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A convergence equivalence for trigonometric series Tan, Jiak-Koon 1971

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A CONVERGENCE EQUIVALENCE FOR TRIGONOMETRIC SERIES  by  -JIAK-KOON TAN  B.Sc., Nanyang U n i v e r s i t y ,  Singapore.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF 7.THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  i n the Department  of -MATHEMATICS  We a c c e p t t h i s required  t h e s i s as conforming  to the  standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1971  In p r e s e n t i n g an  this  thesis  advanced degree at  the  Library  shall  fulfilment of  University  of  make i t f r e e l y  I f u r t h e r agree tha for  the  in p a r t i a l  permission  s c h o l a r l y p u r p o s e s may  by  his  of  this  written  representatives.  available  granted  gain  permission.  Mathematics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  J u n e 22,  1971  for  for extensive by  the  It i s understood  thesis for financial  Department o f  Date  be  British  Columbia  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  (ii) Abstract  Let  W  n  be the  n th  p a r t i a l sum of a Walsh s e r i e s .  There i s  - a necessary and s u f f i c i e n t condition f o r pointwise convergence of the sequence W  2  , (n = 0, 1, 2, • • • ) , namely, l i m W n_  exists  a.e.  2  i f and only i f  T" , (W _ - W . ) < + CD ^n=l 2 2 ^ 2  n  a -  a.e.  •The aim of t h i s thesis i s to formulate and discuss a s i m i l a r convergence -equivalence f o r trigonometric s e r i e s .  (iii) Table of contents  1.  Introduction  p.l  .2. Martingale transforms  p.4  3.  Generalized Rademacher s e r i e s and Walsh s e r i e s  p.9  4.  A-convergence equivalence  p.13  Trigonometric F o u r i e r s e r i e s  p.24  Xacunary trigonometric s e r i e s  p.28  References  p.32  ,5.  .....  (iv) ACKNOWLEDGEMENTS  I am g r a t e f u l t o Dr. J . J . F .  Fournier  f o r h i s guidance i n a r e a d i n g  ^course o f harmonic a n a l y s i s s i n c e l a s t summer, and e s p e c i a l l y f o r h i s -supervision  o v e r my work i n t h i s t h e s i s .  many v a l u a b l e  He p r o p o s e d t h e problem and o f f e r e d  s u g g e s t i o n s from which I b e n e f i t t e d v e r y much.  -,also go t o D r . R.W. S h o r r o c k f o r r e a d i n g t h e f i n a l form.  Thanks s h o u l d  I a l s o want t o  -express my thanks t o The U n i v e r s i t y o f B r i t i s h Columbia, and The N a t i o n a l Research C o u n c i l to Mrs.  o f Canada f o r f i n a n c i a l a i d .  Choo C h i a Y i t S i n f o r t y p i n g  F i n a l l y , my s i n c e r e  the thesis.  thanks  1.  Introduction  JEointwise  convergence o f t r i g o n o m e t r i c  ^mathematics, w h i c h has t h i s type,  l o n g been s t u d i e d .  One  s e r i e s i s an o l d t o p i c i n  o f the famous problems o f  r e l e v a n t to our main theme, i s the c o n j e c t u r e  ^suggested i n 1915,  t h a t the c o n d i t i o n ^ | c  n  |  of L u s i n ,  < + co  2  i s sufficient r  ^ensure the a l m o s t everywhere convergence o f the s e r i e s converse i s f a l s e .  For  instance / t h e  s e r i e s --I| |>2  e  n  •known to converge ( [ 2 0 ] , I , p.183), except p o s s i b l y a t i d n t e g r a b l e sum Lusin's  f(t),  conjecture  and  proof  [1].  2_ n  f&  •  e  l°g  ^h  j I)  L  e  i  n  i s c l e a r l y n o t rthe F o u r i e r s e r i e s o f an  L. C a r l e s o n was  t = 0,  s  function.  remained u n s o l v e d  over  a b l e t o announce a  However, q u i t e a number o f o t h e r r e s u l t s were d i s c o v e r e d  by-products i n p u r s u i t o f a s o l u t i o n to L u s i n ' s p r o p o s a l . going  o n C  0  an  u n t i l 1966,  to  int  0 0  to  i s so deep i n n a t u r e t h a t i t had  a period of f i f t y years,  who  as  Some o f them a r e  to be mentioned i n subsequent s e c t i o n s , where a p p l i c a b l e . In p a r a l l e l with  orthogonal  t r i g o n o m e t r i c s e r i e s , t h e r e i s a n o t h e r c l a s s of  s e r i e s , namely, o r t h o n o r m a l s e r i e s o f independent f u n c t i o n s  on  --which s i m i l a r i n v e s t i g a t i o n s have been p u r s u e d w i t h p r o l o n g e d enthusiasm. Indeed, much e f f o r t was  c e n t r e d a t the q u e s t i o n  s u f f i c i e n t c o n d i t i o n f o r the e q u i v a l e n c e  y__^ k^k a  and (B)  l£=l k a  <  +  0 0  *  e  x  i  s  t  s  and  of f i n d i n g a n e c e s s a r y  of  i s f i n i t e almost everywhere ;  and  - 2 -  where the  a^'s  form a n u m e r i c a l sequence, and the  sequence o f independent  functions,  That  an e a r l y r e s u l t  (B) i m p l i e s  imply  (B),  (A) i s  0, ECffc)  E(f^) =  form an o r t h o n o r m a l =  1»  =  i n p r o b a b i l i t y theory.  can be c o n s t r u c t e d s u c h t h a t  (A) h o l d s , b u t  1»2,«" .  F o r (A)  some a d d i t i o n a l c o n d i t i o n s h o u l d be imposed on the  a counterexample (See  with  fj^'s  f^'s, (B) i s  to  otherwise false  the end of S e c t i o n 2).  As p o i n t e d out i n [4],,at l e a s t  three s u f f i c i e n t  conditions  are  known : (i)  u n i f o r m boundedness  (ii)  uniform i n t e g r a b i l i t y of for  every  such  e > 0,  J  set  The  either  (i)  {t  :  there is  l a s t one i s  f^,  the sequence  f^,  k = 1,2,'•• ; k = l,2,-«»,  t h e r e e x i s t s a number 6*, depending on  I A  f £ dt|  |f£(t)| some  c  > 6}  < e ,  ,  for a l l  k,  where  A  e,  such t h a t  The s t r o n g e s t r e s u l t  E(|fjJ)  >_c > 0,  i n this  uniformly i n  by a f u n c t i o n  v^  t h a t depends  k .  than  d i r e c t i o n was o b t a i n e d by  (k = l,2,-*')> by a sequence of m a r t i n g a l e d i f f e r e n c e s some e x t r a c o n d i t i o n s ,  the  [9].  Gundy [4], who r e p l a c e d the o r t h o n o r m a l sequence o f independent  a^  is  due to M a r c i n k i e w i c z and Zygmund, and i s more g e n e r a l  or ( i i ) .  that s a t i s f y  i.e.  that |  (iii)  o f the f u n c t i o n s  functions  f^,  d^, (k = l,2,-««)»  and r e p l a c e d each n u m e r i c a l  coefficient  on the p r e c e d i n g m a r t i n g a l e  differences  - 3-  Gundy's r e s u l t i s p a r t i c u l a r l y a p p l i c a b l e to Walsh s e r i e s , w h i c h are s i m i l a r  to trigonometric s e r i e s  i n many r e s p e c t s .  c o n s i d e r whether a s i m i l a r s i t u a t i o n a r i s e s series. results  We a r e thus l e d t o  i n the class of trigonometric  I n t h e n e x t two s e c t i o n s , we w i l l g i v e a b r i e f account o f Gundy's ; i n S e c t i o n 4, we w i l l  equivalence  f o r trigonometric series  considerations. problem w i t h  t r y to f o r m u l a t e  The l a s t  convergence  i n the l i g h t o f group -  theoretic  two s e c t i o n s a r e devoted to a d i s s c u s s i o n o f t h e  respect to F o u r i e r s e r i e s  on some c l a s s i c a l r e s u l t s  a similar  and l a c u n a r y  s e r i e s , w h i c h i s based  i n the t h e o r y o f t r i g o n o m e t r i c  series.  - 4 -  2.  Martingale  Let S*  of  S.  (X,S,P), S*,  transforms  (X,S,P)  If f  by  that the conditional expectation of  E(f|s*),  with respect to  of  P  o f t h e s i g n e d measure  with respect to  f  with respect to  fdP r e s t r i c t e d to  X, ' the symbol  S*,  denoted  by  S*.  If A  1(A) w i l l be used i n t h e s e q u e l  d e s i g n a t e t h e c h a r a c t e r i s t i c f u n c t i o n o f A. A  sub-sigma-algebra  i s d e f i n e d t o be t h e Radon-Nikodym d e r i v a t i v e  i s a S-measurable s u b s e t o f to  Consider a  i s a random v a r i a b l e o f f i n i t e e x p e c t a t i o n d e f i n e d on  we r e c a l l  denoted  be a p r o b a b i l i t y space.  The c o n d i t i o n a l p r o b a b i l i t y  P(A|S*),  i s d e f i n e d by  P(A|S*) -  = E(I(A)|S*).  To c l a r i f y  t h e meaning o f c o n d i t i o n a l e x p e c t a t i o n and c o n d i t i o n a l  p r o b a b i l i t y , l e t us c o n s i d e r a p u r e l y atomic sigma-algebra generated by a f i n i t e number o f atoms u n i o n o f t h e E ' s . I n t h i s case,  E^,...,E  E(f|s*)  k  S*,  such t h a t  n  and  P(A|s*)  X  (I(E )/P(E )) J  P(A|S*) =  (I(E )/P(E ))P(A.E ) .  k  k  i s a disjoint  a r e g i v e n by  E ( f | s * ) = l^  =1  which i s  f dP ,  and k  k  k  T h i s s i m p l e example may be h e l p f u l t o u s , because l a t e r , we s h a l l b e concerned mostly with sigma-algebras  o f t h i s k i n d , generated by a f i n i t e number o f  Rademacher f u n c t i o n s o r Walsh f u n c t i o n s , e t c .  Let algebras of  S , (k = 0,1,2,•••)» k  S, and  f  k  b e an i n c r e a s i n g sequence o f sub-sigma-  , (k = 1,2,«»«), be a sequence o f random v a r i a b l e s  - 5 -  o f f i n i t e e x p e c t a t i o n , d e f i n e d on  E  (X,S,P).  We r e c a l l t h a t t h e sequence f , k  (k = 1,2,«>.)>  i s c a l l e d a martingale r e l a t i v e  (^k+ll k^  ^  S  =  f  k  g e n e r a t e d by f^ ^ +  ^ —  o r  Usually,  ' "^\a*  w i t h respect to  a r u  S  * ^  n  t  n  a  S  k  S , (k = 1,2,...), i f k  i s t a k e n t o be t h e s i g m a - a l g e b r a  case t h e c o n d i t i o n a l e x p e c t a t i o n o f  t  i s w r i t t e n as  k  to  E ( f ^ ^ | f-^, • •. , f ) . +  Further, f o r  fc  such a m a r t i n g a l e , we c a n form t h e sequence o f m a r t i n g a l e d i f f e r e n c e s : rn  d^ = f^> ^2 sequence f , k  ^2 ~ ^1 ' *" *  =  s  "  o  ^n  t n a t  Lfc-i  =  •  ^  e  -'--'-  sna  T , (n = 1,2,•••)»' i s a m a r t i n g a l e t r a n s f o r m  where e a c h  v  relative  is  k  to  S , (k = 0,1,2,-..), k  y that the  [4] o f t h e sequence  n  (k = 1,2,...),  s a  i f T  n  = ££  = 1  v d f c  k  ^-measurable.  Now c o n s i d e r t h e f o l l o w i n g t h r e e s e t s respect to the martingale transform  T  n  A, B, C  , ( n = 1,2,..«),  defined with j u s t mentioned  before : A = { t : l i m ^ ^ I _-^  ^C^^Ct)  k  B - £t : I  k = 1  C = it : I^  =1  v (t) k  2  < + co}  (v (t)d (t)) k  k  2  e x i s t s and i s f i n i t e }  ,  ,  < + 00}  .  Gundy has shown i n ( [ 4 ] , S e c t i o n 5, p.731) t h a t :  (1)  I f d , (k = 1,2,...), k  e i t h e r one o f t h e c o n d i t i o n s sets  A, B, C  i s an o r t h o n o r m a l sequence and s a t i s f i e s  (MZG), (MZG)*  l i s t e d below, t h e n t h e t h r e e  a r e e q u i v a l e n t i n t h e s e n s e t h a t any two o f them may d i f f e r  by a t most a n u l l s e t .  ,  - 6Condition  (i)  (MZG) :  E(d^|s _^) = 1  almost everywhere ;  k  (ii)  ( l  p  d k  |  > a Is^^)  uniformly  Condition  k  0 < (E(d2|S _ ))  (ii)  E(|d |  k  b > 0,  < + oo;  %  1  I S ^ ) >  k  The  a > 0,  (MZG)* :  (i)  such that  f o r some c o n s t a n t s  f o r k ^ 1.  c(E(d2|s _ )) k  almost everywhere f o r each of  b > 0  k >_ 1,  1  where  c  i s a f i x e d constant  independent  0 < c <_ 1.  condition  (MZG)* i s a c t u a l l y e q u i v a l e n t t o c o n d i t i o n (MZG)  ( [ 4 ] , C o r . o f Lemma 1, p.728).  I n what f o l l o w s , we w i l l c o n s i d e r t h e  p a r t i c u l a r c a s e o f Gundy's theorem ( 1 ) , t h a t i s most u s e f u l t o us l a t e r .  Suppose  d , (k = 1,2,•••), k  i s an o r t h o n o r m a l sequence o f  independent random v a r i a b l e s .  I t i s known i n p r o b a b i l i t y t h e o r y  sequence  f  (n = l,2,'>-)»  d^ = f p  &2 ~ ^2 ~  n  = d ^ + ••• + d , n  e  t  c  s i g m a - a l g e b r a g e n e r a t e d by  '  ^  o  r a  n  y  f^,'*«,f  s u  k  bset ^,  that the  forms a m a r t i n g a l e  F  of  X,  belonging  such t h a t t o the  we have  E ( d £ | f . . . , f _ ) dP =  I(F)d£ dP  p  F  J  -  ( | 1(F) d P ) ( |  1 dP , F  dl dP)  - 7  i.e.  E ( d | f-p • • • , f ^ j)  = 1  k  is satisfied.  Similarly,  we can check t h a t  almost everywhere, where  F = { Jd | > 6 }  f  dP -  KI  < [( |  Hence, i f  inf  | | d | | ^ > 0, is  If  d ,  functions such that is  inf  || d | | ^ > 0  a  d ^ , •••<L  B - {t  : £ £  C = {t  :  a  c  k  a (t)  k  < + 00}  2  k  (a (t)d (t)) k  dP)]* +6.  2  5  so t h a t  k  2  a (t), k  the  and  a , k  independent (k =  l,2,--0» to  then the t h r e e s e t s :  finite}  ,  ,  < + co}  ,  p.100).  we may take  f (x^,••',x^_^) k  k >_ 1,  e x i s t s and i s  o f measure z e r o ( [ 5 ] ,  some c o n t i n u o u s f u n c t i o n  (F' - X - F ) ,  Is measurable w i t h r e s p e c t  n  ^,  fc  : l i m ^ ^ £ _ ^ k^ ^d (t)  As an example f o r for  for a l l  k  A = {t  d i f f e r e b y a t most a s e t  dP ,  i s an a r t h o n o r m a l sequence of  the s i g m a - a l g e b r a g e n e r a t e d by  1  |djj  A l s o , we have  T h i s enables us to o b t a i n the  any sequence o f f u n c t i o n s such t h a t  =  k  - P(F)  :  (k = 1 , 2 , • • • ) »  k  |d |  1(F) d P ) ( |  also s a t i s f i e d .  f o l l o w i n g c o r o l l a r y from (1)  (2)  f  dP +  6 > 0.  we a r e a b l e to choose  k  c o n d i t i o n (MZG) ( i i )  E(I(F) | f p • • • with  k  f U l k  almost everywhere, so the c o n d i t i o n (MZG) ( i )  of  a (t) = f ( d ^ ( t ) , • • • , d _ ( t ) ) f c  (k-1)  k  variables.  f c  1  - 8 -  The above assertion i s no longer true i f inf || djj.11^ ~ 0 . A simple counterexample was provided by Zygmund ([13], p.73), i n which a sequence of indices  n  F o r such a sequence, £-1 ( W ) %  2  = + co  1  < n  2  <  1^-1 (k) .  was chosen such that I^-^^) * I knjj I () fc  converges almost everywhere, while  0 0  •  -- 9 -  3.  G e n e r a l i z e d Rademacher  We a r e now section  i n a p o s i t i o n to i n t e r p r e t t h e r e s u l t s i n t h e p r e v i o u s  i n terms o f Walsh  Let interval  r n  s e r i e s and Walsh s e r i e s .  functions.  ( t ) be t h e  [0,1]  nth  Rademacher  n  = sgn s i n ( 2  W i t h t h e a i d o f Rademacher w (t), n  (n = 0,1,2,  •)»  n  where  > n  >  2  write  ••• > n w  k  n(t)  functions,  +  n = 0,1,2,". •  we  can d e f i n e  in n  [0,1] .  k  ••• + 2  as a sum o f powers o f  "  r  constants.  c w (t) n  n  t  r  1  t  r  2  w, n  w i l l be r e f e r r e d  In p a r t i c u l a r , i f  t  k  (n = 0,1,2,•••),  c  n  of  f  belonging to  L^"[0,1],  I f we t a k e  X = [0,1],  i s a complete  A s e r i e s o f the  t o as a Walsh s e r i e s , where  f  =  c  n  are  1  f(t)w (t) n  J  some  2,  n ( >- n < >'-- n ( ) •  o r t h o n o r m a l system on t h e u n i t i n t e r v a l ( [ 1 7 ] , p.130). oo _Q  the Walsh  and p u t  The s e t of Walsh f u n c t i o n s  Z  on t h e u n i t  :  t  l  n = 2  ^ 0,  irt),  for a l l  Q  n >_ 1,  n + 1  as f o l l o w s  w ( t ) » 1,  For  defined  by r (t)  functions  function  0  d t , (n = 0,1,2,•••)>  we s h a l l c a l l i t a W a l s h - F o u r i e r  for  series  f .  subsets of  [0,1],  and  S = t h e c l a s s o f a l l Lebesgue measurable  P = the Lebesgue measure on  complete p r o b a b i l i t y space, u s u a l l y  referred  S,  then we o b t a i n  a  t o as t h e Lebesgue p r o b a b i l i t y  - 10  space.  The  Rademacher f u n c t i o n s  v a r i a b l e s on  Let n  c a l l any  (n = 0,1,2,•••),  n  considered  t h e Lebesgue p r o b a b i l i t y space, a r e independent, and  an o r t h o n o r m a l system, but  first  r ,  -  a r e f a r from b e i n g  Q  s e r i e s o f the form  Rademacher s e r i e s .  T  >••^.  Qt  Ly-Q  a  I t can a l s o be  form a m a r t i n g a l e t r a n s f o r m  We  k.( ) k( ) t  r  t  defined  a (t)  s h a l l f o l l o w Gundy [5]  constant  +  1 + £^=0  prescribed  n  the  a  and  generalized  as a s e r i e s whose p a r t i a l sums  of the sequence  A c l o s e r l o o k a t the f u n c t i o n s  etc.).  be a f u n c t i o n depending on  n  Rademacher f u n c t i o n s  they form  complete (e.g. [17] ,'• p.107  ^ ( t ) = f ( r ( t ) , •••,r _^(t)) n  as random  ^(O  > (  n  =  0,1,2, • • • ) .  above w i l l r e v e a l the  fact  t h a t they a r e no more than l i n e a r combinations o f the Walsh f u n c t i o n s . f a c t , we  can  show t h a t any  function  f(r (t),•••,r (t)) Q  n  of  r  »*'*» n r  0  In c  a  n  2n+l_i be  e x p r e s s e d as  set  the l i n e a r c o m b i n a t i o n  o f p o i n t s , by  2^-0  ^k k' w  e x c e  Pt  o  n  a  countable  writing  f(r (t),---,r (t)) 0  n  + 1 ) ( 1 + r ( t ) ) / 2 + f ( r ( t ) , • • • . r ^ C t ) , - 1) (1 - r ( t ) ) / 2  •- f ( r ( t ) , - . - , r _ ( t ) , 0  and we  using  n  1  induction.  n  G i v e n any  generalized  A = { t : l i r n ^ ^ I^_Q  B  d i f f e r by  =  {  t  :  Z*Q  n  In view o f the c o n d i t i o n s t a t e d i n (2) o f S e c t i o n  have i m m e d i a t e l y the f o l l o w i n g  (3)  0  a  k  (  t  )  2  <  k  +  [5] :  Rademacher s e r i e s  a  (t)r (t)  0 0  2,  k  }  I^_Q  e x i s t s and  '  a t most a s e t of Lebesgue measure  zero.  aj (t)r^(t), c  i s finite}  ,  the  sets  - 11 -  We may First  r e p h r a s e t h e above statement i n terms o f Walsh  of a l l , i f  W  =? ly^rO k k ' c  n  sum o f the Walsh s e r i e s  w  Z,_Q  c  k  =  k k»  1  »  =  some p o l y n o m i a l  a (t)  of  n  » " )  denotes t h e  nth  partial  then c l e a r l y  w  V+i" V ^ for  2  functions.  r n ( t )  r (t),"*,r 0  n  '  Cn = 0,1,2  ^(t).  '*"  )  On the o t h e r hand, f o r  nOO  any g e n e r a l i z e d Rademacher  series  ^ ( ^ ^ ( t ) >  a (t)r (t) = n  by t h e p r e c e d i n g that  W  remark.  ( I ^" 2  n  Thus, we  1  w  e  k  k  n  exists  I" i n=l  (W  2 ^-  z. _Q  such  k  t o the f o l l o w i n g ,  (almost everywhere) i f and only i f  n+  a t  ,  c a n form a Walsh s e r i e s  a.e.  L  th  e  n  n  lim W n-*°° 2  e  b w (t))r (t)  ,, - W = a„(t)r„(t) . So (3) i s e q u i v a l e n t 2n+± 2 s t a t e d i n ( [ 3 ] , Cor. ( 3 . 2 ) , p.245) :  (4)  s  )  W  2  2  < + co  a.e.  n  I n o t h e r words, the s e t s A, B d e f i n e d w i t h r e s p e c t  to t h e Walsh  s e r i e s are e s s e n t i a l l y equal.  Remark :  The r e a d e r w i l l n o t e t h a t i n t h e f o r e g o i n g  go i n t o t h e d e t a i l s o f t h e p r o o f  d i r e c t i o n was  didn't  o f Gundy's r e s u l t s ( 1 ) , o r more e x a c t l y ,  Gundy and Neveus' r e s u l t s (Gundy o n l y p r o v e d t h a t The o t h e r  e x p o s i t i o n , we  proved i n Neveu  A  i s contained  in  ( [ 1 5 ] , F r o p . IV. 6.2, p.148).  B. The  - 12 -  r e a s o n i s t h a t t h e t e c h n i q u e s used by Gundy and Neveu a r e a p p a r e n t l y n o t a p p l i c a b l e to trigonometric s e r i e s .  In particular,  the sequence  W  Q  of  p a r t i a l sums o f a Walsh s e r i e s i s a m a r t i n g a l e [3], w h i l e t h e c o r r e s p o n d i n g sequence o f  2 th n  p a r t i a l sums o f a t r i g o n o m e t r i c s e r i e s i s n o t .  - 13 -  4.  A convergence e q u i v a l e n c e .  We have come t o t h e main c o r e o f o u r problem. to e x p l a i n b r i e f l y  L e t us f i r s t t r y  t h e fundamental c o n n e c t i o n between Walsh f u n c t i o n s  w, n  int and t h e t r i g o n o m e t r i c f u n c t i o n s o f t h e form n  i s an i n t e g e r .  exp i n ( - ) : t -—> e  ,  where  From t h e group t h e o r e t i c p o i n t o f view, b o t h o f them a r e  c o n t i n u o u s c h a r a c t e r s on compact  a b e l i a n groups.  The former c o n s t i t u t e t h e  c h a r a c t e r group o f t h e d y a d i c group, w h i c h may be d e f i n e d as t h e c o u n t a b l y i n f i n i t e d i r e c t p r o d u c t o f t h e d i s c r e t e group w i t h elements  0  and  1,  where the group o p e r a t i o n i s a d d i t i o n modulo 2 [2] ; w h i l e t h e l a t t e r form the c h a r a c t e r group o f the u n i t c i r c l e group.  These two groups, namely t h e  d y a d i c group and t h e u n i t c i r c l e group, have a compact metric t o p o l o g i c a l structure.  (and t h e r e f o r e  complete)  The p r o d u c t t o p o l o g y on t h e d y a d i c group may  be m e t r i z e d by t h e homeomorphism ( e . g . [ 1 0 ] , 0, p.165) -n •> 2 I e -3'  (e  n  onto the c l a s s i c a l C a n t o r s e t on t h e r e a l l i n e . similarities  n  = 0, 1)  There a r e many s t r i k i n g  o f the W a l s h - F o u r i e r s e r i e s and t r i g o n o m e t r i c F o u r i e r  f o r which we may r e f e r t o , e.g. [ 2 ] , [ 1 6 ] , [ 1 9 ] .  series,  Thus, i t i s n a t u r a l t o  ask q u e s t i o n s f o r t r i g o n o m e t r i c s e r i e s , s i m i l a r t o the r e s u l t s f o r the Walsh s e r i e s ,  t r e a t e d i n the p r e c e d i n g  sections.  F i r s t , we n o t e t h a t t h e s e t o f Rademacher f u n c t i o n s occurring i n the sets  A, B  2 th n  Walsh f u n c t i o n .  n  ,  (n=0,l,2,•••),  i n ( 3 ) , i s f o r m a l l y a l a c u n a r y sequence o f  c h a r a c t e r s on t h e d y a d i c group i n t h e sense t h a t the  r  r  n  = w  ,  where  w  I n comparison w i t h t h e t r i g o n o m e t r i c system  is  -14 -  exp  (± ±2 t ) , (k = 0,1,2,•••)>  we f i n d immediately n,  between them,  i.e. for n>_l,  another f o r m a l  resemblance  n,  n=2+'--+2  l  s  -,  with  nj > n  2  >• • • • > n , k  we have Wn(t) and  «  r ^ C t ) ••• r  n f c  (t) ,  n n, exp ( i n t ) = exp ( i 2 t ) ••• exp ( i 2 t ) . x  K  So i t may be r e a s o n a b l e t o take t h e t r i g o n o m e t r i c system exp (±i2^t), (k = 0,1,2,•••),  as a n a t u r a l c a n d i d a t e i n d e f i n i n g t h e s e t s  trigonometric series. Rademacher s e r i e s and i f  I _Q l ^ l ^ n  T h i s machinery sometimes works.  I _Q n n ^ ^ c  r  t  n  =  +  0 0  •  converges  s e r i e s i s almost everywhere non-summable  F o r example, t h e  almost everywhere i f  then, whatever t h e method T*  T*  A, B f o r  I _QI nl c  2 < + C D  n  >  o f summation, t h e  ( [ 2 0 ] , I , ( 8 . 2 ) , p.212).  A  s i m i l a r r e s u l t a l s o holds f o r the corresponding lacunary t r i g o n o m e t r i c s e r i e s ( [ 2 0 ] , I , ( 6 . 3 ) , ( 6 . 4 ) , p.203) o r ( S e c t i o n 6 ) .  A more d e t a i l e d examination w i l l  r e v e a l the f o l l o w i n g  characteristics  o f t h e s e t o f Rademacher f u n c t i o n s :  (i)  They a r e independent  f u n c t i o n s , when c o n s i d e r e d as random  v a r i a b l e s on t h e p r o b a b i l i t y space c o n s i s t i n g o f t h e d y a d i c group w i t h t h e n o r m a l i z e d Haar measure.  When c o n s i d e r e d as elements they  form  (ii)  An independent  set ;  (iii)  A dissociate set ;  i n t h e d u a l group o f t h e d y a d i c  group,  - 15 -  (iv)  A Sidon s e t ;  (v)  A s e t o f type  The  notions  explanation.  Let  group G, and  E  for  T  o f l a c u n a r i t y mentioned above may r e q u i r e a l i t t l e be t h e ( d i s c r e t e ) c h a r a c t e r group o f a compact a b e l i a n  be a s u b s e t  l §^ n  ,  either  Then  •••  (j^  <j>  =  K  every  not i n o f type  1 with  then  If  characters  n  k  and i n t e g e r s  n  E  where  k  does n o t c o n t a i n  cj>^, • • •, <f>k >  1,  1  and f o r any  the i n e q u a l i t y  E,  then  A(2),  0,  E  i s said  a  f  to  for a l l  Y  o r e v e r  on  G,  to be a Sidon s e t .  i f there i s a constant  y  {-2,-1,0,1,2}  f i r s t property  K,  having  implies I f there  f ( Y ) = 0,  Finally,  E  f  fora l l Y  i s s a i d t o be  depending o n l y on  trigonometric polynomial  not i n  {$0.} °f c o n t i n u o u s {cf»>  belonging  = 1, then Eo n liys on s a i d E, t o be s ts o c i a t e s e t . K, depending sucha dt ih a  trigonometric polynomial  The set  G.  T,  in  l ^ , <j>j • ••• <J> r 1 >  or  c  nj,  II f 11 2 ^.^H ^Mi » ^ f(Y) =  ^  \  i<j)s a= c o... n s t a=n t for  i s s a i d t o be independent i f ,  <f> ,  ' k *'*=<(>j =1»  t h e i d e n t i t y c h a r a c t e r on  l  E  n  =  f i n i t e number o f d i s t i n c t n  T.  of  any f i n i t e number o f c h a r a c t e r s  nj,***,^ is  A(2) .  on  E, G,  such t h a t having  E .  ( i ) i s t h e s t r o n g e s t one.  In g e n e r a l , i f any  c h a r a c t e r s on a compact a b e l i a n group G s a t i s f i e s ( i ) ,  also satisfies  t h e o t h e r f o u r , o r more p r e i s e l y .  ( i ) =>  ( i i ) =>  I t i s c l e a r from t h e d e f i n i t i o n s  that  ( i i i ) =>  ( i i ) =>  ( i v ) => (v)  (iii).  For  ( i i i ) => ( i v )  - 16 -  and  ( i v ) =>  ( v ) , we r e f e r t o Hewitt [6] and Rudin [ 1 8 ] .  check ( i ) =>  We need o n l y t o  (ii) :  n  Let  7  n,  ••• ^  ~ 1«  First  note that  independent f u n c t i o n s i n t h e sense o f ( i ) .  ^  n.  , •  < | > *• a r e a l s o  I n t e g r a t e t h e above e q u a t i o n  w i t h r e s p e c t t o t h e n o r m a l i z e d Haar measure on  G  to g e t  )  -  1 •  k  I  S i n c e each If  J  <j)j (t ) / 1 J  0  I  f o r some  i i -1 = <(> . (t )<J> , ( t t ) n  ^  1.  1  »  t  w e  i n G,  Q  n  J  J  0  Q  Let  h a v e  f  n  would l e a d t o  I  J  ^  I  =  1  f  o  r  a  1  1  then t h e i n t e g r a t i o n o f  i <j». = 0,  J  =  1  > "» ,  k  <j>j (t) = J  contradiction.  us t u r n now t o t h e t r i g o n o m e t r i c system  exp ( i n t ) ,  (n=0,±l,±2,•••),  w h i c h c l e a r l y c o n t a i n s no sub-system independent i n the sense o f ( i i ) (and k therefore of ( i ) ).  The t r i g o n o m e t r i c system  c o n s i d e r e d b e f o r e does n o t even s a t i s f y  (iii).  exp (±i2 t ) , (k = 0,1,2,•••)» I t i s a Sidon s e t .  for  G = the u n i t c i r c l e  (for  d e f i n i t i o n , see S e c t i o n 6) i s a S i d o n s e t ( [ 1 8 ] , p.127).  group, any l a c u n a r y sequence o f p o s i t i v e  Indeed, integers  This i s the  k i n d o f l a c u n a r i t y we s h a l l use i n S e c t i o n 6. Now i n view o f t h e p r e v i o u s d i s c u s s i o n , we know to what e x t e n t the  two systems, namely, t h e s e t o f Rademacher f u n c t i o n s  t r i g o n o m e t r i c system  exp ( ± i 2 t ) , (n = 0,1,2,•••)» n  a n o t h e r , when c o n s i d e r e d as group c h a r a c t e r s . we may c o n s i d e r a d i s s o c i a t e s e t e.g.  •  k  R  and t h e .  a r e s i m i l a r t o one  Instead of  exp ( ± i 4 t ) ,  r (t),  exp ( ± i 2 t ) , k  (k = 0,1,2,•••).  But,  - 17 -  as f a r as o u r treatment i s concerned, t h e r e s u l t s f o r exp ( ± i 4 t ) k  t u r n out t o be t h e same as those f o r exp ( ± 2 t ) ,  (k = 0, 1, 2, • • • ) •  k  we r e s t r i c t o u r s e l f t o t h e f o r m u l a t i o n o f o u r new problem o n l y w i t h k  to t h e system  exp (±2 t ) ,  Z  oo  -  c_2  e  -2it  +  _  Let  n  c  _ i  e  - i t >  D  k  a  ( t )  We d e f i n e t h e s e t s  be a g e n e r a l t r i g o n o m e t r i c s e r i e s w i t h  D (t)= c , 0  n  °  E  < n < 2  k  C  n  -2- <n<-2k  and  A = {t : l i m  complex  D_^(t) =  ,  2  A  D ^ ( t ) = c^e"^"" + C 2 e ^ , t  0  ^ k-l  =  =  Our  respect  (k = 0, 1, 2, •••) :  Cr,e  0 0  c .  .  So  int  TV  coefficients  will  B  T!  1  e  l  n  ,  t  i f k > 1 ,  ^  '  e x i s t s and i s f i n i t e }  ,  ( k + 1 )  C  n  e  l  n  '  t  ±  f  k  <  by  Dt,(t)  purpose i s t o d i s c u s s t h e e q u i v a l e n c e  of  A  and  B .  I n t h e f o l l o w i n g , we w i l l g i v e some examples.  Example 1. I™_  a, l n | c  Consider 2  < + oo .  a trigonometric series  rioo int 2, _ <» n c  e  n  I n 1924, Kolmogorov showed t h a t t h e  with  n ^ t h p a r t i a l sum  o f such a s e r i e s converges almost everywhere, i f n ^ ^ l n ^ > q > 1 q  ( [ 2 0 ] , I I , (1-17),  p.164).  almost everywhere, s i n c e  F u r t h e r , we have  J^ _^ s  |D (t)| k  2  f o r some  < + oo  - 18 -  J ^ _ „  l%(t>|  d t - ^  2  Thus, we  dt'-  2  < +00  I"  B  :  J K.(t)|  o b t a i n a s i t u a t i o n i n which, b o t h the s e t s  A,  .  B  are equivalent  -the whole o f the u n i t c i r c l e i n the s e n s e t h a t they d i f f e r by  to  a t most a n u l l  .set.  Example 2.  Let  f  be  a  H  •ssee i n n e x t s e c t i o n , the s e t s of  f  are equivalent  "This shows t h a t  f u n c t i o n which i s n o t  1  A  and  B  in  H  .  As we  a s s o c i a t e d w i t h the F o u r i e r s e r i e s J  t o the whole o f the u n i t c i r c l e , where  J _ ^  IcjJ  n  < + ao  i s not  |f(n)|  a necessary condition f o r  r.oo equivalence  of  Example 3.  A  and  a s s o c i a t e d w i t h the s e r i e s  s e r i e s can be  almost everywhere ( [ 2 0 ] , I ,  A  and  B  Example 4 ( i )  are equivalent  G i v e n any  i n c r e a s i n g terms, we  can  -non-zero c o e f f i c i e n t s <the s e t s  =  +co  the  int  2_  „,  n  c  n  e  A  For  (3.6), p.306).  adapted from Kolmogorov's c o n s t r u c t i o n .  an almost everywhere d i v e r g e n t  -sets  2  I t i s known t h a t t h e r e i s a F o u r i e r s e r i e s o f p o w e r - s e r i e s  type, which diverges  of  B  shall  and  B  c  divergent  series  Sn  a trigonometric  satisfy  |c | n  = a  n  associated with i t are n u l l  the c o n s t r u c t i o n , l e t us  of p o s i t i v e i n t e g e r s  M  r  example  s e r i e s f o r which b o t h o f i t s a s s o c i a t e d  to the whole o f the u n i t  construct n  T h i s i s an  Such a  as f o l l o w s  :  circle.  with  positive  non-  s e r i e s such that i t s nth  for a l l  n,  and  such t h a t  sets.  d e f i n e a monotone i n c r e a s i n g sequence  - 19 -  Put such  M  0  = 0.  If  MOJ-'-JM^ ^  have been chosen a l r e a d y ,  take  M  V  that  v-1  and -1  M  'Ut  ,  ,+1 <*«>*• "  1  •  v-1  Let so  6  y  = Tj-i  ( |3)/(Mj - M^_^).  Choose p o s i t i v e i n t e g e r s  1T  k^  inductively  that k -l 2  >  2  >  x  and  Let  lty > k ^  = 2  V  - (M  Q  + 1) ,  + 1, M. -  ( M ^  V  +  1) ,  for  v > 1  1  - M  Then, k -l 2 <  Ic^ + 1 ) , • • •, (My + n^) < 2  v  V  ( M ^  Note t h a t we use t h e c o n v e n t i o n  We w i l l v e r i f y  C=l  2  +  = 0  above.  t h a t the t r i g o n o m e t r i c  Mv  ^n=M  series  ^  + 1 n>'* P (~in6 ) exp i ( n + m ) t (a  ex  v  v-1  y  s a t i s f i e s our requirement.  F i r s t , we want to p o i n t out t h a t i n d e f i n i n g the sequence M Q , M-^, ' • •,  we have made use o f the assumption t h a t  I _^ n  ^  a n t  *  therefore  - 20 C l  ) (  a  i s divergent.  %  n  ^n=M +l v  n  The series  (My - M ^ ) "  ^  -n=M+l  z  -  ^  v  v+l  diverges,  1  ^  u  < " [ 1 + (a^* ] (a^)*  1 +  (My - M ^ ) ^  t  Mv ] I M _ n =  - M^)" ! 1 + (a ) f  1  For any  ^  1  v  % 1 +  l  •  %  x  satisfying |t - 6 |  (u|3)/(My- M  <  V  v-1  )  ,  we have, | D k ( t ) | = | I^!  ( a n ) exp ( - i n e ^ exp i ( n + my)t %  M  +  1  v-1  -1  £  +I  M  (  A  -  )  %  E  X  P  I  N  (  T  -  9  V  )  1  v-1  v-1 My  J,  1 In-M  v-1  1  " 2  +1  ( a  p  My X -M n  n  ) 2 C O S  v-1-+1  ( a n ) 2  ( n  ~V l  )  (  t  ~v e  }  ^  since  -  21 -  Now c o n s i d e r a f i x e d number I  2_  ~  M  2?  v  ^corresponding  t  [0, 2TT).  in  *" d i v e r g e n t , t h e r e e x i s t s a sequence s  to  j = 0 , 1, 2,  1  <  -  v(j) — >  co as  j —>  oo .  | ( t + 2JTT) - S Hence, i n c o n j u n c t i o n w i t h  l %  a  )  v ( j )  t + 2j7r  1 ••I^[ (ir|3)0V- V ^ " )  '  1  From t h i s i n e q u a l i t y , we have  • < '(43)0^ - M ^ ) "  |  the p r e v i o u s  ( t ) |  v =v(j),  such that  I ^ ^ ^ ^ K M v - M ^ ) *  Clearly,  Since the s e r i e s  .  1  result,  = i D ^ . ^ t  +^  l  1  2-1 ' T h i s completes t h e p r o o f  that both  •  £ PkCO  and  J | Dj^Ct) |  2  diverge  everywhere i n [ 0 , 2ir).  (ii)  F u r t h e r , g i v e n any sequence  T i t h non-zero c o e f f i c i e n t and  (b)  |cn|  non-increasing  To  2  - a ^  b  n  C Q can be so chosen t h a t  where  terms, w h i l e  n  —>  oo , t h e  (a) I ^ - ^ l n | / ^ n c  2  i s some d i v e r g e n t s e r i e s w i t h A  and  B  still  remain n u l l s e t s .  c o n s t r u c t t h i s s e r i e s , we c a n assume  f i n i t e number o f t h e f i r s t  0 < b  with  few terms.  b  n  >_ 1  <  +  a  o  '  positive '  by n e g l e c t i n g a  Then p i c k by i n d u c t i o n a sequence o f  - 22 -  n , n^, ••• such that  positive integers  nj - n n  2  -  n  3  - n  Q  >_ 1,  Q  >_  n  x  >^ n  2  and for  n >_ nj  : ( b ) >_ 2 ; 2  n  - n ,  and for n >_ n  — tij^,  and for n >_ n  Q  2  : (b ) ^_ 3 ;  2  n  : ( b ) >_ 4 ; 2  3  n  and so on. We put p  = (v - 1) + (n - n _ ) ( n  n  v  1  - n _ )~  y  v  = <|>(n), for n ^ <_ n < a^,  1  1  v  v = 1,2, ••• clearly,  p  = 0, p — >  Q  = v = <j, (n ), v  v  Pn  co ,  n  -  and 0 < p ^ (b ) . Further, n  n  p ^ = <j> (n ) = v+1  v = l , 2 , - - - . So, P -1  W  =  n  'V-l)  -  >  - 1  0  f  o  r  nv-l < n <_ 0^, v = 1,2, •••  .  From this inequality, i t follows immediately that  p  n ~ P -1  =  ( n  n  v " v-l> n  1  - < v+l " n ) n  1  v  = p  n + 1  - p , for n = n n  v  and Pn - Pn-1  Pn+1 " Pn »  =  Then  J ( p - p _]_)  ;terms.  n  o  r  "v-l < n < n . v  i s a divergent series with positive non-increasing  n  If we set a  f  n  = p  n  - p  n  i n the construction of 4(i), then the  ^resulting series i s the desired one with  I kn! /Vl! 2  (Pn " Pn-1>/Pn  <  +  0 0  •  v  -  23  -  Examples 4 ( i ) , ( i i ) a r e e s s e n t i a l l y due We  have m o d i f i e d  his constructions  s l i g h t l y , as was  From Example 4 ( i ) , ( i i ) , we (a)  g i v e n above.  observe t h a t  the n t h non-zero c o e f f i c i e n t he  to Neder ( [ 1 4 ] , pp.133-136)  c  n  of a trigonometric  chosen to decrease to zero i n a c e r t a i n o r d e r ,  1^1  -  (n)  —3* 2  ,  —P  (n l o g n)  the a s s o c i a t e d s e t s  2  A  ,  (n l o g n l o g l o g n)  and  B  —P  series  e.g. ,  may  taking while  are t r i v i a l l y equivalent  i n the  s e n s e t h a t b o t h of them a r e n u l l s e t s ; (b)  some growth c o n d i t i o n s imposed on non-zero c o e f f i c i e n t s J  a series, like sufficient sets  A  |c | /(log n) 2  n  e  < + co  with  to ensure a n o n - t r i v i a l e q u i v a l e n c e  and  B.  e > 0, o f the  c are  n  of not  associated  - 24 -  5.  Trigonometric  For f o r the  Fourier series.  t h e c l a s s o f F o u r i e r s e r i e s , we  convergence e q u i v a l e n c e  have an almost complete r e s u l t  of their, associated  complex t r i g o n o m e t r i c  ^ s e r i e s i f t h e r e e x i s t s an  2 -_oo n  series  f  : =  c  f(n) =  n  usual, i f  f  is in  L^"[0,1],  Zoo i.e.  nth n  =  The r o l e i n the all  f(k)e  n  Hardy space  c o u r s e of our  a n a l y t i c functions  B.  A  s  id  a  t o be  a Fourier  that  1 i n t  s h a l l use  dt,  0,±1,±2,  n =  the symbol r  of of  and  0  we  c^e  (symmetric) p a r t i a l sum  S [f] = ^ _  s  int  n=—°°  The  f  such  f(t)e~  J  As  ^  e  n  L^[0,1]  in  A  int  r.00  formal  sets  f,  and  S[f]  write  S[f]  represent  v°°  -i  int  S[f] - > ~"n—— i  i s denoted by  to  c-e  00  the symbol  S [f], n  .  i k t  H, p  (l<_p<  + oo),  which w i l l p l a y  an  important  d i s c u s s i o n i n t h i s s e c t i o n , i s the Banach space o f  f , h o l o m o r p h i c i n t h e open u n i t d i s c such  IUI! _ P  H  = l i - ^  =  sup  (l/2*>  0 < r < 1  |f(re  i t :  )|  P  that  dt  0  2ir rzir (( 11//22TTTT)) I  || f£ (( rr ee " )) || x t  P  ]  dt < + GO .  -0  J  The  isometry  to i d e n t i f y  f —> H  P  c o n s i s t s of a l l for a l l  n <  0.  f*,  where  f*  i s the r a d i a l l i m i t o f  w i t h a c l o s e d s u b s p a c e of L  P  L  P  on  f,  permits  the u n i t c i r c l e ,  f u n c t i o n s having the nth F o u r i e r c o e f f i c i e n t  us  which f(n) =  0,  - 25 -  .As  a s t a r t , we d e a l w i t h t h e .H*  S [ f ] - 1^=0  with theory  II Ifc=o (  = 2 _  k  v  *  l k| d  2 )  n  +  ^k=0 I 1  f  belong to  I t f o l l o w s from t h e Littlewood-Paley-Zygmund f,  the sets  A  and  * l ' a . i s ' m a j o r i z e d by  2  c  v  »  e  a n c  *  n  B  associated with i t are  ||  F o r , i f 0 < a < 1,  f||j  ,  where  d  Q  = c  the  ,  D  k » (k = 0, 1, 2, • • • ) , i s a sequence o f i n d i c e s  a^jln^ > q > 1 < +  2  f o r some  everywhere t o = S^ [f]  th  k  ( [ 2 0 ] , I I , ( 5 . 2 ) , p.234).  Consequently,  Furthermore, f o r t h e same  f  p a r t i a l sum  f ( t ) ( [ 2 0 ] , I I , (5.11),  i s a particular  n  q  almost everywhere.  0 0  i t i s a l s o known t h a t t h e n  k  Let  ivt  such that  _.n ,  e i n t  t o t h e whole o f t h e u n i t c i r c l e .  v^k d  n  t h a t f o r such an  both equivalent n o r m  c  space.  p.235).  S ^[f]  converges almost  n  Our sequence  £°  L  functions  P  (1 < p < + oo)  c l a s s o f t r i g o n o m e t r i c s e r i e s f o r which b o t h t h e a s s o c i a t e d s e t s t o t h e whole o f t h e u n i t c i r c l e .  i s t o use t h e f a c t  = Q  D (t)  =  k  case.  The F o u r i e r s e r i e s o f  equivalent  and  form another A,  B  A n a t u r a l way t o g e t t h i s  are result  that  L  P  = H  # H  P  ,  P  f o r 1 < p < + oo ,  where H i s t h e space o f complex c o n j u g a t e s o f H functions vanishing a t t h e o r i g i n ( [ 7 ] , p.150). I n o t h e r words, i f f i s an L function P  P  F  (1 < p < + co) functions  g  with  and  h  such t h a t  T h i s f a c t together with l^k^^l  2  <  +  0 0  , i n t i n t  S [ f ] - 1°^^  3 1 1 ( 1  c e n  c  n  n  e  int  results f o r  £ -_«o l ^ k ^ ^ l  2  n  1^-0  f = = g g + + h h f  u  S[g] - I - g  the previous  the same r e a s o n , we see t h a t  *then - "-i  ,  ^(t)  < + oo and  »  fo or r some some f  , „ r v-1 ^ S[h] ~ Z ^ . ^ r  a n  n  H H  p F  c  n  functions implies almost everywhere.  I^.^  D (t) k  For  and hence  e  int  that  - 26 -  Z _ oo ^ k ^ )  e x i s t s and i s f i n i t e a l m o s t everywhere.  k  a deeper r e s u l t r e l e v a n t  to our q u e s t i o n  h e r e was  which a s s e r t s  t h a t the F o u r i e r s e r i e s o f an  converges to  f ( t ) almost everywhere .  iF  As a m a t t e r o f f a c t ,  announced function  i n [8] r e c e n t l y , (1 < p < + OD )  i The above argument no c l o s e d subspace o f particular,  L  1  ± H  f a i l s i n case of  complementary  to  L  functions  H''"  ( [ 7 ] , p. 154),  f o r which t h e  2  and i n  © HJ .  1  Kolmogorov's c o n s t r u c t i o n can be m o d i f i e d f  f o r there i s  th  n  p a r t i a l sum  S _[f] 2  t o g i v e a summable f u n c t i o n  i s divergent  a l m o s t everywhere  n  on t h e u n i t c i r c l e (3.12), p.308). set  B  ( l i m _ ^ l i m S ^ [ f ] = + oo n  For this  S [ f ] , the s e t  might be l a r g e i n measure.  f  in  L \  b u t not i n  B  associated with  almost everywhere) ( [ 2 0 ] , I ,  n  H"*"  S[f]  and  L  a r e not  As a n o t h e r example,  We  A  i s e s s e n t i a l l y empty.  still  don't know whether t h e r e i s an  (1 < p < + c o ) ,  P  But the  f o r which the s e t s  A,  equivalent.  consider  the f u n c t i o n  f  mentioned i n  S e c t i o n 1, i . e .  f  Note t h a t  (i)  H^" # H^.,  since  f  (  t  is in I _2 n  e  *  )  =  L \ n t  |n|>2  1  /l°g  e  ±  p > 1 ; r  «= + co ,  where  t  /  (  1  0  n  is  n  o  t  t  n  e  ;  1 < p <_ 2.  o f H a u s d o r f f - Y o u n g ( [ 2 0 ] , I I , p.101).  8  l  ;  otherwise, f o r large values • q = p/(p - 1 ) ,  2  but not i n  f u n c t i o n by Hardy's theorem ( [ 7 ] , p.70) for  n  n  |  )  '  (ii) f  i s also not i n  F o u r i e r s e r i e s o f an ( i i i )f  of  r»00  m,  L  This  The s e t  i s not an  > ( l o g n) n=-m •  will A  H"" -  L —Q H  P  function riOO  —J_  > / n — ^n=m  =  c o n t r a d i c t t h e theorem  associated with  this  - 27 -  series i s equivalent s m a l l i n measure.  I  2 **nt )/-'- S t  function f  0  n  F o r a comparison, we f i n d t h a t i t s ^  n  c  l  a  s  s  w  n  n^  f c  ^A°8  ^  n  s  t  n  e  to the u n i t  B  might be  counterpart an i n t e g r a b l e  Walsh-Fourier s e r i e s of  F o r t h i s s e r i e s , we know a l r e a d y  are equivalent  But  o f Walsh s e r i e s a l s o d e f i n e s  L -2  f ( t ) such t h a t  ( [ 1 9 ] , p.235).  A, B  t o the whole o f t h e u n i t c i r c l e .  that i t s associated  sets  interval.  We summarize our r e s u l t s i n t h i s s e c t i o n i n t h e f o l l o w i n g : The function  sets  A  and  (1 < p < + oo )  B  a s s o c i a t e d w i t h a F o u r i e r s e r i e s ^ o f an  are equivalent  For F o u r i e r s e r i e s of the e q u i v a l e n c e o f  A  and  B  still  t o the whole o f t h e u n i t  functions, not i n remains unknown.  or  L?  L  p  circle.  (l<p<+co),  -  -28  Lacunary trigonometric series  A sequence of positive integers be lacunary, i f ther i s a constant  q > 1  k.  e l n t  A trigonometric series J™...^ n c  ..lacunary sequence of indices all  n ^ ±n^ , i n ^ , ••• .  written as  £ __  c  k  n  e  n,  (k = 1, 2,•••••••)» i s said to  k  such that  (v = 1, 2, • • • ) , A, B  the sets |B|  n  > q  c  k  for a l l  n  = 0,  for  Such a lacunary trigonometric series is usually n ^ = -n^ . int  n<»  v  k  n , (k = 1, 2, •••).,• such that  For a lacunary trigonometric serxes • £ n,  n  is lacunary, i f there exists a  » with the convention  k  k  ^]a+±\  v+l| v n  1 > 1>  >  a s  . . associated with  m  i t s lacunary sequence of indices,  are equivalent to one another. A, B  represent, as usual, the measures of  More precisely, i f  |A| ,  respectively, then we have  the following : Either one of the conditions implies that  | n| c  2  <  +  >  0 0  a n d  (i)  |A| > 0  therefore  and  ( i i ) |Bj > 0  |A| = |B| = 1 ,  by  Example 1 i n Section 4 . The statement (i) i s essentially a particular case of the theorem which states that i f n series  J _-^ ( n a  k  c o s k  n  k  k  fc +  th ^n^  A ^(t)  partial sum s  i  n n  k ) t  n  i  s  measure by any T*-summability method, then (6.4), p.203).  of the lacunary trigonometric  summable on a set of positive ^ _^ l n j j c  k  2  < +  0 0  ([20], I,  Our case i s a simple one, for which the proof i s much shorter,  -and so is included for completeness. The f i r s t and crucial step i s to show that i f we are given a subset E  of  (0,2TT]  of positive measure, then there is an integer  M  such that  - 29 -  |E| I  (  kn |  )/2  2  v  <  1  .for .any t r i g o n o m e t r i c p o l y n o m i a l  p(t) =  f  | (t)|  dt  2  P  2|E| I | c |  ,  2  n v  I||>JI V  c  n  For the proof, consider  f  | (t)| P  dt = [  2  JE  (£ C n / V H V  " 14Z l n l  +  c  1 4 Z 1^1  =  c ( n ) = (2ir) ^ I  e *  E c h a r a c t e r i s t i c f u n c t i o n of  IZ ^ U  V  c  n c u  dt  n t  E.  n v  i s the  d  t  e  n c . u  l fr  ^  +  c  V  u  c  dt  E  n  n c u  c(n  n v  - n ) ,  v  u  F o u r i e r c o e f f i c i e n t of the  By t h e Schwarz i n e q u a l i t y ,  c(n  - n ) | <_ (£ | c ^ J  v  2  u  ) ^  <Z l n | ) ( Z ,  |c(n  v  u ?  2  u  2^  2  c  - n )| ) 2  v  |c(nv - n ) | ) 2  v  u  c o n d i t i o n o f l a c u n a r i t y we a c t u a l l y need i s t h a t t h e number o f  -•ways i n w h i c h e v e r y i n t e g e r can be e x p r e s s e d v f u,  2  n th  1  The  Z ^ U  v  where  C n ^ V )  IE  i s bounded by some number  Z  |c(n  N,  u  n  v  - n , u  and t h i s h o l d s i n our c a s e .  - n ) | < 2N( |c(m)| 2  v  i n t h e form  2  +  with So  |c(m + 1 ) | + ... ) , 2  - 30 where  m  Since  n  i s the smallest integer that can be written as n v  - n  we see that  u  >_ n  m  v  - n _^ ^_ n ( l - q -"*") >_ n ( l - q v  v  increases with  I n - - l < >! c  we can pick  M  n  k  n ,  - n , 1 <^ u < v. u  ), for v > u >_ k ,  so i n conjunction with  k  2  v  • ^>  -  1  { lKE)|  dt < 1 ,  2  such that  4TTN( |c(m)| + |c(m + 1)| + . . . ) < |E|/2 < | E | 2  2  .  Hence, | | |p(t)|  2  | | |p(t)|  2  dt|  <  2|E|.I | c |  dt|  > | E | I |c |  2  n v  ,  and  n v  -  2  - |E|I  1*1 I |c | /2 2  nv  |c | /2 2  nv  .  This completes the proof of the technical lemma, due to Zygmund ([20], I, (6.5), p.203). Let us turn now to (i) and ( i i ) , which w i l l follow from the previous lemma by using a standard type argument i n analysis. (i) m = <t  E  :  Since II  k =  _  n  A i s contained i n the union of the sequence of sets D ( 0 Iim, K  for a l l n = 0, 1, 2, • •. } , (m = 1,2,. • •),  R and a subset  we can choose an integer  E of A with  that for a l l t i n E ,  I I =-n k D  k  ( t )  I• 1  R.  for a l l n  | E | > 0 such  - 31 -  Pick  M  corresponding to E. Then, we get  MI  for a l l  |v| >_ M  |v| < M.  Letting  (ii)  E -Rt ,  k  v  k —> ao ,  Z  in„t, R-^ = R + || J c ^ e  where  we obtain  v  v  £  l n | c  m  2  <  +  0 0  v  with  •  Using Egorov's theorem and the inner regularity of Lebesgue  |D (t)| f  <  |n | <_ 2 ,  -measure, we get a compact subset k  I I  2  nv  with  in„t,2 =n e | dt  1  |c | /2  2  E of B  such that  converges uniformly to some function  | E | > 0, and f on E. This function  i s clearly continuous and bounded on E. So,  + oo >  E  f(t) dt  L  Use the lemma to choose  [  ' k D  ( t )  i  d t  ^|k|>N i f  +  i v o i  1+  so large that  |D (t)|  2  k  dt > ( |E| I | c | n v  2  )/2  , for |k| > N  Therefore, we have + co > which implies that  £_  dt  k=-N  E 1 J  k  | c | < + co . 2  m  |D (t)|  n  2  dt + ( |E|. I | c | n v  2  )/2 ,  X  - 32 -  References  [I] . Carleson, L., On convergence and growth of partial sums of Fourier series, Acta Math., 116, (1966), 135-157. [2].  Fine, N.J., On the Walsh functions, Trans. Amer. Math. Soc, 65 (1949), 372-414.  [3].  Gundy, R.F., Martingale theory and pointwise convergence of certain orthogonal series, Trans. Amer. Math. S o c , 124 (1966), 228-248.  [4],  Gundy, R.F., The martingale version of a theorem of Marcinkiewicz and Zygmund, Ann. Math. Statistics, 38 (1967), 725-734.  [5].  Gundy, R.F., On a class of martingale series, 99-102, Orthogonal expansions and their continuous analogues, Southern Illinois Univ. Press, 1968.  [6],  Hewitt, E. and Ross, K.A., Abstract Harmonic Analysis, Vol. II, Springer - Verlag Berlin, 1970.  [7],  Hoffman, K., 1962.  [8].  Hunt, R.A., On the convergence of Fourier series, 235-255, Orthogonal expansions and their continuous analogues, Southern Illinois Univ. press, 1968.  [9].  Kac, M. and Steinhaus, H., Sur les fonctions independantes (II), (1936), Studia Math. 6, 59-66.  [10],  Banach spaces of analytic functions, prentice - Hall,  Kelley, J.L., General topology, D. Van Nostrand, 1955.  [II] . L i t t l e wood, J.E. and Paley, R.E.A.C., Theorems on Fourier series and power series (II), p r o c Lond. Math. Soc, 52-89 (1936). [12],  Loeve, M., probability theory, D.Van Nostrand, 1963.  [13].  Marcinkiewicz, J. and Zygmund, A., Sur les fonctions independantes, 60-90 (1937), Fund. Math., 29  [14].  Neder, L., Zur Theory der trigonometrischen Reihen, Math. Ann., 84 (1921), 117-136.  [15].  Neveu, J., Mathematical foundations of the calculus of probability, Holden-Day, San Francisco, 1965.  -  33 -  [16].  Paley, R.E.A.C., A remarkable series of orthogonal functions, I, Proc. London Math. Soc. 34 (1932), 241-264.  [17].  Renyi, A., 1970.  Foundations of probability, Holden-Day, San Francisco,  [18],  Rudin, W., 1962.  Fourier analysis on groups,  [19].  Yano, S., On Walsh-Fourier series, Tohoku Math. J., Ser. 2, Vol. 3, (1951), pp. 223.  [20].  Zygmund, A., 1959.  Trigonometric  Interscience publishers,  series, Vols. I, II, Cambridge Univ. Press,  

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