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F unc t iona l A na ly s i s and I t s A pp l i ca t i ons , V o l. 32 , N o . 3 , 1998

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

O n S e r i e s w i t h R e s p e c t t o t h e R a d e m a c h e r S y s t e m

i n R e a r r a n g e m e n t I n v a r i a n t S p a c e s " C l o s e " t o L c o

S. V . A s t a s h k i n UDC 5 1 7 .9 8 2 .2 7

L e t

r k ( t ) s i g n s i n 2 k - l r t

b e t h e s y s t e m o f R a d e m a c h e r f u n c t i o n s o n [ 0 , I ].

a c o

F o r a = ( k ) k = I E 1 2 w e d e f i n e t h e l i n e a r o p e r a t o r

c o

T a ( t ) = ~ a k r k ( t )

( k = 1 , 2 , . . . )

(t e [0, 11). (1)k = l

I f I l a l l p = (E =I l ak lP) 1 /p a s u su a l , t h e n i t fo ll o ws f ro m th e K h in c h in i n eq u a l i t y t h a t

I I T a l l L p [ O , J • [ la l 1 2 ( 2 )

f o r 1 _< p < o o ( t h is m e a n s t h e e x i s t e n c e o f t w o - s i d e d e s t i m a t e s w i t h c o n s t a n t s d e p e n d i n g o n p o n l y ) .

M o r e o v e r , w e c a n r e a d i l y v e r if y t h e r e l a t i o n

I I T a l l L | = I l a l l l . ( 3 )

M o r e d e t a i l e d i n f o r m a t i o n o n t h e b e h a v i o r o f s e r ie s w i t h r e s p e c t t o t h e R a c l e m a c h e r s y s t e m c a n b e o b t a i n e d

b y t r e a t i n g t h e s e s e r i e s in t h e f r a m e w o r k o f g e n e r a l r e a r r a n g e m e n t i n v a r i an t s p a ce s .

R e c a l l t h a t a B a n a c h s p a c e X o f m e a s u r a b l e f u n c t i o n s x = x ( t ) o n [0 , 1 ] i s sa id t o b e rea r ran g emen tin v a r i an t ( r . i. s .) i f t h e re l a t i o n s x * ( t ) < y * ( t ) ( t E [0, 1 ]) an d y E X im p ly t h e re l a t i o n ~ x E X an d

[[x[[ < [[YH (w he re z * ( t ) i s t h e n o n in c re as in g rea r ran g em en t o f a fu n c t i o n [z ( t) [ [1, p . 8 3 ]) . F o r an y r .i .s .

X on [0, 1] w e ha ve Loo C X C L1 [1, p. 124].

A n O r l ic z s p a c e is a n i m p o r t a n t e x a m p l e o f a n r . i. s. L e t N ( t ) b e a n i n c r e a s i n g c o n v e x f u n c t i o n o n

[0, co ) su ch t h a t N (0 ) = 0 an d l im, ,_ ~0N ( u ) / u = lim=_~cou / N ( u ) = O . T h e O r l ic z s p a c e L N co n s i s t s o f

a l l m e a s u r a b l e f u n c t i o n s x = x ( t ) on [0 , 1 ] such tha t

H x iI N = i n f { u > O : j oi N k u / ( [ x ( t) l ~ d t < l } < c o .

In p a r t i cu l a r , i f N ( t ) = t p (1 < p < co) , then L N = L p . I n t h e p r o b l e m s d i s c u s s e d b e lo w , a s p e c i a l r o l e

i s p la y e d b y t h e O r l ic z s p a c e L M , w h e r e M ( t ) = e x p ( t 2 ) - 1 , o r , m ore prec ise ly , th e c losur e of Lco in

t h e s p a c e L M , w h i c h is d e n o t e d b y G .

A s w a s p r o v e d i n [ 2] , t h e n o r m o f a n r. i. s. X i s e q u i v a l e n t t o t h e n o r m o f / 2 o n t h e l in e a r s p a n o f t h e

R a d e m a c h e r f u n c t i o n s ( i .e ., a r e l a t i o n s im i l a r to ( 2 ) h o l d s f o r X ) i f a n d o n l y i f X D G .

I n t h i s n o t e w e p r e s e n t r e s u l t s o n t h e b e h a v i o r o f s e r i e s w i t h r e s p e c t t o t h e R a d e m a c h e r s y s t e m i n a n

r .i .s . X t h a t i s i n t e r m e d i a t e b e t w e e n L c o a n d G . T h e m a i n r o l e h e r e i s p l a y e d b y n o t io n s a n d m e t h o d s

o f t h e i n t e r p o l a t i o n t h e o r y o f li n e a r o p e r a to r s .

F o r a B a n a c h p a ir ( X 0 , X 1 ) , x E X 0 + X 1 , a n d t > 0 , w e i n t r o d u c e t h e P e e t r e ~ - f u n c t i o n a l a s fo ll ow s :

J K ( t , x ; X o , X 1 ) = inf{HxoHx o + t H x l i I x , : x = x o + X l , x i E x i } .

S a m a r a S t a t e U n i v e r s it y , e - m a rl : a s t a s h k n ~ s s u . s a m a r a . e m n e t . r u . T r a n s l a t e d f r o m F u n k t s i o n a l ln y i A n a l i z i E g o P r i lo z h e -n i ya , V o l . 32 , N o . 3 , pp . 62 - 65 , Ju l y - Sep t ember , 1998 . O r i g i na l a r t i c l e submi t t ed O c t obe r 7 , 1996 .

1 92 0 0 1 6 - 2 6 6 3 / 9 8 / 3 2 0 3 - 0 1 9 2 $ 20 .0 0 C ) 1 9 9 9 P l e n u m P u b l i s h in g C o r p o r a t i o n

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Le t 310 be a su bsp ace of X0 an d le t 311 be a subs pac e of X 1. A pai r (Y0,311) is ca l led a Jc~-subpair o f

t h e p a i r ( X 0 , X 1 ) i f

y ; Y o , Y 1 ) • y ; X o , x l )

w i t h c o n s t a n t s t h a t d o n o t d e p e n d o n y E Y o + ] I1 a n d t > 0 .

I n p a r t i c u l a r , i f Y i = P ( X i ) ( i = 0 , 1 ), w h e r e P is a b o u n d e d l i n e a r p r o j e c t i o n i n X o a n d X 1 , t h e n

( Yo, Y1) i s a J~ - su b p a i r o f ( Xo , X1 ) [3 , p . 1 3 6 o f t h e R u ss i a n t r an s l a t io n ] . A t t h e sa m e t im e , t h e r e a r em an y ex am p les o f su b p a i r s t h a t a r e n o t Jg : -su b p a ir s ( see [ 3, p . 5 8 9 o f t h e R u ss i an t r an s l a t io n ; 4 ] an d

R e m a r k 1 o f th e p r e s e n t n o t e ) .

C o n s i d e r t h e c a s e i n w h i c h X o = L ~ , X 1 --- G , Y o = T ( l l ) , a n d ]I1 = T ( 1 2 ) , w h e r e T i s t h e o p e r a t o r

g i v e n b y ( 1 ) . I t f o l lo w s fr o m r e l a t i o n ( 3) a n d f r o m t h e a b o v e s t a t e m e n t o f [ 2] t h a t

9 g ' ( t , Ta ; T ( l l ) , T ( 1 2 ) ) • 3 g ' ( t , a ; l l , 1 2 ) . (4 )

I n s p i t e o f t h e f a c t t h a t T ( l l ) i s n o t c o m p l e m e n t e d i n L o o [ 5 ] , t h e f o l l o w i ng s t a t e m e n t h o l d s .

T h e o r e m 1 . T h e p a i r ( T ( l l ) , T ( 12 )) i s a ~ - s u b p a i r o f p a i r ( L oo , G ) . I n o t h e r w o r ds , b y (4 )

. Yd '( t, T a ; L o o , G ) x ~ ( t , a ; l l , 1 2)

(3Ow i t h c o n s t a n t s t h a t d o n o t d e p e n d o n a = ( ak ) k = l E 1 2 a n d t > O .

I n t h e p r o o f o f T h e o r e m 1 , t h e r e s u lt o f [ 6] o n t h e d i s t r i b u t i o n o f R a d e m a c h e r s n m s i s u se d .

R e m a r k 1 . T h e a s s e r t i o n o f T h e o r e m 1 f ai ls i f t h e s p ac e G is r e p l a ce d by L n f o r a n y p < c o .

E v e r y r .i . s. X n a t u r a l l y g e n e r a t e s t h e c o o r d i n a t e s e q u e n c e s p a c e o f t h e c o e f f ic i e nt s o f s e ri e s w i t h r e s p e c t

t o t h e R a d e m a c h e r s y s t e m . T h e o r e m 1 a l lo w s on e t o f in d t h is c o o r d i n a t e s p a c e f or th e c a s e in w h i c h X

i s a n i n t e r p o l a t i o n s p a c e b e t w e e n L o o a n d G ( t h a t i s , L oo C X C G a n d e a c h l i n e a r o p e r a t o r b o u n d e d

i n Lo o a n d G i s b o u n d e d i n X ) .

L e t E b e a B a n a c h i d e a l s p a c e o f t w o - s i d e d s e q u e n c e s s u c h t h a t ( r a i n ( l , 2 k)) ~~ E E . I f (X 0 , X 1 )

i s a B a n a c h p a i r , t h e n t h e s p a c e ( X 0 , X 1 ) ~ c o r r e s p o n d i n g t o t h e r e a l ~ - - m e t h o d o f i n t e r p o l a ti o n c o ns i st s

o f a l l x E X o + X 1 s u c h t h a t

I 1 1 1 = x ; X o , x l ) ) k l l < o o .

I n th e sp e c ia l ca se E = lp ( 2 - k ~ 0 < 8 < 1 , 1 < p _< c~ , we o b ta in th e c l a ss i ca l sp aces ( Xo , X1 ) 0 ,p ( f o r

t h e d e t a i l e d e x p o s i t i o n o f t h e i r p r o p e r t i e s , s e e [7 ]) .

I f a n r . i. s. X i s a n i n t e r p o l a t i o n sp a c e b e t w e e n L oo a n d G , t h e n i t is d e s c r i b e d b y t h e re a l ~ - m e t h o d [8 ].

T h i s m e a n s t h a t

x = ( L o o , ( 5 )

f or s o m e p a r a m e t e r E . T h e r e f o r e , T h e o r e m 1 i m p l ie s t h e f o ll o w in g a s s e rt i o n.

T h e o r e m 2 . L e t r. i .s . X b e a n i n t e r p o l a t i o n s p a c e b e t w e e n L o o a n d G a n d le t r e l a t i o n (5 ) h o l d . I nt h i s c a s e , :f o r t h e s e q u e n c e s p a c e F = ( l l , / 2 ) ~e" w e h a v e

I Irk • I I ( a ) l l F -_ X

T h i s r e s u l t s h o w s t h a t t h e c o r r e s p o n d e n c e e s t a b l i s h e d i n T h e o r e m 2 i s o n e - t o - o n e .

T h e o r e m 3 . L e t r . i .s . 's X o a n d X 1 b e t w o in t e r p o l a t i o n s p a c e s b e t w e e n L o o a n d G . I:f

akrk ~ ~ akrk ,

_ Xo k=l X 1

t h e n X o = X 1 , a n d th e n o r m s i n X o a n d X 1 a re e q u iv a le n t.

193

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T h e a b o v e r e s u l t s a l l o w o n e t o f i n d t h e s e q u e n c e s p a c e s o f th e c o e f f ic i e n ts o f s e r ie s w i t h r e s p e c t t o t h e

R a d e m a c h e r s y s t e m f o r c o n c r e te r . i .s .' s.

I n t h e e x a m p l e s b e l o w , ~ _> 0 i s a n i n c r e a s i n g c o n c a v e f u n c t i o n .

E x a m p l e 1 . T h e L o r e n t z r. i. s. A p ( ~ ) ( 1 < p < c ~ ) c o n s i st s o f a l l m e a s u r a b l e f u n c t i o n s x = x ( s ) s u c h

t h a t

I I = I I , , = d o ( s ) < o o .

I n p a r t i c u l a r , i f ~ ( s ) = l og 2 1- p2 / s a n d 1 < p < 2 , t h e n

akr k x ak p 9

_ "~P,P

E x a m p l e 2 . T h e M a r c in k i e w ic z r .i .s . M ( ~ ) c o n s is t s o f a ll m e a s u r a b l e f u n c t io n s x = x ( s ) s u c h t h a t

HXHMCcP) s u p { ~--~ t) ~0 t }x * ( s ) d s : O < t < _ l < o o .

I n p a r t i c u l a r , i f ~ ( t ) = t l o g 2 l og 2 1 6 / t , t h e n

Hk~_l II { 1 ~ a* )a k r k x JJ (a k )J Jh ( ln ) = sup l og2(2k ) i : k = 1 , 2 , . . . ,

_ M ( ~ ) i = 1

w h e r e ( a ~) i s t h e n o n i n c r e a s i n g r e a r r a n g e m e n t o f a s e q u e n c e (Jakl)k~176.

R e m a r k 2 . T h e o r e m s 2 a n d 3 s tr e n g t h e n r e s u lt s o f [ 2] , w h e r e s im i l a r s t a t e m e n t s w e r e o b ta i n e d f or

s e q u e n c e s p a c es F s a t i s fy i n g m o r e r e s t ri c t iv e c o n d i t io n s . F o r i n s ta n c e , w e c a n r e a d i l y s h o w t h a t t h e n o r m

o f t h e d i l a ti o n o p e r a t o r

f inn --" ( a l , . . . , a l , a 2 , . . . , a 2 , . . . )% 9 9 9

Y

n n

i n t h e s p a c e l l ( l n ) ( s e e E x a m p l e 2 ) i s e q u a l t o n . T h e r e f o r e , c o n d i t i o n ( 1 1 ) o f [ 2] f a i ls f o r t h i s s p a c e , a n d

t h e t h e o r e m s o b t a i n e d i n [ 2 ] c a n n o t b e a p p l i e d t o t h i s o p e r a t o r .

R e m a r k 3 . T h e o r e m s 1 - 3 c a n r e ad i l y b e e x te n d e d t o l a c u n a r y tr i g o n o m e t r i c s er ie s b y m e a n s o f r e su l ts

i n [ 9 ] .

R e f e r e n c e s

1 . S . G . K r e in , Y u . I. P e t u n i n , a n d E . M . S e m e n o v , I n t e r p o l a t i o n o f L i n e a r O p e r a t o r s [ in R u s s i an ] , N a n l ~ ,

M o s c o w , 1 9 7 8 .

2 . V . A . R o d i n a n d E . M . S e m e n o v , A n a l . M a t h . , 1 , N o . 3 , 2 0 7 - 2 2 2 ( 1 9 7 5 ) .3 . H . T r i e b e l , I n t e r p o l a t i o n T h e o r y , F u n c t i o n S p a c e s, D i f fe r e n t ia l O p e r a t o r s [ E n g l is h tr a n s l a t i o n ] , N o r t h -

H o l l an d , A m s t e r d a m - N e w Y o r k , 19 78 .

4 . R . W a l l s t e n , L e c t . N o t e s i n M a t h . , V o l . 1 3 0 2 , 1 9 8 8 , p p . 4 1 0 - 4 1 9 .

5 . V . A . R o d i n a n d E . M . S e m e n o v , F u n k t s . A n a l . P r i lo z h e n . , 1 3 , N o . 2 , 9 1 - 9 2 ( 1 9 79 ) .

6 . S . M o n t g o m e r y - S m i t h , P r o c . A m e r . M a t h . S o c . , 1 0 9 , N o . 2 , 5 1 7 - 5 2 2 ( 1 9 9 0 ) .

7 . J . B e r g h a n d J . L S f s tr S m , I n t e r p o l a t i o n S p ac e s. A n I n t r o d u c t i o n , S p r i n g e r- V e r la g , B e r l i n - H e i d e l b e r g -

N e w Y o r k , 1 9 7 6 .

8 , N . J . C a l t o n , S t u d i a M a t h . , 1 0 6 , N o . 3 , 2 3 3 - 2 7 7 ( 1 9 9 3 ) .

9 . J . J a k u b o w s k i a n d S . K w a p i e n , B u l l . P o l i s h A c a d . S c i. M a t h . , 2 7 , N o . 9 , 6 8 9 - 6 9 4 ( 1 9 7 9 ).

T r a n s l a t e d b y S . V . A s t a s h k i n

1 9 4