SETS OF POEMS GK NON-BMW CCJrfii‘e‘ERGENCE GE ‘FAYLGR SERIES AND TREGGNGMEYREC SERSES Thesis fa: fha Degree of Ph. D. MICHEGAN STAYE UNIVERSITY Dan R. Lick E961 0-169 This is to certify that the thesis entitled SETS 01" POINTS OF NON-UNIFORM CONVERGENCE 0F TAYLOR SERIES AND TRIGONOMEI‘RIC SERIES presented by Don R. Lick has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics gala/g #ve mnjor proiezoy Date May 32 1961 LIBRARY Michigan State University ABSTRACT SETS 0F POINTS OF NON-UNIFORM CONVERGENCE OF TAYLOR SERIES AND TRIGONOHETRIC SERIES by Don R. Lick Let f(2) = § 8 zn be a Taylor series with lim a n30 “ n ->oo “ = 0 and § Ian] = + 00. Problems in sets of convergence n=0 have been studied by Hardy, Lusin, Mazurkiewicz, Nader. Erdgs, Herzog, and Piranian. The latter two also obtained a theorem involving the question of uniform convergence. We will investigate problems involving the question of non- uniform convergence. We define a point 2 on the unit circle C (lzl = l) to be a point of non-uniform convergence if the Taylor series converges at z and if there is no neighborhood of z in which the Taylor series converges uniformly; the set of all such points will be called the get 2£.2212£2.2£ non-uniform convergence and will be denoted by N. A point a on C will be called a 22in; 2; $922; uniform convergence if the Taylor series converges uniformly on some neighborhood of z; the set of all such points will be called the get 2; Don R. Lick points 2; local uniform convergence and will be denoted by Le We shall now describe briefly the results obtained about Taylor series. If N is any closed set on C, then there exists a function f(z), continuous in [a] g 1, whose Taylor series converges everywhere on C and has N as its set of points of non-uniform convergence. By the way L was defined, L is open and so if the Taylor series converges everywhere on C, N is closed. This gives the characteristic proper- ty of the set of points of non-uniform convergence of a Taylor series that converges everywhere on C, namely, the set is closed. The fundamental theorem proved about Taylor series is the following: Let c = N U L U D where, N, L, and D are mutually disjoint, and L and D are open, then there exists a Taylor series which has (i) N as its set of points of non-uniform convergence, (11) L as its set of points of local uniform convergence, and (iii) D as its set of divergence. Let F(9) = E (ancos n9 + bnsin n9) be a trigonomet- n=0 ric series and let I denote the set of real numbers reduced Don R. Lick modulo 2v. Problems in sets of convergence of trigonomet- ric series have been studied by Erdgs, Herzog, and Piranian. Again, we will investigate problems involving the question of non-uniform convergence. Sets of points of non-uniform convergence and sets of points of local uniform convergence for trigonometric series are defined exactly the same as for Taylor series. The results for trigonometric series completely paral- lel those given above for Taylor series. In fact, it is only necessary to change "Taylor series” to "trigonometric series" and "the unit circle C“ to "the set I" and the same results hold. The proof of any of these theorems on Taylor series or trigonometric series is to construct the desired series. These proofs make extensive use of Fejér polynomials and polynomials of the type 1 + z + --- + 2n. SETS or POINTS OF NON-UNIFORM CONVERGENCE 0F TAYLOR SERIES AND TRIGONOMETRIC SERIES BY Don R. Lick A THESIS . Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1961 g"? /"‘I)‘) P 9 ‘ l. ’ (Jew 0/22/95/ DEDICATION To my parents and to Carole ACKNOWLEDGEMENT The author wishes to express his gratitude to Professor Fritz Herzog, who not only suggested the problem but also offered his guidance and friendly criticism during the preparation of this thesis. I. II. III. IV. TABLE OF CONTENTS Page Introduction ..................................... 1 Some Preliminary Lemmas .......................... 7 Taylor Series ................................... 12 1. Sets of points of non-uniform convergence .. 12 2. Sets of_type F0,............................ 30 3. The fundamental theorem .................... 36 Trigonometric Series ............................ 39 1. Sets of points of non-uniform convergence .. 39 2. Sets of type F6,............................ Mk 3. The fundamental theorem .................... M6 BIBLIOGRW 00.0.0000...00......OOOOOOOOOOOOOOOOOOOO ”8 Chapter I. Introduction In this dissertation, we have studied certain proper- ties of Taylor series and trigonometric series. As it turns out, these properties of trigonometric series parallel the properties of Taylor series. And so in the introduction, as well as the rest of this dissertation, Taylor series will be discussed first and then trigonomet- ric series. Let f(z) 8 jfianzn be a Taylor series with lim an n20 n ->oo = O and S [anl a + 00. Problems in sets of convergence _n=0 and sets of divergence have been studied previously by G. B. Hardy [3], Nicholas Lusin [6], Stefan Mazurkiewicz [7], Ludwig Nader [8], and Paul Erdas, Fritz Beraog, and George Piranian [l],[2],[k],[5]. The letter also obtain- ed a theorem involving the question of uniform convergence (h, pp. 532-3]. We will investigate, in connection with Taylor series, problems involving the question of non- uniform convergence. For the purpose of studying non-uniform convergence, ‘we will define a point 2 on the unit circle C (Izl = 1) to be a point of non-uniform convergence if the Taylor series -1- -2- converges at z and if there is no neighborhood of z (i.e., an open arc of C containing 2) in which the Taylor series converges uniformly; the set of all such points will be called the £23.2£.222225 2! non-uniform convergence and will be denoted by N. Also a point s on the unit circle C ‘will be called a point 23 local uniform convergence if the Taylor series converges uniformly on some neighborhood of a; the set of all such points will be called the eee 2; 223222 25 legal uniform convergence and will be denoted by L. The set of all points on the unit circle C at which the Taylor series diverges is called the gee e; divergeece and will be denoted by D. The set N U L of all points en the unit circle C at which the Taylor series converges is ‘called the 322‘s; convergence. We shall now describe briefly the results obtained about Taylor series. First, it is shown that there exists a Taylor series which converges everywhere on the unit circle C, but for which N consists of a single point. Next, by using a sequence of Taylor series each of which has a particular point as its point of non-uniform convergence, it is pos- sible by an appropriate condensation method to construct a new Taylor series with any closed set on the unit circle C as its set N and with D empty. By the way L is defined, if the Taylor series converges everywhere on C, then the set N is closed. These two facts, then give the charac- -3- teristic property of the set N of points of non-uniform convergence of a Taylor series that converges everywhere on the unit circle C. By using a construction method sim- ilar to the one above, it is possible to construct a func- tion f(s), continuous in lzl g 1, whose Taylor series converges everywhere on C but has any given closed set as its set of points of non-uniform convergence. Herzog and Piranian [k, Theorem 1] proved: If M is a set of type F'_on the unit circle, then there exists a Taylor series that converges everywhere on M and diverges everywhere on C - M. This theorem is proved in this dissertation by a construction different from that of Herzog and Piranian: the reason being that the new con- struction can be used in the construction of a certain trigonometric series. A theorem that follows directly from the above theorem is: If F is a closed set on C, then there exists a Taler series that converges uniformly on F and diverges on C - F (See [h, Theorem 2]). This theorem is needed in the proof of the last theorem on Taylor series. The last and most important theorem of the chapter on Taylor series is the following: If the sets N, L, and D are given such that C = LU N U D: L, N, and D are mutual- ly disjoint; and L and D are open, then there exists a Taylor series which has (i) L as its set of points of local uniform convergence, (ii) N as its set of points of non-uniform convergence, and (iii) D as its set of diver- gence. Let f(0) 8 iii (an cos n9 + bn sin n9) be a trigono- n80 metric series and let I denote the set of real numbers reduced modulo 2w. Problems in sets of convergence and sets of divergence of trigonometric series have been stud- ied by Paul Erdgs, Fritz Hersog, and George Piranian [l], [2], but not as extensively as for Taylor series. Again, we will investigate problems involving the question of non-uniform convergence. For the purpose of studying non-uniform convergence, we will define a point 0 in I to be a paint 2f non-uniform coezeggence if the trigonometric series converges at 9 and if there is no neighborhood of 9 on which the trigono- metric series converges uniformly; the set of all such points will be called th°.225.2£.22l2£2.2£ non-uniform convergence and will be denoted by N. Also a point 9 in I will be called a £1.93 2.! loge; uniform convergence if the trigonometric series converges uniformly on some neighbor- hood of O: the set of all such points will be called the 223.22.22l222.2£.l222l uniform convergence and will be de- noted by L. The set of all points in I at which the trig- onometric series diverges is called the 223.25 divergence and will be denoted by D. The set NiJ L of all points in I at which the trigonometric series converges is called the set e; convergenc . -5- ‘We shall now describe the results obtained about trigonometric series. The results for trigonometric series completely paral- lel those given above for Taylor series. In fact, for the theorems involving trigonometric series that converge everywhere on I, it is only necessary to change "Taylor series" to ”trigonometric series" and ”the unit circle C" to "the set I” and the same results hold. The proof of any of these theorems is to construct the proper Taylor series and use essentially the real part as the desired trigonometric series. The main results of the chapter on trigonometric series are as follows: (i) If M is a set of type Fd_on I, then there exists a trigonometric series that converges everywhere on M and diverges everywhere on I - M. (ii) If F is a closed set on I, then there exists a trigonomet- ric series that converges uniformly on P and diverges everywhere on I - F. (iii) If L, N, and D are given such that I I LtJ NlJ D, L, N, and D are mutually disjoint, and L and D are open, then there exists a trigonometric series that has (a) L as its set of points of local uniform con- vergence, (b) N as its set of points of non-uniform con- vergence, and (c) D as its set of divergence. Again, the proofs of these three results follow by constructing a proper Taylor series and then using essent- 1ally the real part of the Taylor series as the desired trigonometric series. Chapter II. Some Preliminary Lemmas This chapter contains all the lemmas that are neces- sary for proving the theorems in this dissertation. Lemma I: If 0 < lel g 1r, then S e"19 gw/Iel. #80 ' inG Proof: E e"19 a l—JTO— . But, ‘1 - eine] g 2. “=0 l-e and by simple geometric arguments ll - eiel = 2 sin(lOl/2) 2" inc 3 1° 3 9 e “,0 l-e 2|0|71r V" . 00 Lemma II: If the terms of a convergent series E Ak kao are themselves finite sums (say, A1‘ = + ... + 00 a , k = 0,1,2,3, ... ), then the new series E a de- “kfl' nvo n duced from All is convergent if the quantity A]; = Ems 4- Max A 'apk‘fl afik+2+...+‘l‘k+}\|,1§)\§uk4‘1-pk has -7- the property that lim A}; = o. k'->oo Proof: Let Sk a A“, s 8 S Ak, and an = E a“. 980 1:80 [i=0 To each n corresponds a definite k, for which pk < n g “ks-1' and so sn = Ska1 + apkfl' + apk+2 4- "- + an. Let 6 > 0; choose in1 so large that. for every k > m1, IS]?1 - s I < 5/2, and choose m2 so large that, for every k>ng la“ +1 + ... apk+A|< 6/2 f0r1§A§Fk+1 ' "k. I! Let m = Max(m1,m2), and k > m, pk < n < “ks-1’ then 4- a < 6/2. Since (sn - s) = |3n - SIC-1| a 'aI‘k4'1 + eee nl (s - sk-l) +(Sk-1 - s), we have that lsn - sl g 00 Is - Sk-1l+'Sk-1 " 3' < E . Therefore, 2: an = 3, n=0 - J - m Lemma III: If 911(2) -- 3: 737 - 3: @13- is the 3:0 3:0 n-th Fejér polynomial, then (i) ‘Pn(s)| < it + 2n for every 2 E C and all positive integers n, and (ii) the modulus of the first n - r terms and the modulus of the last n - r terms is less than 21r/(r+l)|0|, where s = e19 and0< IOI gm -9- For an elementary proof, see [2, pp. h2-k3]. Remar : For convenience, 2n + M will be denoted by A. * Legee IV: If n g 2, then I Pn(z)| < 6 log n, where *Pn(z) denotes any block of consecutive terms from Pn(z). Ezeefe |*Pn(s)l < 2(1 + l/2 + ... + l/h) < 2(1 + log n) = 2(1 + l/log n)log n < 6 log n. 'Legne_1: Let 51,52,0-o ,Bm_1 be real numbers sub- Ject to the restriction 2n - 8 g Bl g 52 g ... g erl g 8) O, and let 3 ,5 , ... , 3 be complex numbers of l 2 m 15 unit modulus, satisfying the condition Siva/Sit 8 e “ (IS 8 1,2; ... .m‘1)e Then, i S“ < Kl/S , Where K1 “=1 is a positive universal constant. For a proof, see [h, pp. 529-30]. Lemma VI: Let gt(z) = l + z + °°° + zt'l, and p be a real number. Then, if -l/ht g 9 g l/ht, at least one of the quantities laedeipgthie)” and Ifleuzngdeie” .1s greater than th, where K2 is a positive universal constant . than 17 fers I either argmm every neare one 0 his -10- Proof: The argument of each term of “(910) is less than 1r/12. Also, at least one of the numbers p and 29 dif- fers from the nearest multiple of 1r by at most 1r/3. So, 1°). either the argument of every term of eipgth or the argument of every term of e gt(e or the argument of every term of both of these expressions differs from the nearest multiple of 1r by at most 517/12. Hence, at least one of the quantities lfldeipgthienl and lflehmpgthienl is greater than t(cos(5w/l2)). Lemma V I: Let 0Pt(z) be the first t terms of the t-th FeJér polynomial and p be a real number. Then if ~1/ht g e 5 I/ht, at least one of the quantities l@e(eipoPt(e1°))l and 'RdeapoPthiGD is greater than K3log .t, where K3 is a positive universal constant. Proof: The proof is similar to the proof of Lemma VI, except t is replaced by log t. Lassa-1111* If -1/h g e g I/h, then the real part of the sum of the first n terms of Paula) is greater than (log n)/2. Proof: The real part of the sum of the first n terms of Pn(e19) is given by l/n + (cos 9)/(n-l) + ... + ‘.-11- (cos n9)/l and this is greater than (l/n + I/(n-I) + + l)cos no, which is greater than (log n)/2. Chapter III. Taylor Series 1- §£§£2£EMQIWWO For the theorems in this section, the set D will be assumed empty (i.e., the Taylor series converges everywhere on C). By the way L was defined, L is open and since C = LlJ N, N is closed. The following theorems, Ia and 11a, are not the best theorems that can be obtained, but are included because of the simple nature of their proofs. Theegem Ia: There exists a Taylor series which con- verges everywhere on the unit circle C and has a single point as its set of points of non-uniform convergence. P1091: Let {An} be any denumerable set of. disjoint open arcs on C such that (i) 8n represents the angular length of An, (ii) “n(“n.#'1) represents the midpoint of An, and (iii) the on approach 1 monotonically. Let Nq tq-l (3.1) Cq(z) 3 (l/tq)z l + s/aq + ... + (z/hq) , >N q+1= q where Nq and tq are positive integers such that N + tq and the tq are such that : (l/tqsq) = T < + co. q=1 -12- -13- The polynomial Cq(z) will be called a circuit, and a ser- ies of circuits will censtitute a Taylor series. Let (3.2) i annzn n=0 be the Taylor series defined by the series of circuits (303) : Cq(2)e q=1 In order to show that the Taylor series (3.2) conver- ges everywhere on C, let a be any point on the unit circle C. Then, either a ¢'AJ for all j, or, z 6 AJ for exactly one 3. (i) If s é'AJ for all j, then by Lemma 1, lcq(s)‘ g 2m/tq87q for all q. Hence, the series of circuits (3.3) converges absolutely. By Lemma I, the modulus of the sum of any block of consecutive terms of Cq(z) is less than 21r/tq8 q’ and by Lemma II, the Taylor series (3.2) conver- ges at 2. (ii) If s 6 A1, then by Lemma I, Icq(z)| g 21r/tq8 q, if q ;! .1 and ICJ(z)| g 1. Hence, the series of circuits (3.3) converges absolutely. In a manner similar to part (i), the Taylor series (3.2) converges at z. Therefore, the Taylor series (3.2) converges for all s 6 C. -1h- In order to show that the point 1 is a point of non- uniform convergence, let P be any neighborhood of 1 on C. Then, there exists an index R such that one P when n > K. But, |Cn(an)| = I, so the Taylor series (3.2) does not con- verge uniformly on P. Hence, 1 is a point of non-uniform convergence of the Taylor series (3.2). In order to show that l is the only point of non- uniform convergence of the Taylor series (3.2), let 2 be any point on C other than 1. Choose 5 > 0, so that the are S 2 {e19: 8 < 9 < 2w - 8} contains 2. Then, there exists an index k =- k(S ) > 0 such that 30 An = if. if n > k. By Lemma I, if 3 is any point of s, then the mod- ulus of the sum of any block of consecutive terms of Cn($ ) is less than 2w/tn8 n when n > 1:. Hence, the Taylor series (3.2) converges uniformly on S. So, if s is any point on C other than 1, z is a point of local uniform convergence. Remark: In the above proof, it has been shown that the Taylor series (3.2) has uniformly bounded partial sums. Indeed, the moduli of the partial sums are'Iess than 21%!“ + 1. Theorem IIa: If N is any closed set on the unit circle C, then there exists a Taylor series which converges -15- everywhere on C and has N as its set of points of non- uniform convergence. Proof: Let N be any closed set on the unit cirlce C. Let {hep} be a sequence of points of N such that the set of limit points of {cup} is N. Let {sq} be the sequence of points defined in the proof of Theorem Ia. Let aqp = cqu, then iii-Igmoqp = top. for all p. Let _ qu tq-I (3.h) qu(s) - (l/tq)z l + s/hqp + '00 +(z/hqp) . where the tq are positive integers such that : (l/tq Sq) q=1 ll *3 < + oo(the 8 q] are defined in the proof of Theorem Ia). and tq > 2q and the qu are integers such that N11 g 0, ”up?" > H ‘ ql = l,q-l when p > 1, and N + t1. q+1.p-1 + 9’1 (The qu are chosen in such a way to ensure non-overlap- ping of the circuits, if the circuits are summed as fol- lows: 011 + 621 + 012 + C31 + 022 + C13 + Chl + ... ), Lot (305) : anpzn n=0 be the Taylor series defined by the series of circuits -16- (306) : cqp(Z)e q=1 lfor each p, the Taylor series (3.5) defined for that p, converges everywhere on C and w is its only point of P :non-uniform convergence. Also, the partial sums of each of the Taylor series (3.5) are uniformly bounded by 2NT+l. Let n (307) i has be the Taylor series defined by the series of circuits (3.8) i Z 2'chptz). m=2 p+q=m Let s be any point on C. The double series (3.8) converg- es absolutely on C, since by rearranging the series of ab- solute values, one obtains }fi 2"p Iqu(z) I, which is ' p=1 q=: iIess than S 2'P(2nr + l) = arr + 1. Hence, the series P31 (3.8) converges everywhere on C. -17- Let .quu) denote any block of consecutive terms a: from the beginning of qu(z). Let C “(2) denote qu(z) - *quu). In order to show that the Taylor series (3.7) converges everywhere on C, let a be any point on C and 6 > 0. Choose po) 0 such that if n > p0, then 2-n < 6. *. By the way the quu) were defined in (3.'+), l qu(z)| g l and so if p > p0, then 2-p [*qu(z)| < 5. As was shown in the proof of Theorem Ia, there exists an index qo > 0 such * . that I qu(s)| < 6 , when q > qo and p 5 p0. Hence, if m x p + q > pa + qo’ than 2-1) l’qu(z)| < 5 and by Lemma II, the Taylor series (3.7) converges at a. To show that every point of N is a point of non- uniform convergence of the Taylor series (3.7). let a be any point of N. Let B be any neighborhood of 2. Then, there exists an we 6 B, for some integer a. Also, there exists an index q2 > 0 such that “qa 6 B, when q > (12. But, 2—a lcqa(a )l = 2'“ and so the Taylor series (3.7) qa does not converge uniformly on B. Hence, any point belong- ing to N is a point of non-uniform convergence of the Taylor series (3.7 ) . 'In order to show that the points of N are the only points of non-uniform convergence, let a be any point of C - N. Let A(z,N) = 38, where A(z,N) denotes the angular -18- distance between the point z and the set N. Let B 5 {e19: A(e1°,z) < 5} . Let S E B and consider SK(S) 3 2 bush. n=K For each X there exists an integer s and an integer N such that - t - -k sxm = 2 8c M_3's($) + 3: 2 CM_k,k(S) k=s+l 'P + , E 2 cqp($ ) mam-1 p+q=m Since the series of circuits (3.8) converges absolutely, .3165) I 3 2-3 lc.M-s,s( 5 )l ” s: 2-]: ICU-k,k(5 ’l k=s+l + i Z 2’Plcqp(5) mam-1 p+q=m 'P Let S > 0; choose p:l > 0 such that Z 1(21rT 4- 1) < 5/3. There exists an integer q1= (11(5 .p1) > 0 such that -19- A(nqp,B) >8 when q > ql and p g p1. Let M > p1 4- ql. Then, ISX($ )l g 2"IC*M_3’S(S )l + :1 2"1‘ |cu_k,k($ )l 4': 2-p qup($)l m=M+1 p+q= = 2‘3 '6',“ !,(S )I +: Z 2 P lcqpm )| man p+q=m pI £- 2-8 10's-. 3(5 ’l * 2-!) j:- Icqp(s )l p=1 q=M-p + ‘2': 2'1’ 2“: |cqp(5) . p=P1+1 qal By Lemma I, lc*qp($ )l < w/tq8 :_<_ 2'q(1r/8) when q > q1 * and p § p1, so 2'8 IC M-s,s(s )l < 2'32'M+8(1r/8) a [MW/8) if s 5 p1. Also 'C.qp($ )l g I, so - s - ‘P 2 3 IC M-s,s($ )l g 2 8 < 2 1 if s > p1. In either case, if M is chosen sufficiently large, 2-s IC‘ (5 ) < 6/3. M-s,s -20- . P1 P Moreover, Z: 2 pi Iqu($ )|< (if: 2"): 2q(V/5 ) p31 q=M -p p=1 q=M-p D1 3 Z 2'”+1(n/8 ) = 2'm1(17/8)p1, since in each case p=1 q > q1 and p 3 p1. If N is chosen sufficiently large, ’1 then X 2-p : Iqu($ )I < 6/3. Finally, since p=1 q=M-p do Z1 lcqp($ )l < 211T + l for each p, : 2-p :Iqu($ )l q: p=P1+1 q=1 2 (2rT + l) a 2 (2NT + l) < 6/3. Therefore, p=P1+1 |sx($')| < 6 when x is sufficiently large. Hence, the Taylor series (3.7) converges uniformly on B and z is a point of local uniform convergence. Therefore, N is the set of points of non-uniform convergence of the Taylor series‘(3.7). Corollagz I: The characteristic preperty of the set of points of non-uniform convergence of a Taylor series that converges everywhere on the unit circle is that the set is closed. -21- The following two theorems give better results than Theorems Ia and 11a, that is the Taylor series constructed is continuous in |z| g 1. Theorem Ib: There exists a function f(s), continu- ous in |s| g 1, whose Taylor series converges everywhere on C and has a single point as its set of points of non- uniform convergence. §£22£3 Let {An} be any denumerable set of disjoint open arcs on C, such that (1)8n represents the angular length of An, (ii) an(an‘#'l) represents the midpoint of An, and (iii) the “n approach 1 monotonically. Leti{hn} be any sequence of positive integers such that 2w/ Snlog hn < 2-“. Let n-l n 2n-l (309) Pn(2)=%+aéi+eee+zl __zi__-...-zn . Let Rn (3.10) Qn(z) = Z Phn(z/Cn)/108 hns where Nh are positive integers such that N1 > O and 8 Nh+1 ; Nh + 2hn. The polynomial Qn(s) will be called a -22- circuit, and a series of circuits will constitute a Taylor series. Now, consider the series of circuits (3.11) i 42,,(2). By Lemma III, IPn(z)| < A, hence, IQn(z)| < A/log hn < A2"n and so the series of circuits (3.11) converges abso- lutely and uniformly and represents a function f(z), con- tinuous in [2] g 1. Let (3.12) 2 ans” be the Taylor series defined by the series of circuits (3.11). In order to show that the Taylor series (3.12) con- verges, let 2 be any point on the unit circle C. Then there exists an integer K > 0 such that z ¢'An if n > K. Let t be any integer such that 0 g t < hn’ then by Lemma III, the modulus of the sum of the first hn-t terms of Qn(z) is less than 21r/(t+1)$nlog hn < 2"", Also, by Lemma III, the modulus of the sum of the first hn+t terms -23- of Qn(z) is less than A/log 15 + 21r/(t+l) Snlog hn < (A + l)2'n. By Lemma II, the Taylor series (3.12) conver- ges to f(z). To show that l is a point of non-uniform convergence of the Taylor series (3.12), consider Qn(an). By Lemma VIII, the modulus of the sum of the first hn terms of Qn(on) is greater than.l/2. But, any neighborhood of the point 1 contains an infinite number of the points a n Hence, the Taylor series does not converge uniformly on any neighborhood of l, and so 1 is a point of non-uniform convergence of the Taylor series (3.12). In order to show that l is the only point of non- uniform convergence of the Taylor series (3.12), let a be any point on C other than 1. Choose 8 > 0 such that the are S 5 {e19: 5< 6 < 21r - 5 } contains 2. Then, there exists an integer r = r(8 ) > 0 such that S n An = E if n > r. If.$ is any point of S and n > r, then by Lemma III the modulus of the sum of any block of consecutive terms of Qn(5 ) is 1688 than 2'n(A + 2m/6 ). Hence, the Taylor series (3.12) converges uniformly on S, and so if z is any point on C other than 1, then s is a point of local uniform convergence of the Taylor series (3.12). Therefore, 1 is the only point of non-uniform conver- gence of the Taylor series (3.12). -2h- Theorem IIb: If N is any closed set on C, then there exists a function f(z), continuous in lzl g 1, whose Taylor series converges everywhere on C and has N as its set of points of non-uniform convergence. Proof: Let N be any closed set on the unit circle C. Let'%A>p} be a sequence of points of N such that the set of limit points of {top} is N. Let {sq} and {hq} be the sequences defined in the proof of Theorem Ib. Let a = a to , then lim a 8 Us . QP q P q ->oo QP P Let "QP ‘/ (3.13) qutz) . z phqtz .qpmor hq where the qu are chosen properly to ensure non-overlap- ping of the circuits (3.13), if the circuits are summed as in the proof of Theorem IIa. Let (301“) : anpzn n=O be the Taylor series defined by the series of circuits (3.15) Z éqpts). q=1 -25- Then for each p the Taylor series (3.1%) represents a con- tinuous function in [2' g l and has “up as its only point of non-uniform convergence. Also, the partial sums are ‘ uniformly bounded by A + 6. Let ( 3 o 16) i bnzn be the Taylor series defined by the series of circuits (3'17) : Z - 2"menu" m=2 p+q8m The double series (3.17) converges absolutely and uniform- 1y on C, since by rearranging the series of absolute val- IIes, one obtains : 2-p Iqu(z)| , which is less than p81 q=1 E 2’pA 8 A. Therefore, the double series of circuits 1p=1 (3.17) represents a function f(z), continuous in |sl 5 1. In order to show that the Taylor series (3.16) conver- ges everywhere on C, let a be any point on C and E > 0. Choose p0 > 0 such that 2’nA < 6 when n > p0. Let .quu) -25- denote any black of consecutive terms taken from the be- ginning of qu(z) and thp(z) denote qu(z) - *qu(s). By the way qu( ) was defined in (3.13), Iqu(s)| < 2'qA. Lemma IV implies that I‘qu(s)| < 6 < A. So if p > p0, ’ then 2-p I*qu(s)| < c. As was shown in the proof of Theorem IIa, there exists an integer qO > 0 such that a qup(Z)l <6 menq>qo andpspo, Hence, 1fm=p+q > po + qo’ 2-p l‘qu(z)l < 6:, and Lemma II implies that the Taylor series (3.16) converges to f(z) at 2. To show that every point of N is a point of non- uniform convergence of the Taylor series (3.16), let a be any point of N. Let B be any neighborhood of 2. Then, there exists an can 6 B for some integer a. Also, there exists an index q2 > 0 such that “qa E B when q > q2. But, Lemma VIII implies that the modulus of the sum of the first hq terms of 2'8Qqa(oqa) is greater than 2'8’1, and so the Taylor series (3.16) does not converge uniformly on B. Hence, any point of N is a point of non-uniform convergence of the Taylor series (3.16). In order to show that the points of N are the only points of non-uniform convergence of the Taylor series (3.16), let a be any point of C - N. Let A(z,N) = 35, -27- and let B 5 {319: A(e19,z) <8}. Let S be any point of B and consider SK($) = : onS“. n81! For each N there exists an integer s and an integer N such that - M- 3155 ) = 2'sq*u-s,s(3 ) + Z z’kqu_k,k( S ) k=s+l If X 2-qup($)’ m=M+l p+q8m Since the series of circuits (3.17) converges absolutely ISK(3 )' g 2-8 lQ‘M-s,“ 5')! I S: 2.1: 'QM-kJ‘S )l k=s+1 + f 2-1, S: Iqu(5 )I * i 2", i qupLC )I. Since p31 q=M+1-p p=M+1 q=1 -p -N f: qup($)l < A for each p, i 2 i |oqpts )|< 2 a. q8l p=M+1 q-l -23- From the fact (5 )| < 2'qA for each p, Iqu M-l - Z 2"k lQN-kdt‘ 3 )l. < E 2'MA < 2'MA(M-s) and k=s+l kzs+l M . 2"p i Iquts )l < :2"9 i 2'41; a p81 q8M+l-p p81 q8M+l-p 2"”); :- 2'MAM. Let 6 > 0; choose p1 > 0 such that ...: p: ’P 2 1A < E . There exists an integer. ql 8 (11(8 .p1) > 0 such that A(aqp,B) > 8‘ when q > q1 and p g p1. Let M > p1 + ql. By Lemma III, q*qp($ )I < 2"“(1 + 21r/8) an“, q > <11 and p g p1. so 2" I Q*M-s,s( S )I < 2'”(A + 21r/8) if s g p1. By Lemma IV, IQ.“($ )' < 6, so 2-s lQ‘M-s,s(s )' - 'P ‘1 < 6-2 3 < 12 1 < 6/2 if s > p1. Let M be so large that 2'“A(2M) and 2'M(A 4- 21r/5) are both less than 6/2. Then, 'SKLS )l < E and the Taylor series (3.16) converges uni- formly on B. Hence, a is a point of local uniform conver- gence of the Taylor series (3.16). Therefore, N is the set of points of non-uniform convergence of the Taylor series (3.16). -29- If N is taken to be the entire unit circle C, then every point on the unit circle is a point of non-uniform convergence, and one obtains the following corollary. Corollary II: There exists a function f(z) which is continuous in |z| g 1 and whose Taylor series converges everywhere on C, but not uniformly on any arc of C. This corollary agrees with Theorem 7 [5, p. 51]. But upon oral communication with the authors, it has been agreed that they have only proved the existence of a func- tion f(s) which is radially continuous on every radius of L lzl g 1 and has the convergence properties mentioned in Corollary II. -30- 2. Sets _o_f type In: Herzog and Piranian [1:] proved that if M is a set of type F6. on the unit circle, then there exists a Taylor series that converges everywhere on M and diverges everywhere on C - M (i.e., N U L 8 M and D 8 C - M). A similar result for trigonometric series will be proved in Chapter IV, section 2, but the method of construction of such a trigonometric series depends on taking essentially the real part of a particular Taylor series. For this purpose, the conStruction of Herzog and IPiranian [h] does not give the desired Taylor series. But, by using a method similar to the construction in [it], and making use of the "doubling" process introduced in sections- 3 and it of [l], the following construction not only gives a proof of this theorem of Herzog and Piranian, but also gives the desired Taylor series that is needed in Chapter IV. Theorem III (Hersog-Piranian): If M 13 a set of type Ff on the unit circle C, then there exists a Taylor series that converges everywhere on M and diverges everywhere on C-M. Pgeof: Since M is of type Ft it is the union of closed sets Pp (p 8 1,2,3, "- ), where Pp Clip+1 and F1 13 not empty. Q For each positive integer q, let nqm = geld-m an, 7*.- -31- (m as 1,2,3. ”t ,hQ), Let flq consist of all the numbers qu which satisfy one of the following conditions 1r/2 < A(nqm.Fq), 1r/22 < anm’Fq-l) g u/z, (3018) "/23 < A(rzqm,Fq-2) g ”/22, see ’ 1’/.2.‘1 < A‘an'Fl) g n/2q'1, twhere the symbol A(z,F) denotes the angular distance be- tween the point z and the set F. Suppose that kq of the frumhers qu belong to the set {Lq3 let them be denoted by wqi (j = 1,2,3, °°° ,kq) in accordance with the ine- quality 0 < 81.8 wa < arg qu < eee < erg quq g 2",. Let I. N R q kg-)- (3019) c (2) a 1+-q Z q i z‘J’l)” Z (Z/UJ )r ‘1 9.1 J31 r80 2N k “L1 q (J‘l)hq r + z :Ef:s 2E::(z/Lqu) J31 r80 wheme the Nq are chosen in such a way as to ensure non- -32- overlapping of circuits and non-overlapping of like powers of 2 within each circuit. Let (3020) i anzn n80 by the Taylor series defined by the series of cirucits (3.21) i Cq(s). (1'1 To show that the Taylor series converges everywhere on M, let 2 be any point in M. Then a is contained in all but a finite number of the sets Pp. Let s be the integer such that 2 lies in F91, but z does not lie in PS. There exists an integer qo > 0 such that if q > qo, the distance between a and any ouq‘1 satisfying one of the last 3 condi- tions of (3.18) is greater than A/2, where A is the pos- itive distance between 2 and the set P3. Also, for all <1 > s, the distance between 2 and any point qu satisfy- ing one of the first q - 3 conditions of (3.18) is greater than 1r/2q. Hence, there exists an integer q1 > 0 such that if q > ql, the distance between a and the set n. q is great- or than 1r/2q. Let arg wqh < arg z < arg quh*1' where -33- l < h g kq. (The case where 2 lies on the are from w to U.) presents no special difficulties in the qkq ql following reasoning if h is taken to be kq). By Lemma V, the modulus of any block of consecutive terms taken from the first h sums of the first ”half” of Cq(z) and the modulus of any block of consecutive terms taken from the first h sums of the second ”half" of Cq(z) is less than Kl/h2q. Also, by Lemma V, the modulus of any block of consecutive terms taken from the last kq - h sums of the first ”half” of Cq(z) and the modulus of any block of consecutive terms taken from the laSt kq ~ h sums of the second ”half” of Cq(z) is less than Kl/w2q when q > ql. Lemma II implies that the Taylor series converges at 2. In order to show that the Taylor series (3.20) diver- ges everywhere on C - M, let a be any point of C - M. Let b1 be any positive integer: there exists an integer c1 c1+l C such that r/é < A(z,Fb1) g n/2 1. Let p1 = b1 + c1, then at least one of the elements ofI).p1 which satisfy th 1 lit ‘/2c1+1 c1 e no ua w < A a) F 2 q 3 ( p1”, b1) g m/ (say'onlnl) -l ”91*2 satisfies the relation A(z,Lop ml) 3 h h . Lemma VI 1 - implies that Cp1(z) has a block of consecutive terms whose -3h- sum has real part with modulus greater than M'ZKZ. Next, let b2 be any positive integer greater than p1: there c2+l C exists a c2 such that 1r/2 < a(z,rb2) g 1r/2 2. Let p = b + c , then at least one of the elements of.fl 2 2 2 p2 c2+1 F which satisfy the inequality 1r/2 < Mo) .1? ) s it 92" b2 - :‘ c 1r/2 2 (say do ) satisfies the relation a(s,w ) 5 2 - I’2‘“ Pan‘s ; -p +2 a 2 . Lemma VI implies that Cp2(z) has a block of ' t‘lt consecutive terms whose sun has real part with modulus greater than h'zxz. By continuing this process, one ob- tains infinitely many values of q for which C§(z) contains a block of consecutive terms whose sum has real part with modulus greater than k'zxz. This establishes the diver- gence at s. A special case of this theorem is when M is closed, in which case all Pp can be taken equal to M. In this case the integer s that occurs in the proof is zero for each s in M, and therefore the Taylor series constructed in the above proof converges uniformly on M and one obtains the following theorem (See [h, Theorem 2]). -35.. Theorem IV (Herzog-Piranian): If F is a closed set on the unit circle C, then there exists a Taylor series which converges uniformly on F and diverges on C.- F. .4 -36- 3. Eye fundamental theorem: The following theorem con- nects the ideas of sets of points of non-uniform conver- gence, sets of points of local uniform convergence, and sets of divergence. fieegem V: Let C = LU N UD where L, N, and D are mutually disjoint and L and D are open, then there exists - a Taylor series which has (i) L as its set of points of local uniform convergence, (ii) N as its set of points of non-uniform convergence, and (iii) D as its set of diver- gence. Proof: Let L U N be the closed set in the proof of Theorem IV. Let qu(z) 8 Cq(z) and Nqo 8 Nq where Cq(s) is defined in (3.19) and N 0 will be defined below. Then, q the Taylor series 00 (3.22) Z bnzn defined by the series of circuits (3e23) : qu(2) q=1 converges uniformly on LiJ N and diverges on D when the -37- Nqo are chosen properly. Let N be the closed set in Theorem IIb. Let the circuits qu(z). p g 1, be defined as in (3.13). The qu, p 2 1, will be defined below. The Taylor series (3.2”) c zr "‘”"‘“_H defined by the series of circuits (3.25) ' i : 2'qup(z) N82 p+q=m p g 1 converges everywhere on C, and has N as its set of points of non-uniform convergence when the N , p 2 l, are chosen up properly. Let (3026) a 23 be the Taylor series defined by the series of circuits 'p (3.27) i Z 2 qu(z). m8l p+q=m (In this sum p g 0 and q > 0). It is assumed that the qu are chosen properly to ensure non-overlapping of circuits and non-overlapping of like powers of s in each circuit. 'The divergence of the Taylor series (3.26) on D follows from the proof of Theorem III: the local uniform convergence on L and non-uniform convergence on N follow in a manner similar to that in the proof of Theorem IIb. Chapter IV. Trigonometric Series 1. Sets 2; points 21 non-uniform convergence. This chap- ter deals with trigonometric series; the notations and symbols used in this chapter are defined in the intro- ] duction. In this section, the set D will be assumed empty (i.e., the trigonometric series converges everywhere on I). By the way L was defined, L is open, and since I = LlJ N, N is closed. Theorem VI: If N is any closed set on I, then there exists a trigonometric series which converges everywhere on I and has N as its set of points of non-uniform conver- gence: also, the sum function of the trigonometric series is continuous on I. 2522;: For the proof of this theorem, map the set N onto the Set NaI on the unit circle in the obvious manner (i.e., 9 in N goes onto e19 in N*). Now, by Theorem IIb, there exists a function‘fl(z) which is continuous in [2' g 1, and whose Taylor series converges everywhere on C and has N‘I as its set of points of non-uniform convergence. By taking essentially the real part of this Taylor series one obtains a trigonometric series with the properties: (i) the trigonometric series converges everywhere on 1, (ii) the trigonometric series has uniformly bounded partial -39- .--.‘ ’ -_-a ‘I 1.5 -ho- sums, and (iii) the sum function in continuous on I. But, there is no assurance that the set N will be the set of points of non-uniform convergence of the trigonometric series, since when multiplying the Fejér polynomial Ph (2) q N by the factor 2 qp, the control of the modulus of the real part of the sum of the first hq terms is lost. To over- come this difficulty, the ”doubling“ process introduced in Sections 3 and h of [1] is used: that is, the circuit qu(z) is replaced by the circuit 2N (z/hqp) + z qp Ph (z/cqp) . N e '1 cm (#.1) qu(z) (log hq) z Phq q where qu g 2hq, and the qu is also chosen so that no two circuits qu(z) overlap. By the proof of Theorem IIb, the Taylor series (152) : ansn n80 defined by the series of circuits (has) i: Z 2'p quu) m=2 p+q=m 1 -hl- has all ofthe properties of the Taylor series (3.16) con- structed in the proof of Theorem IIb. Let (ll-Jr) f (bucos n9 + ensin n9) n80 be the real part of (ins) : ane1n9. n80 Then, the convergence and the boundedness of the partial sums of (k.h) and the continuity of the sum function fol- low from these properties of the Taylor series (h.2). To show that any point of N is a point of non-uniform convergence of the trigonometric series (h.h), let 9 be any point of N and e19 6 N‘. Now, every neighborhood of 19 e contains infinitely many points a , and so by Lemma QP VII, the real part of one of the ”halves” of qu(e19) has a block of consecutive terms with modulus greater than K3, where [(3 is a universal constant. Hence, the point 9 is a point of non-uniform convergence of the trigonometric series (not?) 0 -h2- In order to show that any point of I - N is a point of local uniform convergence, let G be any point of I - N and e10 e c - N’. Then, by the construction used in the proof of Theorem IIb, there exists a neighborhood B. of e19 on which the Taylor series (h.2) converges uniformly: and so the trigonometric series (h.h) converges uniformly on the neighborhood B of 9. Therefore, 9 is a point of local uniform convergence of the trigonometric series (k.k). Remark: Zygmund [9. PP. 326-30, Theorem (3.1)] proved that if (M6) (1/2)Ao 4- i (An cos n9 + anin n0) =1 converges everywhere to an integrable function f(6). then the trigonometric series (h.6) is the Fourier series of f(9). Since the sum function of the trigonometric Series (h.k) is continuous on I, the sum function is integrable on I, and so the trigonometric series (h.h) is a Fourier 88’1630 If N is taken to be the whole set'I, then every point of I is a point of non-uniform convergence and one obtains the following corollary. -h3- Corollary III: There exists a trigonometric series which converges everywhere on I, but not uniformly on any interval of I: also the sum function of the trigonometric series is continuous on I. Using the fact that N is closed, Theorem VI also yields the following corollary. Corollagz IV: The characteristic property of the set of points of non-uniform convergence of a trigonometric series that converges everywhere on I, is that the set is closed. 4.1,. 2. Sets gf type 3‘, In this section, a theorem involving sets of convergence of type Fc,is proved for trigonometric series: this theorem is similar to the one Herzog and Piranian proved for Taylor series. Theorem.VIII: If M is a set of type Fc,on I, then there exists a trigonometric series that converges every- where on M and diverges everywhere on I - M. Proof: Let (‘+.7) : (bucos n9 4- onsin n9) n80 be the real part of the series (ue8) : aneine n80 where the Taylor series i n80 was defined in (3.20), where the set of type F6 on the unit circle is the set N"I defined from the set M in the obvious manner. Then, convergence follows from the -hs- convergence of the Taylor series (3.20); and the diver- gence follows from Lemma VI and the proof of Theorem III. A special case of this theorem is that the set N is closed. Then in a manner similar to the proofs of Theorem IV and Theorem VIII, the following theorem is obtained. Theorem IX: If F is a closed set on I, then there exists a trigonometric series which converges uniformly on F and diverges on I - F. -—‘. - ,n‘ -..;a.'—' up: 1 , -h6- 3. The fundamental theorem. The following theorem con- nects the ideas of sets of points of non-uniform conver- gence, sets of points of local uniform convergence, and sets of divergence of trigonometric series. Theorem X: Let I = LIJ NiJ D, where L, N, and D are mutually disjoint and L and D are open, then there exists a trigonometric series which has (i) L as its set of points f“““ of local uniform convergence, (11) N as its set of points i of non-uniform convergence, and (iii) D as its set of divergence. Proof: Let L, N, and D be mapped, respectively, onto the sets L*, N‘, and D‘ in the obvious manner. Then C = L'U N'U D’. ' Let qu(z) 8 Cq(z) and the integers Nqo 8 N(1 where Cq(z) was defined in the proof of Theorem III (3.19) and the Nqo will be defined below. Let the circuits qu(s). p g 1, be defined as in the proof of Theorem VI (h.1): the N p will be defined below. Let (#09) : anzn n80 be the Taylor series defined by the series of circuits (mo) i: Z 2'9 qu(z). (In this sum p g 0 and q > 0). This Taylor series has the properties of the Taylor series (3.26) constructed in the proof of Theorem V, if the N are chosen preperly. QP Let (1+.11) i (bncos n9 + cnsin n9) n80 be the real part of the series 0+.12) i enema. Now, by the way the Taylor series (h.9) was defined, the trigonometric series (h.ll) has the desired properties. i Witw- l. 2. 3. 5. BIBLIOGRAPHY Paul ErdSs, Fritz Hersog, and George Piranian, fists sf divergence 2f‘geylgg‘gegieg‘ggg‘gg'tgigonometric .22£$22' Mathematica Scandinavica, vol. 2(195h), pp. 262-266. . 9n f“ 222l2£.22£l22t2£ fun°t1°n3 £252£2£ 1£ 92$2£t£2£$222i T Archiv der Mathematik, vol. V(195h), pp. 39-52. g G. H. Hardy, g theorem concerning Taylor's series, The Quarterly Journal of Pure and Applied Mathematics, vol. hh(1913). DP. 1k7-160. Frits Hersog and George Piranian, Sets 2! convergence 2! 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