UBC Theses and Dissertations

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UBC Theses and Dissertations

Random series of functions and Baire category Babinchuk, Wayne George 1975

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RANDOM SERIES OF FUNCTIONS AND BAIRE CATEGORY by WAYNE GEORGE BABINCHUK B.Sc. (Hons), University of Saskatchewan(Saskatoon), 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1975 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h N C o 1 u m b i a , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s re p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 A b s t r a c t I n much o f t h e w o r k d o n e o n r a n d o m s e r i e s o f f u n c t i o n s , l i t t l e a t t e n t i o n h a s b e e n g i v e n t o t h e c a t e g o r i c a l q u e s t i o n s t h a t may a r i s e . F o r e x a m p l e , a common t e c h n i q u e i s t o l e t e = { £ n ' n _ g be a s e q u e n c e o f i n d e p e n d e n t r a n d o m v a r i a b l e s , e a c h t a k i n g t h e v a l u e s ±1 w i t h p r o b a b i l i t y %, 00 a n d t o c o n s i d e r t h e s e r i e s E e c c o s n t ; t h e n o n e c a n s e e k n n n=0 C O c o n d i t i o n s o n t h e c o e f f i c i e n t s {c } n t h a t a l m o s t s u r e l y n n=u g u a r a n t e e t h a t t h e s e r i e s c o n v e r g e s o r t h a t i t b e l o n g s t o a c e r t a i n f u n c t i o n s p a c e . B u t o n e may a l s o a s k i f t h i s s e r i e s c o n v e r g e s f o r a s e t o f e o f s e c o n d c a t e g o r y o r i f i t b e l o n g s t o a p a r t i c u l a r s p a c e f o r s u c h a s e t o f e . T h i s t h e s i s f o l l o w s t h e f i r s t s e v e n c h a p t e r s o f J . - P . K a h a n e ' s b o o k Some Random S e r i e s o f F u n c t i o n s a n d r a i s e s t h e s e k i n d s o f c a t e g o r i c a l q u e s t i o n s a b o u t t h e t o p i c s p r e s e n t e d t h e r e . - i i i -Table of Contents page Abstract i i Acknowledgment i v 1. A b r i e f survey 1 00 2. The radius of convergence of E c r n z 3 n=0 co 3. Pointwise convergence of the series E e c ncos nt 7 n=0 4. Membership i n 14 5. Membership i n A & 18 Bibliography 29 - iv'' -Acknowledgment I would l i k e to express my g r a t i t u d e t o P r o f e s s o r J . J . F . F o u r n i e r f o r h i s generous help and p a t i e n t guidance. I would a l s o l i k e t o thank P r o f e s s o r J . Coury f o r rea d i n g the manuscript and o f f e r i n g many v a l u a b l e comments. F i n a l l y I would l i k e t o thank the N a t i o n a l Research C o u n c i l of Canada and the U n i v e r s i t y of B r i t i s h Columbia f o r t h e i r f i n a n c i a l support. - 1 -1 . A B R I E F SURVEY We g i v e h e r e a s h o r t d i s c u s s i o n o f t h e m a t e r i a l p r e s e n t e d i n t h i s t h e s i s . oo ^ I n S e c t i o n 2 we c o n s i d e r t h e p o w e r s e r i e s E c r z , n=0 n n CO w h e r e { r n } n _ Q i s a f i x e d s e q u e n c e o f n o n n e g a t i v e n u m b e r s a n d t h e n u m b e r s c n a r e c h o s e n i n d e p e n d e n t l y a t r a n d o m f r o m t h e c o m p l e x p l a n e . We show t h a t t h e s e t o f s e q u e n c e s c = { c n }^_Q c a n b e v i e w e d a s a c o m p l e t e m e t r i c s p a c e ; we t h e n u s e t h i s t o show t h a t t h e r a d i u s o f c o n v e r g e n c e o f t h e s e r i e s i s z e r o f o r a l l b u t a f i r s t c a t e g o r y s e t o f c . I n S e c t i o n 3 , p o i n t w i s e c o n v e r g e n c e o f t h e s e r i e s oo E e c c o s n t i s d i s c u s s e d . We f i r s t p r o v e t h a t i f t h e n=0 n n s e r i e s c o n v e r g e s a t a f i x e d t f o r a s e t o f e o f s e c o n d c a t e g o r y , t h e n i t c o n v e r g e s f o r a l l e . L e t t i n g T d e n o t e 00 *^ t h e s e t o f t f o r w h i c h E e c c o s n t c o n v e r g e s a t e , we n=0 n n t h e n o b t a i n c o n d i t i o n s o n T w h i c h r e s t r i c t c i n v a r i o u s e w a y s . 00 M e m b e r s h i p i n L o f t h e s e r i e s E e n c n e i s n=0 d i s c u s s e d i n S e c t i o n 4 . We show t h a t i f t h e s e r i e s b e l o n g s t o IJP f o r a d s e t o f e o f s e c o n d c a t e g o r y , t h e n i t b e l o n g s t o f o r a l l e . A s a c o n s e q u e n c e o f t h i s r e s u l t , we o b t a i n c o n d i t i o n s o n t h e c o e f f i c i e n t s f o r t h e c a s e s 1 — p — 2 a n d p = °° . - 2 -In the l a s t s e c t i o n we d e f i n e the space A a and C O c o n s i d e r membership of E e„c„e i n A„ . We show t h a t n=0 the s e r i e s belongs to f o r a l l e i f i t belongs f o r a s e t of e of second category. We then look f o r c o n d i t i o n s on the c o e f f i c i e n t s t o guarantee t h a t the s e r i e s belongs t o -"A . The c o n d i t i o n ( . E . |c t\'2y 2 = 0 ( 2 - ^ ) i s suggested by 2 D l n < 2 : + l n Kahane's work [6, p. 69] ; i t i m p l i e s t h a t the s e r i e s belongs to A a f o r a l l e i f 3 > h and a — $-% . - 3 -2 . THE RADIUS OF CONVERGENCE OF Z c n r n z n n=0 In t h i s section we discuss the radius of convergence of the power series 0 0 i: c n r n z n , (1) n=0 0 0 where {r } n i s a fixed sequence of nonnegatxve numbers and n n=0 ^ 3 the c n are chosen independently at random from the complex plane . F i r s t we l i s t some conventions concerning terminology. D e f i n i t i o n 2 . 1 . Let , p) be a p r o b a b i l i t y space, and l e t A € A be an event. A i s said to occur almost surely i f p(A) = 1 . D e f i n i t i o n 2 . 2 . Let S be any topological space. A property i s said to hold for most s € S i f the set where the property does not hold i s of the f i r s t (Baire) category i n S. A property holds for many s i f the set where the property holds i s of the second (Baire) category i n S. Our approach i s to view the sequence c as an . 0 0 element of the topological product = Ti. $n , where = <p n=0 n for each n. Let c = {cn}^_Q and d = {cln}^_Q £e elements of (p . Then (p i s a complete metric space with the metric „ 2 _ n | c . - d • P ( C , a* = z 2 V n = 0 1 + jc - d 1 n n - 4 -c a n a l s o b e v i e w e d a s a p r o b a b i l i t y s p a c e . T o s e e t h i s , l e t & b e a a - f i e l d o n (p , a n d l e t £ b e a p r o b a b i l i t y m e a s u r e o n I t f o l l o w s f r o m [6 , p . 4] t h a t ( n ( p , n JX , n p ) i s a u n i q u e p r o b a b i l i t y s p a c e n=0 n n=0 n n=0 n w i t h t h e f o l l o w i n g p r o p e r t i e s : ( i ) n v/fe i s a a - f i e l d w h i c h c o n t a i n s a l l p r o d u c t s oo n=0 II A. • o b t a i n e d b y t a k i n g A 6 Jr\ , w i t h A = C f o r a l l n =0 n n n n n b u t a f i n i t e n u m b e r o f v a l u e s o f n ; oo ( i i ) n p i s a p r o b a b i l i t y m e a s u r e o n CO co I I — 0 CO CO CO ( n <b , n J ^ C ) s u c h t h a t ( n p )( n A ) = n p ( A ) . n=0 n n=0 n n = 0 ~ n n=0 n n = 0 " n n T h e r a d i u s o f c o n v e r g e n c e o f (1) i s t h e e x t e n d e d r e a l n u m b e r r ( c ) = ( l i m - s u p r c ) n + « . n n T h i s i s a m e a s u r a b l e f u n c t i o n o f c , a n d r ( c ) d e p e n d s o n l y o n t h e t a i l o f t h e s e q u e n c e c . T h u s b y t h e z e r o - o n e l a w \jo, p . 6 ] , t h e r e i s a c o n s t a n t R s u c h t h a t r ( c ) = R a l m o s t s u r e l y . I n t h e f o l l o w i n g t h e o r e m we show t h a t r ( c ) = 0 f o r m o s t c . T h e p r o o f i s s i m i l a r t o t h e o n e u s e d b y J . J . F . F o u r n i e r a n d P . M . G a u t h i e r i n [ S ] t o o b t a i n t h e same r e s u l t f o r t h e 00 n s e r i e s E c z n=0 n - 5 -Theorem 2.3. I f r n ^ 0 f o r i n f i n i t e l y many n, then r ( c ) = 0 f o r most c . Proof. L e t N be a p o s i t i v e i n t e g e r , and suppose t h a t 00 00 E c r z converges f o r r > — . Then £ c^r^N n converges n=0 n n n=0 n n and hence, f o r some i n t e g e r M , we have | c n r n | — MNn f o r a l l n. L e t S = { c e<fc°> : lc r n I 1 M n f o r a l l n } . M,N Y 1 n ni Then S M N i s c l o s e d and i t s complement i s dense i n ^ . To see t h i s , suppose t h a t c ( S„ , and e > 0 are gi v e n . M,N Choose m so t h a t 2 - I t l < e and r ¥ 0 . Define d £ (t" by m c i f n ^ m , d n n m 2MNmr ~ 1 i f n = m c - d n • m m , , Since < 1, wer,have 1 + l c - d I 1 m m1 2 _ n | c - d | p(c, d) = ? L_n n l 2 c - d . --nt m 1 + lc - d | 1 m m1 ~-m < 2 < e ? and d ^ S M,N - 6 -T h e r e f o r e \J S i s of the f i r s t c ategory, and M,N=1 M ' N C O hence {c : r ( c ) > 0} C (J S i s a l s o a f i r s t category M,N=1 M ' N s e t . Thus r ( c ) = 0 f o r most c . Remark. F o u r n i e r and G a u t h i e r [5] g i v e a d i s c u s s i o n of the r a d i u s of convergence from the p r o b a b i l i t y viewpoint. For 00 the s e r i e s Z c z n , i t i s shown t h a t almost s u r e l y the r a d i u s n=0 of convergence i s 1. They note t h a t the r e s u l t depends on the measure used, but the c o n c l u s i o n h o l d s f o r many reasonable p r o b a b i l i t y d i s t r i b u t i o n s on ({! . For example, the r e s u l t i s t. t r u e p r o v i d e d t h a t both of the f o l l o w i n g c o n d i t i o n s h o l d : (a) the p o i n t zero does not have mass 1, 00 k (b) £ prob ( |z| > n ) < °° f o r some k. n=0 By an argument u s i n g d i s t r i b u t i o n f u n c t i o n s , c o n d i t i o n (b) i s e q u i v a l e n t to the r a t h e r weak r e s t r i c t i o n t h a t the ^ - t h moment be f i n i t e . A s i m i l a r r e s u l t holds f o r our s e r i e s (1) under 1 /n — 1 c o n d i t i o n s (a) and (b) . I f R = ( Iim sup r ) , n-v°° n then i t can be shown t h a t the r a d i u s of convergence of (1) i s almost s u r e l y R . Thus, i f R ^ 0 , the s e t {c : r ( c ) = 0} has measure zero i n (J:" but c o n t a i n s most c . - 7 -3. POINTWISE CONVERGENCE OF THE SERIES E e c cos nt _ n n n=0 R e s u l t s have been obtained f o r the p o i n t w i s e convergenceeof many types of t r i g o n o m e t r i c s e r i e s ( see f o r example [io] ) . Much of t h i s work i n v o l v e s f i n d i n g c o n d i t i o n s on the c o e f f i c i e n t s t o guarantee convergence of the s e r i e s under c o n s i d e r a t i o n . Here, we assume p o i n t w i s e convergence of co the s e r i e s E e n c n c o s nt , where e n £ {-1, 1} i s chosen a t n=0 random f o r each n, and we make v a r i o u s c o n c l u s i o n s on the c o e f f i c i e n t s c under c e r t a i n c a t e g o r i c a l assumptions on t and E . Most of the r e s u l t s o b t a i n e d a l s o apply to the s e r i e s 00 00 . i n t E e c s i n nt and E e c e ; we comment on these cases n=l n n n=0» n n a f t e r each t o p i c . D e f i n i t i o n 3.1. A f u n c t i o n f ( x ) i s s a i d t o be lower semicontinuous i f and o n l y i f i t i s e x p r e s s i b l e as f( x ) = sup h(x) , where H isssome s e t of continuous f u n c t i o n s , h(x)(H r CO Theorem 3.2. F i x t 6 To, 2-rrl . I f E e c cos nt converges n=0 n n f o r a s e t of e of second category, then i t converges f o r a l l e. Proof. Consider f.(e) = sup | E e c cos nt | . The s e r i e s t k n=0 n - 8 E e^c^cos nt i s a continuous function of e because the _ n n n=0 . :• projection p : { - 1 , 1 } W -* { - 1 , 1 } y i e l d s continuity of each term i n the sum. Therefore f t ( £ ) ^ s lower semicon-tinuous. By [ 4 , v o l . I , p. 1 7 9 ] , i f f i s lower semi-continuous and i f f t ( £ ) < 0 0 f ° r each e i n a nonmeager subset S of { - 1 , 1 } " , then f ^ i s bounded on an open set 00 U of { - 1 , l } " . .Since, by assumption, E e c cos nt n=0 converges for a set of e of second category, we have that 00 f. (e) = E e n c n c o s nt i s bounded on U , that i s , n=0 00 E e c cos nt converges on U . „ n n J n=0 0) The basic open sets i n { - 1 , 1 } are those which leave only a f i n i t e number of coordinates unspecified. Since changing only a f i n i t e number of the e does not a f f e c t the 00 00 convergence of E e n c n c o s nt , i t follows that E E n c n c o s nt n=0 n=0 converges for a l l e i n { - 1 , 1 } W . This proof can be extended to the secies 00 CO Z e n c n s i n n t a n d E e n c n e i n t i n the obvious way. n=0 n=o We are now ready to discuss the main r e s u l t s . For each CO l e t T £ = {t £ [ 0 , 2 IT] : E e c cos nt converges ' P y j y o f ; v n=0 * - 9 -and l e t S always denote a subset of {'-1, 1} W of the second c a t e g o r y . We c o n s i d e r the f o l l o w i n g cases : (1) There e x i s t s a s e t U CZ [0, 2TT] of p o s i t i v e measure such t h a t U C T f o r a l l e i n S ; e (2) T^ has p o s i t i v e measure f o r each e i n S ; (3) There e x i s t s a s e t U C Z [0, 2TTJ of second category such t h a t U CZ T f o r a l l e i n S ; e (4) T^ i s of the second category f o r each e i n S. In each case, we are i n t e r e s t e d i n what c o n c l u s i o n s can be drawn about the c o e f f i c i e n t s c . Cases (1), (2), and (3) are d i s c u s s e d i n the f o l l o w i n g theorem;, case (4) i s c o n s i d e r e d i n Theorem 3 . 6 . Theorem 3.3. L e t (1), (2), and (3) be as above. CO i i ( I ) (1) holds i f and on l y i f Z . c < °=> ; n= o n ( i i ) (2) holds i f and o n l y i f E I c I 2 < °° ; n=0 n ' ( i i i ) (3) holds i f and o n l y i f E | c | < °° . n=0 oo Proof. ( i ) F i r s t we note t h a t E e c cos nt converges i f n=0 n n 00 00 and o n l y i f both E e R e ( c n ) c o s n t and E e Im(c )cos nt n=0 n=0 n converge. A. Zygmund ( see [lO, v o l . I, p. 232] ) proves CO t h a t i f haQ + Ei;( a cos nt + b s i n nt ) converges n=l n «= a b s o l u t e l y i n a s e t of p o s i t i v e measure, then E ( |a | + |b | ) n=l - 10 -converges. Choose t * € U. The s e r i e s E e nRe(c )cos n t * n=0 converges f o r every e i n S and hence, by Theorem 3.2 , CO f o r every e . Choose e* so t h a t E e*Re(c )cos n t * = n n n co n=0 E |;e*Re(c ) cos n t * | . S i m i l a r l y , f o r every t 1 i U we n=0 n oo oo choose e 1 such t h a t E e'Re(c )cos n t ' = E |e'Re(c n)cos n t ' | n=0 n=0 C O By Zygmund' s r e s u l t , we have E. |Re(c ) | < °°. One shows i n a n=0 CO CO s i m i l a r way t h a t E |lm(c ) | < °°; hence E | c | — n=0 n n=0 n CO CO E | Re (c ) | + E |lm(c ) | < 0 0 . n=0 n n=0 n The converse r e s u l t i s t r i v i a l . 00 ( i i ) F i r s t we show t h a t (2) i m p l i e s E |c | 2 < °°. n=0 n The proof i s by c o n t r a d i c t i o n and uses the r e s u l t t h a t i f the s e r i e s % a Q 2 + z ( a n 2 + b 2) d i v e r g e s , then whatever the n=l summation method T*, almost a l l the s e r i e s ± a + ' o co E ±(a ncos nt + b R s i n nt) are almost everywhere nonsummable n=l T* ( [10, v o l . I, p. 214] ) . CO CO Conversely, i f £ |c I2 < °° then E e c cos nt 1 n 1 - n n n=0 n n=0 belongs to L 2 . By [ l ] , the s e r i e s converges almost everywhere w i t h r e s p e c t t o t . / ( i i i ) Zygmund [10, v o l . I, p. 233] shows t h a t the theorem s t a t e d i n the proof of (i) i s v a l i d i s we r e p l a c e "converges a b s o l u t e l y i n a s e t of p o s i t i v e measure" by "converges a b s o l u t e l y i n a s e t of second category" . T h e r e f o r e the c o n c l u s i o n f o l l o w s . - 11 -B e f o r e p r o c e e d i n g w i t h c a s e (4) , we make t h e f o l l o w i n g d e f i n i t i o n s . D e f i n i t i o n 3 . 4 . T h e s y m m e t r i c d i f f e r e n c e o f t h e s e t s A a n d B i s d e f i n e d b y A A B = (A(JB) - ( AR B ) = (A - B) \J (B - A) . D e f i n i t i o n 3.5. A s u b s e t A o f a n y t o p o l o g i c a l s p a c e i s s a i d t o h a v e t h e p r o p e r t y o f B a i r e i f i t c a n b e r e p r e s e n t e d i n t h e f o r m A = G4Af>P, w h e r e G i s o p e n a n d P i s o f f i r s t c a t e g o r y . I t f o l l o w s f r o m [ 7 , p . 19] t h a t t h e c l a s s o f s e t s h a v i n g t h e p r o p e r t y o f B a i r e i s t h e a - a l g e b r a g e n e r a t e d b y t h e o p e n s e t s t o g e t h e r w i t h t h e s e t s o f f i r s t c a t e g o r y . 0 0 T h e o r e m 3 . 6 . C a s e (4) h o l d s i f a n d o n l y i f E | c | < °° . n=0 P r o o f . , D e f i n e t h e p r o d u c t s p a c e { - 1 , I } " .* . [ 0 , 2TT] ='{ ( e , t ) : e e { - l , l } a \ t ( [ 0 , 2 T T ] } : F o r e a c h e i n S , d e f i n e E „ = { ( e , t ) : t £ T } , a n d s e t e e E = I J E . We show t h a t E h a s t h e p r o p e r t y o f B a i r e . eeS e k L e t g ( e , t ) = i n f { s u p | E e c n c o s n t | } ; t h e n N-*-°° Nlm<k <]<n=m 0 0 E e n c n c o s n t c o n v e r g e s i f a n d o n l y i f g ( e , t . ) = 0 . T h u s E n=0 i s t h e s e t w h e r e g ( e , t ) = 0 . k S i n c e EO encn.GOS/,,'nt i s a c o n t i n u o u s f u n c t i o n o f n=m e a n d t , we h a v e f r o m [ 9 , p . 14] t h a t g i s B o r e l m e a s u r a b l e . H e n c e E = g ''"(O) i s a B o r e l s e t a n d t h e r e f o r e - 12 -has the property of Baire. Let X and Y be any two topological spaces, and l e t Y have a countable base. Suppose F d X x Y has the property of Baire. .Then F i s of f i r s t category i f and only i f F = {y^Y : (x,y) £ F} i s of the f i r s t category for a l l x except a set of f i r s t category (see [7, pp. 56-57] ) . I t follows that E i s of second category, since E^ = T^ i s of second category for each E i n S . Since both { -1, 1 }w and [0, 2TT] have countable bases, [7, pp. 56-57] again implies that E^ must be of second category for a set H of t of second category; by Theorem 3.2 , = {-1, 1}W for each such t. We now apply Theorem 3.3 ( i i i ) , with . U = H and 00 S - {-1, 1 }w , to conclude that E |c I < °° ~ 1 n i T r i v i a l l y , the converse i s true. The above arguments for cases (1) to (4) apply as CO 00 . _^ well to the series E e c sin nt . For E e^c e , the -n=l n n n=C> n CO conclusion i n cases (1) , (3) and (4) i s that E | c | < °° n=0 n Indeed, i n each of these cases, i t follows as above that there CO exists t* such that E e c e i n t converges for a l l e . ~ n n ^ n=0 CO r , i n t * M Then E fRe(e c e )•'• converges for a l l e , and, choosing n=0 n n CO i i n t * i e appropriately, we have that E [Re (e c e )| < 0 0 n=0 - 13 -i n t * S i m i l a r l y , X | Im ( e c e )| < » . Hence L \ c | = n=0 n n n=0 , i n t * , * l ^ n 6 I < 00 • n=0 For case (2), we may use the proof o u t l i n e d i n Theorem 3.3 ( i i ) , s i n c e Zygmund 1s and Ca r l e s o n ' s r e s u l t s oo . 1 -, -, J T L N T a l s o h o l d f o r Z e c e n n n n=0 4 . MEMBERSHIP I N L P I n t h i s s e c t i o n we c o n s i d e r m e m b e r s h i p i n Ir f o r t h e same t y p e s o f s e r i e s a s i n S e c t i o n 3 . F o r s i m p l i c i t y , i n t we w o r k w i t h t h e s e r i e s E e c e . S i m i l a r r e s u l t s h o l d n n n n=0 f o r t h e o t h e r s e r i e s . 3 t We a d o p t t h e f o l l o w i n g n o t a t i o n . L e t x ( t ) = e ; CO b y w r i t i n g f = E e c v we mean t h a t f 6 L 1 (-TT , TT) , t h a t n=0 t h e F o u r i e r c o e f f i c i e n t s f ( n ) o f f a r e 0 f o r n < 0 , a n d t h a t f ( n ) = e n c n f o r n i 0 . A s i n [4 , v o l . I I , p p . 5 0 - 5 1 ] , we l e t M d e n o t e t h e s e t o f a l l b o u n d e d (Radon) m e a s u r e s o n t h e u n i t c i r c l e . O u r f i r s t r e s u l t i s s i m i l a r t o T h e o r e m 3 . 2 . °° n P T h e o r e m 4 . 1 . ( i ) I f 1 .< v — m , t h e n £ e c Y € L f o r n=0 p many e i f a n d o n l y i f i t b e l o n g s t o L f o r a l l e . CO ( i i ) E e c x 6 M f o r many e i f a n d o n l y i f n=0 n n i t b e l o n g s t o M f o r a l l e P r o o f . T h e p r o o f o f ( i ) i s s i m i l a r t o t h a t o f T h e o r e m 3 . 2 we d e f i n e a l o w e r s e m i c o n t i n u o u s f u n c t i o n a n d t h e n u s e t h i s f u n c t i o n t o o b t a i n t h e c o n c l u s i o n . - 15 -I f E e c X*1 i L P , d e f i n e f £ = Z e ^ x " . L e t n=0 n=0 G denote the s e t of t r i g o n o m e t r i c polynomials,, g ( t ) ; w i t h | | g | | , 1 1 , where - + = 1 . In view of [4, v o l . I , p . l 9 0 ] P P P and the f a c t t h a t the t r i g o n o m e t r i c polynomials are dense i n the u n i t b a l l , we.have | | f |'| = sup | / f ( t ) g ( - t ) dt | . e P G —TT N i m t L e t g(t) = Z d m e ; then m=-N TT N / f ( t ) g ( - t ) d t = 2TT Z e n c n d n . - IT n= 0 T h i s l a t t e r sum i s d e f i n e d f o r a l l e and i s a continuous f u n c t i o n of e f o r f i x e d g . N L e t <f>(e) = sup | 2TT E e n c g ( n ) | ; c l e a r l y , G n=0 (|>(e) < 0 0 i f e c x € L P . Conversely, suppose t h a t n=0 N j_mt (j)(e) < 0 0 . For a t r i g o n o m e t r i c polynomial g(t) = E d e , m=-N m d e f i n e N L (g) = 2TT E e c d n=0 n n n Then we have |L (g)| — <j)(e) | |g44 i > a n d hence L extends P £ P to a bounded l i n e a r f u n c t i o n a l , of norm <|>(e) i o n L (TT , —T T ) . T h e r e f o r e , there e x i s t s a f u n c t i o n f i n iP such t h a t e IT L £ ( g ) = / f £ ( t ) g ( - t ) d t - T T - 16 -f o r a l l t r i g o n o m e t r i c p o l y n o m i a l s g . T a k i n g g = x* 1 / w e see t h a t f (n) = 0 i f n < 0 and f (n) = e c i f n > 0 , e e n n — C O so t h a t E e c x n € L p . n=0 n n Now <J>(E) i s lower s e m i c o n t i n u o u s s i n c e i t i s a supremum o v e r c o n t i n u o u s f u n c t i o n s o f e . Hence, as i n Theorem 3.2 , <f>(e) i s bounded on an open s e t UCZ {-1, 1 } W , C O w h i c h i m p l i e s t h a t cj>(e) < » f o r a l l e . Thus Z e c x € n=0 n L p f o r a l l e , as r e q u i r e d . P a r t ( i i ) i s s i m i l a r l y p r o v e d by u s i n g t h e d u a l i t y between M and C , t h e s e t o f a l l c o n t i n u o u s , complex-v a l u e d f u n c t i o n s on [0,2TT] s(:s:ee [ 4 , v o l . I I , p. 59] ) . T h i s theorem g i v e s us t h e f o l l o w i n g r e s u l t . Theorem 4.2. ( i ) i f 1 <_ p <_ 2 , t h e n a n e c e s s a r y and C O s u f f i c i e n t c o n d i t i o n f o r Z e c x t o b e l o n q t o iP n n n J n=0 co ( r e s p e c t i v e l y , t o M) f o r many e i s t h a t Z | c | 2 < °° . n=0 n ( i i ) I f p = co , t h e n Z e n c x £ L f o r many n=0 n C O e i f and o n l y i f Z | c [ < °° . - 17 -Proof. Suppose that 1 <_ p ± 2 . Clea r l y , i f oo oo p 0 0 n E lc I 2 < », then Z e c x n E L ( E s c x £ M ) for n=o' n n=0 n n n=0 n n CO a l l e . Conversely, i f £ e c v" £ L P for many e , then n=0 n E e c x € M for many E , and hence, by Theorem 4.1 ( i i ) , n=0 n n co Z E c x n € M for a l l E . By [4, v o l . I I , p. 204] , i f n=0 n n °° i n t the series Z £ c e i s a F o u r i e r - S t i e l t j e s series for a l l n n n=-<=° oo E , then Z | c | 2 < °° . This proves (i) . n = - c o For p = °° , i t follows from [4,vol. I I , p. 204] 00 0 0 - J_ that Z | c | < °° i f the series Z E „ C e i s a Fourier n=0 n n=0 n n series of a function i n L for every E . Thus the condition CO oo n °° I I that E E c x € L for many £ implies that E c I < <=° '.' n=0 n n : n=0 n The converse i s cl e a r . We remark that for the case 2 < p < <» , there seems to be no simple necessary and s u f f i c i e n t condition for 00 n P E E c x to belong to(")L for a l l e . In the next section we n=0 n n give some p a r t i a l r e s u l t s for t h i s case. - 18 -5. MEMBERSHIP IN A a Before proceeding with our discussion, we give some d e f i n i t i o n s . D e f i n i t i o n 5.1. Given a continuous function f on the c i r c l e , i t s modulus of continuity i s defined as u f (h) = sup |f (t) - f ( f ) | ,t-t'l<h' considered as a function of h > 0 / ot D e f i n i t i o n 5.2. We define A„ = { f : w(h) = 0(h ) } . ; a f This i s a Banach space under the norm | | f | | = | | f | | ^ + sup ( OJ (h)h a ) . . h f • De f i n i t i o n 5.3. For 0 < p < °° , H P i s the l i n e a r space of functions f(z) analytic i n |z| < 1 , such that 2 7 7 , i e ,P 1/P "I i X B f . . M (r,f) = { ± X f ( r e ) de } P 2ir D remains bounded as r -»• 1 . P For 0- t p < 1 , H i s not a Banach space but i t i s a complete metric space with the tr a n s l a t i o n - i n v a r i a n t metric d(f,g) = ( lint M (r,f-g) ) P :1 (see [ 3 , p. 32] ) . For r"*"l P convenience, we v/rite | |f | | for d(f,g) . H P ^t;r>; C O In t h i s section we f i r s t show that i f E e c x € A n = 0 n n a - 19 -f o r many e , then i t belongs to A a f o r a l l e . T h i s uses the d u a l i t y between H P and . We then g i v e c o n d i t i o n s t h a t guarantee t h a t the s e r i e s belongs to A . Again, the r e s u l t s 00 OO a l s o h o l d f o r the s e r i e s E e n c n c o s nt and E e n c n s i n n t . n=0 n=l CO Theorem 5.4. I f E e n c n x n ^ A a f o r m a n Y e ' t n e n i t belongs n=0 to A a f o r a l l e . P r o o i . The proof i s s i m i l a r to t h a t of Theorem 4.1 . We co express the A a-norm of E e n c n x i n terms of the m e t r i c on n=0 i t s p r e d u a l H P and show t h a t we have a lower semicontinuous f u n c t i o n . L e t A = { f € C[0,2T T] : f (n) = 0 f o r n < 0 }, and l e t G denote the set of t r i g o n o m e t r i c polynomials;-, g(t) ;.-w i t h | |g| | <_ 1 . By [3, p. 39] , A 0 A i s e q u i v a l e n t H P a to ( H p ) * , where a = i . - 1 and h < p < 1 . Thus i f f 6 A a Q A , I |fI I i s e s s e n t i a l l y equal to sup \2:f f ( t ) g ( - t ) d t | . G 0 s In p a r t i c u l a r , a f u n c t i o n f i n A belongs t o A a i f and on l y i f t h i s supremum i s f i n i t e . CO Suppose t h a t E e n c n x € A f o r many e . By n=0 00 CO I N Theorem 4.2 ( i i ) , we have E jc | < °°, and f = E e c x n=0 n e n=0 belongs to A f o r a l l e . Using the proof of Theorem 4.1 - 20 -and the f a c t t h a t the polynomials are dense i n H P ( [ 2 , p . 1 1 2 ] ) we have 2TT 2 TT sup I / f ( t ) g ( - t ) dt | = sup | / f ( t ) g ( - t ) dt | g| | <1 0 E G 0 e H p N sup I 2TT E e n c n g ( n ) G n=0 N Define <J)(e) = sup | 2TT E e c g (n) | . C l e a r l y <j>(e) < 0 0 G n=0 n n n i f E e n c n x € A • Conversely, we can show (as i n the proof n=0 CO of Theorem 4 . 1 (i) ) t h a t i f r <J>(E) < 0 0 , then E e c y 6 A . . O n n cx _ Since <)> i s lower semicontinuous, we conclude, as i n Theorem 4 . 1 (i) , t h a t <j>(e) < 0 0 f o r a l l e and hence t h a t CO E e n C n x n ^ A a f o r a 1 1 e • n=0 We now i n v e s t i g a t e c o n d i t i o n s under which 00 f = E e c v n ^ A f o r a l l e . Towards t h i s end we make the n=0 n n " « f o l l o w i n g d e f i n i t i o n . D e f i n i t i o n 5 . 4 . I f j = - 1 , l e t J = {0} . I f j > 0 , l e t J = { n : 2^ < n < 2 D + } . I f f = E e c x " oaf; i f — n n n A n=0 c =• . { c N } ™ _ Q , we l e t s_. (f) and s_. (c) denote ( E k j - 2 ) 2 , . f o r j = - 1 , 0 , 1 , . . . B Y [6, p. 69] , a necessary c o n d i t i o n f o r f to belong to A i s t h a t s . ( f ) = 0 ( 2 a-1) . The f o l l o w i n g a J - 21 -t h e o r e m shows t h a t a s u f f i c i e n t c o n d i t i o n f o r m e m b e r s h i p i n A i s t h a t s . ( f ) = 0 ( 2 ~ ( a + 3 s ) j ) . a j Theorem 5.5. I f s • ( f ) = 0 ( 2 J ) , t h e n t h e f o l l o w i n g a r e t r u e : P , ( i ) f 6 L f o r e v e r y e i f 0c.< 8 <_ h a n d L > h - 3; P ( i i ) f € A f o r e v e r y e i f 3 > h a n d a < 3 - h . a — P r o o f . ( i ) F i x p s o t h a t 2 -op < (% - p) 1 , and l e t i . = 1 - — . We s h a l l u s e H o l d e r ' s i n e q u a l i t y t o show t h a t p 1 P E | c n | < °° /-and t h e n u s e t h e H a u s d o r f f - Young t h e o r e m ~, n=0 ( s e e [ 2 , p. 9 4 ] ) tJ t o c o n c l u d e t h a t f £ L P . I n d e e d , we h a v e J ( K 2 - 3 J ; j P ' ( 2^ >*<2-p') . * l c n l < J J  T h e r e f o r e , i °° 1 E | c n | P =. E ( E |c n|P ) n-]> j = 0 l J < E K P Y 3 j P ' + j " ^ P ' ~ 3 = 01 = K P ' E 2 j ( 1 " " 3 p , ) w h i c h c o n v e r g e s f o r 1 - % p 1 - 3p' < 0 , t h a t i s , f o r -1 p i p' > ( h + 3 ) . H e n c e , f b e l o n g s t o L f o r ± > % - 3 CO P s i n c e | | f |' | . * ' <_". •( E \ c~- | P \ i ) : 1 / / P ' f o r 2 £ p <_ «= . P n=0 ( i i ) We show t h a t | f ( t ) - f ( t - h ) I = 0 ( h a ) . F i r s t - 22 -o b s e r v e t h a t • . i * i I , i n t i n ( t - h ) . , f ( t ) - f ( t - h ) | < E | e n c ( e - e ) | n=0 n " i . I i n t i n ( t - h ) , < Z | c J | e - e | n=0 , , , i n t i n ( t - h ) . . < £ , 2 | c n ! | e - e ! + E 2 | c n ! n < T T T T n > " C h o o s e m s u c h h t h a t 2 m < 2 < 2 m + 1 ; t h e n " M E 2 i'Cnl K 2 %J°nl n> 2 n>21Ll o < 2 E :(; E | c n | ) j=m J < 2 £ ( E | c n | 2 ) 2 ( E 1 2 )* j=m J j CO < 2 E K 2 _ e j 2 3 s j j=m = 2K E 2 j ( 3 5 _ e ) j=m * 2K 2 m ( h ~ & ) , since 3 > ^ . N o w 2 m < ^ - < 2 m + 1 implies that 2 " ( m + 1 ) <JiLi<2"m , |h| 2 that i s , ( 2 - < m + 1 V - % < m\^r^)k\rm)^ . ^22 T h e r e f o r e w e h a v e E 2 c < 2K 2 n 1 — n> 2 ! - 2 ^ h - 23 -a l s o , 2 K 2 ( m + l ) ( ^ - B ) 1 _ 2 ( ^ - 6 ) XU>- 2^ 3 ) E |cn||eint-ein(t-hM E |cn||eint-ein(t-h,)' n<-|4r . i ' n < 2 i a + 1 < E (E Ic I l e i n t _ e i n ( t - h ) , " j<m+l J " 1 ' : E (E | c n | | h | |max d__ i n t v |) j<m+l j j d t < E | h | E |cn|2^+1 j<m+l It. j £ i E 2 ^ + 1 | h | K 2 - ^ 2 J 5 J j <m+l <_ 2 I h I K E 2^ ~ &> j <m+l < 2 | h | K 2^0 ' g><m+1> < 2 . | h | K 2 ^ " B ( 2 | h r 1 ) ^ r g 2a;/5»-3 _ x = 2 K.| -h I P 2 ^ - 3 _ x T h e r e f o r e we c o n c l u d e t h a t | f ( t ) - f ( t - h ) I < / 2 H K + 2K \ | h | ' V 2 1 / 5 ' - 3 - l C l ' ' . -t h a t i s , f £ A f o r e v e r y a < 3 - h • - 24 -We show next that the preceding r e s u l t s are best possible. Lemma 5.6. Define Z^  = { c={c n}~ = 0 : S j (c) = 0(2~^®) } . Then | | | | , defined by | |c| | = sup 2®Js . (c) for c € Z , Z j 3 i s a norm on Z^  , and Z i s a Banach space under t h i s norm. Proof. F i r s t we show that || || i s a norm. Suppose 00 CO . 0 0 that c={c n} n_ 0 , d = { d n } n = o a n d a C =^ a Cn^n=0 belong to . (i) Since S (G+d) = ( E f c + d j 2 ) ^ < ( Z|C_|'2 )h + ( E j d P ) k , j j n n i - j 1 m j.' ri we have that ||c+d|| = sup (2 B js.(c+d)) Z J 3 < sup ( 2 6 j S j ( c ) ) + sup (2 B js.(d)) j j 3 - 11 ° 1 I a- -*• I I I I« • ( i i ) l l a c I L = sup (2 B Ds.(ac)) Z j 3 = sup 2 3 j ( E|(ac | 2 ) H j J = M l l c M z • ( i i i ) ||c|| z-> o i f c f 0 Thus [| || i s a norm. Z We now show that Z Q under t h i s norm i s comolete. - 25 -(k) °° (k) °° Let {c }, . be a Cauchy sequence i n Z . Then {c k=l 3 N K — 1 i s a l s o Cauchy f o r each n and converges t o some number, say c n , CO We c l a i m t h a t c = { c n ) n = 0 £ Z^ . To see t h i s , f i x j ; then (k) s•(c) = l i m s . ( c ) . Hence we have 3 k->°° 3 2 3 j s . ( c ) = l i m 2 B J s , ( c ( k ) ) < l i m sup | | c ( k ) | | z • -J k-*00 k->-°° Since t h i s l a s t q u a n t i t y i s independent of j , i t f o l l o w s t h a t c € Z 6 . (k) We show next t h a t c converges i n Z to c . P (k) 00 Since the sequence {c ^ = 1 1 S Cauchy, gi v e n e there e x i s t s M such t h a t | | c ( k ) - c ^ | | < e f o r k,mx >. M . z Thus 2®J ( Z |£'cnk) - c ^ l ) }•)•)** < e f o r a l l j when k,m > M J T h i s i m p l i e s t h a t im 2 3 j ( E |c( k) - c f )h = 2 B j ( E ^ k ) - c j ? 2 ) 1 m->-°° J < e f o r a l l j,k > M . Thus we have z = sup-2< 3 j( E | c ( k ) - c n i h 2 < (k) ~ c - v — ' " - n f o r k > M , p r o v i n g our c l a i m . Hence Z i s complete and i s t h e r e f o r e a Banach 3 space. - 26 -Theorem 5.7. (i) I f 0 < 3 <_ h , there e x i s t s f such t h a t Sj (f) = 0(2" 3) but such t h a t f Jc: L P f o r p <_ h - 3 . ( i i ) I f 0 < 3 < 1 , and a> g-3!/ then there e x i s t s f w i t h s . ( f ) = 0(2~&3) and such t h a t f 4. A . -Proof. (i) F i x 3 w i t h 0 < 3 <_ h , and s e t y = % + 3 ; C O , thus y ± 1 • Let f ( t ) = £ n Y e . W e show t h a t n=l Sj (f) = 0 ( 2 - 3 ^ ) f o r y 1 1 but t h a t f ^ L P f o r p _ 1 <_ h - 3 We have s . 2 ( f ) = Z(n~V „ 2 J ( 2 J ) " 2 Y = 2 J ( 1 _ 2 Y ) ; ^ J t h e r e f o r e s . ( f ) * 2 j ( 3 s " Y ) . Hence s. (f) = 0 ( 2 ~ B : 1 ) . , • • v , • : !, Y < -^ P - 1 Now we show t h a t f £ L i f p <_ % - 3 . We; have V 9 TT oo . ci co - * / | E n e | r d t > / | J n s i n nt| dt , 0 n=l 0 n=l where 0 < s << 1 . I t f o l l o w s from [10, v o l . I, p. 186] °° - -1 t h a t Z- n Y s i n nt 'v. t Y as t ->• 0 + f o r 0 <Y< 2 . Hence n=l S i °° "Y • • iP S ( Y - l ) p / | Z n ' s i n nt[ dt W t ^ dt i s f i n i t e i f and onl y i f 0 n=l 0 (Y - l ) p > -1 , t h a t i s , i f % - 3 < p 1 . Thus we conclude 2 i r , " -y i n t . p n -1 t h a t / | E n e | t d t = ° ° f o r h - 3 1 p 0 n=l - 27 -( i i ) The proof i s by c o n t r a d i c t i o n . F i x 3 and 00 suppose t h a t , f o r some a > & - h , E e c x €.A f o r a l l n=0 n n 0 1 c € Z and a l l e . We f i r s t show t h a t the i n j e c t i o n map <f> p from Z ' i n t o A i s a continuous l i n e a r o p e r a t o r . p a 0 0 n Define <J> : Z -> A by <j> (c) = E c x f o r c £ Z 3 a n=0 n P C l e a r l y $ i s a l i n e a r o p e r a t o r . To prove c o n t i n u i t y , we show t h a t c)> i s c l o s e d and then apply the c l o s e d graph theorem / T o c n - i > T . -i • (m) , (m) , (m) oo (|_8, p. 5 0 J ) . L e t c = l i m c x , where c = {c } t m->co n _ co 00 • . Z f o r every m. L e t f = E c n X n = <|> (c) , f m = E c^1 x n = n=0 n=0 ^ ( c ^ 1 1 1 ^ ) , and suppose t h a t f m -> g i n ; then f m -»• g i n L . Hence we have g(n) = l i m f ( n ) = l i m c ^ = c f o r a l l m n n nH-°° ITB-«> n, and t h e r e f o r e f = g . Thus the map cf> i s c l o s e d . By the c l o s e d graph theorem, <j> i s continuous. We w i l l c o n t r a d i c t the c o n t i n u i t y of <j> by showing ( m) (m) -1 t h a t | | <|> (c ) | | ( | | c | | ) -> co as m -* °° f o r some CX Ll sequence {c ( m )}°° n i n Z . Define c ( m ) = ' { c ( m ) } " n by m=0 3 n n=0 J c . n ( T '• V " ' 0m+l (m) • 1 * f 1 2 < n < 2 (. 0 otherwise C l e a r l y c ( m ) € Z f o r every m s i n c e s ( c ( m ) ) = 22™ , 3 m s m + 1 ( c ( m ) ) = 1 , and S j ( c ( m ) ) = 0 i f j ± m, m+1 . Let f m = <j>(c ( m )) ; then - 28 -2 m + l f - Z X m m n=2 ~ m , ~ m - l 2 = x 2 + 2 < * x n ) ~ „ m - l n=-2 3N = * D N ' m-1 w h e r e N = 2 a n d D„ i s t h e D i r i c h l e t k e r n e l o f o r d e r N N R e c a l l t h a t D ( t ) = ( s i n ( N + % ) t ) ( s i n tot) ) - 1 ( s e e [4 , v o l . I , p . 79] ) , a n d s e t h = -n (N+h)'1 ; f (h) - f (0 )| = |0 - (2N+1)I = 2N+1 . m m 1 I t f o l l o w s t h a t t h e n m a n d h e n c e a . . / u \ ^ / i _ \ — O / i - v i • -i \ l + 0i h a ) £ (h) > (2TT) a ( 2 N + l ) fmMa 1 ( 2 T : ) - a ( 2 N + l ) 1 + a > ( 2 t ) - a 2 m ( 1 + a ) B u t | | c ( m ) | | z < 2 m ( J 2 + 3 ) i m p l i e s t h a t (Mfmlla )( Mc(m)||z ) _ 1 >_ ( 2 , ) - a 2 n i ( a - ^ ) . S i n c e a > 3-J5 , t h e r i g h t s i d e o f t h e l a s t i n e q u a l i t y t e n d s t o i n f i n i t y a s m -> °° , c o n t r a d i c t i n g t h e c o n t i n u i t y o f <|> . T h i s . . c o m p l e t e s t h e p r o o f o f t h e t h e o r e m - . - . 29 -[1 [2 [3 [4 [5 [6 0 [8 [9. B i b l i o g r a p h y L. C a r l e s o n , On convergence and growth of p a r t i a l sums of F o u r i e r s e r i e s , A c t a Math. 116 ( 1 9 6 6 ) , 1 3 5 - 1 5 7 . P.L. Duren, Theory of spaces, Academic P r e s s , New York and London, 1 9 7 0 . P.L. Duren, B.W. Romberg and A.L. S h i e l d s , L i n e a r f u n c t i o n a l s on HP spaces w i t h 0<p<l, J . Reine Angew. Math. 238 ( 1 9 6 9 ) , 3 2 - 6 0 . R.E. Edwards, F o u r i e r s e r i e s (2 volumes), H o l t , R i n e h a r t and Winston, New York, 1 9 6 7 . J . J . F . F o u r n i e r and P.M. G a u t h i e r , Most power s e r i e s have r a d i u s of convergence 0 or 1 , (to appear i n Canad. Math. B u l l . ) . J.-P. Kahane, Some random s e r i e s of f u n c t i o n s , D.C. Heath and Co., Lexington, Mass., 1 9 6 8 . J.C. Oxtoby, Measure and category, S p r i n g e r - V e r l a g , New York, H e i d e l b e r g and B e r l i n , 1 9 7 1 . W. Rudin, F u n c t i o n a l A n a l y s i s , McGraw-Hill,Inc., New York, 1 9 7 3 . W. Rudin, Real and Complex A n a l y s i s , McGraw-Hill,Inc., New York, 1 9 6 6 . [l6] A. Zygmund, T r i g o n o m e t r i c s e r i e s , second e d i t i o n (two volumes), Cambridge U n i v e r s i t y P r e s s , Cambridge, England, 1 9 5 9 . 

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