2o 3.

M. S~ Brodskii, T r i a n g u l a r and J o r d a n R e p r e s e n t a t i o n s of Linear O p e r a t o r s [in Russian], Nauka, Moscow (1969)o D. Sarason, "A r e m a r k on the V o l t e r r a o p e r a t o r , " J. Math. Anal. Appl~ 12, 244~246 (1965)o

4.

B. Sz.-Nagy and C. F o i a s , "Sur les c o n t r a c t i o n s de 1 ' e s p a c e de Hilbert, VII, Triangulation c a n oniques, Fonction m i n i m u m , " Acta Sci. Math., 25, Nos. 1-2, 12-37 (1964).

5.

D. Sarason, " G e n e r a l i z e d interpolation in (1967).

6.

N. Rustffield, "Inner f a c t o r s and Blaschke p r o d u c t s , " P r o c . Amo Math. Soc., 17, NOo 3, 572-579 (1966).

7.

N . K . Nikol'skii, "Five p r o b l e m s on invariant s u b s p a c e s , " Zap. Nauchno Semino LOMI (Leningrad), ~

H~" ," T r a n s . Am. Math. Soc., 127, No. 2, 179-203

115-127 (1971).

8~

Jo-P. Kahane and R. Salem, E n s e m b l e P a r f a i t e s et Series T r i g o n o m e t r i q u e , P a r i s (1963).

9~

No K~ Nikol'skii, "A c r i t e r i o n for weak invertibitity in s p a c e s of analytic functions distinguished by growth bounds," Zapo Nauchno Semino LOMI (Leningrad), ~ 106-129 (1972)o

10o

Do Sarason, " W e a k - s t a r density of p o l y n o m i a l s , " Jo Reine and Angewo Math~ 252, 1-15 (1972)~

TRIGONOMETRIC

APPROXIMATION

IN THE

SPACE

h ~ (R ~, ~) O.

Ao O r e v k o v a

1% Let

be a nonnegative function that is integrable in R~

dimension cL )o We c o n s i d e r the Hilbert space

( R~ is the Euclidean space of

5~(;~,~) of functions q~ such that

!~ i~C:~)Iz~- C ~ } ~ < ~ " The s c a l a r product in this s p a c e is given by the equation

Let

Mc R~

the s p a c e

be an a r b i t r a r y convex s e t having interior points. We denote by Z(M,~) the subset of L~(R~,~)

g e n e r a t e d by the functions exp~(ze, t), teM

o Here ~:,t)=x,t,....+~d

aim of this a~'ticle is to c l e a r up the analytic s t r u c t u r e of the space

o The

ZIM,~} under v a r i o u s a s s u m p t i o n s

about the function ~ ~ The p r o b l e m of d e s c r i b i n g the s p a c e s ZIM, ~) is d i r e c t l y connected with a p r o b l e m in the theory of r a n d o m p r o c e s s e s ~ Let ~(t), teR d , be a homogeneous r a n d o m field, i~

a p a r a m e t r i c f a m i l y of

r a n d o m v a r i a b l e s with a d - d i m e n s i o n a l p a r a m e t e r t satisfying the conditions: its m e a n re(t)= E~(~) Trans]Lated f r o m Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya M a t e m a t i c h e s k o g o Instituta im. V. A. Steldova AN SSSR, Vol~ 39, ppo 94-103, 1974.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, m any form, or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $Z50.

65

and c o r r e l a t i o n f u n c t i o n

We s e t

-~(~)-= 0

S ~,, t D = E [ ~ t t D - ~tL)]E~(t~)-~(tD]

a r e i n v a r i a n t with r e s p e c t to s h i f t s :

o We a l s o r e q u i r e t h a t f o r a n y t, taR s , g~m E Ie~(t* H,)- e, (t))2= O.

In t h i s c a s e the c o r r e l a t i o n f u n c t i o n [5 a d m i t s the r e p r e s e n t a t i o n

B(0c)= !~ e ~ t ' ~ ~-(d, 0a), where

Y(a~} is a f i n i t e m e a s u r e , c a l l e d the s p e c t r a l m e a s u r e of the f i e l d ~ . We l i m i t o u r s e l v e s

to the c a s e w h e n the m e a s u r e derivative

~(ct~) is a b s o l u t e l y c o n t i n u o u s with r e s p e c t to L e b e s g u e m e a s u r e .

Its

~ is c a l l e d the s p e c t r a l d e n s i t y of the h o m o g e n e o u s r a n d o m f i e l d ~ ~ We d e n o t e by H(M)

the m e a n - s q u a r e c l o s e d l i n e a r s p a n of the r a n d o m v a r i a b l e s ~ ( t ) , t e M. It t u r n s out [1 ] t h a t t h e r e e x i s t s an i s o m e t r i c c o r r e s p o n d e n c e b e t w e e n the s p a c e H [ R~) and the s p a c e Lz (R d, ~ ) . I n t h i s i s o m e t r i c c o r r e s p o n d e n c e the s u b s p a c e thespace

H (M ~ c o r r e s p o n d s to the s u b s p a c e z tM.~) . T h e r e f o r e , the p r o b l e m of d e s c r i b i n g

H(M) is e q u i v a l e n t to the p r o b l e m of d e s c r i b i n g the s p a c e Z;H,~) ~ I n t h e c a s e

q u e s t i o n of the s t r u c t u r e of the s p a c e s

Z IM,~) , w h e r e

ct=~ the

M is a s e g m e n t [ - a , 0.] , w a s c o m p l e t e l y s o l v e d

by Mo G. K r e i n [2]~ L a t e r , the w o r k of No L e v i n s o n and Ho M c K e a n a p p e a r e d in [3], w h e r e , t o g e t h e r with o t h e r p r o b l e m s , the s t r u c t u r e of the s p a c e

Z{~-a, a ] j )

is s t u d i e d u n d e r v a r i o u s r e s t r i c t i o n s on

the s p e c t r a l d e n s i t y ~ . H. M c K e a n and Ho D y m [4] a p p l i e d the s p a c e s of e n t i r e f u n c t i o n s c o n s t r u c t e d b y de B r a n g e s [51 to the s t u d y of the s t r u c t u r e of the s p a c e Z ([-a,a],E) . In t h i s p a p e r we s e e k c o n d i t i o n s on the s p e c t r a l d e n s i t y ~ u n d e r w h i c h the s p a c e be d e s c r i b e d with the h e l p of a s p a c e convex set having interior points.)

Z~

of e n t i r e f u n c t i o n s of s p e c i f i c type~ (Here H

F o r a p r e c i s e c h a r a c t e r i z a t i o n of the p h a s e

Z~

Z(H,~)

can

is a b o u n d e d

, we i n t r o d u c e the

following definitions. Definitions= 1o A d i r e c t i o n in R ~ is d e f i n e d to be a v e c t o r

c~= (c~,,...,c~), w h e r e

a,,..., a~ a r e

r e a l n u m b e r s s u c h t h a t a~ * 9 .. + c~= ~ ~ 2~ L e t @ b e a n a r b i t r a r y e n t i r e a n a l y t i c f u n c t i o n on the d - d i m e n s i o n a l c o m p l e x s p a c e ~:a o The P - i n d i c a t o r f u n c t i o n of q~ is d e f i n e d to be the f u n c t i o n ~.~ , ~r

s~p ~|

where

h.r

is d e f i n e d f o r any f i x e d

~c=(~, ..... ~1

and d i r e c t i o n

c~ = (~...... %) b y the f o r m u l a

3o L e t

McR ~

be s o m e c o n v e x s e t , ~ an a r b i t r a r y d i r e c t i o n .

is d e f i n e d to be t h e f u n c t i o n degree.

The c o n v e x s e t

M,

~, (~)= s ~ p (~, $ * . . . + ~ 9 ~ )

~eM

is d e f i n e d to be t h e

f u n c t i o n of t h i s s e t c o i n c i d e s w i t h the direction a.

5. We d e n o t e b y

Z~

The s u p p o r t f u n c t i o n of the s e t

~ 4o L e t @ be an a n a l y t i c f u n c t i o n of f i n i t e

P - i n d i c a t o r s e t of the f u n c t i o n

P -indicator function the s e t of f u n c t i o n s

e n t i r e a n a l y t i c f u n c t i o n of f i n i t e o r d e r in c ~ w h o s e

H

~,

@(z) , ioe.,

~,,r

~ tz), tf the s u p p o r t h,(c~)

f o r any

q~e L~(R~,~) , a d m i t t i n g e x t e n s i o n to an

P - i n d i c a t o r s e t i s c o n t a i n e d in M .

2 ~ D e f i n i t i o n . We s a y t h a t the w e i g h t f u n c t i o n E s a t i s f i e s the c o n d i t i o n ( A ) , o r is a f u n c t i o n of c l a s s A , ff

66

l...I ~'

R'

THEOREM.

r~=4 ( ,1 § ~

:~t

Suppose that the weight f u n c t i o n ~ s a t i s f i e s the c o n d i t i o n A . T h e n Z{~,f)( Z 7

for any bounded convex set

S .

P r o o f . We f i r s t c o n s i d e r the c a s e Let

)

d=2

.

cV be a n a r b i t r a r y f u n c t i o n in Z ( ~ , ~ ) .

T h e p r o o f that

q~ b e l o n g s to the s p a c e

5~

is

b r o k e n up into two p a r t s : f i r s t it is n e c e s s a r y to find an e n t i r e a n a l y t i c f u n c t i o n cp(z)= q~(z~,z~) , w h e r e z=(z~, z~)e c '~ , s u c h t h a t on the r e a l p l a n e it c o i n c i d e s a l m o s t e v e r y w h e r e with to p r o v e that in ~

r

r

, and then

is an e n t i r e a n a l y t i c f u n c t i o n of f i n i t e o r d e r with P - i n d i c a t o r s e t c o n t a i n e d

~ To find the a n a l y t i c f u n c t i o n

cp (z,, z~) we u s e the f a c t that r

c~'~ e~p ~(t,~ x~+t

w h e r e q~(x,, ~:.)= Z

x~)

F o r any n the f u n c t i o n ~(x,,x~) order

q~(z,, z~), w h o s e

:c )=

,

r

~ - . ,. . . .

(x,, ~ ) , )e I"1

and

O--K--~

c a n be e x t e n d e d in C~ to an e n t i r e a n a l y t i c f u n c t i o n of f i n i t e

P - i n d i c a t o r s e t is c o n t a i n e d i n M . We fix a r b i t r a r y

cp~,(~) = qv (z,, z D o We show that the s e q u e n c e function

~

rt~

cp:'(z~) c o n v e r g e s p o i n t w i s e in C' to a n e n t i r e a n a l y t i c

q~s (z~) ~ T h e n , a r g u i n g s i m i l a r l y , we p r o v e that f o r any z , , z ~ 6 r

~p~" (z,)= ~

c o n v e r g e s point-wise in C' to an e n t i r e a n a l y t i c f u n c t i o n

(z~, z~)

z,, z~ e C~ . We s e t

, the s e q u e n c e of f u n c t i o n s

q~:~(z,)

of one v a r i a b l e ~

Since, f o r any ,~ and f o r all (z,,z~)eC ~ , ~'(z~)~q~(z,,z~)=~(z,) , it follows that

and the f u n c t i o n

q~'(z~) c o i n c i d e s a l m o s t e v e r y w h e r e (with r e s p e c t to L e b e s g u e m e a s u r e ) with

(P(:c,,z~ , s i n c e the f u n c t i o n s where.

cp.(z,,~c,) c o n v e r g e to

T h u s , we m u s t show that f o r any

e n t i r e a n a l y t i c f u n c t i o n (P~' (z~) LEMMA. z,, ~,r r

r

p o i n t w i s e and to

z~ the s e q u e n c e

r

q~(r

almost every-

c o n v e r g e s p o i n t w i s e in C' to the

~ F o r t h i s , we need the following 1 e m m a .

Suppose that the weight f u n c t i o n f (~%, :%) is a f u n c t i o n of the c l a s s h ; t h e n f o r

, we c a n c o n s t r u c t f o r the f u n c t i o n

f(x,, x~)

any

a n o n n e g a t i v e i n t e g r a b l e f u n c t i o n A~'(z~),

s u c h that

M o r e o v e r , if

~t~,

J.~(R~, ~) , t h e n f o r any space

is a s e q u e n c e of t r i g o n o m e t r i c p o l y n o m i a l s in

z,, z~eC' , the s e q u e n c e of f u n c t i o n s c p ~ ' ( ~ = q~(z~,x~),n~ , c o n v e r g e s i n the

B~(R', a ~' (:%) . P r o o f of L e m m a .

We c o n s i d e r the c a s e w h e n

p o s i t i v e i m a g i n a r y p a r t (if be c o n s i d e r e d s e p a r a t e l y ) . ~ e R,

L~(R~, ~) t h a t c o n v e r g e s in

z, e C'

is a n a r b i t r a r y c o m p l e x n u m b e r with

Y~ z, < 0 , t h e n s i m i l a r a r g u m e n t s a r e r e q u i r e d ; the c a s e Since

~ (x,, :r

is a f u n c t i o n of the c l a s s

Jm z, :0

will

A , we have f o r a l m o s t e v e r y

that

67

(*) We r e d e f i n e the f u n c t i o n holds f o r any

(*)

~(~,, ~c,) on a s e t of z e r o m e a s u r e in s u c h a way that the c o n d i t i o n

~c~e~~ ~ We fix an a r b i t r a r y

~c~ ~' . The f u n c t i o n

~. ~,)= ~(~c~,~c~) s a t i s f i e s the

condition:

C o n s e q u e n t l y , it c a n be w r i t t e n in the f o r m of a f u n c t i o n

~

b e l o n g i n g to the s p a c e

~ (~c,)= I~**(x,~l ~ , w h e r e

q~

a r e the b o u n d a r y v a l u e s

H ~ in the u p p e r h a l f - p l a n e o We take the weight ~ ' ( : ~

to be the f u n c t i o n

We v e r i f y that f o r any fixed

z~=x/~?

tu~,~C) we have

& (:c~)dxz
F o r t h i s , we p r o v e the i n e q u a l i t y 1'

(**)

R

For any

x,e ~

the f u n c t i o n

~(z 3

c a n be r e p r e s e n t e d in the f o r m of the P o i s s o n i n t e g r a l of its

boundary values:

F r o m the l a s t e q u a l i t y we get

The i n e q u a l i t y

!, Jen's~'c=~)l,4:~ ~;~ ~ ~,co is p r o v e d s i m i l a r l y .

To p r o v e that the s e q u e n c e of f u n c t i o n s L~C;~~, ~'/~%)),

q~(x~)= q~ (z~, ~c=), ~-_ 4 , c o n v e r g e s in the s p a c e

we need to u s e the i n e q u a l i t y

It is p r o v e d j u s t as the i n e q u a l i t y (* *) w a s p r o v e d . If

68

Vrn z~ < 0

F o r the c a s e

, t h e n the proof is a n a l o g o u s to the one g i v e n .

~m z~ ~ 0 the l e m m a is p r o v e d .

~mz~=O . To r e d u c e it to the p r e c e d i n g one we fix an a r b i t r a r y

We c o n s i d e r the c a s e w h e n number

V , 0 and c o n s t r u c t a s e q u e n c e of t r i g o n o m e t r i c p o l y n o m i a l s ~c~c~, =c0

=

~cD, ~

%.(~.,+r,~,

=

~, Z , . . .

9

We t a k e the w e i g h t f u n c t i o n c~to be the f u n c t i o n c~
bHR~,~)

/A3 , and the s e q u e n c e

. I t h a s a l r e a d y b e e n p r o v e d t h a t f o r any

{~,~t,.~ c o n v e r g e s in the s p a c e

I~(R ',

s t r u c t e d with r e s p e c t to the f u n c t i o n d ( x , , ~ )

z,=~+~$,

~'I~0,~ , w h e r e

just as

o It is e a s y to v e r i f y

e~(x~)

{~}~ converges

suchthat

~o

the

is the w e i g h t f u n c t i o n c o n -

A~'{~c~) is c o n s t r u c t e d with r e s p e c t to the

function ~. z,

Since ~ ( z ~ , ~ D = cp.(~+~,~)~ , if f o l l o w s t h a t Y:C~) = r f o r a n y =,, x~eR' , the s e q u e n c e

2)

for

z~=~,-~

o Consequently,

{cp~ t~*., c o n v e r g e s in the s p a c e L~(~, ~'(=~0) , w h e r e A~'/x~)= 0~"~ c ~ ) .

The l e m m a is p r o v e d . Contim~ing w i t h the p r o o f of the t h e o r e m , we a p p l y to the s e q u e n c e

O:'(~c~ the t h e o r e m of

L e v i n s o n and M c K e a n , w h i c h s a y s t h a t f o r a n o n n e g a t i v e i n t e g r a b l e f u n c t i o n

A s a t i s f y i n g the c o n -

dition

we h a v e the e q u a l i t y ~~ z ( [ - a - ~, ~ a + ~~] , a ) = Z ( [ - a , a ]. ,. a. .~. = . Z~ F r o m the d e f i n i t i o n of the f u n c t i o n s s p a c e Z(I, A~') , w h e r e r

t h a t i s i t s l i m i t in

I

it f o l l o w s t h a t the s e q u e n c e

is s o m e b o u n d e d i n t e r v a l on t h e l i n e .

Z(I, 5 ~' )

r

in

a l s o b e l o n g s to Z(I, 5 " ) = Z ~ ,

C

p o i n t w i s e and to

r

{@2' }~,

, s i n c e A "'

T h i s m e a n s t h a t the f u n c t i o n

r

. On the r e a l p l a n e the f u n c t i o n

e v e r y w h e r e w i t h r e s p e c t to L e b e s g u e m e a s u r e w i t h

a, a e ( 0 , ~ )

~

b e l o n g s to the

F r o m t h i s it f o l l o w s t h a t the f u n c t i o n

I

of t h e t h e o r e m of L e v i n s o n and M c K e a n . entire analytic function

q~:'

f o r any n u m b e r

$(~,, ~,~ , s i n c e

s a t i s f i e s the c o n d i t i o n

c a n b e e x t e n d e d to an q~(z,,z,)

cP~x,,~,)

concides almost

c o n v e r g e s to cp|

almost everywhere.

It is nc,w n e c e s s a r y t o p r o v e t h a t the

P - i n d i c a t o r s e t of the f u n c t i o n

cp~Cz,,z,)

is c o n t a i n e d

in the s e t H , t h a t i s , the i n e q u a l i t y

where Let

}, (k)

is the v a l u e of the s u p p o r t f u n c t i o n of the s e t

k=~k.k,} b e an a r b i t r a r y

d i r e c t i o n , and l e t

a rotation c~rrying the arbitrary point We s e t ~,~/~,~0~ r

x~)

~=(~,z~)

~'=(x$, ~)

M , holds for all directions

~ ={k,, k,)

b e an a r b i t r a r y p o i n t in ~ o L e t

.

A~ be

into the p o i n t A ~ =~,~, x ~ ) = c ~ , k , + ~ , ~ , ~ - ~ , ~ , l

~ In t h e f o l l o w i n g a r g u m e n t s it is i m p o r t a n t t h a t

and c o n s i d e r the s e q u e n c e of t r i g o n o m e t r i c p o l y n o m i a l s

69

The s e q u e n c e

l~V.}~.< converges in the space

On t h e o t h e r hand, the e q u a l i t y V~(z)= r

L'(R', ~ " ( ~ , +=)) , and

z, =:.~,z) m e a n s that the t r i g o n o m e t r i c p o l y n o m i a l s

~ ( z ) , ~ % t . . , h a v e the f o r m :

~(z)= Z

d:e:~P~(k,t::'+Lt~'~)z, cp (z,, z~), a r e c o n t a i n e d , b y d e f i n i t i o n in the

where d ,(~' = c'~eoc , p ~(t~, '~):c,* § c~, '~' ~ * ) o The p o l y n o m i a l s

space

Z(M, ~)

t h a t i s , the p o i n t s

t"'=~.'~' t '~) (~K~K'~' n=~,. )

and f r o m the definition of the support function at the point -X,,(-L)-- k, t,, + L,t:~ ~F,,(L), (here

- L = (~L,-L,))o T h e r e f o r e , the s e q u e n c e

;% (~)], ~

(~,+~)) o F r o m t h e s e q u e n c e

Lo=- ~c,~(~-~.~-~

~,~(R,,a~(a ~)) to a f u n c t i o n

function

aP~(z)

e ~to~

is c o n t a i n e d in the s p a c e

we p a s s to the s e q u e n c e

e '~o~ ~

From this

L=(~. L,) it follows that

By the t h e o r e m of L e v i n s o n and M c K e a n , the s e q u e n c e

in the s p a c e

H

O~K~K ~''~, n,= O, 4 , . , . .

f~(z)}

~}~.,

b e l o n g to the s e t

(~r

Z([-)%(~t),

te'~'~c:r

, where

te ~to= ~ ( ~ }

converges

t h a t c a n be e x t e n d e d to an e n t i r e a n a l y t i c

of f i n i t e o r d e r , and (;~)§ IZL~r

F r o m t h i s it f o l l o w s t h a t

~

is an e n t i r e a n a l y t i c f u n c t i o n of f i n i t e o r d e r , and

It r e m a i n s to r e m a r k t h a t ~ Remark.

( i ~)= ~ (x, + L~, ~, ~% ~ ~t~ x ) o The t h e o r e m i s p r o v e d .

We c a n p a s s to the c a s e

d~ z

b y i n d u c t i o n on

~ The p o s s i b i l i t y of the i n d u c t i o n

s t e p is p r o v e d j u s t a s in the t h o e r e m . 3 ~ Definition.

By z(M +, ~)

we d e n o t e the s e t

Z(M+,~)= ~ ZCM+K~,~)

w h e r e K~=tt=(t,,..

ta):

4

-~--tz--~- , ~= q d } T H E O R E M 2. Suppose t h a t the n o n n e g a t i v e f u n c t i o n f h a s the f o l l o w i n g p r o p e r t y : 9 c~=

~

~ (x)

the f u n c t i o n

~ e R~, s a t i s f i e s the c o n d i t i o n

R'

Then

z I M*, ~) = Z~ Proof.

f o r any b o u n d e d c o n v e x s e t g w i t h n o n e m p t y i n t e r i o r ~

F o r a p r o o f of the t h e o r e m we u s e a f u n c t i o n of S. N. B e r n s h t e i n t h a t is d e f i n e d in the

f o l l o w i n g w a y [8, po 376]. F o l l o w i n g B e r n s h t e i n , we s p e c i f y a s e q u e n c e 4

70

tt~l . . . .

t~ > 0

, for which

and we s e t

k=l

4 +

is ~:a e n t i r e a n a l y t i c f u n c t i o n of type on

~0oo)

-

7 z -

Z ~

~

.

_~ ~ . G i v e n a m o n o t o n i c a l l y d e c r e a s i n g f u n c t i o n c}

t]hat s a t i s f i e s the c o n d i t i o n (*) and c o n v e r g e s to 0 a s

~+oo

, Vo A. M a r c h e n k o in

[9] c o n s t r u c t s a B e r n s h t e i n f u n c t i o n B such that f o r any ~ , ~ - o , and any n a t u r a l n u m b e r s the following i n e q u a l i t y h o l d s :

Cs(~} 9(u)

B(t~)~ (4+M~}s-,

(**)

~ e ( 0 , oo).

We p a s s f r o r a the B e r n s h t e i n f u n c t i o n to the f u n c t i o n of 4 v a r i a b l e s

..

B(z~,...,

zd.)=

~1

.

Z~

Z~ ...+ Zd

F r o m the s p e c i f i c a t i o n of the f u n c t i o n ~ and the i n e q u a l i t y (* *) it follows that for any i n t e g e r -~ ~-~C(~)o F u r t h e r , we s e t Let

~ be art a r b i t r a r y f u n c t i o n in

f u n c t i o n q~B~ b e l o n g s to t a i n e d in

I~(R~, # )

z~

Bjz}= B ( ~z,, . . . ,

o We now apply a n o t h e r t h e o r e m of P o l y a and P l a n c h e r e t [7, po 401]. cp B~ b e l o n g s to the s p a c e

We fix a n a r b i t r a r y n u m b e r

~pr

e, e ~ o

~rn ~

I

/4-B(;~*)I-~

T M

for

I,I
The i n c l u s i o n Z (~fl+, ~) ~ Z ~

. C o n s e q u e n t l y , cp B~ b e l o n g s

I m I:~} t~ ~-lsc} 4 :r.. < ~.

FV;~)I-- N ~ The f u n c t i o n B is c o n t i n u o u s and B(0)= ~ . We c h o o s e

r

B~).

is c o n t i n u o u s , h e n c e , on the c o m p a c t s e t

I~l<~

Z{m + ~

According

o We find f o r it a n u m b e r A s u c h that

I~b.A

for

i n c l u s i o n ~ / ~ + , ; ) ~ z~

o F r o m the i n e q u a l i t y j u s t p r o v e d it follows t h a t f o r a n y r~ the

to Z r M + K , ~) o It r e m a i n s to p r o v e that

II-B~I-<@

o Weprovethe

~/IR~,~I , and, by a t h e o r e m of P o l y a and P l a n c h e r e l [6, p~ 150], it is c o n -

to t h i s t h e o r e m , the f u n c t i o n

The f u n c t i o n e

~ -~)

~= ~

t~: Ixl~ At

its m o d u l u s is b o u n d e d :

, so t h e r e e x i s t s

~, ~ > 0 , s u c h that

o F o r any ;% ~,~(0,~D , we have the i n e q u a l i t y

o We e s t i m a t e the n o r m

llq:-qVB II f o r

r~-[-~/]+~

:

o

is provedo The c o n v e r s e i n c l u s i o n follows f r o m the t h e o r e m of Seco 2 ,

w h i c h is a p p l i c a b l e h e r e in an o b v i o u s way. 4 ~. H e r e we p r o v e a t h e o r e m that shows that the s i t u a t i o n d e s c r i b e d in the p r e c e d i n g s e c t i o n s is not g e n e r a l . T H E O R E M 3. Suppose that the f u n c t i o n ~ s a t i s f i e s the c o n d i t i o n

71

i

I,'' R'

I e~X~l~l l~~

~ , ~>0,~=I,...,~ , T h e n

for some

Z(M,~)=L~(R r

f o r any b o u n d e d c o n v e x s e t M c o n t a i n i n g

interior points~ Proof.

L e t

~ (R~, ~)

r~ of the f u n c t i o n ,~ c a n be e x t e n d e d to a f u n c t i o n

o We p r o v e that the F o u r i e r t r a n s f o r m a t i o n

~. (z)

that is h o l o m o r p h i c in the p o l y d i s k

Indeed, the i n t e g r a l I

e "~'::'

~P(:~)~(~c)d.s~

c o n v e r g e s a b s o l u t e l y and u n i f o r m l y on any c o m p a c t s e t

R~

~ c D'

, s i n c e f o r any

z

, z e ~< ,

R' 4

~(x, ..... m~)~m,.., c~m~)=-oo .

Rj,., I e

We r e c a l l the following d e f i n i t i o n . The r e g i o n if it has the f o r m :

~={=: ~

z e~}

, where

Hc C~ is s a i d to be a - c o n v e x t u b u l a r r e g i o n

~ is a convex r e g i o n in

H . We c o n s i d e r the c o n v e x t u b u l a r r e g i o n

~ ' with b a s e the s e t

a t h e o r e m of Io Vo O s t r o v s k i i [10, po 230], the f a c t that function

Pc(z)

Rd , c a l l e d a b a s e f o r the r e g i o n

t z : Rez=O,

P~{z) is h o l o m o r p h i c in

I~zl

< ~-t

~ By

~)' i m p l i e s that the

c a n be e x t e n d e d to a f u n c t i o n h o l o m o r p h i c in H ' .

We c o n s i d e r a n a r b i t r a r y bounded c o n v e x s e t that is o r t h o g o n a l to the s u b s p a c e

McRd ~ L e t

Z(M,t1 o T h i s m e a n s that f o r any f u n c t i o n V in

z (M, f).

we

have

P a s s i n g to the F o u r i e r t r a n s f o r m a t i o n , we o b t a i n

R a-

for any function and, s i n c e that

,~(%

'

that is s q u a r e i n t e g r a b l e on the s e t M . C o n s e q u e n t l y ,

McH' , w h e r e

q)(:~}~(:r.)=O

I'1

Pr

is a n a l y t i c , it follows that

a l m o s t e v e r y w h e r e in R~ o If

is c o n c l u d e d . We s u p p o s e t h a t

P,(t)=O

for

teM

fC~c)>0 a l m o s t e v e r y w h e r e on R~ , then the p r o o f

~(~)-=o on a s e t of p o s i t i v e L e b e s g u e m e a s u r e ; t h e n we i n t r o d u c e a

new s p e c t r a l d e n s i t y

t~(:c) , if ~(~):~0 ~(x)= @-~-'" , if We apply the above a r g u m e n t s to the f u n c t i o n e ~c~'~'

72

~m ~'

>- c'$'

O~KeK~

'"

~(~r

,

V.[z)=-O in H' o F r o m this it follows

~(sc)= 0 .

. T h i s m e a n s that f o r any '~

t e M

we have

Then e 'r

= s

7__

c':' e0cp L(t.'~', ~c),

since 9(~),~c~) for all ~ . The theorem is proved.

LITERATURE

CITED

Io

A.M. Yaglom, "Some classes of random fields in ~ -dimensional space that are related to stationary random processes," Teor. Veroyatn. Ee Primeneno, 2, No. 3 (1957).

2.

Mo G. Krein, "On the basic approximation problem in the theory of extrapolation and filtration of stationary random processes," Dokl. Akado Nauk SSSR, 94, No. 1 (1954)o

3~

No Levinson and H. P. McKean, Jr., "Weighted trigonometrical approximation on R I with application to the germ field of a stationary Gaussian noise," Acta Math., 112, 99-143 (1964)o

4.

H. Dym and Ho Po McKean, Jr., "Application of de Branges spaces of integral functions to the predictions of stationary Gaussian processes," Illinois Jo Math~ 14, No~ 2 (1970).

5o

Lo de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, (1968).

6.

S.M. Nikol'skii, Approximation fin Russian], Moscow (1969).

7.

B.A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex [in Russian], Fizmatgiz, Moscow (1962).

8.

No Io Akhiezer, Lectures on Approximation Theory [in Russian], Moscow (1965)o

9.

V . A . Marchenko, "On certain questions for the approximation of continuous functions on the whole real line, III," Soobshch. Kharko Mat. Obshch., 22 (1950)o

10.

Yu. V. Linnik and I. V. Ostrovskii, Expansion of Random Variables and Vectors [in Russian], Moscow (1973)o

A OF

CONTINUOUS G.

ANALOGUE

Englewood

Cliffs, New Jersey

of Functions of Several Variables and Embedding

OF

A

Theorems Variables

THEOREM

SZEGO V. N.

Solev

1o I n t r o d u c t i o n Let ~ be a nonnegative integrable function defined on the unit circle

tz:tzl=41 , ~ i s its

k-th F o u r i e r coefficient x

~,=z--~ I ~r(z)z-~cLX (z=e ~', K=O,+-t,...),

(1)

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 39, pp. 104-109, 1974.

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73