S. V. Astashkin- On the Selection of Subsystems Equivalent in Distribution to the Rademacher System

8/3/2019 S. V. Astashkin- On the Selection of Subsystems Equivalent in Distribution to the Rademacher System

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M a t h e m a t i c a l N o t e s , V o l. 6 6, N o . 3 , 1 9 9 9

B R I E F C O M M U N I C A T I O N S

O n t h e S e l e c t i o n o f S u b s y s t e m s E q u i v a l e n t i n D i s t r i b u t i o nt o t h e R a d e m a c h e r S y s t e m

S . V . A s t a s h k i n

KEY WORDS: ran dom variable, lacunary series, Peetre /C-functional, Sidon system.

I n t r o d u c t i o nW e s a y t h a t s y s t e m s { f n } ~ =l a n d {gn}~=, o f r a n d o m v a r i a b l e s d e f i n e d o n p r o b a b i l i t y s p a c e s ( ~ , E , ? )a nd ( ~ ' , E ' , I P ') , r e spe c t ive ly , a r e equivalent in distribution ( de n o te d { f n} s { gn} ) i f t he r e e x i s t s a C > 0s uc h t h a t f or a r b it r ar y m E N , a,~ 6 ~ ( n = l , 2 , . . . , m ) , a n d z > 0 w e h av e

I I m [ } {[~--1 angn (w') } {~--1 }- l ? a nA ( ) > C z > z >C- z " ' n - - - - I - - - -

I f o n l y t h e s e c o n d i n e q u a l i t y h o ld s , t h e n w e s a y t h a t t h e s y s t e m { gn }n ~1 76 s ma jorized in distributionb y t h e s y s t e m { fn },~ 1 F o r t h e c a s e o f f i n it e s y s t e m s { f n } N =l a n d N9 { gn} ,~ = l t he se no t io ns a r e de f ine d ina s im i l a r wa y .I n [1] i t i s s h o w n t h a t i n a n y u n i f o r m l y b o u n d e d s e q u e n c e { f,~ }n ~_ l o f r a n d o m v a r i a b le s ( r .v . 's ) o n ec a n s el e c t a s u b s e q u e n c e { f nk } ~= 1 t h a t i s m a j o r i z e d i n d i s t r i b u t i o n b y t h e R a d e m a c h e r s y s t e m { r n } ~ = , ,rn(x ) = s igns in ( 2n - 17 r x ) ( x 6 [0 , 1 ] ) .I n t h i s n o t e w e p r e s e n t s o m e a s s e r t io n s c o n c e r n i n g t h e p o s s i b i li t y o f c h o o s i n g a s u b s e q u e n c e e q u i v a le n tto t h e sy s t e m {r,~}n~__x un de r t he s a m e c ond i t i ons . M o r e ove r , we f ind ne c e s sa r y a n d su f f i c i e n t c ond i t i on sf o r a s y s t e m o f r. v . 's t o c o n t a i n a s u b s y s t e m e q u i v a l e n t i n d i s t ri b u t i o n t o t h e R a d e m a c h e r s y s t e m . I n t h ec a s e o f f in i t e s y s t e m s , w e a r e i n t e r e s t e d i n t h e d e n s i t y o f th e i r s u b s y s t e m s w i t h a s i m i l a r p ro p e r t y 9T h e p r o b l e m s c o n c e r n i n g t h e s e l e c t i o n o f " n e a r l y i n d e p e n d e n t " s u b s y s t e m s ( i .e ., p o s se s s in g s o m e p r o p -e r t y t h a t i s c h a r a c t e r i s t i c f o r s y s t e m s o f i n d e p e n d e n t f u n c t i o n s ) w a s s t u d i e d i n d e t a i l in t h e w e l l- k n o w np a p e r o f V . F . G a p o s h k i n [2 ]9 L e t u s r e c a l l t h e m o s t i m p o r t a n t r e s u l t s c o n c e r n i n g i n t e g r a b i l i t y a n d a b s o l u t ec o n v e r g e n c e o f t h e s u m o f a s e r ie s c o n s t r u c t e d f r o m a g i v e n s y s t e m o f f u n c ti o n s .B y t h e c l a ss i c al i n e q u a l i t y p r o v e d b y A . Y a . K h i n c h i n [ 3 ], t h e s u m o f t h e s e ri e s )-~,~176 anr,~(x) , w h e r e(x 6 [0 , 1] ) , be lo ngs to a l l spac es L p (p > 2) i f th e seq uen ce of coef f ic ients a = (an) ,~__l be lon gs to 12 .I n t h e c a s e o f l a c u n a r y t r i g o n o m e t r i c s e ri e s, a s i m i l a r r e s u l t w a s o b t a i n e d b y A . Z y g m u n d [ 4] 9 I n t h i sc o n n e c t io n , s o m e w h a t l a t e r t h e n o t i o n o f S p - s y s t e m a p p e a r e d i n t h e w o r k s o f S. B a n a c h a n d S . S id o n . As e q u e n c e o f r . v . ' s { fn}~=l , f n 6 Lp ( p > 2 ) , i s c a l l e d a Sp-sys tem i f we ha v e

E a n f n < a n ," n = l " P _ 2

w h e r e t h e c o n s t a n t K p > 0 i s i n d e p e n d e n t o f m 6 N a n d a n e Ir ( n = l , . . . , m ) . I f { f ~ } i s a S p -s y s t e m f o r a n y p > 2 , t h e n i t is c a l l ed a S~-sys tem. A c la s s i c a l r e su l t on s e l e c t ing S p - subsys t e m sb e l o n g s t o S . B a n a c h ( [5 ] o r [6 , 7 .2 ] ). I t i m p l i e s , i n p a r t i c u l a r , t h a t a n y u n i f o r m l y b o u n d e d o r t h o n o r m a ]Translated from Matematichesk'ie ametki,Vol. 66, No. 3, p p. 47 3-477, Septemb er, 1999.Original article submitted April 25, 1999.

384 0001- 4346 /1999 /6634- 038422.00 C)20 00 Kluwer Academic/Plenum Publishers


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s y s t e m o f f u n c ti o n s c o n t a i n s a n S e e - s u b s y s t e m . A m o r e p r e c is e r es u l t i n th i s d i re c t io n w a s p r o v e d b yS . B . S t e c hk i n [ 2, T he o r e m 1 . 3 .1 ] , i .e . , a s e que n c e o f f unc t i ons {f,~ }n~ 1 on [ 0, 1] c on t a i n s a n S p - s u bs y s t e m( p > 2 ) i f a n d o n l y i f t h e r e e x i s t s a s u b s e q u e n c e { f - k } s a t i s f y in g t h e f o l lo w i n g co n d i t io n s :

1 ) I I f~ [ E p < - D ( k = 1 , 2 , . . . ) ;2 ) f ~ k --> 0 w e a k l y i n L 2 .A n o t h e r i m p o r t a n t p r o p e r t y o f t h e l a c u n a r y s e r i e s i s a b s o l u t e c o n v e r g e n c e . F o r t r i g o n o m e t r i c s e ri e s, t h e

c l a ss i c al S i d o n t h e o r e m o n a b s o l u t e c o n v e r g e n c e o f a l a c u n a r y ( in t h e s e n s e o f H a d a m a r d ) F o u r i e r se r i es o fa b o u n d e d f u n c t i o n i s w e l l k n o w n . A . Z y g m u n d p r o v e d a l o c a l v e r s i o n o f t h i s t h e o r e m [ 7 ]. M o r e r ec e n t ly ,s i m i la r r e s u l t s w e r e e s t a b l i s h e d f o r t h e R a d e m a c h e r s y s t e m a n d s o m e o t h e r s p e ci fi c l a c u n a r y s y s t e m s o ff unc t i ons [ 8 ] . A s a na t u r a l ge n e r a l i z a t i on , t he f o l l ow i ng no t i o n a p pe a r e d : a s e que n c e o f r . v . ' s { f,~ },,~ --1 isc a l l e d a Sidon system i f

l a l _ < c l l a n I . l [ ,n = l n = l

w h e r e t h e c o n s t a n t C > 0 is i n d e p e n d e n t o f m 6 N a n d a , 6 I~ ( n = 1 , 2 , . . . , m ) .I n [2 , T h e o r e m 1 .4 .1 ] V . F . G a p o s h k i n o b t a i n e d b r o a d s u f fi c ie n t c o n d i t i o n s f o r t h e p o s s i b i li t y o f s e l e ct i n g

S i d o n s u b s y s t e m s . S u p p o s e t h a t a s y s t e m { f- },~ -- 1 o f m e a s u r a b l e f u n c t i o n s o n [ 0, I] p o s s e s s e s t h e f o l lo w i n gp r o p e r t i e s :

1 ) 1 l 1 1 2 = 1 ( n = 1 , 2 , . . . ) ;2 ) I f ~ (x ) ] _ < V ( n = l , 2 , . . . ; x E [ 0 , 1 ] ) ;3 ) t he r e e x i s t s a s ubs e qu e nc e { f , ~ } C { f ,~ } s uc h t h a t f , ~ --~ 0 w e a k l y i n L 2 ; t he n { f- }, ,~ -- 1 c on t a i n s

a S i d o n s u b s y s t e m .T h e a b o v e d e f in i ti o n s a n d t h e p r o p e r t i e s o f t h e R a d e m a c h e r s y s t e m i m p l y t h a t i f { f n } ~ { r n } , t h e n

t h e f o l lo w i n g a s s e r t io n s h o l d :1 ) { f n } i s a n S o ~ - s y s te m , w h e r e t h e c o n s t a n t o f p - l a c u n a r i t y h a s th e s a m e g r o w t h r a te w i t h r e s p e c t

t o p a s i n t h e c a s e o f t h e R a d e m a c h e r s y s t e m , i .e ., K p ~ v ~ ;2 ) { f n } i s a S i d o n s y s t e m .

I n t h is c o n n e c t i o n , t h e a s s e r t i o n s s t a t e d i n w e n s u r e " g o o d " p r o p e r t i e s o f t h e s e l e c te d s u b s y s t e m inb o t h s e n s e s, r e f in i n g a n d s t r e n g t h e n i n g t h e r e s u l t s m e n t i o n e d a b o v e .

A l o n g w i t h t h e q u e s t i o n o f s e l e c ti n g i n f i n it e s u b s e q u e n c e s , o t h e r p r o b l e m s , i .e ., c o n c e r n i n g f in i te s y s t e m so f r. v . 's , a r e o f c o n s i d e r a b l e i n t e r e s t . S u p p o s e t h a t { f,~ }N = 1 is a f a m i l y o f r . v . 's . W e s e e k t h e m a x i m a ln u m b e r s = s ( N ) w i t h t he f o l l ow i ng p r o pe r t y : t he r e e x i s t s a f a m i l y { f ,~ ,} i~ x C {f,~}g=1 s a t i s f y i ngs o m e g a p c o n d i t i o n w i t h t h e c o n s t a n t i n d e p e n d e n t o f N . H e r e w e o n l y m e n t i o n a re s u lt o n th e d e n s i t y{ fn}n= l i s anf f i n i t e S i don s ub s y s t e m s p r o ve d by B . S . K a s h i n ( s e e [9 ] o r [10 , T h e o r e m 8 . 9 ] ) , i .e . , i f xo r t h o n o r m a l f a m i l y o f f u n c t i o n s d e f i n e d o n [ 0, 1 ] , ] f , ( x ) ] _ D ( n = 1 , . . . , N ; x E [ 0, 1 ] ), t h e n t h e r e i sa s u b s y s t e m { f - , } ~ = l ( 1 < n l < - - - < n ~ _< N ) s u c h t h a t s > m a x { [ 1 / 6 1 o g 2 N ] , 1 } a n d f o r a n y a i E I~( i = 1 , . . . , s ) w e h a v e

D -1 E a i fm < ]all < 4D a i fm" i=1 cr i=1 =

I n w w e s t a t e a s i m i l a r a s s e r t io n o n s e l e c ti n g a s u b s y s t e m { f n , } i ~ x a l s o o f " l o g a ri th m i c " d e n s i ty , b u ts a t i s f y i n g a s t r o n g e r c o n d i t i o n , i . e . , e q u i v a l e n t t o t h e s y s t e m { r i } i S l w i t h t h e c o n s t a n t d e p e n d i n g o n l yo n D .

I n w h a t f o l lo w s , b y w r i t i n g F 1 F 2 w e m e a n t h a t C-1F1 < F2 < CF1 f o r s o m e C > 0 , w h e r e t h ec o n s t a n t C i s, a s a ru l e , i n d e p e n d e n t o f a l l o r p a r t o f t h e a r g u m e n t s o f F 1 a n d F ~ . I f 1 < p < c ~ ,f = f (~ v ) is a r . v . o n t h e p r o b a b i l i t y s p a c e ( f l , E , P ) , a n d a = ( a , ) , = x i s a n u m b e r s eq u e n c e , t h e n t h en o t a t i o n s E f t ) , I l f l l p , [ l a l lp ( r e s p e c t i v e l y Lp, lp) a r e u n d e r s t o o d i n t h e s t a n d a r d w a y , I E I i s t h e L e b e s g u em e a s u r e o f t he s e t E C [ 0 , 11.

w S y s t e m s o f r . v J s t h a t a r e e q u i v a l e n t in d i s t r i b u t i o n t o t h e R a d e m a c h e r s y s t e mH e r e w e s t a t e a c r i t e r io n f o r a n a r b i t r a r y s y s t e m o f r . v . 's t o b e e q u i v a l e n t i n d i st r i b u t i o n t o t h e

R a d e m a c h e r s y s t e m , a n d a l so p r e se n t i t s c o n s e q ue n c e s.3 8 5


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I n w h a t f o ll o w s , a n i m p o r t a n t p a r t i s p l a y e d b y t h e P e e t r e M - f u n c t i o n a l f r o m t h e i n t e r p o l a t i o n t h e o r yo f o p e r a t o r s [ 1 1 ]. I f ( X 0 , X 1 ) i s a B a n ac h p a i r , x E X 0 + X 1 , t > 0 , t h en b y d e f i n i t io n w e h av e

/ C ( t , x ; X 0 , X , ) = i nf {l lx 0 1 IX o + t I I x : t l x , : x = x o + x x , x o e X o , x: e X , } .W e s h a l l o n l y n e e d t h e f u n c t i o n a l / g l , 2 (t , a ) = / g ( t , a ; l l , 12) c o n s t r u c t e d b y t h e B a n a c h p a i r ( 11 ,1 2 ).

In [12] i t is pro ve d t ha t 11 ~-~~176a~rn[l t ~ K : l , 2 ( v / t, a ) w i t h t h e co n s t an t i n d ep en d e n t o f t E [ 1, o o ) , an da = (an)n=1~17612 ( i m p l i c i t l y t h i s r e l a t i o n i s a l r ead y co n t a i n ed i n [ 13 ]) . T h e f o l l o w i n g a s s e r t i o n s h o w s t h a t ar e l a ti o n s i m i l a r t o t h e o n e a b o v e d e t e r m i n e s s y s t e m s o f r . v .' s e q u i v a l e n t i n d i s t r i b u t i o n t o t h e R a d e m a c h e rs y s t e m .

T h e o r e m 1 . L e t {fn},~__~ be a sys te m o f r .v . ' s de f ined on a probab i l i t y space ( f~ , ~ , P ) . The fo l lowin gcond i t ions are equ iva len t :

1 ) { f n } s { r n } ;2 ) the equ iva lence~ anTn

- - / : I I n = 1 l i t

h o l d s w i t h t h e c o n s t a n t i n d @ e n d e n t o f t ~ [ 1 , ~ ) , m ~ N , a n d a ,, ~ ~ ( n = 1 , . . . , m ) ;3 ) the equivalenceI I , r l , = 1 I 1 $

h o l d s w i t h t h e c o n s t a n t i n d e p e n d e n t o f t E [1, c~) a n d a = (an)n~176 E 12 .L e t u s a p p l y t h i s t h e o r e m t o t w o m o r e s p e c if ic c a s e s.

ooR e c a l l t h a t a s y s t e m o f r . v . 's { f n } , , = : i s cal led mul t ip l i ca t ive i f f o r a n y p a i r w i s e d is t i n c t n l , n 2 , . . . , n k( k E N ) w e h av e E ( f , ~ l f n ~ - - - f n k ) = 0 . I f w e a d d i t i o n a l l y h a v e E ( f m f n 2 . . . f n k f 2 ) = 0 f o r t h e s am en : , n 2 , . . . , n k a n d n y~ n ~ ( s = 1 , . . . , k ) , th e n t h e s y s t e m { f,~ } is c a l le d s t rong ly mu l t ip l i ca t ive .

A n i m p o r t a n t e x a m p l e o f a s t r o n g l y m u l t i p l i c a t iv e s y s t e m i s t h e s e q u e n c e { f n } ~ = : o f i n d e p e n d e n t r . v .' ss uc h t ha t f ~ E L 2 a n d E ( f ~ ) = 0 ( n = l , 2 , . . . ) .F r o m t h e r e s u l t s o f [ 14 ] a n d T h e o r e m 1 w e o b t a i n t h e f o ll o w i ng a s se r t io n :

T h e o r e m 2 . S u p p o s e t h a t {f ,~}~=l i s a s t r o n g l y m u l t i p l i c a t iv e s y s t e m o f r .v . ' s su c h t h a t [f,~(ca)[ < D( n = 1 , 2 , . . . ; w E ~2) a n d d = in fn= : ,2 . .. . E ( f 2 ) > 0 . T h e n { f n } { r n } , w h e r e t h e c o n s t a n t o f t h i se q u i va l e n ce d e p e n d s o n l y o n D a n d d .

C o r o l l a r y 1 . Ea c h s e q u e n c e o f i n d e p e n d e n t r . v. 's { f n } ~ = l s u c h th a t f n E L 2 , E(f,~) = 0, If ,~(w)l < 19( n = 1 , 2 , . . . ; w E f~ ) , a n d d = inf ,~=:,2 . .. . E( f~ ) > 0 i s e q u i v a l e n t i n d i s t r i b u t i o n t o t h e Ra d e m a c h e rs y s t e m .

N o w l e t u s c o n s id e r a n o t h e r s i t u a t i o n . S u p p o s e t h a t G i s a c o m p a c t A b e l i a n g r o u p , F i s i t s g r o u p o fc h a r a c t e r s , a n d # i s t h e H a a r m e a s u r e o n G . I n a c c o r d a n c e w i t h t h e d ef i ni t io n a b o v e ( se e t h e i n t r o d u c -t i o n ) , t h e s e t F C F i s c a l l ed a S i d o n s e t i f f o r s o m e C = C ( F ) we hav e ~-'~Ter I ] ( 7 ) t < c I I f l l ~ r fo r anyf E C ( G ) s u c h t h a t ] ( 7 ) = f c f ~ / d ~ = 0 ( 7 ~ F ) .G . P i s i e r p r o v e d t h a t f o r an a r b i t r a r y S i d o n s e t F = { %,} C F w e h av e

mI I o o o K I II Z a o r ~n= l n= l

t _ l

( t h e e q u i v a l e n c e c o n s t a n t d e p e n d s o n l y o n t h e S i d o n c o n s t a n t C ( F ) ) [ 1 5 ]. H e n c e b y T h e o r e m 1 w ei m m e d i a t e l y o b t a i n t h e f o ll o w i n g a s s e r ti o n , w h i c h w a s p r o v e d i n [ 1 6 ] i n a d i ff e re n t w a y ( b u t a l s o w i t h t h eh e l p o f t h e t h e o r e m o f G . P i s i e r) :38 6


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T h e o r e m 3 . A n y i n f i n i t e S i d o n s y s t e m F = {7n}~=1 o f charac ter s de f ined on a compact Abel iangroup is equ iva len t in d i s t ribu t ion to the Ra dem ach er sys t e m. Here the equ iva lence cons tan t depends on lyo n th e S i d o n c o n s t a n t C ( F ) .

C o r o l l a r y 2 . The sequences f ,~(x) = s in(27rknx) and gn(x) = c o s ( 2 . k ~ z ) , where x E [0, 1] andk n + x / k n _>/~ > 1, are equ iva len t in d i s t r ibu t ion to the Rademacher sys t em.

R e m a r k 1 . N o t e t h a t i t w a s t h e an a l o g y i n b e h a v i o r b e t w e e n t h e R a d e m a c h e r s e ri es a n d l a c u n a r yt r i g o n o m e t r i c s e ri e s ( s ee t h e i n t r o d u c t i o n ) t h a t s e r v e d a s t h e s t a r t i n g p o i n t i n d i s c o v e ri n g t h e i r d e e p e rr e l a t i o n s h i p a n d l e d t o s t u d y i n g s y s t e m s eq u i v a l e n t i n d i s t r i b u t i o n t o t h e " m o d e l " R a d e m a c h e r s y s t e m .

w S e l e c t in g s u b s y s t e m s e q u i v a l e n t i n d i s t r i b u t i o n t o t h e R a d e m a e h e r s y s t e mT h e m a i n r o l e in t h e p r o o f o f th e p o s s i b i li t y o f s e l e c t i n g s u b s y s t e m s e q u i v a le n t i n d i s t r i b u t i o n t o t h e

R a d e m a c h e r s y s t e m i s p l a y e d b y t h e f o l lo w i n g a s s e r t io n :T h e o r e m 4 . Le t a sy s te m of r .v . 's {f ,~}n~=l def ined on a probabi l i ty space (I2, E , ]P) conta in a subse-

quence { fnk }k~_-I sa t i s f y ing the fo l lowing cond i t ions:i) I f ~ ( w ) l < D ( k = l , 2 , . . . ; w E D ) ;2) d = infk = l ,2 . .. . I I f~1t2 > 0;3 ) f ,~ --+ 0 weakly in L 2 .T h e n t h e r e i s a s u b s y s t e m {~i}i~=l C {f~},~--1 s u c h t h a t

I C l , 2 ( v q , a ) ( a = e 1 2 , t > _ 1 )_ t

wi th the cons tan t depend ing on ly on D and d .R e m a r k 2 . A n e s s e n ti a l p a r t o f t h e p r o o f c o n s is t s i n fi n d i ng u p p e r b o u n d s f o r L q - n o r m s (q > 1 ) o f

t h e m o d i f i e d R ie s z p r o d u c t s ( i n s e l ec t in g a S id o n s y s t e m , i t su f fi c es t o e s t i m a t e t h e L l - n o r m [2 , T h e o -r e m 1 .4 .1 ]). T h e l a t t e r h a v e a b l o c k s t r u c t u r e , w h i c h f o l l ow s b y a p p l i c a t i o n o f a n a p p r o x i m a t i o n f o r m u l af o r t h e / C - f u n c t i o n a l / C l ,2 ( t , a ) d u e t o S. M o n t g o m e r y - S m i t h [13].

T h e f o l lo w i n g tw o a s s e r t i o n s a r e d i r e c t c o n s e q u e n c e s o f T h e o r e m s 1 a n d 4 .T h e o r e m 5 . A s y s t e m {f~}n~r of r . v . ' def ined on a probabi li ty space (12, E , P) conta ins a subsys-t e m { ~ } ~ equ iva len t in d i s t r ibu t ion to the Rademacher sys t em on [0, 1] if and only i f there exis ts .a

subsequenc e { fn~ } C { f n } sa t i s f y ing cond i t ions 1 ) - 3 ) o f Theorem 4 . Bes ides , t he equ ivalence cons tan tdepends on ly on D and d .

T h e o r e m 6 . I f {f~}=~--1 i s an o rthon orm al sequence of r .v . 's def ined on a probabil ity space ( f~, E, ]~) ,I f,~(w)[ C log 2 N ) e q u i v a l e n t i n d i s t r i b u t i o n t o t h e s y s t e mo f t h e f i rs t s R a d e m a c h e r f u n c ti o n s , w i t h t h e c o n s t a n t i n d e p e n d e n t o f s .

T h e o r e m 7 . L e t Nf n } n = l b e a n o r th o n o r m a l f a m i l y o f f u n c t i o n s d e fi n ed o n [0 , 1] , l f~(x) l < D (n =1 , 2 . . . . , N ) . T h e n t he r e e x is ts a f a m i ly

{ f n , } i ~ l , 1 O d ep en din g o nl y o n D , f o r a l l a ~ E ] ~ ( i = l , 2 , . . . , s ) a n d z > O w e h av e

c - l { ~ > C z } z } l < _C _ a i r i( x ) > C - l z } .

387


5/5

R e m a r k 3 . T h e e x a m p l e of t h e tr i g o n o m e t ri c s y s t e m s h ow s t h a t T h e o r e m 7 (j us t as t h e t h e o r e m o fB . S . K a s h i n m e n t i o n e d i n t h e i n t r o d u c t i o n ) i s u n i m p r o v a b l e w i t h r e s p e c t t o o r de r . I n d e e d , a s w a s p r o v e db y S . B . S t e c h k i n [ 18 ], i f t h e s e q u e n c e { v ~ c o s 2 7 r n ~ x } ~ = l ( x E [ 0, 1 ]) i s a S i d o n s y s t e m , t h e n w e h a v e~ ' ~k : n k< g 1 < C l n N ( N = 2 , 3 , . . . ) .

T h e a u t h o r w i s he s t o t h a n k B . S . K a s h i n f o r s e t ti n g t h e p r o b l e m w h o s e s ol u t io n i s p r e s e n t e d i n wR e f e r e n c e s

1. S. V. Astashkin, Mat . Z ametk i [Math. Notes], 65, No. 4, 483-495 (1999).2 . V. F. Gaposhkin, Uspekhi Mat. Nauk [Russian Math. Surveys], 21, No. 6, 3-82 (1966).3 . A. Khintchine , Math. Zeitschr. , 18, 109-116 (1923).4 . A. Zygmund, J. Lond. Math. Soc. , 5, No. 2, 138-145 (1930).5. S. Banach, Bull. Acad. Polon. , 149-154 (1933).6 . S. Kaczm arz and H . Steinhaus , Theorie d e r Orthogonalreihen, Chelsea Publishing Company, New York (1951).7. A. Zygmund, Trans. Amer. Math. Soc. , 34, No. 3, 435-446 (1932).8 . S. Kaczm arz and G . Steinhaus , Stud ia Math . , 2, 231-247 (1930).9. B. S. Kashin, Trudy Mat. Ins t . Steklov [Proc. Steklov Inst. Math.], 145, 111-116 (1980).10. B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, M oscow (1984).11. J . Bergh and J . Lofs trom, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York (1976).12. E. D. Gluskin and S. Kwapien, Studia Math. , 114, No. 3, 303-309 (1995).13. S. Montgomery-Smith, Proc. Amer. Math. Soc. , 109, No. 2, 517-522 (1990).14. J . Jakubowski and S. Kwapien, Bull. Polish Acad. S ci. Ma th. , 27, No. 9, 689-694 (1979).15. G . Pisier, "Le s indgalit~s de Ka han e-K hintc hin d'aprds C. Borell," in: Sdminaire sur la gdometrie des espaces de Banach.Exposd Nos. 7. 1977-1978.16. N. H. Asmar and S. Montgomery-Smith, Arkiv . M ath. , 31 , No. 1, 13-26. ]~cole Polytechnique, Palaiseau (1993).17. S. B. Stechkin, I z v . Akad . Nauk SSSR Ser . Mat. [Math. USSR-Izv.], 20, 385-412 (1956).

SAMARA STAT E UNIVERSITY

Mathematical N otes, Vol. 66, No. 3, 1999

O n I n t e r p o l a t i o n i n t h e C l a s s e s E PD . Y a . D a n c h e n k o

KEY WORDS: inter pol ation problem, Jord an dom ain, A hlfors class, Smirnov class, c~-separated sequence.

w W e s ay t h a t a s im p l y c o n n e c t e d J o r d a n d o m a i n G o n t h e c o m p l e x p l a n e C b e l o n g s t o t h e c la ss A( A h l f o r s cl a ss ) i f it s b o u n d a r y ~ / = O G i s l o c a l ly r e c t if i a b l e a n d s u p { m e s l ( ' y ~ d ) / d i a m ( d ) } < o o , w h e r et h e s u p r e m u m i s t a k e n o v e r a l l o p e n d i s k s d C C ( s ee , f o r e x a m p l e , [ 1, 2 ]) . L e t A - - { z j } ~ = 1 b e as e q u e n c e o f p o i n t s i n G . T h i s s e q u e n c e is c a l l ed i n t e r p o l a t i n g ( i n t h e s e n s e o f C a r le s o n ) i f f o r a n y f a m i l yo f co m p l e x n u m b e r s { a j } ~ = 1 e 1~ 1 7 6 i t h [ [ aj [ [l c r --- s u p { l a j l : j e N } 0 a t a f i xe d p o i n t v E G . F u r t h e r , s u p p o s e t h a t t h e p r o d u c t B x ( z ) : = Y I j = I ~ j ( z ) c o n v e r g e si n G . F o r j e N s e t B x j ( z ) : = B ~ ( z ) A o j ( z ) . I t i s k n o w n t h a t t h e p r o p e r t y I B x j ( z j ) [ > 5 ( V j e N ) f o ra f ix e d ~ - - 5( A ) > 0 i s c h a r a c t e r i s t i c f o r a s e q u e n c e A t o b e i n t e r p o l a t i n g ( C a r l e s o n ' s t h e o r e m , s e e [3 , 4 ] ).

Trans la ted from Matematicheskie Zametki, Vo i. 66, No. 3, pp. 477-480, September, 1999.Original ar t ic le subm it ted Janu ary 12, 1998; revis ion subm it ted O ctober 27, 1998.3SS 0001-434611999/6634-0388 $22.0 0 (~)2000 Kluwer Acad emic/Plenu m Publishers

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