f3|f(x+t)—f(x)-tf1,p as h ~ 0. The number f1 p(x) is called the first Lp derivative of f at x. For its application to Fourier analysis it suffices to consider p 3 1, but for establishing properties of the derivatives that arise, O < p < 1 may also be considered. All of these methods for the first order differentiation are contained in one that is more general. It is called approximate differentiation and is defined in the following manner. Definition 1.9. Let E be a measurable set and let x be a real number. Define m(E n [x-h,x+h]) ,E = lim , d(x ) h-«O 2h m(E n [x,x+h]) ,E = lim ———————-——--» d+(x ) h-*0+ h m(E n [x-h,x]) d_(x,E) = Iim + \ . h-0 h If d(x,E) = 1 then x is called a point of density of E; if d(x,E) = 0 then x is called a point of dispersion of E. If d+(x,E) = 1 then x is called a right-hand point of density of E; if d+(x,E) = 0 then x is called a right-hand point of dispersion of E. Similarly, if d_(x,E) = 1 (0) then x is called a left-hand point of density (dispersion) of E. Definition 1.10. A function f, defined on I = [a,b], is said to have an approximate derivative at a point x E I if there exists a number fép(x) and a measurable set E having 0 as a point of denstiy such that f(x+h)-f(x) E-lim —— = f5p(X)- h-oO h Ne call fép(x) the approximate derivative of f at x. Remark. In Definition 1.10 if x = a (x = b) then the expression, there exists a measurable set E having 0 as a point of density, is understood to mean that E E [0,w) (E E (-w,x]) and 0 is a right-hand (left-hand) point of density of E. This same convention will be adopted in Definition 1.13 Since approximate differentiation is more general ' ' ‘ ' tiation, than ordinary differentiation and first Lp differen 0 < P < w, every property possessed by the approximate deriv- atives is also possessed by the ordinary derivatives and the first Lp derivatives, 0 < P S ”' In 1960, C. Goffman and C. J. Neugebauer [4] gave concise proofs of the facts that every approximate deriv- ative is a fuction of Baire class one and possesses the Darboux property. These facts were first established by G. Tolstoff [8]. The Denjoy property for an approximate derivative was established by S. Marcus [5] and C. E. Neil [10]. Neil, in addition, verified the Zahorski property in [10] and property Z in [11] for approximate derivatives. Another important property of an approximate derivative, a proof of which can be found in the paper of Goffman and Neugebauer, is that if an approximate derivative is bounded above or below on an interval, then it is an ordinary derivative on that interval. Even more can be said for kth order differentiation. If a function f has a kth ordinary derivative at a point x, then by Taylor's theorem k _ f(x+h)-f(x)—hf’(x)-...-Ejf( )(x) — o(h as h-oO. In many instances it was only the existence of such a polynomial that was ever needed, even though the function was assumed to have k derivatives at x. This led th to the introduction of the following so called k Peano derivative. Definition 1.11. A function f, defined on I, is said to have a kth Peano derivative at a point x E I, k = 1,2,..., if there exist numbers f1(x), f2(x),..., fk(x), such that f(x+h)-f(x)-hf1(x)-...-—.fk(x) = 0(hk) as h-rO. The number fk(x) is called the kth Peano derivative of f at x. That every kth Peano derivative has the Darboux property, is a function of Baire class one, and possesses the Denjoy property was first proved by H. N. Oliver [7]. Neil in [10] also gave an independent proof of the Denjoy property together with a proof of the Zahorski property; and in [11], he gave a proof of property Z for the kth Peano derivative. Oliver also showed in his paper that if a kth Peano derivative is bounded above or below on an th interval, then it is an ordinary k derivative. Recently, S. Verblunsky [9] showed how to prove this last property from the definition without using the Darboux property or other properties as Oliver had done. Proceeding in the same fashion, as in the case of . th - - the first ordinary derivative, the k Peano derivative may be generalized by means of the Lp-norm, 0 < p g m. Ne thus have the following definition. Definition 1.12. Let f be a function defined on I. LEt X E I and let k be a positive integer. If there exist (x), f2 m(x),..., fk,m(x) such that numbers f1 w , k ess sup|f(x+t)-f(x)—tf1 0°(x)-...-E—,fk 0°(x)| = o(h k) t e <0,h) then f is said to have a kth Lon derivative at x. The number fk,m(x) is called the kth L0° derivative of f at x. If th ' ere eXIst numbers f1,p(x)’ f2,p(x), , fk,p(x), where O < p < m, such that k 1/p I h t p _ k [hf0[f(x+t)-f(x)-tf1’p(x)-...-k—!fk,p(x)| at] — 0(h) then f is said to have a kth L derivative at x. The number f th k,p(x) is called the k Lp derivative of f at x. This concept was first introduced by A. P. Calderén and A. Zygmund [I] but their interest was only in the case p 3 1. In a manner analogous to the way the approximate derivative was introduced, the kth approximate Peano deriv- ative may be defined. This type of differentiation is more general than kth Peano differentiation and kth Lp differ— entiation where 0 < P S ”- Definition 1.13. A function f, defined on I, is h approximate Peano derivative at a point = f(x), said to have a kt X E I, k = 1,2,..., if there exist numbers f(0)(x) f(1)(X),..., f(k)(x), and a measurable set E having 0 as a point of density such that hk k f(x+h)‘f(X)'hf(1)(X)—..."k—!'f(k)(X) ) 10 as h-o and h e E. The number f(k)(x) is called the kth approximate Peano derivative of f at x. This latter generalized derivative was first studied by M. J. Evans [3] where he showed that every such derivative is a function of Baire class one. He was also able to estab- lish the other properties (Darboux, Denjoy, and Zahorski) but only for the kth th Lp derivatives with p 3 1. Property Z for the k Lp derivatives can be established in the same way as Evans established the Zahorski property for these derivatives. h In Chapter II we prove that kt approximate Peano differentiation is a true generalization by giving an example of a function having a kth approximate Peano derivative at 0 but no kth Lp derivative for 0 < p 5 m, at 0. Ne also show, given two real numbers p,q, with O < p < q f w, how to con- th th struct a function having a k Lp derivative at 0 but no k Lq derivative at 0. In Chapter III we prove that if 0 is a point of density of a measurable set E then there exists a sequence of positive real numbers {An}:=1, strictly increasing to 1 (strictly decreasing to I), so that O is a point of density of the set H:=1AnE, where AnE = {Anx x 6 E}, n = 1,2,.... This result will play a key role in Chapter IV where we th . prove our major theorem that when a k apprOXImate Peano ' ‘ is an derivative is bounded above or below on an interval, 1t ordinary kth derivative. The Darboux and Denjoy properties follow easily then from known theorems. In Chapter V, our final chapter, we give a proof of property 2 for the kth approximate Peano derivatives. The weaker Zahorski property then follows for the kth approx- imate Peano derivatives. CHAPTER II EXAMPLES Let f be a function, defined on I, possessing a kth approximate Peano derivative at a point x E I. Then there exist numbers f(1)(x), f(2)(x),..., f(k)(x), and a measurable set E having 0 as a point of density such that (2 1) f(x+h)-f(x)-hf(1)(x)-...-E7f(k)(x) = o(h as h-aO and h E E. ..., , easil The numbers f(1)(x), f(2)(x), f(k)(x) can y be shown to be unique and for each n, n = 1,2,...,k, (2.1) can be rewritten as f(x+h)-f(x)-hf(1)(x)-...-hh-!f(n)(x) = o(h"). Thus f has an nth approximate Peano derivative f(n)(x) at x for n = 1,2,...,(k-1), and f(1)(x) = fép(x), the first . th approximate derivative. Moreover, if f has a k Peano derivative at x then f1(x) = f’(x), the ordinary first . th . . derivative. Notice that if f has an ordinary k derivative f(k)(x), at x, then Taylor’s theorem Shows that f(k)(x) exists and equals f(k)(x). However, as we shall show, . - ' x existing for f(k)(x) may exist at a POIOt X WithOUt fk,p( ) any P O < p < m, at x. In [3], M. J. Evans has shown that 12 13 if a function f, defined on 1, possesses a kth Lp derivative fk,p(x) at a point x E I, where 0 < p f m, then f has a kth approximate Peano derivative f(k)(x) at x and furthermore, fk,p(x) = f(k)(x). Let k be a positive integer and let p be a positive real number. Ne now show how to construct a function which th has a k Lp derivative at 0 but not a kth L derivative at q 0, where p < q 5 w. Example 2.2. Let k be a positive integer and let p be a positive real number. Suppose q is a real number such that p < q < w. Set c = 1+k+% and M = é¥% . Let 1 1 1 I = {—3 _' + _] a n 2n 2n 2qu n = 1,2,..., and E = I-u:=lin, where I = [0.1]. Let us first show d+(0,E) = 1. Let 0 < h < 1, and choose the positive integer N so that Now it can be easily shown that [E 0 [0,h]| < < ZNIMq-1)(2Mq,1) - h As h-°0+, N-+w, and since Mq-l > 0 it follows that IE n [0,hll im + —-——-—-—~——'= 1 h-o h Thus, d+(0,E) = 1. 14 Define a function f on I as follows: 2"" . if x e 1 , f(x) = n 0 , if x 6 E. Since _ k f(h) - 0(h ) as h-O+, h E E, f has a kth approximate Peano derivative f o g = ) = 9 9‘0"° (k)(0) at 0 Furthermore f(n)(0) O for n 0 1 k If f has a kth Lr derivative at 0, fk r(0)’ where O < r f m, then as was mentioned in the beginning of this k,P(0) (0) does not exist. To show chapter fk,r(0) = f(k)(0) = 0. Ne first show that f exists then we show that fk q fk p(O) = 0 it suffices to show I/p . 1 1 h P _ lim ——[—] |f(t)| at] _ 0. h-*0+ hk h 0 LBt O < h < I, and choose the nonnegative integer N so that < h < —— . 2N+1 ‘ 2N , 1 For notational convenience set am — 2fii+ 2qu a m Then 1/p a 1/ flatware] _2mm)[zrlmrenpdtl A 1 2k(N+1)[2N+IZ:=N IInITIt)lpdtI Mpn l/p k N+1 N+1 w Z___ S 2 ( )[2 Xn=N Zqu J 15 S 2k(N+I)[2N+1Zoo 1 ]1/p 2k(N+1)[2N+1 2Cp Jl/P 2cpN(2cp_1) 1 2c+p +k 1 < o ' (2Cp-1)”p 2N(c-k-p-1) 2C+p-1+k 1 < § o _ - (2cp_1)1/p 2N As h-oo+, N-w, and Jfi-o. Thus fk (0) = o. 2 ’p Ne now show that fk q(O) does not exist. Assume to the contrary that f (0) exists. Then fn (O) = O. for k,q 9q n = O,1,...,k. Thus, 1/9 . 1 1 0 q _ (2 3) Iim ——[—j lf(t)l dt] — o. h'*0+ hk h 0 However, if we let 0 < h < 1, and choose the nonnegative integer N so that then l/q 2Nk[2Nf:N+2lf(t)qut] IV 1 J; 1 h q hk[hf0|f(t)| at] IV Nk N w q l/q 2 [2 2...... iinwtn dt] 1/q Nk N w qu ] 2 2 [2 2n=N+2 [In 2 dt 16 > 2Nk 2N2“ gig: l/q - n=N+2 2qu This contradicts (2.3) and thus fk q(0) does not exist. Remark. From the nonexistence of f (0) we further k,q have the nonexistence of fk(0) and fk 0°(0). Thus, kth Lp differentiation is more general than kth Lq differentiation, where 0 < p < q f w, and kth Peano dif- ferentiation. Example 2.4. Let k be a positive integer. Here we construct a function having a kth approximate Peano deriva- tive at 0 but no kth Lp derivative at 0, for O < p f w, and no kth Peano derivative at 0. Let 1 1 1 I=[-9_+—]9 n n n 2n n = 2,3,.... Define a function f on I = [0,1] as follows: 2 n , if x E In, f(X) = where E = I-Un=21n' Again it can be shown that d+(0,E) = 1 and that f( )(0) = O, for n = 0,1,...,k. Let p be a real number such Let 0 < h < 1 and choose N a positive integer n that o < p < w. so that 1 l WEh 0 be given. Choose N, a positive integer, such that for every m 2 N 20 1 E n — I [0.m]I 5 6 EC <=> x e A(E ). Lemma 3.8. Let E and F be two measurable sets. (i) If d+(0,E) = 0 and d+(0,F) = 0 then d+(0,E 11 F) = 0. - C (ii) d+(0,E) = o if and only 1f d+(0,E ) — 1. Proof. Follows easily from Definition 1.9. Lemma 3.9. Let E be a measurable set and let A be a real number, A > 0. If d+(0,E) = 0 then there exists a co . . sequence of positive integers {an}n=1’ increaSing to ”’ SUCh that for every A 3 A |AE n [0,h]] 1 __________——-< —— 2 h n whenever 0 < h <-l-. an 23 Proof. Let c > 0 be given. Since d+(0,E) = 0 there exists a 6 > 0 such that [En [0.1:]! h whenever O < h < 6. For 0 < h < A6 and A 3 A IAE n [0,h]| |E n [o.%h]| = 1 < e h Ah . 1 h Since X—h 5 K < 6. The proof of the lemma may now be com— pleted by letting a progress through the numbers 1/n2 and choosing an so that 1/an is smaller than the corresponding A6 and also the an's increase to m. Lemma 3.10. Let E be a set of finite measure and let 6 > 0 be given. Then there exists a 6 > 0 such that IAE-EI < 8 whenever |1—A| < 6. Proof. If [E] = 0 then the result is obvious. Thus assume |E| > 0. First assume E = (a,b). Then it is easy to see that there exists a 6 > 0 such that |A(a,b)—(a,b)| < 5 whenever ll—Al < 6. Assuming E = U:=1(an’bn)’ then again it is obvious that there exists a 6 > 0 such that IAE-El < 6 whenever |1-A| < 6. Now assume E = U:=1(an’bn)' Choose N such that E IUn=N+1(an’bn)l < 4 24 Sett' = N - ” ing F Un=1(an,bn) and H — Un=N+1(an’bn) we have E = F U H. Choose 0 < 6 < 1 so that for all A, ll-AI < 6, _ 2 [AF F] < 2 . Since AE-E E AE-F = (AF U AH)-F E (AF-F) U AH, and 0 < A < 2 lAE-El 5 |AF-F| + IAHI < § + 11H] < 5 whenever |1-A| < 6. Finally, assume E is a set of finite measure and let G be an open set such that E E G and lG—El < % . Choose 6, 0 < 6 < 1, so that for all A, ll-Al < 6, lAG-G| < g . Since (AE-E) g AG-E g (AG-G) u (G-E) mm lAE—El 5 [AG-GI + |G-E| < E + whenever ll-Al < 6. Lemma 3.11. Let E be a set of finite measure. Let {an}::1 be a sequence of positive integers such that lim an = w. Then there exist two sequences of positive n-om (X) 00 . I . real numbers {an}n=1 and {8n}n=1’ with the an s strictly increasing to 1 and the Bn's strictly decreasing to 1, such that for each n, n = 1,2,..., 1 1 1 1 _ _ ___ - n o,— < ——— [(anE E) n [0,m]l < m2" and [(BnE E) [ mll m2" whenever 1 < m < an. Proof. We first show the existence of the sequence {an}n=1' By Lemma 3.10 choose a1, 0 < a1 < 1, such that 25 _ _L_ lalE E] < a12 . Then for each m, 1 < m < a1, [(a E-E) fl 0,1 < - —l— 1 1 [ m]l - l0‘1": 5' < a12 5 a? ‘ B Lem . y ma 3 10 choose a2 such that max{a1,1—%} < 02 < 1 and a22 Then for each m, 1 f m < a2, 1 2 5_1_. l(a2£—E) n 10%]! s lazE-El < 2 m2 2 a2 Inductively define an as follows: By Lemma 3.10 choose a n such that max{an_1,1-%} < on < 1 and 1 an2 [anE-El < Then for each m, 1 5 m f a n 151. an2n m2n [(anE-E) n [o,%]] 5 lanE-El < The existence of the sequence {an};1 thus follows by induction. A proof similar to the one given above can be given to show the existence of the sequence {8n}:=1. Theorem 3.12. Let E be a measurable set and let d+(0,E) = 0. Then there exist two sequences of positive real numbers {01"}n:1 and {8n}n=1’ With the an s strictly increasing to 1 and the Bn's strictly decreasing to 1, such that (X) d+(0,Un=1anE) = o and d+(0,un=18nE) 26 Proof. We first prove the existence of the sequence {an}:=1. Set F = E n J, where J = [0.1); then d+(0,F) = o. By Lemma 3.9, there exists a sequence of positive integers {an}:=1, where the an's increase to m, such that for each A, 1 f E A < 1 [AF n [0,h]] (3.13) ———————————-< J? h n whenever 0 < h < 5L: By Lemma 3.11 there exists a sequence n of positive numbers {an}n=1’ where we may assume that for each n, % 5 an < 1, strictly increasing to 1 and correspond- co _ , such that for each n n-l ing to the sequence {an} |( F-F) n [0 1—]| < ~1— (3.14) an ’m m2n whenever I < m < an. Given an e > 0, choose a positive integer k so that 1 1 + —— < a E 2k Set 6 = aL-and let 0 < %-< 6. Choose j, j 3 k, so that k 1.1.51. aj+1 1 Now A ' 1 l[U:=1(anF-F)] n [o.%]l - Ifi=1|(anF-F) n [0.fiil + X:=j+1|(anF-F) n [0.%]l. Moreover by (3.13) 1 | [01 F n [O’I‘fi’] ' 1 j _]_-, l J __i__—————— < —-‘lz = —.- Zn=1l(anF‘F) n [0am]l S mzn=1 % mj m3 27 and by (3.14) 2m - I(a F-F) n 0 l m 1 n=3+1 n I .m]| < 2 =. ___ = ___ n 3+1 m2" [112‘] Therefore, ” 1 |[Un=1(“nF’F)] n [o’fi]| < 1 + 1 1 1 1 T _T S — + —- < e. a J 2.] k 2k Thus, ” 1 11m IIUn=1(anF-F)] n [0,fill _ 0 m-ocx) l — I'll Therefore by Lemma 3.4, d+(0,U::1(anF-F)) = 0. Furthermore, Since F n [U:=1anF] E F we have 00 n=1anF]) = 0‘ d+(0,F n [u Therefore, by Lemma 3.8(i), d+(0,Un=1anF) = 0. Now anE n alJ E anE n anJ, for n = 1,2,...; and since (Un=1anE) n [0,a1) = Un:1(anE n alJ) Un=1(anE n anJ) Ifl Un=1 lfl anF it follows that d+(0,Un=1anE) = 0- The proof for the existence of the en's is analogous except we take J = [0,%), and we choose the Bn's so that 1 < 8n f % for each n. This completes the proof of the theorem. 28 By Lemma 3.7 and Lemma 3.8(ii), Theorem 3.12 can be stated in the following form which will play a key role in the next chapter. Theorem 3.15. Let E be a measurable set. If d+(0,E) = 1 then there exist two sequences of positive real numbers {an}n=1 and {8n}n=1’ with the an s strictly increaSing to 1 and the Bn's strictly decreasing to 1, such that d+(0,nn=1anE) = d+(0,fln=lsnE) = 1. CHAPTER IV THE MAJOR THEOREM In this chapter we deduce the fundamental result stated in the following theorem. Theorem. Suppose f possesses a kth approximate Peano derivative f(k) everywhere on the interval [a,b]. (i) If f(k) > O on [a,b], then f(k-l) lS continuous and increasing on [a,b]. (ii) If f(k is bounded either above or below on [a,b], then f(k) = f(k) on [a,b]. The proof of this theorem will require some additional definitions and lemmas. Lemma 4.1. Assume f to have a kth approximate Peano derivative f(k) for each point in [a,b] and that f(l) is increasing in [a,b]. If k > 2 furthermore assume — = : : 0. f(2)(a) — f(3)(a) ... f(k_l)(a) Then (f(1))(k_1)(a) = f(k)(a), that is, there exists a measurable set E E [0,1] having 0 as a point of right-hand density such that E-lim f(1)(a+h)'f(1)(a) = 11121531 _ h-'0 hk‘1 (MM 29 30 Proof. By subtracting from f a multiple of x, we may assume that f(1)(a) = 0. By hypothesis there exists a measurable set F E [0.1] having 0 as a point of right-hand density and such that (4.2) F-lim -l;{f(a+h)-f(a)-Ahk} = 0. = l where A f(k)(a)/k.. By Theorem 3.15 there exist two sequences of positive real numbers {9E}m=1 and {6n}n=1 such that . * = . = 0 $wm6m hymen and m - = ” F = 1. d+(0’”m (1 6;)F) d+(0,n (1+en) ) :1 nzl {00 Let E = F n [lm:1(1-6;)F] n [n n=1(1+en)F]. By Lemma 3.8 d+(0,E) = 1. To complete the proof of the lemma we show f 1 :th) = Ak 3 f(k)(a) . J-l—r _ E-l' th (k-l)! 0 Let 6 > 0 be given. Choose an and a; such that if 6n 3 _ 6; = 9 " _ * “ 1+9n 1 am then ‘+ k ’-1 e k k j-l < E . AX§=2(’1)J 1(j)aJ > ’2 and AXJ=2(J)B 2 Set 58 60. E. = min 21(1+s)k+11 ’ 21(1-a)k+11 By (4.2) there exists a 6' > 0 such that 31 |f(a+h)-f(a)-Ahk| < e'hk whenever 0 < h < 6‘, h 6 F. If t1 and t2 are values of h such that 0 < t1 < t2 < 6' and t1,t2 6 F then k k . k k I[f(a+t2)-f(a+t1)]-A(t2-t1)[ < e (t2+t1). Hence k k k k k k k k (t -t ) (t +t ) f(a+t )-f(a+t ) (t -t ) (t +t ) A 2 l _ C. 2 1 < 2 1 < A 2 1 + €.__§__l_ t2"1 t2"‘1 t2"1 t2't1 t2‘t1 Since f(l) is increasing on [a,b] and f(1) = fap we have f(1) = f' on [a,b] (see [4]) and hence f(a+t2)-f(a+t1) ___________————— f +t . f(1)(a+t1) 5 t _t S (1)(a 2) 2 1 Thus, whenever 0 < t1 < t2 < 6' and t1,t2 E F (t -tf) (t§+tf) (4 3) f(1)(a+t1) < A + e' t2-t1 tZ-tl and (t;_t§> (t§+tf) (4.4) f(1)(a+t2) > A-—————— — e'————;—— . t2"1 t2“ 1 Set 6 = min{6'/(1+B),6'(1-a)} and let h 6 E such that 0 < h < 6. Since h E (1-6;)F, there exists a t2 6 F such that h = (1-e;)t2. Hence * em h .. = + t2 — (1 + 1'9$)h (1 B) and h < t2 < 6'. Thus from (4.3) we have 32 f ( +h hk 1 k_ k k k k (4.5) (I):1 ) < A[ ( +8) h ] + e'[h (1+8) +h ] h Bhk Bhk [(1+B)k-1] [(1+8)k+1] < A + E. 8 B < Ak + A2§=2(§)83‘1 + g < Ak + 6. Moreover, since h C (1+en)F, there exists a t1 6 F such that h = (1+6n)t1. Hence en t1 = (1-1+e )h = (1—a)h n and t1 < h < 6'. Thus from (4.4) we have k f(l)(a+h) [hk-hk(1-a)k] [hk+hk(1-a) 1 (4,5) k-l > A k _ e' k h ah ah k k [1-(1-G) I [1+(1-G) I > A - E' a a .+ ._1 > Ak + Az§=2(-1>J 1<§ia3 N|m > Ak - 5. Thus from (4.5) and (4.6) we have f 1 (a+h) Ak - e < —i)k 1 < Ak + e h whenever O < h < 6 and h E E. Hence f(l) 6+h) Ak f(k)(a) E-lim __, 44— = = . h—oo hk_1 (k-l)! 33 Corollary 4.7. Assume f to have a kth approximate Peano derivative f(k) for each x 6 [a,b], and that f(l) is increasing on [a,b]. If k > 2 furthermore assume f(2)(b) = f(3)(b) = ... = f(k-1)(b) = 0. Then (f(1))(k-1)(b) = f(k)(b). Proof. Define a function g on [—b,-a] as follows: g(x) = f(-x) for each x e [-b,-a]. Then g(k)(x) exists for each x E [-b,—a] and _ _ n _ g(n)(x) ‘ ( 1) f(n)( X) for n = 0,1,...,k. where f(0)(—x) = f(-x). Now it is easily shown that 9(1) is increasing on [-b,-a]. Also, if n > 2 then 9(2)('b) = 9(3)(‘b) = --~ = 9(k_1)(‘b) = 0- By Lemma 4.1 there exists a measurable set E E [0,1] such that 0 is a point of right-hand density of E and (-b+h)-g (-b) 9 (-b) E_,,m 911) k 111. z 1E1 h~0 h'1 (bIH Hence E_]im f(l)(b—h)-f(1)(b) : f(k) h-oo (-h)k‘1 (k-l)! that is, (f(1))(k_1)(b) = f(k)(b)- Corollar 4 8 Assume f to have a second approximate y . . a . f Peano derivative f(z) for each pOint in [a,b], and that (1) = for is increasing on [a,b]. Then, (f(1))(1)(x) f(2)(x) each x 6 [a,b]. 34 Proof. Follows immediately from Lemma 4.1 and Corollary 4.7. Lemma 4.9. Suppose f has (k-l) derivatives at the point x, then for each sufficiently small non-zero h, there is a 6, 0 < 0 < 1, depending on h such that “£32 {f(x+h)-Z:;é 2:f(n)(x)} = f(k'2)(x+eh)—f(k'2)(x) — 0hf(k'1)(x) where f(0)(x) = f(x). 3599:; Let (4.10) g(t) = f(x+t)-z:;$ %;f(n)(x). Then 9 is (k-2) times differentiable around 0 and . . ._ n . (4,11) g(J)(t) = f(J)(X+t)'Z:_—-_g 1 fijf(n+J)(x) for j = 0,1,...,(k-2). By the extended mean value theorem for each sufficiently small h there exists a 6, 0 < e < 1, depending on h so that hk-2 k-2 g(h) = 2:;3 h—.g‘"'<0) + k., ,g‘ ) = f(k-2)(x+8h)-X:“=O .142? ”f(k+n'2)(x) = f(k’2)(x+eh)—f(k'2)(X)-6hf(k-l)(X)~ Definition 4.13. A function f defined on an interval is said to be convex if for every pair of points P1, P2 on the curve y = f(x) the points of the arc P1P2 are below, or on, the chord P1P2. The following lemma is due to S. Verblunsky [9]. Lemma 4.14. Let f have a finite derivative at each point of (a,b). Suppose that, for each x0 6 (a,b) there are, in every neighborhood of (xo,f(xo)), points of the graph of f above the line y = f(xo)+f'(xo)(x-xo). Then f is convex in (a,b). Proof. If possible suppose that there are points c,d, a < c < d < b, such that the arc y = f(x) (c f x f d) has points above the chord joining (c,f(c)) and (d,f(d)). Let f(d)-f(c) d-c K— Now the function f(x)-f(c)—K(x-c) is continuous and so it will attain its maximum at some point v in [c,d]. By our assumption c < y < d. Let 36 f(Y)-f(C) y-c u : Then u > K. Since (f(x)-f(c))/(x-c) is a continuous function on [Y.d] it will attain all the values between H and K some- where between y and d. Let T be such that K < T < u. Then there exists a w, y < w < d, such that f(w)-f(C) T =._________ w—c Now the function (4.15) g(x) = f(x)-f(c)-T(x—c) is continuous and so it will attain a maximum at some point 5 in [C,w]. Since u > T, we conclude c < E < w. Also from (4.15), T = f’(E). Now choose 6 > 0 such that for each x E (E-6,E+6), g(x) 5 g(g). This implies using (4.15) that 1’(X) f f(€)+f'(£)(X-€) for x E (E-6,E+6). Hence the line y = f(E)+f’(E)(x—£) has the property that there exists a neighborhood of the point (E,f(E)) such that no point of the graph of f is above the line. This, however, contradicts our hypothesis. Definition 4.16. Let f be a function defined in a neighborhood of x. Then define f(x+h)+f(x-h)-2f(x) 0 f(x) = lim sup 2 2 h-oo h 5f(x) is called the upper symmetric second derivative of f at x. 37 Remark. It can easily be shown that if f"(x) exists then sz(x) = f"(x). However, the upper symmetric second derivative may exist without the second derivative existing. A proof of the following lemma can be found in [13]. Lemma 4.17. A necessary and sufficient condition for a continuous function f to be convex in (a,b) is that sz(x) 3 0 for each x in (a,b). Lemma 4.18. Suppose f has a second approximate Peano derivative f(z) at a point x 6 (a,b). Then there exists a measurable set E E [0,1] such that 0 is a point of right-hand density of E and f(x+h)+f(x-h)-2f(x) f ( ) E-l' — x . hTO h2 (H Proof. By hypothesis there exists a measurable set F having 0 as a point of density and such that 2 h = F—lim l—2{f(x+h)-f(x)-hf(l)(x)-§—f(2)(x)} 0. h-90 h Set F1 = F m [0,1]. Then 0 is a point of right-hand density of F1. Also, 0 is a point of left-hand density of F 0 [-1,0]. Setting F2 = {h —h e F n [—1,0]}, then 0 is a right-hand point of density of F2. It follows from Lemma 3.8 that 0 is a point of right-hand density of E = F1 0 F2. Let c > 0 be given. Then there exists a 0 < 6 < 1 such that 38 f(x+h)—f(x)-hf (x) f (X) (1) 2 h? __(_:_ mum whenever h E F and |h| < 6. Let h E E so that 0 < h < 6. Then h e F1; thus f(x+h)—f(x)—hf(1)(x) _ f(2)(x) < E h2 2 2 ’ and h 6 F2, that is, -h 6 F; thus f(x-h)-f(x)+hf(1)(x) _ £12)(X) < E . h2 2 2 Hence for 0 < h < 6, h E E f(x+h)-f(x-h)—2f(x) ( ) - f h2 (2) X f(x+h)—f(x)—hf(1)(x) f(2)(x)l < _ - h2 2 f(X-h)-f(X)+hf(1)(X) f(2)(x) + _ h2 2 < % + g = 8 Thus f(x+h)+f(x-h)-2f(X) E-lim 2 = f(2)(x)i h-*0 h Suppose f has a second approximate Corollar 4.19. Peano derivative f(z) at each point in (a,b), and f(2) 3 0 0" (a,b). Then sz(x) 3 0 for each x 6 (a,b). Proof. Follows immediately from Lemma 4.18. 39 In what follows we shall use without specific reference several well known results. We list these results here without proof. Let g be a function defined on an interval J and let 9 have an ordinary derivative 9’ on J. If g is convex on J then 9’ is increasing on J. Let g be a function defined on [a,b]. If g is monotone on (a,b) and has the Darboux property on [a,b] then 9 is monotone on [a,b]. Let g be a function defined on an interval J. If g is monotone on J and has the Darboux property on J, then 9 is continuous on J. Let g be a function of Baire class one on [a,b]. Then every non-empty closed set F, contained in [a,b], contains points of continuity of 9 relative to F. Let g be a function defined on an interval J and assume gap exists at each point in J. Then the following are true (see [4]): 1) g' is a function of Baire class one on J, ap 2) g’ has the Darboux property on J, ap 3) if g’ is bounded above or below on J then a .4 40 Lemma 4.20. Let f be a function satisfying the following two conditions on [a,b]: (i) fép(x) exists for each x in [a,b], (ll) 52f(x) 3 0 for each x in (a,b). Then fép is continuous and increasing on [a,b]. Proof. Let G be the set of all points x in [a,b] with the property that there is a neighborhood of x on which fép is bounded. Then G is an open set. Let (c,d) E G. Then a simple compactness argument shows fép is bounded on [c',d'], where c < c' < d' < d. Hence fép = f’ on [c',d']. Therefore it follows that fép = f’ on (c,d). Since f is continuous on (c,d) and 02f(x) 3 0 for each x E (c,d), f is convex on (c,d) by Lemma 4.17. Hence fép is increasing on (c,d). Moreover since fép has the Darboux property on [c,d] it follows that fép is continuous and increasing on [c,d]. In particular, fép is continuous and increasing in the closure of each component of G. To complete the proof of the lemma we show G - [a,b]. Let H = [a,b]-G. From above H is a closed set having no isolated points. Suppose H is non—empty. Then H is a perfect set. Since f5 is a function of Baire class one on . . . s at [a,b] there eXists an x0 6 H such that fép lS continuou Hence there exist numbers M 3 0 and 6 > 0 Let x0 relative to H. so that lfép(x)| f M for each x 6 [xo—G,Xo+5] “ H- a' = min{x : x E [xO—6,x0] 0 H}, b' = max{x : x E [xo,xo+6] n H}. 41 Note that since H is perfect a',b' E H and a‘ < b'. Also, if x E [a'.b'] H H then |fép(x)| 5 M. Let x E (a',b')-H. Then there exists a component of G, say (6,8), where o,B E H, such that x 6 (o,8) E (a',b'). From the first part of the proof fé is increasing on [0,8]. Hence P -M 5 fap(a) 5 fap(x) 5 fap(8) s M. Thus for each x 6 (a',b'), |f;p(x)| f M and so (a',b') E G. First assume x0 6 (a',b'). Then from above x0 6 G, which contradicts xO being contained in H. Secondly, assume X 0 M I a'. Then (x0-6,x0) E G and there exists a number M', I V 0, so that fép is bounded by M on [x0-6,xo]. In the last paragraph we showed f’ was bounded by M on (xo,b'). aP Thus fép is bounded by the max{M,M'} on (xO-6,b'), and again XO 6 G which is a contradiction. In a similar fashion a contradiction is obtained, if x0 = b'. Thus H must be empty. Therefore G = [a,b] and the proof of the lemma is complete. h approximate t Theorem 4.21. Suppose f possesses a k Peano derivative f(k) everywhere on an interval [a,b]. (i) If f(k) > 0 at each point in [a,b], then f(k-l) is continuous and increasing on [a,b]. (ii) If f k is bounded either above or below on [a,b], then f(k) = f(k) on [a,b]. Recall that Proof. Consider first the case k — 1. = ’ ' > 0 on a,b then f = f’ on 1) fap. Thus, if f(l) [ l (1) [a,b]. Thus, f(o) f< = f is continuous and increasing on [a,b]- Moreover, if f(l) is bounded either above or below on [a,b] 42 then f(1) = f’ on [a,b]. Thus the theorem holds when k = 1. Secondly, consider k = 2. By Corollary 4.19 and Lemma 4.20 the proof of (i) is immediate. Turning to case (ii), it is no loss of generality to assume f(z) > 0 on [a,b]. From (i) it follows that f(l) is increasing on [a,b]; hence f(1) = f’ on [a,b]. By Corollary 4.8, (f')(1) = f(2) on [a,b]. Moreover by assumption (f’)(1) > 0 on [a,b]; hence (f')(1) = (f')' = f(z). Therefore f(2) = f(z) on [a,b]. We may now assume that k > 2, and we can complete the proof by induction. We therefore assume the following: If f possesses a (k-l)th approximate Peano derivative everywhere on an interval [a,b], then for 1 g n f (k—l) (i) if f(n) > 0 on [a,b], then f(n-l) is continuous and increasing in [a,b], (ii) if f(n is bounded either above or below on [a,b], then f(n) = f(n) on [a,b]. Let k > 2 and assume f(k) > 0 at each point in [a,b]. Let G be the set of all points x of [a,b] with the property that there is a neighborhood of x on which f(k-l) is bounded. Obviously G is open. Let (c,d) E G. If c < a < B < d, then a simple compactness argument shows f(k-l) is bounded o?k 1) [a,B]. By (ii) of the induction hypothesis, f(k-l) = f - (k-2) . M eover 0" [a,B] and therefore f(k-Z) — f on [a,B] or = ' on these relations hold on (c,d). Thus f(k-l) f (k—2) (c,d) and f(k 2) is continuous on (c,d). If x 6 (c,d) then . . ' f there exists a measurable set E such that 0 is a pOint o 43 density of E and hk k- h”f ( _T (n) x)+Ej(f(k)(X)+€(x,h)) WW = 2.1. where E-lim e(x,h) = 0. From Lemma 4.9 for each suffi- h-0 ciently small non-zero h E E there is a 6, depending on h, between 0 and 1 such that g ) n :k§2!{f(x+h)'z:;é%1f(n)(x)}= f(k_2)(X+6h)'f(k_2)(X) Hence 2 h f(k_2)(x+eh) = f(k_2)(x)+6hf(k_1)(x)+ETI:T7{f(k)(x)+g(x,h)} > f(k_2)(x)-+0hf(k_1)(x) for all sufficiently small non—zero h E E. Thus, it follows by Lemma 4.14 that f(k-Z) is convex on (c,d); hence f(k_1) is increasing on (c,d). Choose y between c and d. Then is bounded below on [y,d]. Applying (ii) of the f( on the interval k-l) induction hypothesis to the function f(k-l) _ (k—l) [i,d], it follows that f(k_1) — f on [y,d]. and has the Darboux property Now since f is increasing on (y,d) (k-l) . 0 on [y d] we have that f(k 1) is continuous and increaSing on [Y,d]. Similarly, since f(k-l) is bounded above on [c,v], ' ‘ ‘ reasin on [c,v]. we deduce that f(k—l) is continuous and inc 9 44 Thus it follows that f(k-l) is continuous and increasing on [c,d]. In particular, f(k-l) is increasing and continuous in the closure of each component of G. To complete the proof of (i) we show G = [a,b]. Let H = [a,b]-G. From above H is a closed set having no isolated points. Suppose H is non-empty. Then H is a perfect set. Since f(k-l) is a function of Baire class one on [a,b] (see [3]), the same type of argument given in the proof of Lemma 4.20 shows H is empty. Hence G = [a,b] and the proof of (i) is complete. Consider, finally, (ii) for k > 2. It is no loss of generality to suppose that f(k) > 0 on [a,b]. By (1), f(k-l) is increasing on [a,b] and by (ii) of the induction hyPOtheSiS f(k-l) = f(k-l) on [a,b]. Thus it follows that f(1) = f' on [a,b]. We shall prove that (f(1))(k-1) = f(k) 0” [a,b]. It will then follow by the induction hypothesis (ii) applied : (f )(k'l) = f(k). to f(l) that in [a,b], f(k) = (f(1))(k-1) (1) th It suffices to prove that in [a,b) the (k—l) approximate Peano derivative of f(l) on the right, equals f(k)‘ For’ applying Corollary 4.7, it will follow that in (a,b] the (k-l)th approximate Peano derivative of f(l) on the left equals f(k)‘ Without altering f(k) by adding to f a suitable polynomial of degree less than n, we may assume that f(.)(a) = 0, for j = 2,3,...,(k-1). Note, since J k-l) f(k'1)(a) = 0 and f(k'l) is increasing on [a,b], f( 2 0 on [a,b], Now for each h, 0 < h < (b-a), there exists by + the extended mean value theorem a number 5, a < E < a h 45 such that k-3 f(2)(a+h) z (ELSTif Hence f(z) 3 O in (a,b). Thus f(l) is increasing on [a,b]. (k'l'm. By Lemma 4.1, (f(1))(k_1)(a) = f(k)(a). Since a may be replaced throughout by any a 6 [a,b) the proof of the theorem is complete. In [6], C. J. Neugebauer proved that if g is a function of Baire class one on an interval J, then 9 has the Darboux property on J if and only if for each real number A, the sets EA = {x : g(x) 3 A} and EA = {x : g(x) f A} have closed connected components. We thus have the following corollary to the last theorem. Corollary 4.22. If f possesses a kth approximate Peano derivative at each point of an interval [a,b], then f(k) has the Darboux property on [a,b]. Proof. Since f(k) is of Baire class one on [a,b] (See [3]), in order to show f(k) has the Darboux property we need only show that the connected components of the sets A E = {x : f( A} are closed IA k)(x) 3 A} and EA = {x : f(k)(x) for every real number A. So suppose f(k)(x) 3 A for all x in the interval (a,8). We must show that f(k)(a) 3 A and ' ' bounded below on (a,B), f f(k)(B) 3 A. Now Since f(k) is is bounded below on [a,B]. ThUS by Theorem 4'21’ f(k) = on [a,B]. Since f(k) has the Darboux property on [a,B], TTTTT_T—T___—_T__"_"""""""'TIIII 46 k f( )(a) 3 A and f(k)(8) 3 A. Hence, f(k)(a) 3 A and also f(k)(B) 3 A. Thus the connected components of EA are closed. Similarly the connected components of EA are closed. Thus, f(k) has the Darboux property on [a,b]. In [10], C. E. Weil proved that a function g of Baire class one has the Denjoy property on an interval J if, for every subinterval L of J on which 9 is bounded either above or below, 9 restricted to L has the Denjoy property. Using this result along with the facts that a kth approximate Peano derivative is a function of Baire class one and that an ordinary kth derivative has the Denjoy prop- erty, we also have the following corollary to the last theorem. h approximate Corollary 4.23. If f possesses a kt Peano derivative at each point of an interval [a,b], then f(k) has the Denjoy property on [a,b). 47 CHAPTER V PROPERTY Z FOR APPROXIMATE PEANO DERIVATIVES In this our final chapter we prove property Z for the kth th approximate Peano derivatives. To prove that every k approximate Peano derivative has property Z we first prove a lemma, which is a modification of a lemma proved by C. E. Weil [11]. Before proving the lemma some facts are established that will be needed. Let A be a measurable set and [c,d] a closed interval. Define a function on [c,d] as follows: F(x) = m(A n [c,x]) = f: x(s)ds where X denotes the characteristic function of A. Then F is absolutely continuous and F’(x) = x(x) a.e. on [c,d]. More- over, almost everywhere 6")? Fj+1= (1+1)F'x s N+1)”- Consequently, ' 1 j+1 12mm [c.s1)>~"ds zWWA n [c.dm . Likewise if we define on [c,d] d G(X) = m(A n Ix,d]) = [X x(S)ds, then G is absolutely continuous and G’(x) = -x(x) a.e. on [c,d]. In addition almost everywhere 48 d '+1 . ' , ~ 3; 63 = (3+1)GJ(-x) z -(1+1)GJ. Therefore, Iii-(m(A n [5,d]))‘jdsf 37,11 (m(A n1c.d1>)j+1. Lemma 5.1. Suppose f is a function whose kth derivative exists and is nonnegative on the interval [a,b], and let A = {x 6 [a,b] : f(k)(x) 3 e} where c is a fixed positive number. Then there exists a partition {a = t0 < t1 < ... < tg = b} of the interval [a,b] with z 5 2k and such that for each i = 1,2,...,k, with x,y E [ti-l’ti] and x f y k |f(y)-f(X)| 2 (6%)(m(A n [x,y])) . Proof. It will be shown by induction that for each integer j = 1,2,...,k, there is a partition of [a,b] {6:12 r 2 <§%))j and f(k'j)(y) 5 0. 49 (A) A (_J v ‘9) A 7K” I (_A. V A x V I —+‘ A 77 I L; v A ‘< v IV (m(A n [x,y1)>j L4 and f(k'j)(x) 5 0. 4(1): f(k‘“(x)-f“"j)(y) > (—%—)(x) = If f(k)(s)d5 3 cm(A fl [x,y]). In this case 1(1) holds for each x,y E [a,b] with X E Y if we take t = a and t1 = b. 0 If t = b, then since f(k—l) is nondecreasing on [a,b], f(k'1)(b) f O and for each x,y E [a,b] with x f y, f(k‘1)(y) 5 o and f(k‘1’ 0 since g'(x) > 0. Hence 9 is nondecreasing and continuous on [c,d]. Let |g| attain its minimum value on [c,d] at t E [c,d]. There are as before three cases to consider: t = c, t = d, and c < t < d. ,4 If t = c, then g(c) 3 0 and for each x,y E [c,d] with x 5 y, g(x) 3 0 and g(y)-9(X) = I; 9'(S)ds IV If (§%)(m(A n (x.s)))jds 2 (fi—mmw n [x,y1))j'1. (The last step was made by using one of the facts established previous to the lemma.) Thus, 1(j+1) holds for each x,y in [c,d] with x 5 y. If t = d, then g(d) 5 0 and for each x,y E [c,d] with x 5 y, g(y) 5 o and g(y)-g(X) = f; g’(slds 1i (5%)(m(A n (x.s1))jds IV L4 2 (TgffyT)(m(A n [x.y1))J+1. Thus, 2(j+1) holds for each x,y in [c,d] with x 5 y. If c < t < d, then g(t) = 0 and it can be Shown by arguments like the two just given that 2(j+1) holds for each x,y E [c,t] with x 5 y, and that 1(j+1) holds for each x,y E [t,d] with x 5 y. Second, assume that 2(j) holds for all x,y E [c,d] with x 5 y. Then g'(x) s (3%)(m(A n [x.y1))j + g'(y) 3%)(m(A n (x.y1))j IA 0 IA 52 since g'(y) 5 0. Thus, 9 is nonincreasing and continuous on [c,d]. Let lg) attain its minimum value on [c,d] at t. There are three cases. If t = c, then g(c) 5 0 and for each x,y E [c,d] with x 5 y, g(x) 5 0 and 9(y)-9(X) = fy g’(s)ds 5 1i (3%)(m(1 n [S,y]))jds _ (Hajj—now) r) [x,y])rl”. Thus, 1+1. g(X)-g(y) (WHMA fl [x,y] )) Therefore, 3(j+1) holds for each x,y in [c,d] with x 5 y. If t = d, then g(d) 3 O and for each x,y E [c,d] with x 5 y, g(y) 3 0 and g(y)-g(X) = If: g’(s)ds 51% (gr-EMMA 1‘) 1s.y1))st _ (finmw n [x,y])r'”. Thus, 1+1. (WHMA I) IXaYIH Therefore, 4(j+1) holds for each x,y in [c,d] with x 5 y. 9(X)-g(y) If c < t < d, then g(t) = 0 and using the same reasoning as above it can be established that 4(j+1) holds for all x,y E [c,t] with x 5 y, while 3(j+1) holds for all x,y E [t,d] with x < y. (It should be observed that cases 3 and 4 arise from 2 and thus are essential.) ..4 53 Third assume that 3(j) holds for all x,y E [c,d] with x 5 y. Then g'(y) IA (4%)(m(A n (x,y)))j + g'(x) (.1 5 (;%)(m(A fl [x,YIllj L4 5 0 since g’(x) 5 0. Hence 9 is nonincreasing and continuous on [c,d]. Let lg) attain its minimum value on [c,d] at t. If t = c, then g(c) 5 0 and for all x,y E [c,d] with x 5 y, g(x) 5 0 and g(y)-g(X) = If g'(S)ds 1y (;%)(m(A n (x.s1))jds IA (..1 1+1 (WWW) n [x,y])) Thus, 1+1 g(x)-g(y) . Ccféfyf)(m(A O [x,Y])) Therefore, 3(j+1) holds for each x,y E [c,d] with x 5 y. If t = d, then g(d) 3 0 and for all x,y E [c,d] with x 5 y, g(y) 3 0 and g(y)-g(X) = If: g’(S)ds 5 If (3%)(m(A n [x.s)))jds - (ijii71) 1’( (£001) (A n (s 11))jds 3+1 IV (qu )(m (An [x.y1)) Thus, 1(j+1) holds for each x,y E [c,d] with x 5 y. If t = d, then g(d) 5 0 and for all x,y E [c,d] X S y, g(y) s 0 and g(y)-9(X) = [i g'(S)ds I; (le(m(A 0 [5,YI))jds IV e j+l Z (TFITTT)(m(A n [x,YI)) Thus, 2(j+1) holds for each x,y E [c,d] with x 5 y. If c < t < d, then g(t) = 0, and proceeding as has already been demonstrated it can be established that 2(j+1) holds on [c,t] and 1(j+1) on [t,d]. This completes the proof of the lemma. 55 Theorem 5.2. If f has a kth approximate Peano derivative f(k) everywhere on [a,b] then f(k) has property Z on [a,b]. Proof. Let x be contained in [a,b] and e > 0. It suffices to Show that if given an n > 0 there exists a 6 > 0 such that if the closed interval [a,B] is contained in (x-6,x+6) 0 [a,b], x E [6,8] and f(k)(y) 3 f(k)(x) for each y E [a,B] or f(k)(y) 5 f(k)(x) for each y E [a,B] then m{y 6 [6,8] 3 Ika)(Y)'ka)(X)| > E} - < n. (B-a) + dist.(x,[a,B]) (5.3) Let n > 0 be given and set U g(y) = for)" Lill— 1mm. n=0 n! Then g(k)(y) exists for each y E [a,b] and furthermore g(k)(Y) = f(k)(YI‘f(k)(X)- From the existence of f(k)’ there exists a 6 > 0 and a meas- urable set E E [a,b] such that x is a point of density of E, and so that k 6(0/2) k (5-4) [9(Y)) E k(k+1). ly-XI k!-2 for |y-x| < 6 and y E E, (5,5) m(J (1 EC) 5 m(J)°12'- ) 0 [a,b] and x E J, where EC = [a,b]—E. Let [a,B] be a closed interval contained in (X-6,x+6) 0 [a,b] such that x E [a,B]. First assume that 56 f(k)(Y) f( % :(k)(x) for each y E [a,B]. By Theorem 4.21, k k) f 0" [0,8]. Applying Lemma 5.1 to the function g, which satisfies (“(y) = f(k)(y)-f(k)(x) for each y E [a,B], there exists a partition of [a,B] {a = to < t1 < ... < t£ = B} with 2 5 2k such that for each i = 1,2,...,1, and each s,w E [ti-l’ti] with s 5 w (5.6) |g(w)-g(s)l 3-5%(m(A O [51W]))k where A = {y E [a,B] : [g If f(k)(y) 5 f(k)(x) for each y E [a,B], then consider -g and apply Lemma 5.1 to obtain precisely the same inequality (5.6). We first obtain an estimate for m(A 0 E). For this purpose assume [ti-l’ti] 0 E f 0. Let t 5 t. < t9 < t. I i-l l - i - l with t;.t} e E. Then by (5.6) and (5.4) m(A 01t[,t'11])5 ("E—H1”)<1(t';.)-g(1;.)I“k \ “fig—)1” ()g(t';)l“" + )g(t;))“") I /\ (lg—)1”(e(12‘-)k/k!~2k(k+1))“"‘(lt';-xl+lt;-xl) —”— [dist.(x.[a,81) + (B-aH 2-2k I A < %% [dist.(x,[a,B]) + (B-a)]- ..1 57 If V) II lanCi : t! e [ti-l’ti] 0 E} and M II supIti : ti E [ti-l’ti] n E} then it follows from the above inequality that m(A n E n [ti_1,ti]) m(A n E n [51,53]) IA m(A n1s;.s';)) IA %% [dist.(x,[a,8]) + (B-a)]. Clearly the same estimate holds if [ti-1’ti] n E = 0. Hence = ile m(A n E n [ti_1,ti]) < Z§=1 %% [dist.(x,[a,B]) + (B-o)] IA % [dist.(x,[a,8]) + (B-o)]. Secondly, we obtain an estimate of m(A 0 EC). Let J be the smallest closed interval in [a,b] containing both x and [a,B]. Using (5.5) we have the following estimate (5.8) m(A n EC) 5 m(J n EC) 5 rum-121. Note that m(J) = dist.(x,[a,B]) + (B-a). Therefore by (5.7) and (5.8) m(A) = m(A n E) + m(A n EC) 5 m(A n E) + m(J n EC) 5 [dist (x,[a,Bl) + (B-a)]°% + m(J)-% = [dist.(xa[a,8]) + (B-a)]°n , 58 and (5.3) holds. Thus f(k) has pr0perty Z on [a,b] and the proof is complete. As was mentioned in the introduction, C. E. Weil introduced property Z in [11]. He further showed in [11] that if a function g has the Darboux property and property Z on an interval J then 9 has the Zahorski property on J (an example of a function having the Darboux property and the Zahorski property but not pr0perty Z can also be found in [11]). Thus in the class of functions having the Darboux property, property Z is strictly stronger that the Zahorski property. Thus, by Corollary 4.22 and the previous paragraph, we have the following corollary to the last theorem. Corollary 5.9. If f possesses a kth approximate Peano derivative f(k) everywhere on [a,b], then f(k) has the Zahorski property on [a,b]. BIBLIOGRAPHY 10. 11. 12. 13. BIBLIOGRAPHY P. Calderon and A. Zygmund, ”Local properties of solutions of elliptic partial differential equations”, Studia Math., 20 (1961), pp. 171-225. Denjoy, "Sur une propriété des fonctions dérivées exactes”, L'Enseignement Mathématique, 18 (1916), pp. 320-328. J. Evans, "L derivatives and approximate Peano derivatives", Trans. Amer. Math. Soc., 165 (1972), pp. 381-388. Goffman and C. J. Neugebauer, "0n approximate deriv- atives”, Proc. Amer. Math. Soc., 11 (1960), pp. 962-966. Marcus, ”On a theorem of Denjoy and on approximate derivatives", Monatsh. Math., 66 (1962), pp. 435-440. J. Neugebauer, "Darboux functions of Baire class one and derivatives", Proc. Amer. Math. Soc., 13 (1962), pp. 838-843. W. Oliver, "The exact Peano derivative", Trans. Amer. Math. Soc., 76 (1954), pp. 444-456. Tolstoff, ”Sur la dérivée approximative exacte", Rec. Math.(Mat. Sbornik)N.S. 4 (1938), pp. 499-504. Verblunsky, ”0n the Peano derivatives", Proc. London Math. Soc., (3) 33 (1971), pp. 313-324. E. Weil, "0n properties of derivatives", Trans. Amer. Math. Soc., 114 (1965), pp. 363-376. ”A property for certain derivatives", to 3 appear in Indiana 1. Math. Zahorski, "Sur la premiere dérivée", Trans. Amer. Math. Soc., 69 (1950), pp. 1-54. Zygmund, Trigonometric Series, 2nd edn., Cambridge, 1959. 59 _4 LIBRQRIES IIHIWHI[lllllllllllll II | | | I H l MICHIGAN STATE UNIV 31293 65184