Approximation Theory and Functional Analysis 1977: International Symposium Proceedings 0444852646, 9780444852649

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APPROXIMATION THEORY AND FUNCTIONAL ANALYSIS

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NORTH-HOLLAND MATHEMATICS STUDIES

35

Notas de Matematica (66) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Approximation Theory and Functional Analysis Proceedings of the International Symposium on Approximation Theory, Universidade Estadual de Campinas (UNICAMP) Brazil, August 1-5, 1977 Edited by

Joio B. PROLLA Universidade Estadual de Campinas. Brazil

1979

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

0

OXFORD

0 North-Holland Publishing Company, 1979

All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 7204 1964 6

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors for the U.S.A. and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

I n t e r n a t i o n a l Symposium on Approximation Theory, Universidade Estadual d e Campinas, 1977. Approximation t h e o r y and f u n c t i o n a l a n a l y s i s . (Notas de matemdtica . 66) (North-Holland mathematics s t u d i e s ; 35j Papers i n English o r French. Includes index. 1. Functional analysis--Congresses. 2 . Approximation theory--Congresses. I. P r o l l a , Joao B. 11. Universidade E s t a d u a l de Campinas. 111. T i t l e . I V . S e r i e s , QAl.N86 no. 66 [QA3201 510'.8s [ 5 1 5 ' . 7 1 78-26264 ISBN 0-444-85264-6

PRINTED IN THE NETHERLANDS

FOREWORD

T h i s book c o n t a i n s t h e P r o c e e d i n g s of t h e I n t e r n a t i o n a l Sympo-

sium on Approximation Theory h e l d a t t h e U n i v e r s i d a d e Campinas (UNICAMP), B r a z i l , d u r i n g August 1 - 5 ,

1977.

Estadual

de

Besides

the

t e x t s of l e c t u r e s d e l i v e r e d a t t h e Symposium, it c o n t a i n s some papers by i n v i t e d l e c t u r e r s whowere u n a b l e t o a t t e n d t h e m e e t i n g . The Symposium w a s s u p p o r t e d by t h e I n t e r n a t i o n a l Union, b y t h e Fundaqao d e Amparo 5 P e s q u i s a do E s t a d o

Mathematical de

,550 P a u l o

(FAPESP), by German and S p a n i s h government a g e n c i e s , and by

UNICAMP

itself. The o r g a n i z i n g committee w a s c o n s t i t u t e d by P r o f e s s o r s Machado, Leopoldo Nachbin, Joao B . P r o l l a ( c h a i r m a n ) ,

Silvio

and

Guido

Zapata. W e would l i k e t o t h a n k P r o f e s s o r U b i r a t a n D’Ambrosio, d i r e c t o r

of t h e I n s t i t u t e of Mathematics o f UNICAMP, whose s u p p o r t

made

the

m e e t i n g p o s s i b l e . Our s p e c i a l t h a n k s a r e e x t e n d e d to Miss E l d a M o r t a r i who t y p e d t h i s volume.

Joao B . P r o l l a

V

This Page Intentionally Left Blank

TABLE O F CONTENTS

R. ARON,

J.

.

1

. . .

13

. . . . . .

19

P o l y n o m i a l a p p h o x i m a t i o n and a q u e o t i a n 06 G.E.Skieov. A n a l y t i c h y p o e l l i p t i c i t y 0 6 a p e h a t o h n 06 paincipae type . . . . . . . . . . . .

BARROS NETO,

. .

.

.

. . .

H . BAUER,

Kahawkin apphoximatian i n dunctian npacen.

K.

an compact n e t n , a p p h a x i m a t i a n a n p h o d u c t n c t n , and t h e apphoximation phopehty . . , . . . , . . ,

D. BIERSTEDT,

A hemath a n v e c t a h - v a l u e d apphaximatian

.

B.

.

.

. . .

T h e c o m p l e t i o n 0 6 p a h t i a l L y a h d e t e d wectah dpacen . . . . . . . . . . . , . and KOhOWhin'b t h e o h e m

BROSOWSKI,

. .

63

. . . . .. . . .. . ..

71

. . .

121

de wahiabLe.4

..

133

Mehamokphic unidahm a p p h a x i m a t i a n a n c e a s e d n u b n e t s a d o p e n Riemann nuhdacen . . . . . . .

. .

139

. . .

159

P . L . BUTZER,

.

R. L .

STENS and M.

WEHRENS,

g e b h a i c canvalLLtian i n t e g a a l o

A p p h a x i m a ~ a nb y d-

Nan-ahchimedean w e i g h t e d a p p h o x i m a t i o n

J. P.

Q. CARNEIRO,

J. P.

FERRIER, T h z o k i e

P . M.

GAUTHIER,

n p e c t h a l e en une i n d i n i t E

.

.

C. S . GUERREIRO, W h i t n e y ' n n p e c t h a l n y n t h e b i b t h e o h e m

. . . . .

d i n i t e dimennionn G.

37

G. LORENTZ a n d S . D.

,

RIEMENSCHNEIDER,

Bihkhodd i n t e k p o L a t i a n

Rec.ent

phogenn

.

-

* .

.

in

.. . . . . . . . . . . .. . .

P. MALLIAVIN, A p p k o x i m a t i o n poLynamiaLe p o n d e k z e e t C a f l O f l i Q U U . .

i n in-

. . . . . . . .

. . . .

vi i

187

phoduitn

. . .. - .- .. . -

237

viii

TABLE OF CONTENTS

R . M E I S E , Spacen a d d i d d e h e n t i a b l e d u n c t i o n n and

t i o n phapehty.

. . . . .

,

.the u p p o x h a -

. . . . . . . . . . . . . .

L . NACHBIN, A l o o k a t a p p h o x i m a t i a n t h e o h y

. . . . . . . . . .

,

309

. .

333

. . .

343

L . N A R I C I and E . BECKENSTEIN, Banach a l g e b h a ovm valued ~L&dh

P h . NOVERRAZ, A p p h o x i m a t i a n a d p L u h i n u b h a k m o n i c d u n c t i o n n . 0. T . W.

PAQUES, T h e a p p h o x i m a t i o n p h o p e h t y d o h c e h t a i n npacen

. . . . . . . . . . . . . . .

351

. .

371

. . . . . . . . .

383

..

409

. . . .

421

.. .... . . . . .. . .. .

429

......

445

o d h o l o m o h p h i c mappingn. J . B . PROLLA,

The a p p h o x i m a t i a n p h o p e h t y d o h Nachbin n p a c e n .

I . J . SCHOENBERG,

I)n c a h d i n a l n p L i n e n m o a t h i n g

0 6 e c h e l o n KB,#~e-Schwahtz npacen

M. VALDIVIA, A c h a h a c t t h i z a t i o n

D . WULBERT, T h e h a t i a n a l ? a p p h o x i m a t i o n a d h e a l d u n c t i o n n G.

ZAPATA,

lndtx.

263

Fundamental? neminahmn

. . . . .

,

.

,

. . . .

,

. . . . . . . . . .

Approximation Theory and Functional Analysis J. 8. Prolla led. I 0 North-Holland Publishing Company, 1979

POLYNOMIAL APPROXIMATION AND A QUESTION OF G. E.

SHILOV

RICHARD M. ARON

I n s t i t u t o de Matemztica Universidade Federal

do Rio de J a n e i r o

Caixa P o s t a l 1835, z c - 0 0 2 0 . 0 0 0 Rio de J a n e i r o , B r a z i l

and School of Mathematics University

ABSTRACT

Let

s p a c e . For

of

Dublin

39 T r i n i t y

College

Dublin

Ireland

2,

E be an i n f i n i t e d i m e n s i o n a l r e a l o r complex

n =0,1,2,.

.. , m ,

let

Banach

a n ( E ) be t h e a l g e b r a g e n e r a t e d

by

a l l c o n t i n u o u s polynomials on E which a r e homogeneous o f d e g r e e ( n . u n ( E ) with respect t o s e v e r a l

W e d i s c u s s t h e completion of

natural

t o p o l o g i e s , i n t h e r e a l and complex c a s e . I n p a r t i c u l a r , weprove that when

i s a complex Banach s p a c e whose d u a l h a s

E

T~ - c o m p l e t i o n of

property, then the

t h o s e holomorphic f u n c t i o n s compact

-+

Q:

approximation

whose d e r i v a t i v e

with

a f : E + E l is

.

Let

ball

f :E

the

a 1( E ) c a n be i d e n t i f i e d

B1.

E

be a Banach s p a c e o v e r

For e a c h

n

c o n t i n u o u s polynomials s u p { I I P ( ~ ) I:I x

E

E

IN

,

P :E

let -+

B ~ )( P ( O E , F )

IK= IR o r

a:, w i t h c l o s e d

-

unit

P(nE,F) be t h e s p a c e o f n-hctruxJeneous F, normed by E F).

P E P(nE,F)

P(E,F) is

11 P 11

the s p a c e of mcon-

t i n u o u s l y F r g c h e t d i f f e r e n t i a b l e f u n c t i o n s from E t o

F and

H(E,F)

i s t h e s p a c e o f holomorphic mappings from E t o F , where E and are complex Banach s p a c e s . Throughout, i f t h e 1

range

space

F

F

is

ARON

2

HE)= H(E,C).

F = IK i s u n d e r s t o o d ; t h u s f o r example

suppressed, then

I n t h i s p a p e r , w e c o n s i d e r v a r i a t i o n s on t h e f o l l o w i n g problem posed by G . E . S h i l o v [ 8 ]

.

F o r each

n = 0,1,2,.

.., ,

b e t h e a l g e b r a g e n e r a t e d by t h e c o l l e c t i o n o f f u n c t i o n s CL)

5 n; thus

j

a (E) =

( ~ " ( E ) , T ) " of

topology

P(jE),

E P ( E ) . Then, what is t h e

E lN,

completion

a n ( E ) w i t h r e s p e c t t o some s p e c i f i e d l o c a l l y

on

T

P("E)

n@j:

an ( E )

let

03

convex

a n ( E ) ? I n t h e r e a l c a s e , t h i s problem h a s been con-

s i d e r e d by many a u t h o r s . I n S e c t i o n 1, w e b r i e f l y o u t l i n e some recent r e s u l t s i n t h i s c a s e . When E i s a complex Banach s p a c e ,

the

above

problem h a s a p p a r e n t l y n o t been s t u d i e d . I n S e c t i o n 2 , w e d i s c u s s the c o m p l e t i o n of

.

1

u ( E ) and

a m ( E ) f o r s e v e r a l c o m o n t o p o l o g i e s on the

(Related r e s u l t s w i l l a l s o appear i n [ 1 1 .) I n p a r t i c u l a r , we c h a r a c t e r i z e t h e completion o f A 1 ( E ) a s a s p a c e o f anaH(E)

space

l y t i c f u n c t i o n s h a v i n g weakly uniformly c o n t i n u o u s d e r i v a t i v e s ,

and

i n t e r m s o f compact holomorphic mappings. Some o f t h e r e s u l t s i n t h i s p a p e r were o b t a i n e d w h i l e t h e

au-

t h o r was a v i s i t o r a t t h e I n s t i t u t o d e Matemstica, U n i v e r s i d a d e

Fe-

d e r a l d o Rio d e J a n e i r o ,

s u p p o r t e d i n p a r t by t h e CNPq and FINEP, t o

which t h e a u t h o r e x p r e s s e s h i s g r a t i t u d e .

SECTION 1.

Among t h e most n a t u r a l , and so f a r u n s o l v e d , v e r s i o n s of

t h e q u e s t i o n of S h i l o v i s t h e f o l l o w i n g . Given E , d i m e n s i o n a l Banach s p a c e , l e t

0

T~

a

real i n f i n i t e

d e n o t e t h e t o p o l o g y on um(E) = P(E)

g e n e r a t e d by t h e f a m i l y o f norms

where Bm = {x A

F ( E ) 'b

0

E

E :

.

o f am(E)

[ 1x1 I 5 m

.

Then, c h a r a c t e r i z e t h e c o m p l e t i o n

W e r e c a l l t h a t t o e a c h polynomial P

E

P("E)

c o r r e s p o n d s a u n i q u e symmetric c o n t i n u o u s n - l i n e a r mapping A : E x E x Z

Ax".

...

x E +

K , v i a t h e t r a n s f o r m a t i o n P(x) = A ( x ,

Thus, s i n c e

..., x)

POLYNOMIAL APPROXIMATION AND A QUESTION OF SHILOV

3

1 + ...

for

P

E

P(nE)

element i n

I

x, y

P(E)

,

E Bm

,

and a c o n s t a n t

Cm

,

w e c o n c l u d e t h a t every

and hence e v e r y e l e m e n t i n ( P ( E ) , T : ) ~

c o n t i n u o u s on bounded s u b s e t s o f Nemirovskir and Semenov [ 6 1

E

.

is uniformly

However, i t h a s been shown

by

t h a t f o r any i n f i n i t e d i m e n s i o n a l Banach

s p a c e E l t h e r e always e x i s t s a u n i f o r m l y c o n t i n u o u s f u n c t i o n on

B1 B1 by p o l y n o m i a l s . Incon-

which c a n n o t b e a p p r o x i m a t e d u n i f o r m l y on

n e c t i o n w i t h t h i s , w e remark t h a t i n many Banach s p a c e s

E

I

t h e norm

f u n c t i o n (which i s o b v i o u s l y u n i f o r m l y c o n t i n u o u s on bounded sets) is n o t t h e u n i f o r m l i m i t of p o l y n o m i a l s on bounded s e t s . T h i s was s e r v e d by Kurzweil [ 4 ]

,

( r e s p . Rp 1

-

1 5 p,

(resp. [ p ] -)

noted, i f

inf

who showed t h a t , f o r example i n

ob-

E = C [ 0,1]

p n o t e v e n ) I t h e norm i s n o t t h e u n i f o r m l i m i t of d i f f e r e n t i a b l e f u n c t i o n s . I n p a r t i c u l a r , as Kurzvd.1

IP(x) 1

:

IIx 11 = 1 } = 0 for e v e r y

P E P("E)

and n E l N ,

t h e n , t h e norm c a n n o t be u n i f o r m l y a p p r o x i m a t e d byplynosnials on balls; t h i s c o n d i t i o n i s c l o s e l y c o n n e c t e d w i t h t h e u n i f o r m c o n v e x i t y of the space [ 5

I

.

F o r a r b i t r a r y r e a l Banach s p a c e s E 1

a (E) w a s d i s c u s s e d i n [ 2

1

. We

f :E

-+ F

the

0

T~

- completion

of

b r i e f l y s k e t c h t h e p r o o f of a g e n e r -

a l i z a t i o n o f t h i s r e s u l t . Given a f a m i l y tion

,

P C P(E)

, we

say t h a t a func-

i s P - u n i f o r m l y c o n t i n u o u s on bounded subsets of

E

ARON

4

( a b b r e v i a t e d "P-continuous") there is if

6

x, y

i f f o r each

E

some

n , then

f(y)ll


hi,

h;

,... ,h;l

...,h n'

and

h = sup ( h i ,

...,h;)

f

be a f u n c t i o n i n

p. Compactness

of

'2 .

and

in

-

such t h a t t h e

X

h = i n f (h;,

...,h i )

Then

S then leads,

t o t h e e x i s t e n c e o f f i n i t e l y many

a g i v e n number

0,

Let

for

functions

two

functions

satisfy

and h(x)

- h(x)


o

E

S

X

C

33

will be called bfittrUng& 6 > o

there exists a

d:-de-

such that

the

implication

f E 8. Obviously, strongly L-determining impliesl-de-

holds for all

termining. A closed set if the map

: d:

ps

+

d:

S

C

X

is strongly E-determining if andonly

defined by restricting a function

S ,

f E d:

to the set S, is bijective and open. of

We have seen that the closure &determining.

If S is 6-determining and if

then, by the open mapping theorem, d:

S

is

aEX

d:

S

always

is closed in C ( S )

is strongly E-determining.

If

has finite dimension n then there exists astrongly L-determining

set S of cardinality n. It suffices to choose a base of

strongly

d:

.

, ... , fn

A simple induction argument then yields the existence of

xl,. ..xn E X

points

fl

n

such that

det (fi(xj))

*

0.

s = ~xl,...,xnl is E-determining and by the preced-

Consequently,

ing argument strongly E-determining. In particular, if 6 .is the set of real polynomials of degree 5 n [ a,b ] C

IR , a

*

restricted to a compact

b, every set of n + 1

interval

different pints xl,. ..,xn+,E[a,b]

is strongly determining. Therefore,in Example 1 the set S ={-l, - 2 ' 1) 2' E is strongly E-determining and contained in a x X . 1

A simple revision of the proof of Proposition 3 now leads

1

to

the announced improvement:

THEOREM 3 : fion

f E E

Let

S

be a h t t O n g C y

hatisdying

E - d e t e h m i n i n g h e x . T h e n euehy dunc-

BAUER

i n i n Kor(Jf,L). Since f

E

E

ascX

i s t h e i n t e r s e c t i o n of a l l sets

h

If = f

with

}

V

E , we o b t a i n

COROLLARY: Kor(X,E) = E

id

E aJcx

cantainh a btkongty

L -detehmining

bet.

This c o r o l l a r y s e t t l e s Example 1. I t contains t h e

corollaries

of Proposition 3 as s p e c i a l c a s e s .

For t h e case of a b s o l u t e Korovkin approximation, t h a t

e

=

C(X)

,

Theorem 1 s t a t e s t h a t

Kor (JC,E)

is

equals ?E , We have seen

t h a t i n t h e r e l a t i v e theory o n e cannot expect a s i m i l a r r e s u l t o u t an a d d i t i o n a l assumption on

f . . For

for

c

with-

s t a t e space 1 S ( C ( X ) ) , defined i n Chapter 11, i s t h e convex compact s e t M + ( X ) of

a l l (Radon) p r o b a b i l i t y measures on

X

,

= C(X)

the

hence a simplex ( i n t h e sense

of Choquet). I t has been proved r e c e n t l y by Leha and Papadopoulou [81 t h a t t h e corresponding property f o r general

d: l e a d s t o t h e complete

g e n e r a l i z a t i o n of Theorem 1. Continuing t h e discussion i n t h e general case of t h e theory,

relative

l i s c a l l e d b i m p L i c i a l i f t h e s t a t e space S(f) i s asimplex.

The r e s u l t then i s :

The proof given i n Lazar [ 6 1

[ 8

1 makes use of t h e s e l e c t i o n theorem of

f o r ( m e t r i z a b l e ) simplexes. A n immediate consequence

is

35

KOROVKIN APPROXIMATION IN FUNCTION SPACES

t h e n t h e f o l l o w i n g r e s u l t which c o n t a i n s Theorem 2 as a s p e c i a l c a s e :

-

aSx c aJCd: X.

F o r t h e r e m a i n i n g p a r t o f t h e p r o o f w e o n l y have that

to

observe

i s c o n t a i n e d i n t h e i n t e r s e c t i o n of a l l t h e s e t s

{f = f )

with a r b i t r a r y

aEx

Since

Gd: =

f

E

c

E ajcx

A

V

d:.

is equivalent to

we

aJCX = a E X

a

obtain

p a r t i a l c o n v e r s e t o C o r o l l a r y 2 of P r o p o s i t i o n 3:

COROLLARY:

aJCX = a E X

t o a b i m p t i c i a t Apace

h o t d n id

i h a K a J w v h i n hpace W i X h

JC

kehpect

E.

We a r e now i n t h e p o s i t i o n t o f i n i s h t h e d i s c u s s i o n o f Example 6: i s s i m p l i c i a 1 s i n c e e v e r y c o n t i n u o u s real f u n c t i o n

2. Here

a E X = ] 0,1] i s t h e r e s t r i c t i o n of a f u n c t i o n

compact subset o f d: ( c f . [ 5 ]

,

aEX =

p. 1 6 9 ) . From

X = [ O,l]

But a f u n c t i o n

f E 6: \ Jc

for a l l

a c c o r d i n g t o Lemma 1.

x E X

s e n t i n g measure f o r d e f i n i t i o n of

JC.

x =

cannot be

0;

however,

We t h u s o b t a i n

l~ =

f fdp

Kor(JC,E)

*

monic in

E ) i s a n JE-repre-

1

.

u

c

X be

d',

n

2

the

t h e closure 2.

Define

U

a n d 6: as t h e s e t o f f u n c t i o n s f E C ( X ) which a r e h a r 6: U . Again aJCX C U" where U* denotes t h e topolcJgical bound(and X )

.

Furthermore

ajcX = e x X

and

aE X

= U* s i n c e a l l

boundary p o i n t s of t h e convex s e t U are r e g u l a r ( c f . [ 2 d:

Mx(JC)

=

f ( 0 ) according to

= JC

3-f.

-

JC = A ( X )

a r y of

Mx(JC)

+

Example 3 c a n be g e n e r a l i z e d as follows. L e t

of a n o p e n , convex, r e l a t i v e l y compact set

;6: =7C

A

T ( E ~ , ~

in h

it f o l l m s that

Jc-affine s i n c e 1

a

on

is s i m p l i c i a l s i n c e e v e r y f u n c t i o n

f E C(U*) i s t h e

1 , p. 127). restriction

BAUER

36

of a function in 1:. It follows from the preceding Corollary andCorollary 2 of Proposition 3, or from Theorem 5, that JC space with respect to d:

ex X

if and only if

=

is a Korovkin

U”, i.e. if and only

if U is n t t r i c t L y c o n v e x .

REFERENCES [ 11

E. M. ALFSEN, C o m p a c t conucx s e t s and boundcay d. Math. 57, Springer-Verlag (1971).

[ 21

H. BAUER. Silovscher Rand und Dirichletsches Problem, Ann. Inst. Fourier 11 (1961), 89 - 136.

[ 31

H. BAUER, Approximation and abstract boundaries, Amer.

hLtqhd5,

Ergebnisse

Math.

Monthly (to appear). [ 41

H. BAUER and K. DONNER, Korovkin approximation in Co(X), Math. Ann. (to appear).

[ 51

G. CHOQUET, L e c t u h e A o n a n a L y s i n , vol. I1 (Repeoenhtion theohy), W. A. Benjamin, Inc. (1969).

[ 61

A. LAZAR, Spaces of affine continuous functions on simplexes, Trans. Amer. Math. SOC. 134(1968), 503 -525.

[ 71

G. LEHA, Relative Korovkin-Satze und Rsnder, Math. (1977), 87 - 95.

[ 81

G. LEHA and S . PAPADOPOULOU, Nachtrag zu “G. Leha: Relative Korovkin-Satze und RZnder ” Math. Ann. 233(1978) , 273-274.

91

Ann.

229

.

Y. A. ;ASKIN, The Milman-Choquet boundary

and approximation theory, Funct. Anal. Appl. 1(1967), 170 -171.

Approdmation Theory and Functional A ~ ~ ~ l y e i e J.B.

ProlZa ( e d . )

0 North-Holland Publishing Company, 1979

A REMARK ON VECTOR-VALUED

APPROXIMATION ON COMPACT

SETS, APPROXIMATION ON PRODUCT SETS, AND THE APPROXIMATION PROPERTY

KLAUS

-

D.

BIERSTEDT

FB 1 7 d e r GH, Mathematik, D2-228 Warburger S t r . 1 0 0 , P o s t f a c h 1 6 2 1 D-4790 Paderborn Germany (Fed. Rep.)

INTRODUCTION

A f t e r Grothendieck [ 211

,

a l o c a l l y convex ( 1 . c . )

space

s a i d t o have t h e apptoximation phopehty ( f o r s h o r t , a . p . ) i f the identity

idE

precompact s u b s e t of

of E

E

is

E

i f andonly

can be approximated u n i f o r m l y

on

by c o n t i n u o u s l i n e a r o p e r a t o r s from

every into

E

E of f i n i t e r a n k ( i . e . w i t h f i n i t e d i m e n s i o n a l range).lvlany " c o n c r e t e "

1.c. s p a c e s are known t o have t h e a . p . , (1972)

, with

b u t a countehexampLc?

s u b s e q u e n t r e f i n e m e n t s due t o Figiel,Davie, and Szankmski,

shows t h a t t h e r e a r e even c l o s e d subspace o f each

EndLo

06

lP w i t h o u t

a . p.

for

p 2 1, p # 2 . I n connection with t h e a.p.,

a c r i t e r i o n due

to

L.

Schwartz

1 2 6 1 i s v e r y u s e f u l : Schwartz i n t r o d u c e s f o r two L . c . s p a c e s E and

F

t h e i r E-ptroduc-t by E E F := Le(FA

where

Fk i s t h e d u a l of

on precompact subsets of

,E ) ,

F w i t h t h e topology of uniform convergence F and where t h e s u b s c r i p t e on t h e 37

space

BIERSTEDT

38

E(F;,E)

of a l l c o n t i n u o u s l i n e a r o p e r a t o r s from FA i n t o E i n d i c a t e s

t h e t o p o l o g y of uniform convergence on t h e e q u i c o n t i n u o u s s u b s e t s of F'

.

F are q u a s i - c o m p l e t e ,

E and

If

o n e c a n e a s i l y show E E F S F E E ,

E E F o f t w o complete s p a c e s E and F i s oanplete

and t h e € - p r o d u c t

( c f . [26]). Moreover, t h e E - t e n n o h p h o d u c t

[21 1 i s a t o p o l o g i c a l s u b s p a c e o f ctitenion

60t

t h e a.p.

I26

E BE F

of

Grothendieck

E E F. W e c a n now f o d a t e SchwatLtz'b

, Proposition

11, c f . a l s o 131, I,

3.9,

and [ 8 ] ) :

THEOREM (L. Schwartz) :

id and o n l y id L.c.

bpace F

T h e quahi-complete L . c .

i n denbe i n

E 0 F

equivalently,

(at,

and F ahe complete l . c . get:

bpaCeb

E EF

doh

bpace E ha4 t h e a . p .

d o h each ( q u a s i - ) c o m p l e t e

each Banach Apace F ) . S o id

buch t h a t E o h F han t h e a . p . ,

V

E E F = E BE F, t h e c o m p l e t i o n 06 t h e E - . t e M b O t phoduct

( w h i c h we w i l l

UehO

caLC,

doh

E

we

E QE F

b h a h t , c o m p l e t e E-tenboh p h o d u c t ) .

I n f a c t , t h e a p p l i c a t i o n s of t h i s theorem, s a y , i n t h e c a s e o f f u n c t i o n s p a c e s E d e r i v e from t h e remark t h a t t h e "abstract"operator space

E

E

F

c a n u s u a l l y be i d e n t i f i e d w i t h a

F-valued f u n c t i o n s " o f t y p e E "

. And

E QE F

"concrete"

i s t h e s p a c e of

responding" f u n c t i o n s w i t h f i n i t e dimensional ranges i n proof of t h e a . p .

of

E

space

F.

of

"cor-

Hence

is t h e n e q u i v a l e n t t o t h e approximation

a of

c e r t a i n F-valued f u n c t i o n s by f u n c t i o n s w i t h v a l u e s i n f i n i t e dimens i o n a l s u b s p a c e s o f F f o r e v e r y ( q u a s i - ) complete L . c . o n l y f o r e v e r y Banach s p a c e F ,

space

F

or

a r e s u l t which i s o f i n t e r e s t i n b o t h

directions.

I n t h i s a r t i c l e , w e w i l l g i v e some ( r a t h e r s i m p l e ) new examp.h o f how t o a p p l y S c h w a r t z ' s theorem t o f u n c t i o n s p a c e s

more

general

t h a n , b u t e s s e n t i a l l y s i m i l a r t o t h e well-known u n i f o r m a l g e b r a s H(K) and

A ( K ) on compact s u b s e t s

K of

CN (N '1).

More p r e c i s e l y , we deal

h e r e w i t h s p a c e s of c o n t i n u o u s f u n c t i o n s on a compact

set K

which

VECTOR-VALUED

APPROXIMATION O N COMPACT SETS

39

e i t h e r are u n i f o r m l y a p p r o x i m a b l e by f u n c t i o n s b e l o n g i n g ,

t o a g i v e n bubbheah

sets U c o n t a i n i n g K ,

F of t h e s h e a f

c o n t i n u o u s f u n c t i o n s o r have r e s t r i c t i o n s b e l o n g i n g t o terior

$

of

on

open

C of a l l

F on t h e i n -

K.

In

The genehue d i t u a t i o n i s t h e s u b j e c t of s e c t i o n s 1 a n d 2 .

s e c t i o n 1, the v e c t o r - v a l u e d case i s c o n s i d e r e d , w h i l e s e c t i o n 2deals w i t h "slice product''

-

r e s u l t s (on p r o d u c t s e t s ) . F i n a l l y , i n s e c t i o n

3 , w e look a t some o f t h e m o t i v a t i n g exampeed and s u r v e y

the

known

r e s u l t s ( a n d their r e l a t i o n s ) i n t h i s case.

So, i n a s e n s e , t h i s p a p e r i s b a s e d on a g e n e r a l i z a t i o n o f t h e author's old article ( 2 1

and m o t i v a t e d , among o t h e r t h i n g s , by

the

more r e c e n t a r t i c l e [27] o f N . Sibony: W e show t h e c o n n e c t i o n of sane of Sibony's r e s u l t s with topological tensor product theory and t h e a . p . o f t h e s p a c e s of s c a l a r f u n c t i o n s i n q u e s t i o n . The o f t h i s p a p e r w i l l be combined w i t h t h e t e c h n i q u e o f o f t h e a.p.

with

results

"localization"

f o r s u b s p a c e s of w e i g h t e d Nachbin s p a c e s ( c f . [ 5 1 and [lo])

i n a s u b s e q u e n t p a p e r t o y i e l d new examples o f f u n c t i o n s p a c e s mixed t y p e " w i t h a . p .

"of

and t o demonstrate a p p l i c a t i o n s of t h e l o c a l

-

i z a t i o n p r o c e d u r e i n some c o n c r e t e cases.

ACKNOWLEDGEMENT:

The a u t h o r g r a t e f u l l y acknowledges

,support

under

t h e GMD/CNPq a g r e e m e n t d u r i n g h i s s t a y a t UNICAMP July-September1977 w i t h o u t which i t would n o t h a v e been p o s s i b l e t o a t t e n d t h i s Confere n c e i n Campinas. I would a l s o l i k e tothank J. B . P r o l l a f o r h i s

con-

s t a n t i n t e r e s t i n my c o n t r i b u t i o n t o t h e s e P r o c e e d i n g s . A s everybody can see i m m e d i a t e l y , p a r t o f t h e r e s u l t s i n t h i s a r t i c l e d a t e s

(at

l e a s t ) back t o t h e t i m e when t h e j o i n t p u b l i c a t i o n [lo 1 was p r e p a r e d . So t h e a u t h o r t h a n k s B. Gramsch and R. Meise f o r many v e r s a t i o n s and remarks i n t h i s c o n n e c t i o n .

helpful

con-

EIERSTEDT

40

CASE

1. THE GENERAL VECTOR-VALUED Let

and

X be a c o m p l e t e l y r e g u l a r ( H a u s d o r f f ) t o p o l o g i c a l

space

F a c l o h e d .LocaL.Ly convex ( L . c . 1 bubdhead of t h e s h e a f Cx of a l l o r complex

continuous ( r e a l open s u b s e t

v a l u e d ) f u n c t i o n s on

C ( U ) w i t h t h e compact-open

f i c i e n t to r e q u i r e

t o p o l o g y c o . I n f a c t , i t would be

F to be a

p t e a h e a 6 o n l y , and w e p r e f e r

presheaf n o t a t i o n throughout t h i s paper. compare [ 9 1 and [ 101

Let

.A

+.

of our

F a s above was called "ahead

sheaf

use

notation 06

F-matpkic

E always d e n o t e a q u a s i - c o m p l e t e locally convex ( H a u s d o r f f ) W e w i l l always assume t h a t

C).

t h a t any f u n c t i o n

f : X

( F o r some

to

suf-

I.)

space ( o v e r R o r i.e.

foreach

i.e.,

X, F ( U ) i s a c l o s e d t o p o l o g i c a l l i n e a r subspaceof

U of

dunc-tianh" i n [ 9

X,

f : X

+.

X

IR ( o r , e q u i v a l e n t l y ,

i s a kR-space, any

function

Y, Y any c o m p l e t e l y r e g u l a r s p a c e ) i s c o n t i n u o u s i f and only

i f the r e s t r i c t i o n of

t o e a c h compact s u b s e t o f

f

X

i s continuous.

(Each l o c a l l y compact o r m e t r i z a b l e s p a c e , a n d , more g e n e r a l l y , e a c h k-space is also a KIR-space,

km-space.)

c

U C X

c f . B l a s c o [12], and hence t h e s h e a v e s

p l e t e , i.e. the spaces

u

Then each open

( C ( U ) , C O ) and

Cx

is

again

a

and F are com-

F ( U ) a r e complete f o r e a c h open

x. Under t h e s e a s s u m p t i o n s , t h e r e e x i s t s ( c f . 110 1,1.5) the '!E-vdutd

ahead

FE

06

=

F", namely, f o r any open

U in

X,

t h e s p a c e o f a l l c o n t i n u o u s E-valued f u n c t i o n s which s a t i s f y e ' o f with t h e topology

subsets of

U

E

F ( U ) f o r each

e' E E ' ,

f

on U

endowed

c o of uniform convergence on ccmpct

( c f . 1 3 ) and

151 ),

and t h e c o n o n i c a l r e s t r i c t i o n mappings of t h e s h e a f

FE a r e j u s t t h e

VECTOR-VALUED APPROXIMATION ON COMPACT SETS

o r d i n a r y r e s t r i c t i o n s o f f u n c t i o n s . FE sheaf

:C

41

i s a c L a b e d subsheaf of

of a l l c o n t i n u o u s E-valued f u n c t i o n s on

X.

I n o u r d e f i n i t i o n and i n some of o u r r e s u l t s below, h e l p f u l t o keep t h e f o l l o w i n g m o t i v a t i n g examples F-morphic f u n c t i o n s i n mind ( c f . a l s o [ 9

( i i ) X open i n

of

it may

be

F of

sheaves

1 and [lo] for mre examples) :

1. EXAMPLES: ( i )X = complex monifold or j u s t o f holomorphic f u n c t i o n s on

the

11, F=O=sheaf

CN (I?

XI

(n 2 1) , L = P(x,D) a ( l i n e a r ) h y p o e l l i p t i c

IRn

d i f f e r e n t i a l o p e r a t o r w i t h Cw-coefficients,and F = t = s h e a f of n u l l s o l u t i o n s o f L , i . e .

f o r any any open

U i n X.

N ~ ( u )= I f

and by

C"(U)

c"(u); (LI

U)frOI

(The c l o s e d graph theorem

F r g c h e t s p a c e s i m p l i e s t h a t , on N,(U), duced by

E

for

t h e topologiesin-

c o c o i n c i d e and hence t h a t N ( U )

L i s a c l o s e d t o p o l o g i c a l l i n e a r subspace o f (CCU), c o ) .)

E s p e c i a l l y , the sheaf

X

of harmonic f u n c t i o n s on

IRn

satisfies

a l l a s s u m p t i o n s o f 1. (ii)above, and a l s o t h e "harmonic s h e a v e s " o f a b s t r a c t p o t e n t i a l t h e o r y are s h e a v e s of F-morphic f u n c t i o n s .

All

t h e s h e a v e s of example 1. a r e (FN)-sheaves.

2.

For a compact s u b s e t K o f

DEFINITION: (i)

X I we d e f i n e :

C ( K , E ) := t h e s p a c e o f a l l c o n t i n u o u s E - v a l u e d

functions

on K w i t h t h e topology of uniform convergence on K , (ii) A F ( K , E )

:= i f E C ( K , E ) ;

i.e. ( i i i )H F ( K , E )

{f

E

:=

e'of

I

I f(EFE(Ei)r

f

K

E

the closure i n

C(K,E);

(depending on

0

F ( K ) f o r e a c h e'E E ' } , and C ( K , E ) of

t h e r e e x i s t s an open neighbourhood f ) and a f u n c t i o n

g

c o n t i n u o u s and e ' o g E F(U) for any

e'E

E

U of

K

E

F ( U ) [ i . e . g: U + E

El] such t h a t g

iK

=f

1.

BIERSTEDT

42

h o l d s , and b o t h are closed s u b s p a c e s of C(K,E) which

C AF(K,E)

HF(K,E)

w e endow w i t h t h e topology o f uniform convergence on K ( i n d u c e d C(K,E)).

If

E =

IR o r

by

w e w r i t e C ( K ) , A F ( K ) , and H F ( K ) , r e s p e c -

C,

tively. NOW, of c o u r s e , i f

and

HF(K,E)

i s complete, a l l t h e spaces C(K,E), AF(K,E),

E

are complete, t o o . The e q u a t i o n

quasi-complete

E i s well-known

(cf. [ 3

for

= E EC(K)

C(K,E)

1 ) , and, once t h i s e q u a t i o n is

w e l l - u n d e r s t o o d , t h e proof of t h e f i r s t p a r t of t h e f o l l o w i n g r e s u l t

i s c l e a r (see e . g .

1 or

[ 3

a r b i t r a r y subspace of

f o r a d e s c r i p t i o n of

[5]

C(K),

an

E EF, F

from which o u r r e s u l t below

is

easily

derived, too) :

3 . THEOREM:

(1) A F ( K , E )

AF(K,E)

Hence

(2)

(oh,

= E

m

V

aPEA F ( K ) h o l d s do& a&? complete

equiuaeently,

doh

t.c.

a l e 8 a n a c h J Apace4 E id and o n l y

hub t h e a.p.

AF(K)

id

= E EAF(K)

For t h e second p a r t of 3, S c h w a r t z ' s c r i t e r i o n for t h e a . p . t h e i n t r o d u c t i o n ) i s needed. I n o t h e r words, A F ( K ) h a s t h e and o n l y i f , f o r a r b i t r a r y Banach space with e ' o f on

K,

1

it

E

0

F ( K ) f o r any e'

each f u n c t i o n f E C ( K , E )

E,

may be approximated, uniformly

E E'

E

t h a t s a t i s f y e' o g

I

have t h e form g(x) =

E

if

by c o n t i n u o u s f u n c t i o n s g on K w i t h v a l u e s i n f i n i t e dimen-

s i o n a l s u b s p a c e s of

n

a.p.

(in

n

C eigi(x) i =1

IN f i n i t e (depending on g ) , ei

E

f o r complete t . c . E . )

E

F(I?),

for all

E, and

(Remark t h a t such a n approximation w i t h p o s s i b l e by t h e a.p.

K

gi

gi

x

E

oney

o f C ( K ) and by t h e e q u a t i o n

t o o , and

E

hence

K;

AF(K)

, i = l , ... , n . is

UeWayb

C(K,E) = E

aE C(K)

E

C(K)

V

VECTOR-VALUE0 APPROXIMATION ON COMPACT SETS

As t o t h e a . p . o f t h e c o r r e s p o n d i n g s p a c e

43

HF(K), the situation

t h e r e i s , i n some s e n s e , j u s t t h e o p p o s i t e :

We U b b U m e t h a t , d o h some b a b i b

4.

THEOREM:

K,

F ( U ) hub t h e a . p .

d o t each

U E UL

. [ Fah

le

neighbouhhoodb

06

06

I I ] b e L o w , we couLd a l b o

i n b t e a d t h a t E hub t h e a.p.1

UbbUme

Then

(1)

06

E QE H F ( K ) i b a denne topoLogicnL [ i f l e a n .

and hence

HF(K,E),

hoedo w h e n e u e t E (2)

compLete.

i b

has t h e a s p . id and onLy id, doh each

ConsequentLy

HF(K)

compeete L . c .

( o h each B a n a c h ) bpace

HF(K,E)

= {f E C ( K , E ) ;

thehe exints

nubbpace

UM

d o h each

e'

open n e i g h b v u h h o o d

E

E

,

E'

and e a c h

U = U(e',E)

E

06

> 0

Kaod

g = g ( e ' , E ) E F ( U ) buch t h a t

a 6unction

E BE C ( K ) i s a t o p o l o g i c a l l i n e a r subspace o f C ( K , E )

and

PROOF:

As

as t h e

E - t e n s o r p r o d u c t p r e s e r v e s t o p o l o g i c a l l i n e a r s u b s p a c e s , only E Q HF(K)

d e n s i t y of

s e r t i o n . So l e t f E HF(K,E). function

g

s&watz's

must b e v e r i f i e d f o r t h e f i r s t a s -

p be a c o n t i n u o u s seminorm

E

FE(U)

such t h a t

= E

on compact subsets o f

a,

HF(K,E)

on

E,

By d e f i n i t i o n , t h e r e e x i s t s a n open s e t

definition, FE(U)

U E

in

E

F(U)

m~ p ( f ( x )

-

g(x))


0

u 3 K and $. B u t , a g a i n

and

a by

( w i t h t h e t o p o l o g y o f uniform convergence

U). W i t h o u t l o s s o f g e n e r a l i t y , w e may assume

and hence t h e a.p.

of

F ( U ) or o f

E and o n e

theorem from t h e i n t r o d u c t i o n imply t h a t

direction

of

E 0 F ( U ) i s dense

44

EIERSTEDT

5.

E 4 F ( U ) w i t h s u p p(g(x)- h ( x ) ) < xCK Now h l K E E d H F ( K ) h o l d s and s u p p ( f ( x ) h ( x ) ) < E , which p r o v e s XEK t h e r e q u i r e d d e n s i t y of E @ H F ( K ) i n HF(K,E).

i n FE(U). Therefore we can f i n d h

E

-

( 2 ) i s t h e n c l e a r from S c h w a r t z ' s c r i t e r i o n because t h e

on t h e r i g h t hand s i d e of t h e e q u a t i o n i s n o t h i n g b u t

a close look w i l l i m m e d i a t e l y r e v e a l .

E

E

space

-

HF(K)

as

0

I n other wordsl i t i s adwayd t r u e ( u n d e r t h e a s s u m p t i o n of t h a t a function

f E C ( K , E ) which can be a p p r o x i m a t e d u n i f o r m l y on K

FE

by f u n c t i o n s e x t e n d i n g t o e l e m e n t s o f K may a l s o b e a p p r o x i m a t e d u n i f o r m l y on

h(x) =

n

Z

i=l

n E IN f i n i t e (depending on But t h e a . p .

eihi(x)

on open neighbourhoods of

K by f u n c t i o n s of t h e form

for a l l

x

E K;

..., n .

ei E E l a n d

h)

gi E HF(kI1 i =1,

HF(K) is equivalent to the f a c t t h a t , f o r a r b i t r a r y

of

Banach s p a c e E l e a c h f u n c t i o n given any

4)

e' E Eq1 e' o f

f E C(K,E) with the property

K

by

( s c a l a r ) f u n c t i o n s b e l o n g i n g t o F on open sets c o n t a i n i n g K i s

al-

ready an element of

may b e a p p r o x i m a t e d u n i f o r m l y

i . e . can be approximated u n i f o r m l y

HF(K,E),

K by E-valued f u n c t i o n s b e l o n g i n g t o Or,

F

E

on open s e t s c o n t a i n i n g

t o p u t i t this wayl H F ( K ) h a s t h e a.p.

Banach s p a c e E and an a r b i t r a r y f u n c t i o n e x i s t s f o r any

E

e' o g o E F(Uo)

I (e'

0

REMARK:

f ) (x)

> O , unidahmly f o r a l l U

E l I an open s e t

0

3

K

for each

- (e'

and a f u n c t i o n

e ' E Ei

o g o ) ( x ) ]
0

E

and a f u n c t i o n

x K2

W i t h o u t l o s s of g e n e r a l i t y w e may assume

g E (F1

U = U

E

x U2

F 2 ) ( U ) s u c h that

with

Ui E UI

( i= 1 / 2 1 , a n d h e n c e

F1(U1)

by S c h w a r t z ' s t h e o r e m , b e c a u s e there exists

h

E

F1(U1)

8 F2(U2)

F2(U2)

or

f E A F ~F,(K~

XK2)

,

y i e l d c o n t i n u o u s l i n e a r mappings of The c h a r a c t e r i z a t i o n o f 6 implies

v

I1(gl)

C

I1 : t K

A F ~F , ( K ~ x

A F ( ~K 2 )

-

and

h a s t h e a.p. fien

such t h a t

( 2 ) Notice t h a t , by t h e i d e n t i t y C ( K 1 x K 2 )

for arbitrary

+

=C(K1,C(K2))=C(K2,C(Kl))r

f ( t , * ) resp.12:x

0

12(K2)

C A F ~ ( K ~ a) n d

0

(1) 14

A F , (Ki) 1

= H

Fi

t h e b e b p a c e n hub -the a . p . ,

f(.,x)

hence also

C

8. COROLLARY:

+

r e s p . K 2 i n t o C(K2) resp. C(K1). 1 K2) a t t h e e n d o f t h e proof of

and 1 2 ( K 2 ) C A ( K 1 ) . So, f o r f a t s e t s K1 AF ( K 2 ) 2 F1 t h e a s s e r t i o n f o l l o w s i m m e d i a t e l y from 6 . (1).o

I1(K1)

1

(Ki)

(i = 1 , 2 ) hoLdb a n d

t h e n we o b t a i n

and K 2 ,

one

04

VECTOR-VALUED APPROXIMATION ON COMPACT SETS

nets (2) H

(K1

F1

K2.

and

K1

x

han t h e a . p .

K2)

F2 have -the a . p .

( 3 ) 1 6 K1

have t h e a . p . ,

AFl

wheneve& both H

F1

(K ) and H ( K )

F2

id b o t h

ahe 6 a t and

K2

and

61

F2(Kl

AF2 (K2)

han t h e a s p . , t o o .

xK2)

(1) i s clear from S c h w a r t z ' s t h e o r e m , 6 . ( 3 ) , a n d 7 . ( 2 ) .

PROOF:

(2)

and ( 3 ) f o l l o w from 7 by a i d of t h e r e s u l t ( S c h w a r t z [ 2 6 ] , P r o p o s i t i o n 11, C o r o l l a i r e 2 ) t h a t t h e € - p r o d u c t of two compete L . c .

a.p.

spaces with

a l s o e n j o y s t h e a.p. I n d u c t i o n on I a n d 8. (1) u s i n g , among o t h e r ( o b v i o u s ) t h i n g s ,

t h a t f i n i t e E - p r o d u c t s are U h A o C i a t i W e a n d t h a t E - p r o d u c t s of carrplete

spaces w i t h a . p . are a g a i n spaces w i t h a.p.

9 . COROLLARY: with

x1

nheaveb

...

x

OA

xn

x

-

c xll..

hen p e c t i v e l y

.

.,Xn

X1,..

Let

be c o m p l e t e l y h e g u l a h (H~~.4li0h,46) bpaCeA

a klR-npace,

,

'CXn

d o h each

F1,

bet

.. ,Kn

K1,.

and

(1) L e t , d o h name banid Ui have t h e a.p.

y i e l d s now:

, . . , Fn compact

hubbe&

06

06

neighbouhhoodn 0 6

Ki

Ui

E

Uli

( i=1,

a t mod2 one i) o h l e t a l e b u t one have t h e a . p . H

F1

E

...

Fn

(K1 x

H

name h o l d b d o h LCZ U e e t h e b e t 6

A

F1

E

...

H

Fi

... , n

X1 ,

...,Xn

,

Fi(Ui)

except ( i=1,

(Ki)

...

604

n)

Then

... x K H

i n t h u e , and id aLl

(2)

be c l o n e d l . c . nub-

Fn

(K1

F1

E

Fi

(Ki)

...

K1,... x

=H

...

Frl ,Kn

x

F1

(K1)

( i= 1 . . . , n )

(K1

x

.. .

have t h e a . p . , Zhe

x Kn)

.

be bat. Th e n

Kn) = A F (K1)

1

E

...

E

AFn(Kn)

BIE RSTEDT

62

hoRdn t h u e , and id aLL t h e bame h o l d s d o h

E

E

(Ki)

(i = 1 ,

...

i=l,.

...

Fn

(K1

X

. ..

x

= H

F1

E

mod?

...

have t h e u . p . ,

x Kn).

AFi ( K i ) =HF i (K.11-

be dat and

Ki

Kn)

... , n )

..-

at.! t h e d e dpaced ( e x c e p t do& at

16 t h e n

F1

A Fl

Fi

..,n ,

( 3 ) L e t , doh each

A

A

oneJ have t h e a.p.,

Fn

(K1 x

.. . x Kn)

ib valid, too. F o r t h e c o r r e s p o n d i n g s p a c e s o f f u n c t i o n s w i t h values i n a quasicomplete .t.c. s p a c e E

(1) L e t

1 0 . COROLLARY: A

F1

E

...

s e c t i o n 1), we g e t e . g . :

(see

,...,K n

K1

Fn

(K1 x

be dat. Then

. .. x K n l E )

= E € A F (K1)

1

E

.. .

in t t u e . bouhhoods

06

.., n ) .

( i=1,.

H F1 i d

m i 06

E be compeete and l e t , d o h borne babio

( 2 ) Let

E

...

Fn

K~

Fi(Ui)

I

neigh-

h a v e t h e a s p . doh e a c h

Ui

E

Lzi

Then

(K1 x

.. . x K n , E )

=E

./eE H F

U

(K1) BE

1

. .. BE H Fn(K V

)

uaLid.

( 3 ) L e t E be c o m p l e t e , l e X Ki b e bat and A

.., n .

e a c h i =1,.

have t h e a . p .

T h e n id a L l t h e b p a c e b

( i =1,.

.. , n )

(Ki) Fi

A

Fi

= H F . (Ki) doh 1

(Ki)

=HF

i

(Ki)

I

holds, too.

PROOF:

(1) is a consequence of 3 . (1) and 9. ( 2 ) . Let u s remark t h a t ,

under the h y p o t h e s i s of ( 2 1 ,

(F1

E

.. .

E

Fn) ( U ) ( a s c - p r o d u c t of ample&

VECTOR-VALUE0 APPROXIMATION ON COMPACT SETS

spaces with a.p.) up

s a t i s f i e s the a.p. :=

IU,

of neighbourhoods of

x

... x Un

K1

;

Ui

... x K n

x

f o r each

i n the basis

( i =l,.

Uli

6

U

53

., n ) 1

Hence ( 2 ) follows from 4 . (1) and

9 . ( 1 ) . F i n a l l y ( 3 ) i s i m p l i e d by 9 . ( 2 ) , ( 3 ) and by t h e remark a t t h e

v e r y end o f s e c t i o n 1.

0

3. DISCUSSION OF THE MOTIVATING EXAMPLES

I n t h i s f i n a l s e c t i o n , w e w i l l look a t some o f t h e known results i n t h e case of o u r m o t i v a t i n g examples o f s h e a v e s F ( c f . 1 above) a n d

w i l l p o i n t o u t t h a t , between s o m e theorems i n t h e l i t e r a t u r e , s t r o n g It is

r e l a t i o n s f o l l o w from o u r p r e v i o u s d i s c u s s i o n . here to survey

not

intended

aLL t h e r e l e v a n t a r t i c l e s , b u t we w i l l r a t h e r i l l u s -

t r a t e some of t h e ideas which m i g h t p l a y a r b l e , when one t r i e s

to

a p p l y t h e r e s u l t s of s e c t i o n s 1 and 2 , by s p e c i f i c examples. P e r h a p s t h e case most p e o p l e have b e e n i n t e r e s t e d i n i s

F

=o,

the nucLeah F r g c h e t s h e a f o f holomorphic f u n c t i o n s on a complex manifold

X . F o r s i m p l i c i t y , however, w e w i l l o n l y d e a l w i t h h o l o m o r p h i c

f u n c t i o n s on of sheaves

X = CN (N 2 1) h e r e .

I t i s c l e a r t h a t f i n i t e ~-prcducts

I) are n o t h i n g b u t t h e c o r r e s p e n d i n g s h e a f

u c t a n d that, f o r a n y q u a s i - c o m p l e t e L.c.

s p a c e E , OE

F

s h e a f o f E-valued holomorphic f u n c t i o n s . When for short, A(K,E)

, H(K,E)

i n s t e a d of

0 on t h e prod-

AF(K,E)

,

=o,

is j u s t

we w i l l

the

write,

HF(K,E), respectively.

F = O , some o f the r e s u l t s i n s e c t i o n s 1 and 2 are

appar-

e n t l y p a r t o f t h e “ f o l k l o r e “ of t h e subject, b u t u s u a l l y n o t

easily

For

e have a l r e a d y p o i n t e d o u t i n t h e i n accessible i n the l i t e r a t u r e : W t r o d u c t i o n t h a t this p a p e r i s b a s e d on a g e n e r a l i z a t i o n of t h e “ o l d ”

article [ 2 1 .

L a t e r on ( i n [ 1 ]

c l o s e d s u b s p a c e s of

C(K)

,

K

,

s e c t i o n 1), 0. B. Bekken l o o k e d

at

compact, w i t h the so-called “Afice p t o p U t y ”

64

BIERSTEOT

A f t e r the p r o p r change

and showed t h a t t h i s p r o p e r t y i m p l i e s t h e a . p .

of n o t a t i o n and some i d e n t i f i c a t i o n s ( u s i n g t h e f a c t t h a t e a c h Banach space is a c l o s e d subspace o f

f o r some compact K ' ) h i s r e s u l t s

C(K')

there a r e q u i t e s i m i l a r t o o u r theorem 3 ( f o r Banach s p a c e s E ) . s e c t i o n 3 of [ 1 ]

,

(making u s e o f t h e n u c l e a r i t y o f

In

Bekken obtains

(1)

a p r o p o s i t i o n r e l a t e d t o ( b u t somewhat weaker t h a n ) o u r theorem

4.

For a d e t a i l e d account of t h e r e l a t i o n of t h e slice p r o p e r t y w i t h t h e a . p . and t h e consequences of a theorem o f Milne i n t h i s see a l s o [ 6

connection,

I. we

A s u s u a l w i t h s p a c e s o f holomorphic f u n c t i o n s ,

s p l i t up o u r d i s c u s s i o n f o r t h e cases

N =1 and

N22. If

must

now

i.e.

N = l

K i s a compact subset of t h e complex p l a n e , t h e problem i s completely

s o l v e d : A ( K ) and

H ( K ) have t h e n atLoayn t h e a . p .

(whereas i t r e m a i n s

a n o p e n p h o b l e m w h e t h e r e v e n t h e Banach a l g e b r a

H m ( D ) o f a l l bounded

D e n j o y s t h e a . p . Fanark

holomorphic f u n c t i o n s on t h e open u n i t d i s k t h a t the a.p. i a l !).

of t h e d i s k a l g e b r a

A(;)

=

H(6)

is really quite triv-

T h i s i n t e r e s t i n g r e s u l t i s due t o t h e j o i n t e f f o r t o f several

p e o p l e ( a n d a l s o , u n f o r t u n a t e l y , n o t e a s i l y a c c e s s i b l e i n the l i t e r a t u r e i n i t s f u l l generality) : E i f l e r [171 6 for

H ( K ) , and Davie [151 f o r

A(K)

r e s u l t s ) . More g e n e r a l l y , Gamelin

,

Gamelin-Garnett [19],secticn

u e c t o h - UaLued

( t h e y a l l use

[ 181, s e c t i o n 12

has pointed

out

t h a t t h e c o n s t r u c t i v e t e c h n i q u e s (and t h e a p p r o x i m a t i o n scheme)

of

Mergelyan and V i t u s h k i n show t h a t t h e s o - c a l l e d "T-inuahiant"algebras have the a . p . A s t o

A ( K ) = H ( K ) i n t h e c a s e o f one v a r i a b l e l a neced-

s a h y and b u d d i c i e n t

c o n d i t i o n ( i n v o l v i n g COntinuouA andyx%

was g i v e n by V i t u s h k i n , see e . g . For

[19]

and [ 2 9 ] .

N z 2 , t h e r e are o n l y p a h t i a l r e s u l t s . Remark f i r s t

by a n example o f D i e d e r i c h and F o r n a e s s , there e x i s t s compact domain G of holomorphy i n A(K)

#

H(K)

capadty)

for

K =

c.

1CN

with

a

Cm-boundary

that,

relatively such t h a t

For a s u r v e y o f some r e l a t e d r e c e n t work

on

VECTOR-VALUED

t h e q u e s t i o n when

APPROXIMATION ON COMPACT SETS

55

A ( K ) = H ( K ) i n s e v e r a l complex v a r i a b l e s , we r e f e r

t o B i r t e l [111, and f o r r e s u l t s i n “ 6 i n i t e S. P a . C . m a n i d o e d n ” Rossi-Taylor [ 25

1.

I t i s known now t h a t

A(K)

lowing t y p e s of compact s e t s (i)

to

K c

f o r the fol-

( o r H ( K ) ) has t h e a.p.

cN:

i s t h e c l o s u r e of a a t f i i c t ~ yp a e u d o c o n v e x k e g i o n w i t h

K

s u f f i c i e n t l y smooth ( s a y , C 3 -1

boundary, o r :

i s t h e c l o s u r e of a heguLan WeiL p o t y e d e h .

(ii) K

Both c o n d i t i o n s imply K f a c t ( t r i v i a l l y ) , and

(in

A(K) =H(K)

c a s e ( i ), t h i s approximation theorem i s due t o Henkin-Lieb -Kerzman, i n c a s e ( i i ) ,i t i s a r e s u l t of P e t r o s j a n ) .

( i )was proved

e.g.

in

Bekken 11 1 , s e c t i o n 2 , a p p l y i n g a v e c t o r - v a l u e d v e r s i o n of Henk n ’ s s e p a r a t i o n o f s i n g u l a r i t i e s r e s u l t . I t a l s o f o l l o w s from Sibony P r o p o s i t i o n 4 ( i n view of o u r C o r o l l a r y 5 ) . Sibony [ 2 7 ]

,

p. 1 7 3

a l s o remarked t h a t P e t r o s j a n ’ s arguments may be m o d i f i e d A(K,E)

= H(K,E)

f o r each F r s c h e t s p a c e E i f

K

to

is the closure

271, has

yield of

a

r e g u l a r W e i l p o l y e d e r , and hence ( i i ) f o l l o w s a g a i n from ourCorollary 5.

The method of ‘ Y o c a L i z a t i a n

REMARK:

tio n spaces ( c f . [ 5 t h e a.p.

1 and

06

t h e a.p.”

for certain

[ 1 0 ] ) may be used t o show t h a t

f o r compact sets K ’ t h a t a r e “ s u f f i c i e n t l y w e l l ”

func-

A ( K ’ ) has

didjoint

UMionb of s e t s K a s above and t h a t some r e l a t e d f u n c t i o n s p a c e s h a v e the a.p.

,

too ( c f . [ 5 1 ,

Corollary 15)

,

b u t w e w i l l n o t go i n t o

de-

t a i l s here. L e t u s now e x p l i c i t l y s t a t e what w e g e t from t h e p r e c e d i n g res u l t s by a p p l y i n g o u r C o r o l l a r i e s 9 and 1 0 :

14. THEOREM: ( i = 1,.

.. , n )

(i)

(1) H(K) hub t h e a . p . id

eithen. a n y compact n u b b e t

06

C

oh

K = K1

x

... x K n

with

Ki

BIERSTEDT

56

(ii) t h e C l o A u h e a d

a b t h i c t C y pdeudoconvex k e g i o n w i t h

6 i c i e n t L g nmooth boundahg o h (2)

A ( K ) had t h e a . p .

06

a t e g u l a h Weil polyedeh.

undeh t h e name c o n d i t i o n n

i n ( 1 ) ( i ) ,a d d i t i o n a l t y , Ki t o be h a t . " " = A ( K ~ )aE ... QE A ( K n ) i n t h e n t h u e . (3)

H ( K ) = A ( K ) holdd

doh K = K 1

t i t h e h ( i )a 6a.t compact an

x

det

nub-

... x K n

heqLLitled

And

W,h%

i n a: w i t h

4 one

A(K)

=

...,n )

Ki ( i= 1 ,

H(Ki)

= A(Ki)

oh

i n ( I ) (ii) a b o v e .

L e t .then E b e an a h b i t k a h g c o m p l e t e 1 . c . n p a c e . (4)

Undeh t h e ahnumptionn o d ( 2 1 ,

(5)

Undek t h e annumptionb

06

( 3 1 , we h a v e

A(K,E)

= H(K

, E ),

too.

11. ( 3 ) i s r e l a t e d t o a r e s u l t of Weinstock [30 I ,

p . 812, where,

i n s t e a d of the assumption of a smooth boundary i n 11. (1)( i i ) ,he needs o n l y t h e s o - c a l l e d " n e g m e n t p h a p e h t y " o f K.

( W e i n s t o c k ' s methods a r e

q u i t e d i f f e r e n t , however.) A t t h i s p o i n t , a few remarks on p a p e r [ 271 a r e a l s o i n o r d e r ( i n c o n n e c t i o n w i t h

OLX

Sibony's

preceding results):

P r o p o s i t i o n 1 o f [ 2 7 1 i s , i n some s e n s e , e a s y , i f n o t t r i v i a l ,

a s o u r theorem 4 . ( 1 )

(and i t s simple p r o o f ) demonstrates: I t is

n e c e s s a r y t o invoke G l e a s o n ' s theorem a t t h i s p o i n t ; t h e w e l l n u c l e a r i t y ( o r even t h e a . p . )

of

not

- known

0 and s i m p l e t e n s o r p r o d u c t a r g u

-

ments s u f f i c e ! C o r o l l a i r e 3 of [ 2 7 ] c o r r e s p o n d s w i t h 7 . (1) and l0.Q) i n t h i s p a p e r . A s we have a l r e a d y n o t e d above,

however,

Sibony's

p r o p o s i t i o n 4 i s r e a l l y a Mon-thiWial? r e s u l t b a s e d on H e n k i n ' s mthod and i m p l i e s t h e a . p . of

A ( K ) i n c a s e ( i ) .Hence, by o u r C o r o l l a r y 5,

it i s ( e s s e n t i a l l y ) e ~ u i ~ a t e n t ot theorem 2.4 o f Bekken [ 1 1. Finally,

Corollaire 8 of [ 2 7 ] c o r r e s p o n d s w i t h o u r theorem 1 1 . ( 5 ) .

It should

p e r h a p s be p o i n t e d o u t t h a t , whereas p a r t o f S i b o n y ' s p r o o f s looks as

VECTOR-VALUED APPROXIMATION ON COMPACT SETS

57

though t h e y a r e b a s e d on theorems and methods which a r e j u s t t r u e i n h i s g i v e n d p e c i a l s i t u a t i o n , i t t u r n s o u t from o u r d i s c u s s i o n t h a t what i s r e a l l y needed i s o n l y a p r o o f o f t h e a . p .

above

o f A(K) ( = H ( K ) )

t o make e v e r y t h i n g work, even i n many u t h e h cases.

W e turn t o sheaves

F of harmonic f u n c t i o n s o r , more g e n e r a l l y ,

of n u l l - s o l u t i o n s o f h y p o e l l i p t i c d i f f e r e n t i a l o p e r a t o r s w i t h Cm-meff i c i e n t s now. These a r e a g a i n n u c l e a r F r g c h e t s h e a v e s , a n d h e n c e o u r

F h a s t h e a . p . i s c e r t a i n l y s a t i s f i e d . For nuclearity

assumption t h a t

of the sheaves i n axiomatic p o t e n t i a l theory, c f . Cornea 1 1 4 1 ,

Constantinescu-

5 11.

I n t h i s case, w e w i l l assume f o r t h e moment t h a t

s e t K i s t h e c l o s u r e o f some open s u b s e t

U of

X

for

AF(K):

F

,

i . e . f o r each with

f E AF(i) L :f

+

f

I au

g E C(aU) there e x i s t s

I f w e suppose that

i s b i j e c t i v e from

f o r functions i n

AF(K) onto

A F ( K ) w i l l imply t h a t

c e r t a i n l y h a s t h e a.p.

result that

L :f

+

dpace w i t h a . p . o f closed set

K'

621/2,

the

function

C(3I.l) and h e n c e y i e l d s a

C aU)

L

( A maximum p r i n c i p l e

i s e v e n a n i d u m e t h y . ) Then

I n f a c t , i t would be enough f o r s u c h

flK,

is bijective from

C(K')

f o r some c l o s e d subset K '

.

Let f o r instance t i o n s on

unique

to

f l a u = g , t h e c o n t i n u o u s l i n e a r r e s t r i c t i o n mapping

t o p o l o g i c a l isomorphism of t h e s e Banach s p a c e s .

AF(K)

a

fat).

comp.leteLy x7~Lv.L-

U is a hegulak set for t h e DihiehLet phoblem with r e s p e c t

sheaf

compact

(and hence

A v e r y n i c e phenomenon may o c c u r h e r e which y i e l d s a a t s o l u t i o n t o t h e question of t h e a.p.

the

F be t h e s h e a f

AF(K) onto

of

a cLodsdbubK

(say,

JC of ( r e a l ) haamonic

R" ( n 1. 2). W e refer e . g . t o Ho-Van-Thi-Si

a

a

func-

[ 2 2 ] , p. 617/8,

6 2 6 , 637 f o r c o n d i t i o n s c o n c e r n i n g , s a y , t h e e q u a l i t i e s (i)

= HJC(K)

+(K)

(ii) +(K)

I aK

ciple,

,

and

( o r , e q u i v a l e n t l y , by t h e maximum prin-

= C ( aK)

L : +(K)

+

C ( 3 K ) b i j e c t i v e and i s o m e t r i c ) .

BIERSTEDT

68

L e t u s o n l y n o t e t h a t i n g e n e r a l a s u i t a b l e ( o u t s i d e ) cone con-

d i t i o n i m p l i e s b o t h ( i ) and ( i i )and t h a t , i n t h e case

n =2,

(ii)are v a l i d for a compact set K such t h a t e a c h p o i n t

x

E

( i )and aK

is

a boundary p o i n t of a c o n n e c t e d component o f t h e complement o f K . So then

= HX(K)

Ax(K)

has the a.p.

W e a l s o r e f e r t o Weinstock [ 311 f o r r e s u l t s on

f o r sheaves

AF(K)

F = NL (on Rn) o f n u l l s o l u t i o n s of ( l i n e a r )

partical differential operators

L of o r d e r

elliptic

m with constant coeffi-

c i e n t s i n t h i s c o n n e c t i o n and t o Vincent-Smith [ 2 8 ] f o r i n t h e s e t t i n g o f harmonic s h e a v e s

= HF(K)

AF(K) = H F ( K )

F of a x i o m a t i c p o t e n t i a l t h e o r y .

I t would l e a d us too f a r a f i e l d e v e n t o g i v e o n l y c o m p l e t e he@~encu

for a l l interesting relevant results i n t h i s direction. Another argument t h e n y i e l d s t h e a . p . of

AF(K)

and

HF(K)

e v e n i n a much more g e n e r a l s e t t i n g :

1 2 . THEOREM:

(n

1. 2 ) and

L e t again K

JC

be t h e nhead a d haamonic dunctiono o n R n

an a h b i t h a h y compact nubhet

(1)

Then b o t h

(2)

Hence

Ax(K,E)

dpace

E , a n d , doh duch an

+(K,E)

PROOF:

=

E

GE

Hx(K)

alwayn have t h e a . p . h o l d n d o h each c o m p l e t e l . ~ .

+(K)

=Hx(K)

m a y s himpfiu

p. 6 2 1 , 634 shows, b o t h

A = Hx(K)and

E , +(K)

= Hx(K,E).

As Ho-Van-Thi-Si

A = Ax(K)

and

&(K)

1221,

are h i m p . t i c i a l s p a c e s , i . e . t h e null measure i s t h e

A - m a x i m a l measure ( o r , e q u i v a l e n t l y , measure

Choquet boundary of 116 1 , p.

A)

99) t h a t t h e s t a t e space

C(S).

concentrated

only

in

the

o r t h o g o n a l t o A . T h i s means (cf. Effros-Kazdan S = S(A) i s a A i m p L e x and t h a t

i s order isometric t o t h e Banach s u b s p a c e tions in

Wn.

06

A(S)

o f a l l addine

A

func-

However, i t i s well-known t h a t e a c h s u c h A h f l e x Apace

A(S) h a s t h e a . p . :

In f a c t , A ( S )

i s an a b d t k a c t

(L)

- apace.

( This

68

VECTOR-VALUED APP ROXlMATlON ON COMPACT SETS

argument can be found e . g . i n t h e p r o o f o f C o r o l l a r y 2 . 6 , Namioka-Phelps

( 2 ) f o l l o w s from (1) and 3 . ( 2 ) ,

l231.1

p. 4 7 7

of

5 above.

For t h e c o n n e c t i o n between s i m p l i c i a l s p a c e s a n d t h e of " w e a k PihichLet p t o b t e m n " see Effros-Kazdan 1161 :

solution (say) is

+(K)

s i m p l i c i a l i f and o n l y i f e a c h c o n t i n u o u s f u n c t i o n d e f i n e d on a comp a c t s u b s e t o f t h e Choquet baundaty of element of

A X ( K ) may b e e x t e n d e d t o

an

o f t h e s a m e norm.

+(K)

But now w e g e t t h e a . p .

of

A F ( K ) and

f o r many

HF(K)

sheaves

F o f a x i o m a t i c p o t e n t i a l ? t h e o h y a n d aLl? sets K = c l o s u r e o f a r e l a t i v e l y compact open s e t

U:

I n f a c t , under c e r t a i n

u n d e r l y i n g hahmonic npace ( X IF)

,

i t i s known t h a t

axioms

on

the

AF(K) resp. HF(K)

i s a g a i n b i m p L i c i a L , and t h e n we may p r o c e e d a s i n t h e p r o o f o f t n e orem 1 2 t o c a r r y t h e c o r r e s p o n d i n g r e s u l t s o v e r t o t h i s

(much

more

g e n e r a l ) s e t t i n g . For t h e r e l e v a n t axioms needed here and t h e AF(K) resp. H (K)

F

[ 1 6 ] , Cor. 4 . 3 ,

i s a s i m p l i c i a l s p a c e , we

p . 1 0 8 and Cor. 4 . 2 ,

s u f f i c i e n t condition for

orem 4 . 4 ) .

I n [16 ]

,

r e f e r t o E f f r o s -Kazc*.

p. 112.

(For a n e c e s s a r y

and

A F ( K ) = H F ( K ) i n t h i s s e t t i n g see [ 1 6 ] , t h e -

t h e axioms s t i l l e x c l u d e d genehat

sets

open

U

f o r d e g e n e t a t e e l l i p t i c e q u a t i o n s , b u t t h e c o r r e s p o n d i n g problem was s o l v e d a f f i r m a t i v e l y by B l i e d t n e r - H a n s e n [ 1 3 ] ,

and w e r e f e r t o

f o r t h e m o s t g e n e r a l r e s u l t s on s i m p l i c i a l s p a c e s

[13]

AF(K).

I n concluding, we should p o i n t o u t t h a t t h e €-product

E

Jfl

o f two s h e a v e s o f harmonic f u n c t i o n s i n a x i o m a t i c p o t e n t i a l

X2

theory

y i e l d s n o t h i n g b u t t h e ( m u L t i p L y r e s p . ) b e p a h a t e e y h a h m o n i c functions of

Gowrisankaran [ 201

resp.

Reay [ 2 4 1 . W e l e a v e i t t o t h e reader to

combine o u r p r e c e d i n g remark on t h e a . p .

of

AF(K) resp.

in

HF(K)

above

to

o b t a i n , s a y , theorem 11 and lemma 2 3 of [ 2 4 ] w i t h o u t any e f f o r t .

Of

axiomatic p o t e n t i a l theory with t h e r e s u l t s i n s e c t i o n

c o u r s e , w e could also immediately s t a t e r e s u l t s f o r holomorphic

- harmonic

sheaves

0

E

JC

etc.

,

2

"mixed"

(say)

b u t t h e p r e c e d i n g examples

BIE RSTE DT

60

and a p p l i c a t i o n s may s u f f i c e .

REFERENCES

[ 11

0. B. BEKKEN, The a p p r o x i m a t i o n p r o p e r t y f o r Banach spaces

of

a n a l y t i c f u n c t i o n s , p r e p r i n t (19741, u n p u b l i s h e d . [ 21

K.-D.

BIERSTEDT, F u n c t i o n a l g e b r a s a n d a t h e o r e m o f f o r vector

- valued

Papehs

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Mergelyan

t h e Summeh

6hom

Gathehing o n Function A l g e b k a h , A a r h u s , V a r i o u s . P u b l i c a t i o n S e r i e s 9 (19691, 1 - 1 0 . [

31

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BIERSTEDT,

Gewichtete

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Raume

stetiger

vektorwertiger

das i n j e k t i v e T e n s o r p r o d u k t

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J.

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[ 41

K.-D.

BIERSTEDT, I n j e k t i v e T e n s o r p r o d u k t e und

Slice

- Produkte

g e w i c h t e t e r Raume s t e t i g e r Funktionen, J. reine Math, 266 (1974) [ 5]

K.-D.

angew.

121-131.

BIERSTEDT, The a p p r o x i m a t i o n p r o p e r t y f o r w e i g h t e d

func-

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Bonn 1974) , Bonner. Math. [ 6]

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BIERSTEDT, Neuere

S c h r i f t e n 81 (19751, 3-25; 26-48.

Ergebnisse

von Banach-Grothendieck

,

zum A p p r o x i m a t i o n s p r o b l e m

J a h r b u c h U b e r b l i c k e Math. 1976,

B1, 45-72. [ 71

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BIERSTEDT a n d R. MEISE, L o k a l k o n v e x e Unteraume l o g i s c h e n Vektorraumen und d a s

Math. 8 (1973) , 1 4 3 - 1 7 2 .

I 81

K.-D.

E -Produkt,

BIERSTEDT and R. MEISE, Bemerkungen tionseigenschaft

i n topo-

Manuscripta

iiber d i e Approxima-

lokalkonvexer Funktionenraume,

Ann. 209 (19741, 99 - 1 0 7 .

Math.

VECTOR-VALUED APPROXIMATION

[ 9

1

K.-D.

BIERSTEDT, B .

ON COMPACT SETS

61

GRAMSCH a n d R . MEISE, Lokalkwexe Garben und

g e w i c h t e t e i n d u k t i v e L i m i t e s F-morpher F u n k t i o n e n , Func-

t i o n Spacen and Denbe Appno ximation ( P r o c . Conference Bonn 1974) , Bonner Math. S c h r i f t e n 8 1 (19751, 59 - 7 2 . [10 ]

.

K -D.

BIERSTEDT , B

.

GRAMSCH a n d R . MEISE , A p p r o x i m a t i o n s e i g e n -

schaf t, L i f t i n g

und

KO

- homologie

bei

lokalkonvexen

P r o d u k t g a r b e n , M a n u s c r i p t a Math. 1 9 (1976) , 319

Ill]

[12]

- 364.

F. T . B I R T E L , Holomorphic a p p r o x i m a t i o n t o b o u n d a r y v a l u e g e b r a s , B u l l . h e r . Math. SOC. 84 ( 1 9 7 8 ) , 4 0 6

J . L. BLASCO, Two p r o b l e m s o n k m - s p a c e s ,

- 416.

al-

(19771, t o

preprint

a p p e a r i n A c t a Math. S c i . Hungar.

[13]

J . BLIEDTNER and W .

HANSEN, S i m p l i c i a 1 c o n e s i n p o t e n t i a l t h e -

o r y , I n v e n t i o n e s Math. 29 ( 1 9 7 5 ) , 8 3

[14]

C. CONSTANTINESCU a n d A.

- 110.

CORNEA, P o t e n t i a l t h e o r y

on h a r m o n i c

s p a c e s , S p r i n g e r G r u n d l e h r e n d e r Math. W i s s .

Band

158

(1972). DAVIE, The a p p r o x i m a t i o n p r o p e r t y o f A ( K ) o n p l a n e sets ,

[ 15 1

A.

[ 16 ]

E . G. EFFROS a n d J . L . KAZDAN, A p p l i c a t i o n s o f Choquet

M.

p r i v a t e communication ( 1 9 6 9 ) , u n p u b l i s h e d .

simplexes

t o e l l i p t i c and p a r a b o l i c b o u n d a r y v a l u e p r o b l e m s , D i f f . E q u a t i o n s 8 ( 1 9 7 0 ) , 95 - 1 3 4 .

J.

[17]

L. EIFLER, The s l i c e p r o d u c t of f u n c t i o n a l g e b r a s , Proc. Amer. Math. SOC. 2 3 ( 1 9 6 9 1 , 559 - 5 6 4 .

1181

T. W.

GAMELIN, Uniform a p p r o x i m a t i o n o n p l a n e sets, /\pphoxima-

t i o n Theohy (1973), 1 0 1

[19]

T. W.

( E d i t o r : G.

- 149.

G. L o r e n t z )

GAMELIN a n d J. GARNETT, C o n s t r u c t i v e

,

Academic

Press,

techniques

i n ra-

t i o n a l a p p r o x i m a t i o n , T r a n s . Amer. Math. SOC. 143 (1969) ,

187

- 200.

62

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K.

GOWRISANKARAN, M u l t i p l y h a r m o n i c f u n c t i o n s , Nagoya Math. J . 28 ( 1 9 6 6 1 , 2 7 - 4 8 .

[211

A.

GROTHENDIECK, P r o d u i t s t e n s o r i e l s topologiques

e t espaces reprint

n u c l 6 a i r e s , Memoirs Amer. Ma th. SOC. 1 6 ( 1 9 5 5 ) , (1966).

F r o n t i g r e de C h o q u e r dans les espaces de f o n c t i o n s e t approximation d e s f o n c t i o n s h a r m o n i q u e s , B u l l . SOC. Roy S c i . L i s g e 4 1 ( 1 9 7 2 ) , 6 0 7 - 6 3 9 .

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1231

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sets, P a c i f i c J. Math. 31 ( 1 9 6 9 ) , 4 6 9 - 4 8 0 . [241

I . REAY, S u b d u a l s

[25]

H.

and t e n s o r products o f spaces of harmonic f u n c t i o n s , Ann. I n s t . F o u r i e r 24 ( 1 9 7 4 ) , 1 1 9 1 4 4 .

-

ROSS1 and J . L. TAYLOR, On a l g e b r a s o f h o l o n o r p h i c f u n c t i o n s on f i n i t e pseudoconvex m a n i f o l d s , J. F u n c t i o n a l Anal.24 ( 1 9 7 7 ) , 11 - 3 1 .

126 1

L. SCHWARTZ, T h g o r i e des d i s t r i b u t i o n s 5 v a l e u r s

vectorielles

I , Ann. I n s t . F o u r i e r 7 ( 1 9 5 7 ) , 1 - 1 4 2 .

[271

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SIBONY, A p p r o x i m a t i o n de f o n c t i o n s 5 v a l e u r s d a n s u n F r 6 c h e t p a r d e s f o n c t i o n s holomorphes, Ann. (1974) , 1 6 7 - 179.

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G.

Inst. Fourier

F. VINCENT-SMITH, U n i f o r m a p p r o x i m a t i o n s of t i o n s , Ann. I n s t . F o u r i e r 1 9 ( 1 9 6 9 1 , 339

[291

A.

G.

- 157.

B. M. WEINSTOCK, Approximationbyholomorphic f u n c t i o n s o n cert a i n p r o d u c t sets i n 811

[31]

harmonic func-

- 353.

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(30 1

24

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CN

,

Pacific

J . Math.

43 (1972) ,

B . M. WEINSTOCK, U n i f o r m a p p r o x i m a t i o n b y s o l u t i o n s o f e l l i p t i c

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SOC. 4 1 ( 1 9 7 3 1 , 5 1 3 - 5 1 7 .

Approximation Theory and Functional Analyeie J . B. Prolla (ed. ) 0 North-Holland Publishing Company, 1979

THE COMPLETION OF PARTIALLY ORDERED VECTOR SPACES AND KOROVKIN S THEOREM

BRUNO BROSOWSKI

Johann Wolfgang Goethe U n i v e r s i t a t F a c h b e r e i c h Mathematik Robert Mayer-Str. 6-10 D-6000 F r a n k f u r t / Main, Germany

I n t h e p r e s e n t p a p e r w e w i l l g i v e a new p r o o f of a g e n e r a l i z a t i o n o f K o r o v k i n ' s theorem u s i n g t h e completion of a g a r t i a l l y o r d e r e d v e c t o r s p a c e by Dedekind-cuts.

The g i v e n proof works n o t o n l y i n t h e

case of C[0,1] b u t also f o r c e r t a i n p a r t i a l l y o r d e r e d realvector spaces

where a mode of convergence i s d e f i n e d , which i s c o m p a t i b l e w i t h t h e

l i n e a r s t r u c t u r e and t h e p a r t i a l o r d e r i n g o f t h e c o n s i d e r e d

linear

space. L e t X b e a r e a l vector s p a c e w i t h a p a r t i a l o r d e r i n g d e f i n e d b y K , t h e s e t o f a l l p o s i t i v e e l e m e n t s of

a cone space X

i s c a l l e d Dedekind-complete

X including

i f e v e r y non-empty

. The

0

s u b s e t which

i s bounded f r o m above h a s a supremum and i f e v e r y non -empty

subset

which i s bounded from below h a s a n infimum. I n t h e f o l l o w i n g w e

as-

sume t h a t t h e p a r t i a l o r d e r i n g i s Archimedean which i s d e f i n e d by

f o r a l l elements

u , v E X.

T h m we have t h e f o l l o w i n g

THEOREM:

Let X be a p a f i t i a g l y ohdefied h e a l u e c t o f i o p a c e , w h i c h 63

id

64

BROSOWSKI

Ahchimedean. Then we have: T h e h e i n a uni que detehmintd Dedekind - cornpLete p a h t i a L L y

dehed heaL v e c t o h Apace (i)

oh-

6x w i t h t h e BoLlowing p n o p e h t i e d :

x 06

Thehe e x i b t b a dubbpace

6 X duch t h a t

x ahe

X and

ihom okphi c. (ii)

~ v e h ye k e m e n t

x#

E

6x

hatis die4

6~ L A c a l l e d t h e V e d e k i n d - c o m p l e t i o n

X i s directed i.e.

If i n a d d i t i o n t h e o r d e r i n g i n

then

06 X .

6X is also a lattice. For a proof of t h e theorem compare LUXEMBURG, ZAANEN [ Z

DEFINITION:

A subspace

c a l l e d Dedekind-denhe i n

1.

X of a p a r t i a l l y o r d e r e d B - v e c t o r s p a c e is Y

iff

x c

Y C 6X.

For s t a t i n g t h e g e n e r a l i z e d Korovkin-theorem w e have t o d e f i n e t h e mode of convergence i n a p a r t i a l l y o r d e r e d v e c t o r s p a c e . W e some r e s u l t s of BANASCHEWSKI A subset

[ 11 :

E C K \ {O} d e f i n e s a convergence g e n e r a t i n g s e t i n X

i f E s a t i s f i e s t h e following conditions:

REMARK:

S i n c e w e assume X t o be Archimedean w e have

111.

use

inf E = 0 .

COMPLETION OF PARTIALLY ORDERED SPACE5 AND KOROVKIN'S THEOREM

85

Now w e d e f i n e a mode of convergence a s f o l l o w s : A sequence (xn)

x

c

converges t o an element

I n t h i s case we w r i t e

xn

+

E

lowing p r o p e r t i e s :

Z

x

iff

-

x. T h i s mode of convergence h a s t h e fol-

(a)

C o n s t a n t s e q u e n c e s are c o n v e r g e n t .

(b)

If

(x

E

converges t o

converges a l s o t o

x.

G I t h e n e v e r y subsequence of ( x n )

F u r t h e r w e assume

(e)

L e t ( x n ) b e a sequence such t h a t

and such t h a t

x (f)

Let

*

n E

i n f (x,) e x i s t s I t h e n inf(rcn).

(3:n ) be a sequence s u c h t h a t XI

2 x2 2 x3 5

and such t h a t

---

s u p ( ~ ~e )x i s t s , then

xn * s u p ( x n ) . E

Now w e can s t a t e t h e g e n e r a l i z a t i o n of K o r o v k i n ' s theorem:

THEOREM I:

LeZ

Y be a p a h t i a l l y atdehed

b e a COi?vehgenccL g e n e h a t i n g 6 e Z i n Y

.

W - w e c t o h & p a c e and LeZ

E

Fuhtheh L e t X be a n k c h e d e a n

BROSOWSKI

66

p a h t i a L L y o h d e f i e d a - v e c t a h pace, Luhich

Let

( L ) be

a oequence Ln:Y

06

i 6

Vedekind-dende i n Y.

monotonic opetlatohn

Y

+

buch t h a t

A : Y + Y

i b

a monotonic o p c h a t a h ouch t h a t t h e h e d t h i c t i o n A

map o d

X o n t o X and

-LA

06

,x

i d u

bijtdue

mona.tonic t y p e l i . e . A ~ ~ ( ~ ~ ) & *AZ ,I =< ~ z( 1,z ~ ) 2

T h e n rue have

PROOF:

For t h e proof l e t

u L

F o r each

u

E

U

Y

Y Y

and

y

E

Y C 6X

be given. Then d e f i n e t h e s e t s

:= I u E

x

:= { I

X I 2 5 - y).

I

E

E

L

Y

I y

5 ul,

w e have

2 5 Y ( U . Since L n

and

A

are monotonic w e have Ln(I)

and

5

Ln(Y)

5

Ln(u)

COMPLETION OF PARTIALLY ORDERED SPACES A N D KOROVKIN'S THEOREM

For a b b r e v i a t i o n w e set 1 , := L , ( Z ) , W e now p r o v e :

(y,)

y,

:= LJy),

un := L J u L

converges t o a n element y

0

.

S i n c e by a s s u m p t i o n

w e have

From t h i s w e c o n c l u d e t h a t t h e e l e m e n t s

e x i s t and a l s o s a t i s f y t h e r e l a t i o n

C o n s e q u e n t l y w e have

where

-

i := s u p { i n )

T h i s i s t r u e for e v e r y

E E

and

-

s

:= i n f

isn}.

E; t h u s w e have a l s o

67

68

BROSOWSKI

iqow l e t

u E

U-

i

. Then w e

7 5

have

u

and by ( " 1

S i n c e A i s of monotonic t y p e and b i j e c t i v e w e have

and c o n s e q u e n t l y

A

-1

(u) E U

Y

.

From t h i s w e c o n c l u d e

and hence

Now l e t

.

u E UA(Y)

I = A-'A(Z)

W

l € L

and consequently

and hence

A

u E U-

i

-1

,

Then w e have

z

(u) E U

i.e.

=

2

u

and

2 A- 1 (u)

Y

Y

. Using

'A(y)

c U-

S i m i l a r l y one c a n p r o v e L - = L A ( y t h i s w e conclude:

A($)

s

From t h e r e l a t i o n s

i

= A(y).

(*)

we c o n c l u d e

. Consequently we

have

y)

.Using

COMPLETION OF PARTIALLY ORDERED SPACES AND KOROVKINS THEOREM

Since

E E

was a r b i t r a r y w e have

E

Let

REMARK:

68

C [ a , b 1 be t h e v e c t o r l a t t i c e o f a l l r e a l - v a l u e d con

-

t i n u o u s f u n c t i o n s on [ a, h 1 under t h e o r d e r i n g d e f i n e d by "f 0.

1 , is

whether

n c a n be c o n s t r u c t e d which

gives

The n a t u r a l e x t e n s i o n of t h i s p r o b l e m , p o s e d i n [ 8 a n a l g e b r a i c p o l y n o m i a l of d e g r e e

u n i f o r m a p p r o x i m a t i o n t o t h e associate order

0 (n-l-a)

f

on t h e w h o l e [ - 1 , 11 w i t h

p r o v i d e d t h e d e r i v a t i v e f'

L i p l ( a ; C) ,

belongs t o

o 0, p

E

by h y p o t h e s i s , g E M

f

r,

wl,

such t h a t

k = 1 f max I w . ( x ) I . Then 3 ' which p r o v e s t h a t

€

E

... ,wm

p [ f

(XI

p[wj(x)(f(x)

zA(M).

-

x

€

W,

-

g (XI 1
~ j .

aGj

i s t r a n s v e r s a l t o each

aPk

and

aG. n

aQk

3 By a r e s u l t o f S c h e i n b e r g [ 1 7 , Theorem 3 . 2 ]

aPk

and

are i s o l a t e d sets.

,

each of t h e R i e

-

mann s u r f a c e s

G j U Q1 U Q2 U

... " k'

a d m i t s a compact e s s e n t i a l e x t e n s i o n . Thus, by t h e s p e c i a l Theorem 1, t h e r e i s a f u n c t i o n

T h e r e e x i s t s .a f u n c t i o n holomorphic on

El

U

P1

.

Set

ml E M(GZ

pl E M(R)

U

Q1)

case

of

with

such t h a t

m1

-

p1

is

151

MEROMORPHIC APPROXIMATION ON CLOSE0 SUBSETS OF RIEMANN SURFACES

-

ml

-

f

p1

p1

on

G1

on

F2

u

F1 ,

By t h e s p e c i a l c a s e of Theorem 1, t h e r e i s a f u n c t i o n g2EM(G U Q U Q ) 3 1 2

such t h a t

.

Set

m2 = g2 + p1

Set

f l = f . Then, w e may p r o c e e d i n d u c t i v e l y t o c o n s t r u c t a s e q u e n c e

m' j

s a t i s f y i n g for

Then,

E

M(Gj+l

j =2,3,..

.

U

...

Q1 u

U Qj)

, J

E

c

Imj(z)

- f(7.11

Im.(z)

- mj-l(z) I


01

and

I n i n f i n i t e d i m e n s i o n s , W h i t n e y ' s theorem i s f a l s e i n formulat i o n 1, even i n t h e c a s e

U = H , a real s e p a r a b l e H i l b e r t s p a c e , and

m = l . We p r e s e n t an example of t h i s i n s e c t i o n 2 . I n f o r m u l a t i o n i t i s t r u e , w i t h r e s p e c t t o t h e u s u a l compact-open

case

m =1 w i t h some r e s t r i c t i o n s . The case

2

topology, f o r the

m 1. 2

r e m a i n s a n open

problem and o u r g u e s s i s t h a t t h e theorem i s f a l s e i n t h i s c o n t e x t . Two o t h e r d i r e c t i o n s a r i s e n a t u r a l l y i n i n f i n i t e dimensions:the f i r s t one i s t o c o n s i d e r subspaces o f d i m e n s i o n s , w i t h t h e whole s p a c e new t o p o l o g y i n

am(U)

g r n ( U ) which c o i n c i d e , i n f i n i t e

am(U);

t h e second i s t o l o o k f o r a

which c o i n c i d e s , i n f i n i t e d i m e n s i o n s ,

with

t h e u s u a l one. I n s e c t i o n 2 w e c o n s i d e r t h e c o n c e p t o f d i f f e r e n t i a b i l i t y type, which g i v e s u s a u n i f i e d way t o d e a l s i m u l t a n e o u s l y s u b s p a c e s of

several

with

grn(U).

I n [ 1 2 ] R e s t r e p o s t u d i e d t h e c l o s u r e o f t h e a l g e b r a of

poly-

n o m i a l s of f i n i t e type i n a Banach s p a c e o f a c e r t a i n k i n d , f o r t h e topology o f t h e uniform convergence of t h e f u n c t i o n and i t s d e r i v a t i v e on bounded s u b s e t s . I n [ l ] Aron and P r o l l a e x t e n d e d t h i s r e s u l t

to

a more g e n e r a l c l a s s of Banach s p a c e s , c o n s i d e r i n g t h e case m 2 2 and polynomial a l g e b r a s o f v e c t o r f u n c t i o n s weakly u n i f o r m l y

continuous

on bounded s u b s e t s . I n s e c t i o n 3 w e s t u d y i d e a l s of f u n c t i o n s weakly u n i f o r m l y c o n t i n u o u s on bounded s e t s , w i t h r e s p e c t t o t h e t o p o l o g y of t h e u n i f o r m convergence o f order m on bounded sets. I n s e c t i o n 4 , w e c o n s i d e r t h e topology

in

[lo].

T

C

introducedbyProlla

WHITNEYS SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS

161

F i n a l l y , i n s e c t i o n 5, we use t h e r e s u l t s of s e c t i o n 4

t o es-

t a b l i s h some f a c t s a b o u t modules. The r e s u l t s o f t h i s p a p e r a r e t a k e n from t h e a u t h o r ' s D o c t o r a l D i s s e r t a t i o n a t t h e U n i v e r s i d a d e F e d e r a l d o R i o de J a n e i r o ,

written

under t h e guidance of P r o f e s s o r J . B . P r o l l a .

1. PmLIMINARIES

I n t h e sequel

stands

N

for

{0,1,2,...1,

m s t a n d s f o r a n e l e m e n t of

e l e m e n t s of

N.

Let

E

cal d u a l s E ' E' 8 F 9 8

a n d F'

For

E E +

X

IN

U

natural

{ml

and

respectively,

9(x)v

E

U

for

v

applications

F.

E

a real H a u s d o r f f l o c a l l y c o n v e x s p a c e , a function

unique) such t h a t , f o r

x E U,

Df(x)y = l i m

X

E

uniformly with respect to

Df : U

+

f :U + X

6(E;X) ( n e c e s s a r i l y

IR,

f ( x + XY)

A+O

A

-

f(x)

y o n e a c h bounded s u b s e t of

I n t h e same way, w e d e f i n e c - d i 6 6 e h e n t i a b i l i t y by

i, j,k

a non-empty open s u b s e t ,

s p a n n e d by t h e

d:(E;F)

F, p E E ' ,

E

C

i s c a l l e d b-diddehentiabLe i f there i s

b

integers

a n d F b e r e a l normed l i n e a r s p a c e s w i t h t o p o l o g i -

# 0

t h e l i n e a r s u b s p a c e of

v: x

set of

the

c a n d bounded by compact. W e o b s e r v e t h a t i f

space, b - d i f f e r e n t i a b i l i t y i s Frechet

E

E .

by

replacing

is

a

normed

d i f f e r e n t i a b i l i t y and c - d i f -

f e r e n t i a b i l i t y i s Hadamard d i f f e r e n t i a b i l i t y (Nashed [ 9 1 1 . Let

gy

T~

b d: ( E ; X )

denote the space

S ( E ; X ) endowed w i t h t h e t o p o l o -

of u n i f o r m c o n v e r g e n c e on bounded s u b s e t s o f

denote the space

f(E;X)

endowed w i t h t h e t o p o l o g y

E 7

C

and of

LC(E;X)

uniform

may d e f i n e c o n v e r g e n c e o n compact s u b s e t s of E . By i n d u c t i o n w e b k b b k-1 d:'(OEIF) = F a n d , f o r k 2 1, d: ( EIF) = d: (E;d: ( E I F ) ) . I n t h e same

162

GUERREIRO

way, replacing b by c, we have

LC(kEIF). Furthermore, let C(U;X)

denote the vector space of all continuous functions from U endowed with the compact-open topology The space

.

0

7

Gbm(U;F) and its topology

T~~

will

be

to

X,

defined

inductively as follows: For if

m = O , gbo(U;F)

=

C(U;F),

T~~

0

and we denote D f = f ,

= '7

f E C(U;F). gbl U;F) as the vector space of all € b

For m = 1, define

which are b-differentiable and such that

rbl

pology

Df

E

E

C(U;F)

C(U;d: (E;F)). The to-

is defined as the topology €or which the isomorphism

f E gbl(U;F)

+

(f,Df)

C(U;F)

x

C(U;Lb (E;F))

is a homeomorphism. For uniformity of notation, D1f = Df. Suppose we had already defined Eb(k-l) (U;F), 'Ib (k-1) Dk-l , &b(k-l) (U;F)

-+

C(U;lb(k-lEIF)), for some

and

2.

k

8b(k-1) (U;F) b k such that Dk-lf is b-differentiable and D(Dk-lf) E C(U;I: ( EIF)) Define Dk: gbk(U;F) -t C(U;eb(kEIF)) by Dkf = D(Dk-'f) and the toDefine

pology

-rbk

gbk(U;F) as the vector space of all

f

E

.

as being the only one for which the isomorphism

is a homeomorphism. Finally, define

ab"(U;F) =

n

kslN

the topology for which the isomorphism

is a

homeomorphism.

sbk(U;F) and consider as

b-

7

WHITNEY'SSPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS

F = IR, w e w i l l w r i t e

For t h e c a s e The s p a c e

BCrn(U;F) and i t s t o p o l o g y

163

Ebm(U;F) = 8bm ( U ) .

i s defined

T~~

t i v e l y i n t h e same way, by j u s t r e p l a c i n g b by

induc-

c i n t h e above defi-

nition. k k There i s a n a t u r a l i d e n t i f i c a t i o n between L ( EIF) and L ( E;F) , t h e v e c t o r s p a c e o f c o n t i n u o u s k - l i n e a r maps from Ek t o F. b k

t h e r e i s a homeomorphism between b k d: ( E;F) ( r e s p e c t i v e l y

and

w i t h t h e topology

d: ( EIF)

d : C ( k E I F ) ) ,t h e space

(respectively

T'

(respectively

T

C

I n fact,

d:C(kEIF))

d:(kE;F)

endowed

1.

On t h e o t h e r hand, t h e n a t u r a l isomorphism between

Xs(kE;F)

t h e vector s p a c e of c o n t i n u o u s symmetric k - l i n e a r maps f r o m k

F , and

P ( E ; F ) , t h e s p a c e of c o n t i n u o u s k-homogeneous

from E

i n t o F,

Ek

,

to

polynomials

i s , a c t u a l l y , a homeomorphism, i f w e endow both spaces

w i t h t h e topology

T~

or both with t h e topology

T

C

.

cm

Moreover, g i v e n f b e l o n g i n g t o Cbm(U;F)or 8 (U;F), x E U, k z m , k k w e may a s s o c i a t e D f ( x ) w i t h a n e l e m e n t d k f ( x ) o f gs( E;F) which k may be i d e n t i f i e d w i t h a p o l y n o m i a l a k f ( x ) of P ( E ; F ) . bm I n t h a t case, t h e T t o p o l o g y may be d e f i n e d i n gbm(U;F) by t h e f a m i l y o f seminorms o f t h e form

K

C

U

a compact s u b s e t , k 5 m. cm

The t o p o l o g y

T

may be d e f i n e d i n

o f seminorms :

K

C

U, L C E

compact s u b s e t s ,

k

F o r d e t a i l s , see Nachbin [ 8

5 m.

1

.

LCm(U;F) by t h e

family

164

GUERREIRO

2 . IDEALS AND DIFFERENTIABILITY TYPES The c o n c e p t of holomorphy t y p e f o r complex f u n c t i o n s i s already

w e l l known (Nachbin [ 7 1 1 . The same d e f i n i t i o n may be a p p l i e d t o real s p a c e s (Aron and P r o l l a 11

DEFINITION 2.1:

P + II PII,

Pek (E;F) k

, which

E INl

t h e norm on e a c h b e i n g

denoted

s a t i s f i e s the following conditions:

i s t h e normed s p a c e o f a l l c o n s t a n t functions fran

Peo(E;F)

i)

F is asequence

A di6dekentiabiLity type dhom E ,to

of Bnnach s p a c e s by

1 1.

to F, i d e n t i f i e d w i t h F ; 8k ii) each P ( E ; F ) i s a v e c t o r s u b s p a c e o f E

iii) t h e r e i s a r e a l number

x E E

DEFINITION 2.2:

Let

0 b e a d i f f e r e n t i a b i l i t y t y p e from E

E

Pek(E;F) imply

j, k E IN

P

,jc

k,

i J P ( x ) E P e J ( E ; F ) and

pern(U;F) a s t h e v e c t o r s u b s p a c e of

t o F.We

gbm(U;F)

of

such t h a t , f o r x E U, k 5 m , w e have 2 f ( x ) EPek(E;F) -k Bk x E U + d f ( x ) E P (E;F) is c o n t i n u o u s .

f

and t h e mapping

W e endow

u 1. 1 s u c h t h a t

and

d e f i n e t h e space a l l functions

k P (E;F);

sem(U;F) w i t h t h e topology

em

d e f i n e d by t h e fam-

‘I

ily of seminorms; -i

p K I k ( f ) = sup IIld f ( x ) l l e ; x E K , 0

where

K

C

U

i s a compact s u b s e t and

I n t h e case

F = IR w e w i l l w r i t e

W e remark t h a t t h e s p a c e definition.

k E IN

5

i

5 kl,

I

k

5 m.

Eern(U;F)

= tZ e m ( U ) .

sbrn(U;F) i s a p a r t i c u l a r case of t h i s

WHITNEYS SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS

DEFINITION 2 . 3 (Aron a n d P r o l l a [ 1 1 ) : from E

A d i f f e r e n t i a b i l i t y type

-3

F i s c a l l e d compact i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i -

to

t i o n s f o r each i)

166

k E IN:

k

Pf (E;F)

,

t h e v e c t o r s p a c e of c o n t i n u o u s k-homogeneous p l y -

n o m i a l s of f i n i t e t y p e , i s d e n s e l y c o n t a i n e d i n

v E F

ii) f o r e a c h

q

+

i s continuous

qk 8 v

from

* I l l t o ( Pkf ( E ; F ) , 11 * I l e ) ;

(E',lI

iii) i f

t h e map

Pek(E;F);

P E E ' 8 E,

then

For each

k E IN,

Q O P E Pek(E;F) f o r a l l

6k (E;F)

Q E P

and

EXAMPLES 2.4: k P f (E;F) i n

let

PCk(E;F)

k

be

the closure

of

6 = c i s a compact dif-

P (E;F) f o r t h e u s u a l norm. Then

f e r e n t i a b i l i t y t y p e c a l l e d cukhenZ compact t y p e . I f we c o n s i d e r , f o r each

k E IN, P N k ( E ; F ) , t h e Banach to

o f a l l n u c l e a r c o n t i n u o u s p o l y n o m i a l s from E

I/*l l N ,

n u c l e a r norm

F , endowed w i t h t h e

E h a s t h e approximation p r o p e r t y ,

then

i s a compact d i f f e r e n t i a b i l i t y t y p e called nuceeah type (see[ 2 1 ) .

9 = N

PROPOSITION 2.5:

Xy

and i f

space

.type ghom 16

P

Bk

E

Let

F b e a Banach npace and

F. k

to

(E;F) = P ( E ; F ) , k E I N , k

6 a di66eaentiabili-

5 m, t h e n

gbm(U;F) = Egm(U;F)

topoLogicalLy.

PROOF:

map

As w e h a v e ( P e k ( E ; F ) , 11. 11 ) a Banach s p a c e a n d t h e i n c l u s i o n 8

k

Pek(E;F) C P (E;F)

e q u i v a l e n t norms.

COROLLARY 2 . 6 :

is continuous, then

11

I1

and

11

- It6

are

0

Let E be a 6 i n i t e dimension nohmed bpace and

compac2 d i 6 6 e h e n t i a b i l i t g t y p e daom E t o

F.

9

a

GUERREIRO

166

k Pf(E;F) =

PROOF:

DEFINITION 2 . 7 : A

C

P 9k (E;F)

k

= P (E;F),

k E IN.

0

8 b e a d i f f e r e n t i a b i l i t y t y p e from E t o F and

Let

a e m ( U ; F ) a non-empty s u b s e t .

W e define:

i= where

n {A

+

m; k

I ( a , k ) ; a E U, k

I ( a , k ) = { f E Eem(U;F); $ f ( a )

=

E

0, 0 5 i

IN}

5

and:

k)

PROPOSITION 2 . 8 : A C

1 6 0 i d a di6dexentiabiLity t y p e 6hom em Eem(U;F) a n o n - e m p t y d u b b e t , t h e n in T - c l o d e d .

PROOF:

Fix

If every

g

f

a E U, k

E A,

T

t o F

and

E,

for

and c o n s i d e r

9 B(a,k) there is

E >

0

such t h a t

p(f

-

g) 2

where

Consider

em

5 m

E

V = {h E gem(U;F); p ( f

-neighborhood o f I f there exists

-

h)
0

1

vi(yl

Let

B

< 6

and

,

p1

, ...

I

(pk E

1 5 i 5 k , imply

E'

Ilf(x)

-

be a d i f f e r e n t i a b i l i t y t y p e from

We d e f i n e :

i s wucbs,

k E IN

, k 5 ml.

such

and that

f ( y ) i I < E.

E

to

F.

WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS

W e endow

gem(E;F) w i t h t h e

d e f i n e d by t h e fam-

+:-topology,

W

175

i l y o f seminorms:

B

C

a bounded subset,

E

W e remark t h a t

k E IN

,

IF(E;F) = I

k

5 m.

bm

(E;F)

f o r a n y compact

type

of

0 , whenever E i s a f i n i t e d i m e n s i o n a l s p a c e . No-

differentiability

t i c e a l s o t h a t f o r 8 a compact d i f f e r e n t i a b i l i t y t y p e from E t o F , Om Pf ( E ; F ) C Iw ( E ; F ) (see Aron and P r o l l a [ 1 ] 1 .

DEFINITION 3 . 3 :

Let

b e a d i f f e r e n t i a b i l i t y t y p e from E

O

A C 8 p ( E ; F ) a non-empty

subset. W e define:

i n a s i m i l a r way a n d , by i n t r o d u c t i n g t h e na-

We may d e f i n e

t u r a l m o d i f i c a t i o n s i n Example 2 . 1 9 , +We m - c l o s e d subset of f o r any

A C

t o F and

is n o t always a

A -4

On the o t h e r h a n d , A

&:(E;F).

em Iw (E;F) a

w e see t h a t

non-empty

is

+:-closed

s u b s e t . The p r o o f o f t h i s f a c t

is

similar t o 2.8.

PROPOSITION 3 . 4 :

16

0

id

a di6dekentiabiLity t y p e 6kom

o a t i d 6 y i n g ( i i i ) 0 6 U e d i n i t i o n 2 . 3 , .then att

to

F

S p ( E ; F ) O P C S:(E;F),

doh

P E E ' @ E.

PROOF: then

E

Let

f E & p ( E ; F ) and

i k ( f o P ) ( x ) = ikf(Px) oP.

P E E ' 8 E.

If

k E IN, k

5 m, x

E

E,

176

GUERREIRO

Let

b e a bounded s u b s e t and

B C E

bounded subset, t h e r e are

- vi(Py)

Ivi(Px)

x,y E B,

IIGkf

(Px)

and

6 > 0

I

< 6 , 1:

-

i k f (Py) II

i

ql,...,ps

5
0. A s

E

Sr

P(B) E

is a

C E

such t h a t

E'

imply

€/I1 P I1k .

Then :

which p r o v e s that

x

E E

Let

DEFINITION 3 . 5 :

+

€I be

hk(f

oP) (x) E Pek(E;F)

i s wucbs.

a d i f f e r e n t i a b i l i t y t y p e from

s a t i s f y i n g (iii)of D e f i n i t i o n 2 . 3 , and l e t G

C

0

E

to

F

E ' 8 E and A C &$(E;F)

be non-empty subsets.

W e s a y t h a t ( A , G ) 6 a t i h d i e n c o n d i t i o n (L) i f g i v e n

have

A og

POlogY

C

em

A,

t h e closure being considered with respect t o t h e to-

'Iw *

{Pn ; n E IN }

C

E' 8 E

such t h a t

Let

IR nuch t h a t

& r ( E ) i6 an a l g e b h a and L e t

Suppobe thehe

v

o Pn

+

i b

a nequence

I

C

Then

ib

for all 9 EE'.

G = { P n ; n E IN } C E ' 8 E

ha4 p h o p e h t y (B*) N i t h h t h p e c t t o G ;

?

,

hatib6ie6

the

c o n d i t i o n (L).

Tp-tLobuhe

06

I i n

For t h e proof w e need t h e f o l l o w i n g lemmas:

.lE(:&

E

t o

8 P ( E ) b e an i d e a l .

that:

ii) ( 1 , G )

9

be a compact d i d b e h e n t i a b i l i t y Xype dhom

0

THEOREM 3 . 7 :

i) E

se-

W e s a y t h a t E had phopehty ( B * ) i f t h e r e i s a

DEFINITION 3 . 6 : quence

we

g E G

nuch

WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS

Let

LEMMA 3.8:

be a d i d i e h e n t i a b i e i t y .type daom

8

& P ( E ) in an aegebaa and

.that

R : g E Iwe m ( E l

16 i d e a l ad

I

+

C

C

m 1;

E

Thehe

id

Muheouek,

R ( 1 ) i n an

f

E

E P ( E ) , and

in

f E

I,

then

Rf

Sbm(El).

8 a compact d i d d e h e n t i a b i l i t y t y p e d h o m

id

IN, k 5 m , B C E

u bounded n u b n e t ,

E

henpect

E

t o

F,

> 0.

no E hl d u c h t h a t

S e e Aron a n d P r o l l a [ 1]

PROOF OF THEOREM 3 . 7 : Conversely, l e t

.

I t i s clear t h a t f E

:,

B

E

C

-

Y

I C I.

a bounded subset,

k

5 m,

and

be given.

By Lemma 3 . 9 ,

there is

no

E N

such t h a t

pBIk(f - f oPn) < ~ / 3 , n Fix

n

1. n 0

and l e t

P = P

a n d r e s u l t s from Lemma 3 . 8 , Rf in

connideh

l e t E b e a b p a c e 6 a t i b B y i f l g phopehty (B*) w i t h

f E s ~ ( E ; F ) ,k E

> 0

and

Analogous t o 2 . 1 3 .

t o {Pn ; n

E

nuch

EP(E).

g F ( E ) i n an i d e a e Xhen t h e rbm-cLonuhe 06

Ebm(E1).

LEMMA 3 . 9 :

PROOF:

rn

t o

glEl E gbm(El).

6eLongn t o t h e ~ ~ ~ - c k ? o b u06h e R ( 1 )

PROOF:

E

b e a d i n i t e dimenbionae llubbpace

El C E

Let

E'

I??

n

.

If

n0

.

= P(E)

belongs to t h e

Sbrn(El). F u r t h e r m o r e , P ( B ) C El

,

by u s i n g n o t a t i o n

r b m - c l o s u r e of

R(1)

i s a bounded subset, then a rela-

t i v e l y compact s u b s e t , a n d t h e t o p o l o g y by t h e f a m i l y of seminorms:

El

2

T~~

may be defined i n gh(El)

178

GUERREIRO

L C El

a compact subset, j

So, t h e r e i s

g

such t h a t :

E I

I i i i ( R f ) (Px) O P x E B ,

El, j 5 m.

E

-

ii(Rg) (Px) o P l l , < E / 3 ,

O ( i ( k ,

and u s i n g t h e f a c t t h a t ( 1 , G )

s a t i s f i e s c o n d i t i o n ( L ) , t h e r e is h E I

such t h a t

'B,k

(4 O P

-

h) < ~ / 3 .

Then :

x E B, 0

4.

-

i < k , which c o n c l u d e s the p r o o f .

IDEALS OF

0

Ecm(U)

DEFINITION 4.1:

For

A C ECm(U;F) a n o n - e m p t y

= n {A+I(a,k,L,E); a

E

U, k 5 m , L

C

subset

E compact,

E

we

define

> 0)

where I ( a , k , L , ~ l = { f E g C m ( u ; F ) ; I I i i f ( a ) v I I < E , v E L, 0

5 i 5

kl.

WHITNEYS SPECTRALSYNTHESIS THEOREM IN INFINITE DIMENSIONS

The d e f i n i t i o n o f

e x t e n d s n a t u r a l l y and obvious modifica

-

may b e f a i l t o b e TCm-closed. Bycontrast,

t i o n s i n 2.19 show t h a t

i s always

179

Tcm-closed.

The d e f i n i t i o n of c o n d i t i o n (L) f o r a p a i r ( A , G ) ,

G

C

E' 8 E

a non-empty s u b s e t , i s n a t u r a l l y e x t e n d e d t o o .

THEOREM 4.2:

be a n k k d and buppobe Ahetre 0 G

I C Ecm(U)

Let

C

E' 8 E

buch t h a t

i)

iE, t h e i d e n t i t y

E

06

,

betungb t o t h e ctobuhe

06

i n

G

F(E;E) ; ii) (1,G) b a t i b 6 i e b c o n d i t i o n ( L ) Then

LEMMA 4.3: VeCtOh

i b

Let

t h e Tcm-c.labuhe

I

U n El

C

16 we c a n b i d e h 06

06

then

Bcm(U1).

i,

K

C

giUl E gCm(U1)aU'l .the Tcm-dClbWze

06

acm(U),

f E

id

R(1)

i n

f E

1,

gCm(U1).

gbm(U1) = BCm(U1)

W e j u s t remark t h a t

I t i s clear t h a t

PROOF OF THEOREM 4.2: f E

dimevlshnd

to-

is a f i n i t e d i m e n s i o n vector space.

p o l o g i c a l l y b e c a u s e El

Let

+

Moheoveh,

Rf b e l o n g b t o t h e Tcm-C.tObWLe

PROOF: Analogous t o 2.13.

a dinite

E

C

a non-empty open bubbet.

Scm(U)

R :g

R ( 1 ) i b an i d e a l

ECm(U).

i n

be an i d e a l , El

C Ficm(U)

a u b b p a c e , U1

06 I

.

U

and

L

C E

i

C

i.

compact s u b s e t s , k

By Lemma 3 . 1 , P r o l l a a n d G u e r r e i r o [ l l ] , t h e r e are

5 m,

u E G

E

> 0.

and V C U

a non-empty open s u b s e t s u c h t h a t

Consider

El = u ( E ) , U1= E l

11 U,

K 1 = u(K) a n d

L1 = u ( L ) .

By

GUEAREIRO

180

u s i n g n o t a t i o n and r e s u l t s f r o m Lemma 4 . 3 , t h e r e i s

g

On t h e o t h e r hand (1,G) s a t i s f i e s c o n d i t i o n ( L ) h

E

I

acm(U)

-

THEOREM 5.1: C

aLL

so

there

is

L, 0 5 i 5 k .

x

T h i s shows t h a t

W

such t h a t

such t h a t

(x,v) E K

5.

E I

f E

7.

0

SUBMODULES OF

tCm(U;F)

Let F be a bpace with t h e apphoximation phopehtg

BCm(U;F) an

8m(u)-submodule s a t i n d y i n g :

(v

o W) 8 v c

tp E F', v E F .

Suppose thehe is i)

iE

G C E' Q E

duch t h a t :

6eLongn t o the ~ L o d u h e06 G in

LC(E;E);

if

and doh

WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS

Then

06

i b t h e Tcm-ctobuhe

W in

The p r o o f of 5 . 1 u s e s t h e f o l l o w i n g W

C

GCm(U;F) i s a n

T h e v e c t o h bubhpace

LEMMA 5.2:

. Moheoueh,

ECm(U)

&Cm(U)-submodule a n d

(9 o

W,G)

SCm(U;F). two

lemmas,

both ,

In

9 E F'.

9 o w = {p

hatib 6ieb

181

o g; g E W }

ad

an id&

i b

condition (L), id ( W , G )

hatib-

iieb condition (L).

PROOF:

If

h E gcm(U) a n d

= h(p og) E 9

g E W, then

O W . Therefore

9

OW

a n d , so

hg E W

i s an i d e a l .

Suppose now t h a t (W,G) s a t i s f i e s (L) a n d l e t be a non-empty open s u b s e t s u c h t h a t and

L C E

compact s u b s e t s , f

E

9 o (gh) =

g

and

E G

V C U

g ( V ) C U . I f we c o n s i d e r K

W , k 5 m,

E

> 0,

is

there

h

C

V

E

W

such t h a t

Then :

This proves t h a t

Suppobe t h a t

LEMMA 5.3:

G

doh borne

16

C

f E

(9 o

W) o ( g !V) C (9 o W l V )

.

0

iE beLong6 t o t h e c t o b u k e

G

06

in EC(E;E),

E' 8 E , and t h a t ( W I G )batid6ieb condition ( L ) .

GI

then 9 o f

beLong6 to t h e r c m - c l o b u h e

06

9 o

W

in

FhCrn(U).

PROOF:

Consider

f E

5,

a

E

U, k 5 m ,

E >

0

and

L

C

E

a

compact

182

GUERREIRO

s u b s e t . There i s

y

E

L, 0 5 i

5

g E W

such t h a t

9 o f E

k , which p r o v e s t h a t

, S i n c e Lemma 5.2

(9 o W)'

e n a b l e s u s t o a p p l y Theorem 4 . 2 , w e c o n c l u d e t h a t t h e TCm-closure of

q oW

PROOF OF THEOREM 5.1:

sets, k 5 m,

Then

E

>O

in

acm(U)

f

Let

E

i;,

and d e f i n e f o r

A = U {Ai;O

5

i

5 k)

a p p r o x i m a t i o n p r o p e r t y , t h e r e are that:

E

W

5

K C U, L i

5 k

be compact sub-

C E

t h e set

i s a compact s u b s e t of n

to

E

N,

'jEF',

"j

F.

E F

By t h e such

belongs t o

9.o f 3

9 .OW, 1

so

t h e r e are

w.

Consider

such t h a t

'K,L,k where

belongs

0

let 0

o f

n

By Lemma 5 . 3 , e a c h gj

.

q

E~

Let

= ~ / 3(1 +

(9.o f 7

-

9 . 09.) < E 3 3 1

n I: II vj 1 1 ) . j=1

n By h y p o t h e s i s , h = .E ( 9 . 09.)8 V ]=I 1 3 j'

t E W such t h a t

h

E

WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS

183

-

P ~ , ~ , k ( ht) < E/3.

Then:

E/3

n

+

I:

(x,v)

E

This proves t h a t

p

+

EIIIV.lI

3

j=1 K

x

L,

KiLik

(f

4 3 < E,

0 5 i

- t)
-(

r + l n - r

=

A s a corollary, we obtain

(2.2 * 10)

D(E,X) =

f

D(E1,X)D(E2,X).

Hence ,

THEOREM 2.3:

.LA h e g u d a h

( A t k i n s o n and Sharma [ 1 1

.L6 and o n l y i6 b a t h

2.3. REGULAR MATRICES

a6

i t b

A decompobable rncukix E =El @ E2

components ahe h e g u l a h .

By Theorem 2.2, t h e P 6 l y a c o n d i t i o n

(2.2.5)

i s n e c e s s a r y f o r r e g u l a r i t y . To o b t a i n a w o r k a b l e s u f f i c i e n t

t i o n , w e need t h e f o l l o w i n g n o t i o n . By a s e q u e n c e i n a r o w

condi-

i of t h e

m a t r i x E , w e mean a c o n t i n u o u s b l o c k of o n e s e ik =

(2.3.1)

,.. -- eill

= 1

which i s maximal. T h e r e f o r e , f o r a s e q u e n c e e i t h e r e i ,k-1

= 0,

and e i t h e r

d

=

n

or

ei,l+l

= 0

.

k = 0

A sequence

or is

else odd

LORENTZ and RIEMENSCHNEIDER

194

(or euen) i f i t h a s a n odd ( e v e n ) number o f o n e s . A sequence (2.3.1) i s b u p p a a t e d i f t h e r e e x i s t two o n e s

t o the

NW and

SW of

,

positions (il,kl)

in

E

eik = 1, i n o t h e r words, i f t h e r e a r e o n e s i n

(i2,k2) i n

Already G . D . B i r k h o f f [ 3

E with

il < i , k

k; i 2> i tk 2 0.

q i m p l i q u e que

est c r o i s s a n t e .

implique que

Q(z) l a transformge de M e l l i n de q .

Soit

mellement l e s deux membres de 2 . 1 . 2 .

1

m

0

l o g 11

-

uI uZ-ldu =

-

Transformons

for-

p a r M e l l i n , remarquant que

cotg

-

712

Z

1 < Rez < 0

on o b t i e n t

2.1.3.

donne, n o t a n t p a r

h ( x ) l a fonction ayant

Notons p a r

%,

M l a transform6e de M e l l i n - S t i e l j g s de

-

< Rez
0

2.1.4.

2.1.5.

06

L

(Indgalitd fondamentale)

%,a

2.1.6.

h E Lm

1,a

mesure

243

- -'2

< a < -1

- -l

< a 2 -1

2

2 -

fa

2

ddsigne l'espace des fonctions sommables par rapport 5 la -a l'espace des fonctions borndes par x .Ceci xa-1 dx; a

dtant, justifions les opsrations formelles effetudes ci-dessus. Posons :

dm

Alors

E

Mc1

espace des mesures sommables pour xa

d'oii d'aprss 2.1.5.

2.1.8.

est bien d6fini et

C o m e d'autre part

duit de composition k

- -

2

h*dm

:=

2.1.7.

< B < 0.

=

log11

log 11

-

- XI

XI

%,a

LlIa, - 1 < a < 0, le

E

*

s

1 Si a > - 7 ; - - 1< a < - .1 2 2

4

est bien d6fini et

On a enfin

K(z) =

- cotgnz

. 6(z)

si

Re2 = 6,

-

1 7

< B < 0,

et

(z) K ( z ) =- M 2

k(t) =

jot

les deux membres &ant

dx

et

presque partout

continus ceci vaut partout d'oc 2.1.2.

E

proL

1, B

MALLlAVlN

244

Posons

il rgsulte de 2.1.8. que

lim r(x)

2.1.9.

existe

X=m

Nous allons monter un lemme glbmentaire sur l’allure d’un potentiel d’une mesure portbe pour l’axe ri?el.

lim y=o

exibte e t

+ iy)

b0it

dinie

.

ALohd o n u

lim U” (xo + iy)

y=o

PREWE:

=

up(x0).

Up (x) est semi-continue infbrieurement donc

D‘o6 l’intggrale

- .f log11

-

xot-1I d p (t) est convergente. ReMlrquant

que les points rgguliers de E lim h(x y =o d’oG en utilisant 2 . 2 .

+

iy) = q(x)

APPROXIMATION POLYNOMIAL€ PONDIRE€ ET PRODUITS CANONIQUES

en tous les points rsguliers de E l tout dense sur E et

246

ceux-ci formant un ensemble par-

Wp(x) 6tant semi-continue supgrieurement, q(x)

continue, on obtieni

3.

3.1.

Nous nous proposons dans ce paragraphe de d6montrer 6nonc6s 1.2.

THEOREME:

u n e mebuhe dX

DEMONSTRATION:

Si

H(E,

- logp)

e b t non

v i d e , aeohn o n p e u t &ouve,t

a y a n t pouh buppotrt u n enbemble d i b c h e t

Soit

H ( E , -1ogp)

El

# 6. I1 existe d'aprgs

I1 r6sulte du fait que cette int6grale est >

-

m

que

t

C E,Z&

que

1.1.

dp=p(t) est

0

une fonction continue. Soit n(t)

=

et soit exp [

-

partie entigre de

1

log(1

-

zt-')dn(t)]

~(t)

= F(z).

F ( z ) est une fonction m6romorphe n'admettant que des pzles simples.

D'autre part, posons s(t) = II

3.1.3.

j log11 - zt-l

MALL1AVlN

246

=

a/x

Jo

Lx 1/2

+

2

+

J1/2

+

Jim.

La premisre int6grale est i n f 6 r i e u r e 5

L a seconde d

+

log x

.

0 (1)

.

La d e r n i s r e d

(1 (1)

R e s t e d 6 v a l u e r l a 3sme i n t 6 g r a l e s = s

1

+ - -1

I

1 Isl! 2 T

La p r e m i s r e i n t 6 g r a l e
A

+

2, bl,

. . , ,br

,r

a l o r s on p e u t t r o u v e r une f r a c t i o n r a t i o n n e l l e pour p 6 l e s s i m p l e s e t t e l l e que

F

1

(2)

= F(z) H(z)

v6rifiera

p o i n t s d e E distincts; H ( z ) a y a n t les

bk

APPROXIMATION POLYNOMIALE

PONDEREE ET PRODUITS CANONIOUES

247

3.1.3.

On a

OG

E R6sidus de F1(x) < t. e Le m- r6sidu a v6rifiant

p(t)

=

D‘autre part on a

-

d’oc .f

06 yn

t2


0.

D'autre part

0 , on o b t i e n t

= R

est a t t e i n t sur l'axe

qu'il

existe

une

suite

APPROXIMATION POlYNOMlALE PONDERBE ET PRODUITS CANONlClUES

R + k

261

telle que

D’autre part on a sur E

d’oG en remarquant que Wr et W’

sont hmniques dans { z ; [ z / 0

such t h a t f o r any

t

E

IR, with

ti 5 6 1 1

s u p q ( t ( f ( a + t h )- f ( a ) - u ( t h ) ) 2 1). he S Obviously y-

u i s uniquely determined by

d e h i v a t i v e ob

f i n a. W e w r i t e

f and a ; u i s c a l l e d

f ' ( a ) i n s t e a d of u .

t h e system of a l l bounded ( f i n i t e ) s u b s e t s of ( G i i t e a u x - ) di66ehentiable at

if

a. f

f i s y - d i f f e r e n t i a b l e a t any

E

,f

If

the y

is

is c a l l e d Fhzchet-

i s c a l l e d y-diddehentiable o n

51,

a E 52.

For G l t e a u x - d i f f e r e n t i a b l e f u n c t i o n s t h e r e e x i s t s e v e r a l g e n e r a l i z a t i o n s of t h e c l a s s i c a l mean v a l u e theorem (see e . g .

Yamamuro

SPACES OF OlFFEAENTlABLE FUNCTIONS AND THE APPROXlMATlON PROPERTY

[24I

,

27 1

1 . 3 ) . We s h a l l u s e t h e f o l l o w i n g o n e , which i s a consequence o f

t h e Hahn-Banach theorem and a r e s u l t o f c l a s s i c a l c a l c u l u s .

2. LEMMA:

l e t E and F be l . c . b p a c e b , 51 an open bub6e.t i n

a,b E R

let

.

Abbume

fitiabte at any

x E s

tained i n R g ( t ) := f ' (a

S [ a , b l : = {a t t ( b

b e buch t h a t

+

t(b

f(b)

-

-

f : S2

duhthehmohe t h a t

+

-

a ) I t E [0,11 1

F

and

E

con-

i h

Gzteaux- d;ddmen-

i h

and t h a t t h e mapping g : [ 0 , 1 ] * L a ( E , F ) , [a,bl a ) ) , i d c o n t i n u o u s . T h e n t h e doU0wing hoLh .thue: 1 f'(a

f(a) =

+

t(b

-

a))[b

-

aldt.

The f o l l o w i n g lemma i n d i c a t e s t h a t y - d i f f e r e n t i a b i l i t y

of a function

f i s a l r e a d y i m p l i e d by Gzteaux d i f f e r e n t i a b i l i t y and

a

continuity

p r o p e r t y of t h e derivative (see also K e l l e r [ 1 8 1 , 1 . 2 . 1 a n d Y a m a m u r o [24

1 , 1.4.4).

3. LEMMA:

L e t E and

F be L.c.

dpacen, 51 an open b u b b e t

f : S2 + F Gzteaux didbetentiable on

t i n u o u b , &en f PROOF:

i b

f'

16

: 52 +

Ly(E,F)

i b

con-

y-diddehentiable o n R .

L e t a be any p o i n t i n

bounded subsets of

$2.

and

E

06

S any e l e m e n t of t h e s y s t e m

S2,

y

of

E and l e t q be any c o n t i n u o u s semi-norm o n F . Py

Uleoontinuity of f ' in a, f o r

E

> 0 t h e r e e x i s t s a convex b a l a n c e d n e i g h -

bourhood U o f zero i n E s u c h t h a t

a

+

U

C 51

and s u c h t h a t f o r any

x € a + U

S i n c e S i s bounded i n E 2 we

have f o r any

, we

t with

can f i n d

0
0 6

with

and any

6s

C

h E S:

U.

By lemna

272

MElSE

This implies

Hence

f is y-differentiable a t a .

Let

4 . DEFINITION: E and

-

a system of bounded subsets of

y

n E mm(:=

U (

1

#

E and F be 1 . c . s p a c e s , s2 E

which

we d e f i n e t h e s p a c e o d

)

n

if

:R

-+

j E

F I f o r any

covers

t.imea

y - d i d 6 u e n t i a b L e dunctions o n R w i t h vaLuea i n

c ~ ( P , F ) :=

$ an o p e n s u b s e t o f

F

m0

cantinuouaLy

a6

with

0

~ < jn + l

) : = f ) and f o r any f . E C ( C ~ , L ~ ( E , F )(fo 1 Y

with

0

5j

on R and

A e t d 06

R

i s f . Gsteaux

3

f; = f j + l

j

E

06

. This

- differentiable

I .

Cn(B,F)

topology i s given by t h e system {pLrKrSrq 1

5

e

+

of semi-

norms, where

L

s u b s e t of

S is any e l e m e n t of y and q is any c o n t i n u o u s

norm on F ,

ill

lNo

i s endowed w i t h t h e t o p o l o g y od unidohm Y t h e dehiuatiweb up t o t h e ohdeh n a n t h e compactaub-

The v e c t o r s p a c e convehgence

s

i s compact i n EK and E .

Wo b e a convex b a l a n c e d neighbourhood of z e r o

Now l e t f o r which

+

KO

Wo

C

KO C K , t h e r e e x i s t s

n. s

l i m um = i d K i n

Since

Lo := Ls

J

and

E

since

m*m 0

E IN

s u c h t h a t f o r a n y s ? s o a n d a n y x E KO

us(x)

Put

C(K,E)

in

-

x

E

wo.

then it follows

0

LoCKo+WoCR.

S i n c e Lo tion

is a compact subset o f

0 2 1 ( 1 t h e func-

f ( j ) : 51 + L A ~ ( E , F )i s c o n t i n u o u s , t h e r e e x i s t s a convex

a n c e d neighbourhood any

51 and s i n c e f o r

j with

U of z e r o i n

0 < j 5 l , any

x

E

E with

Lo

Lo

and a n y

+

bal-

U C s2 s u c h t h a t

z E E

with

for

x-zEU

t h e f o l l o w i n g estimate h o l d s

For

1 5 j 5 !k t h e s e t

f (j) (Lo)

i s compact and hence

bounded

in

= L c o ( E r L f ~ l ( E r F1). E i s b a r e l l e d by h y p o t h e s i s , hence fJ(Lol j- 1 i s e q u i c o n t i n u o u s i n Lco(E,Lco (E,F)). T h i s i m p l i e s t h a t t h e r e i s a

L:O(E,F)

MEISE

280

neighbourhood W and any

j

of z e r o i n

y ' E Lj-l

E such t h a t f o r any

f o r any

y = (ylI...,yj)

o n e of t h e yk i s i n The s e t

Now w e d e f i n e

W

e

x

El, where

E

Lo,any y1 E W

j

,

t h i s means t h a t we have

j

- 1 of

t h e yk a r e i n

L and

j '

-1 ( n ( W . j, j=1 J

hence t h e r e e x i s t s

E

w e have

f (1) (x) i s symmetric f o r any

Since

x

fl

U) i s a neighbourhood of z e r o i n

s E IN w i t h

s

our construction we get

u(x)

E KO

,

2 s0 s u c h t h a t

and o b s e r v e t h a t b y t h e c h o i c e of

u := us

EK

+

U C B

f o r any

x

E

s

and by

Ko(from now

on l e t us omit t h e map j, l i . e . w e r e g a r d u as mapping from E i n t o E). Then t h e s e t w := u-1 (s1) i s an open neighbourhood of KO and on w w e c a n d e f i n e t h e mapping

is e a s y t o see t h a t with

j

f

0

n t 1 and any

f

0

u : w

+

F. By o u r d e f i n i t i o n 2 . 4

u E Cgo(w,F) and t h a t f o r any y E Ej

x

E

w,

any

it j

t h e following holds

I n o r d e r t o prove t h e d e s i r e d estimate, w e o b s e r v e f i r s t t h a t we have

u(Ko) (1):

C

Lo, and t h a t f o r any

x

E

KO

,u(x) -

x E U. Hence w e g e t fran

SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY

Then w e o b s e r v e t h a t f o r any

x

hence f o r any

KO

E

j

and any

with y

E

291

1 5 j 5.e w e have u(Q,) Q:

1

C

L

1’

w e g e t from (11,( 2 ) and ( 3 )

< (e+1).L + 1 -

By ( 4 ) and (S), t h e proof of t h e lemma i s complete.

4.

L e t us r e c a l l from B i e r s t e d t and Meise 1 6

REMARK:

d u c t i v e i n j e c t i v e system

s u b s e t of

Ea

~

o A f 1 . c . s p a c e s Ea

(CFA)

. Then

i t i s obvious t h a t any 1 . c . s p a c e E which c a n b e

E A of

lEa I

. Hence i)

is called

E = ind E

r e p r e s e n t e d a s an i n d u c t i v e l i m i t o f a compactly r e g u l a r system

in-

i s Hausdorff And i f f o r any acanpact a+ a E t h e r e e x i s t s a E A such t h a t K i s a l r e a d y a ccnpact

campactLy heguLah, i f s u b s e t K of

{ E a I*j u BIa

1 t h a t an

( F ) - s p a c e s Eu w i t h a.p.

has

inductive

the

property

i n any of t h e f o l l o w i n g c a s e s E h a s (CFA):

E i s a (F)-space with a.p.

i i ) E = i n d En I where { E n l j n m } i s a s t r i c t i n d u c t i v e n + of ( F ) - s p a c e s En w i t h a . p . i i i ) E = in$ En

n

I

system

where { E n l j n m } i s a compact i n j e c t i v e induc-

t i v e system of

( F ) - s p a c e s En w i t h a . p . For b r e v i t y w shall

c a l l any s p a c e o f t h i s t y p e (DFSA)-space. Using a t r i c k which g o e s back t o Aron and S c h o t t e n l o h e r [ 4 ],we can now prove t h e d e s i r e d r e s u l t on t h e a . p .

of

Czo(Sa).

MElSE

232

5. THEOREM:

L e t E be an in Lemma 3 and

and

y = yco

n E INm aLl t h e hypothedeb

a t e d a t L d 6 i e d . Then

PROOF:

in

of

n Cco(Q)

identify

c a n b e proved by showing t h a t C z o ( Q ) 8 F is dense

Czo(R)

E

F

f o r any Banach s p a c e F.

C:o(Q)

E

F

in

Q,

ma 3 , t h e r e e x i s t s such t h a t Let

f

0

us define

fo E Cao(Qo,F)

E , any

u

E

e

C g o ( Q , F ) , a n y compact subset KO o f

+ I, and

0

and E ~ f o := f

(finite

CEO (Eo)

i s dense

in

Czo(R)

3

,

such t h a t

g : = hou E C a o ( E ) 8 F, and f o r any

Y

E

E ’ 8 E and an open neighbourhood

u(Ko) C Q n Eo = Qo, and s i n c e

there exists

any

f E

u E CZo(wlF) s a t i s f i e s t h e estimates g i v e n i n lemma 3 .

CZo(QotF) = Ccn0(Q) that

that

C Z o ( Q t F ) f o r any Banach s p a c e F .

To do t h i s , l e t any any compact

By c o r o l l a r y 2 w e mayand shall

CZo(R,F). Hence w e o n l y have t o show

and

i s dense i n

Czo(Q) 8 F

6 a h a n y open d u b n e t R a 6 E.

had t h e a . p .

Cgo(Q)

c0(Q)

theohem 1 o n E and

06

i s quasi-complete by h y p o t h e s i s . Hence, by theorem 1.7

Cgo(Q)

t h e a.p.

adburnt duathehmohe t h a t doh

x E KO

,

any

x

E KO

j with

15 j

5 L

and

SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY

Hence w e have shown of

mm

(f

-

5

g)

2s

,

which proves t h e d e n s i t y

a F in c ~ ~ ( . Q , F ) .

c:~(E)

A l l t h e hypotheses of theorem 6 a r e s a t i s f i e d

6 . REMARK:

n E

K~ ,Q,

293

s2 of

and any open s u b s e t

E,

if

E is either

w i t h a . p . o r a (DFSA)-space. T h i s follows from 2 . 5 ,

for

any

an (F) - s p a c e

2 . 6 andremark 4 .

We s h a l l show now t h a t f o r Frgchet spaces E w i t h a . p . t h i s r e s u l t i s optimal.

7. THEOREM: a)

Fax a F h e c h e t Apace E t h e doLLowing axe e q u i u a e e n t :

C:o(Sl)

Q

# $

has t h e a . p . 06

that

c)

E

n E INw

and a n y o p e n n u b s e t

E-

Thexe exidt

b)

d o h any

n E INm and a n open n u b b e t

czo(a)

had

R # pl

ad

E

nuch

t h e a.p.

had t h e a . p .

(a) * (b): t r i v i a l

PROOF:

(b)

=.

( c ) : By 2 . 8 ,

ELo = E ’ i s a continuously p r o j e c t e d topo-

l o g i c a l l i n e a r subspace of Frgchet space E t h e a . p . of

C

C : o( . Q ) ,

hence

EA has t h e a.p.

But f o r a

EA i s e q u i v a l e n t t o t h e a . p . of Elhence

E has t h e a . p .

( c) * ( a ) : This is clear according t o t h e remark 6.

REMARK:

For Banach spaces E theorem 7 was shown by Bombal

Gorddn

294

MEISE

and Gonzslez Llavona [lo] f o r

51 = E . Again f o r Banach s p a c e s

s l i g h t l y d i f f e r e n t version (using

[ 201 and a l s o by Aron [ 3

topology

.

I

T h e h e C X i 4 t A a n (FS)-npace

8. COROLLARY:

06

the

a

C z o (51) ) of theorem 7 w a s p r e s e n t e d by P r o l l a andGuerreiro

i n d u c e d by

t h e a . p . go& a n y

n o t have

Cf: ( 0 ) endowed w i t h

E

E huch t h a t

doeA

Czo(51)

mm a n d a n y n o n - e m p t y o p e n nubnet

n E

R

E.

T h i s i s a consequence of theorem 7 and t h e e x i s t e n c e of (FS)-

PROOF:

s p a c e s w i t h o u t a . p . The e x i s t e n c e of s u c h (FS) - s p a c e

follows

from

E n f l o ' s c o u n t e r e x a m p l e , a s Hogbe-Nlend p r o v e d i n [ 1 6 1 . Because of lemma 3 , t h e method a p p l i e d i n t h e proof o f theorem 5 c a n be used a l s o t o d e r i v e some f u r t h e r d e n s i t y r e s u l t s

just

by

" l i f t i n g " d e n s i t y r e l a t i o n s known i n t h e f i n i t e d i m e n s i o n a l case. Bef o r e s t a t i n g them l e t u s r e c a l l t h a t a c o n t i n u o u s n-homogeneous p o l r nomial

p on E i s c a l l e d Ainite, i f t h e r e

exist

y i ,...,y;

E

E'

such t h a t n p(X) =

By

n

j =1

f o r any

(y;,X)

x E E.

P f ( E ) w e d e n o t e t h e l i n e a r h u l l o f a l l c o n t i n u o u s n-homogeneous

p o l y n o m i a l s on E

,

9. THEOREM:

E be a q u a A i - c o m p k k t e b a h a L t e d

Let

(CFii). Then doh 0(#

0)

PROOF:

06

E

n

any

E

n E

t h e space

L e t any

p a c t s u b s e t Q of

f

INo.

I t i s e a s y t o see t h a t

and 1 . c . Pf(E) @ F

,

E CZo(QIF)

E , any

m

Pf (E) C C c o ( E l .

L.c.

pace F a n d a n y

i n dense i n

Apace open

With

oubaet

Czo(51,F).

a n y compact s u b s e t K of

51 , a n y can-

1 < n +1, any c o n t i n u o u s seml-norm

q onF,

296

SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY

and

> 0

E

be g i v e n . W e s h a l l show t h a t t h e r e e x i s t s

g

E Pf(E)

€3 E

with

Let

F

9 and l e t

d e n o t e t h e c o m p l e t i o n o f t h e c a n o n i c a l normed s p a c e

F/ker q

d e n o t e t h e c a n o n i c a l c o n t i n u o u s l i n e a r map.Since q II 0 f E Cn ( Q I F ) , a c c o r d i n g t o lemma 3 t h e r e e x i s t s u E E' 8 E s u c h co 9 that a : F

+

F

Now we p r o c e e d as i n t h e proof of theorem 5 and d e f i n e

no :=

Ci n Eo and f o : (IT

0

f ) ICio.Then f o E C,"o(Qo,F

S i n c e t h e p o l y n o m i a l s o n Eo are d e n s e i n in F

q i

and s i n c e

ho

E

)

= CEo(Qo)

gE

Fq.

CEo(Qo) ;since II(F) is dense

no ( t h i s was shown

u(K) i s c o n t a i n e d i n

p r o o f of lemma 3 ) , t h e r e e x i s t s

4

:= I m u I

Eo

P(Eo) 8

IT

(F) =

Pf (Eo)

in @

IT (

the F)

such t h a t

Assume t h a t

i=ll...,m.

ho = Then

m -..

Z pi 8 r ( y i ) , where i=1 h :=

Z piOu8yi

i =1

pi E Pf (Eo) and

is i n

Pf(E) 8 F

yi E F f o r and

as

in

t h e p r o o f of theorem 5 i t f o l l o w s

PL,KIQ,q

Hence we have shown t h a t

(f

-

h) 5 2~

Pf (E) €3 F

.

is d e n s e i n

Czo(Q,F).

The following c o r o l l a r y i s a n immediate consequence of theorem 9.

286

MElSE

10. COROLLARY:

Let E be a q u a b i - c o m p l e t e b a w i e l l e d l . c . npace w i t h

(CFA). T h e n

any

n ( # 0 ) 06

604

.the a p a c e

E

m m , any

n E

l . c . b p a c e F , and a n y o p e n n u b a e t

63 F

C:o(E)

CEo(n,F).

dense i n

i b

Looking a t theorem 5 and c o r o l l a r y 1 0 and t h e i r p r o o f

in

f i n i t e d i m e n s i o n a l c a s e one h a s t h e i m p r e s s i o n t h a t c o n d i t i o n ( o r more o r less t h e a . p . )

t o g e t h e r w i t h f i n i t e dimensional

the (CFA)

results

c a n b e u s e d i n s t e a d o f C m - f u n c t i o n s w i t h compact s u p p o r t . T h e f o l l o w i n g theorem i s o f t h e s a m e n a t u r e . B e f o r e w e s t a t e i t , l e t us remark E be any 1.c. space

t h a t a n e a s y c a l c u l a t i o n shows t h e f o l l o w i n g : L e t and l e t

d e n o t e i t s ( c o n t i n u o u s ) d u a l . For any system y o f bounded

E'

subsets of

( c o v e r i n g E ) and any

E

m

C y ( E ) . Using t h i s and t h e c l a s s i c a l theorem

belongs to

Wiener-Schwartz

11. THEOREM:

0) 0 6

denbe

ifl

E

Paley

L e t E b e a q u a b i - c o m p l e t e b a a a e l l e d l . c . bpace n

a n y 1 . c . b p a c e F , and a n y o p e n

E INm,

t h e L in ea h hue1

06

the net

Ie,

-

*

with bubbet

f I y E E', f E F)

LA

Cgo(Q,F).

4 . A KERNEL THEOREM FOR FUNCTIONS OF CLASS

CEO

I n t h i s s e c t i o n w e s h a l l show ( u n d e r a p p r o p r i a t e t h a t any f u n c t i o n s i n m

of

t h e proof o f theorem 9 a l s o g i v e s

(CFA). T h e n d o h a n y

fi(#

y E E', t h e f u n c t i o n

m

Cco(Ql

x

hypotheses)

Q 2 ) c a n b e r e g a r d e d as a n e l e m e n t o f

m

C c o ( Q l , C c o ( ~ 2 ) ) and v i c e v e r s a . Using theorem 3 . 5 t h i s a l s o

a tensor product representation f o r

m

Cco(Ql

x Q,)

.

B e f o r e w e c a n prove

o u r r e s u l t w e need s e v e r a l lemmas. The f i r s t lemma i s consequence o f d e f i n i t i o n 2 . 4 .

implies

an

immediate

SPACESOF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY

1. LEMMA: 06

Let E,F and

b e L . c . b p a c e b , l e t R be an o p e n

G

y b e a covetring b y b t e m

E l leX

297

0 6 bounded b u b b e t

dubbet

let

E , and

06

u E L(F,G) b e g i v e n .

any

a)

Foh

b)

Fotr a n y

f E Cm(51,F) t h e d u n c t i o f l

Y

f E Cm(R,F) a n d a n y

belongs t o c)

2.

06

subbet

x1

1e.t

LEMMA:

Ei

PROOF:

t o p o t o g i c a l bubbpace

and

doh i =1,2.

F

The mapping

m

+

tonuous and j - l i n e a r ,

El

(ii)* :

( i d ) * ( m ) [ x ]= m ( [ i d ( x ) ] ) .

t h e n by lemma 1 . b )

,

j

f o r any

c o n t i n u o u s l i n e a r map

1.5,and l

(ii)*

( i a )*

o

f'j)

any j E IN

0

6

be an

open

,F)

by

g E C(R1,Cco(Q2,F)1 .

i2

d e f i n e d by

ii

: E';

IN. Thus

eEo( (El x

(0,x2)is

=

(El x E 2 ) j i s con-

+

gives rise

ia

E2)JrF)

( x2 )

to

a

eEo(E);,F), d e f i n e d

-+

I f now f i s a n y e l e m e n t of C~o(511xR2rF)r m

m l.c), f ( 1 ) is i n C c o ( R l ~ ~ 2 , ~ ~ o ( (E2)jrF). E1~ m

(EJrF)), (i$*o f ( j ) i s in Cco (511X 512' Ls co 2

L3 (E ,F)). L e t u s d e n o t e t h e f u n c t i o n 512' co 2 m R2,F) t h a t f o r Then it f o l l o w s from f E Cco(R1x

Cco(nl

f (1) by

Cco(R,G).

m

x E2

Then i t f o l l o w s from lemma 1.a) that hence

m

x R2

f E Cco(Rl

T h e n dotr any

i2: E 2

b e l o n g n to

.then

F,

06

be 1 . c . hpaceb and l e L Qi

o b v i o u s l y l i n e a r and c o n t i n u o u s , hence

by

f(1)

Y

1 , o n e de,4inen a 6 u n c X i o n

f (xlf

+

Y

E2

El

t h e dunction

E INo

f E Czo(RrF) m i t h f ( R ) C G

any

Y

C m ( R , L j (E,F)1 .

G i b a closed lineah

76

j

Y

beloflgb to C m ( R r G ) .

uo f

gj

.

and any x1 E

R1 t h e f u n c t i o n

gj(xlr

) :

R2

-+

Lio(E2rF)

i s G z t e a u x - d i f f e r e n t i a b l e and t h a t i t s Gsteaux-derivative is g j + l ( y , * 1. T h i s p r o v e s t h a t f o r any m

x1 E Q1

Cco(512,F) , hence t h e f u n c t i o n

t h e function

g : nl

f(xlr

) belongs

Cco(n2,F) , g(Xl) = f 00

-+

to

( X l l o ) Can

be d e f i n e d . I n o r d e r t o show c o n t i n u i t y o f E

> 0 , and any c o n t i n u o u s

be given. Since

g

j

g on

semi-norm on

R1

, let

any

x1

6

61,

any

m

Cco(S22rF) of the f o m p

l r 5 r Q 2 r ~

i s uniformly c o n t i n u o u s on {x,} x K 2 f o r any

j,

MElSE

258

t h e r e e x i s t s a neighbourhood f o r any ( x l , x 2 ) any j w i t h

(where

0

{xll

E

and any ( h l , h 2 )

d e n o t e s t h e semi-norm

I

Pj,Q2,s

E

V1

El

w e have

V2

x

such t h a t

x E2

for

hl

E

u

+

s u p . q(u(y)) on L20(E,F)).

FQ;

V1

g i s continuous.

3 . PROPOSITION: 604

of zero i n

x V2

5 j5 l

T n i s i m p l i e s f o r any

hence

K2

x

V1

Which

and L e t

i =1,2

Fok

(Ei)A

Let

C o m p L e t e and w h i c h eQllaeA (Ei)AA

i d

be a n o p e n h u b b e t a d

sli

b e a quabi-compLete L . c . bpuce

Ei

t#pOfOgiCUk?Ly,

Abdume 6uhthekmohe t h a t

Ei.

E2

in

a k I R - b p a c e . Then t h e h e e X i b t h a continuoub Lineah a n d i n j e c t i v e map 03

A : Cco(Ql 6oh

any

PROOF:

tion

x

f

E

Q2)

m

+

m

Co(Q1,Cco(~Z)),

m

Cco(nl

x

dediMed b y

-+

f

(xl,

: x1

*

f(X1,

1

a,)

t h e func

-

Q2).

L e t u s show f i r s t t h a t for any

g : x1

A(f)

)

belongs to

m

Cco

f E CEo(fil

x

(Ql,c;o(i22)1 *

L e t il d e n o t e t h e l i n e a r c o n t i n u o u s mapping i l : E 1 El x E2, m i l ( x l ) = (xl,O) a n d l e t f E Cco(Ql x Q,) b e g i v e n . As i n t h e p r o o f -+

of lemma 2 one shows t h a t f o r any

j E IN

t h e mapping

~p :=

j

( i i ) * o f(1)

SPACES OF OIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY

m

belongs t o g j : x1

+

Cco ( i l l

x

R2,L~o(El,IR1 ) .

29%

Hence, by lemma 2 t h e

mapping

i s i n C(Rl,C~o(R2,L~o(E1,1R))). Now o b s e r v e

9 . (xl,

3

i t f o l l o w s from 3.2

by o u r h y p o t h e s e s

and g e n e r a l

results

that

on t h e

€ - p r o d u c t t h a t w e have n a t u r a l isomorphisms

Using t h i s isomorphism, w e g e t from f o r any

j E IN

. Obviously

g.7 the napping

g. E C(Ql,L~o(E1,C~o(Q2)) 7

go = A ( f ) , and w e s h a l l p r o v e

now t h a t

i s t h e G s t e a u x - d e r i v a t i v e of g I n o r d e r t o do t h i s , gj+1 j ’ (k) = (‘j * 0 f f i r s t remark t h a t f o r any k E IN w e have

l e t us and

5)

‘Oj

that

f o l l o w s f r o m t h e proof of lemma 2. Hence w e g e t

Now l e t

R2

x1 E Q1

hl E E l ,

any compact Q2 o f

any

L

E M

, any

E 2 , any compact s u b s e t Q1 o f

b e g i v e n . W e have t o p r o v e t h a t t h e r e e x i s t s any

t

with

0
0,

6

By (1) and 1 . 5 w e have t o e s t i m a t e f o r

0

5

k

5 R

El

K2 o f

and

such t h a t

E

> O for

MEISE

300

-t ( f ( j + k )(xl

+

-

thl,x2)

f

(X1'X2)

f ( j + k + l ) i s c o n t i n u o u s on

Since

R~

x

a2 ,

-

it i s uniformly continu-

ous on a s u i t a b l e neighbourhood o f t h e compact set

uniform c o n t i n u i t y o f

Cx,)

element o f

.

By

g = go

isan

.

m

m

K2

f ( j + k + l ) a n d ( 3 ) i t i s clear t h a t t h e r e exists

s a t i s f y i n g ( 2 ) . C o n s e q u e n t l y w e h a v e shown t h a t

6 > 0

x

Cco (Ql,Cco (Q,)

L i n e a r i t y and i n j e c t i v i t y o f

A are o b v i o u s . C o n t i n u i t y

of

A

follows i m m e d i a t e l y from (1) a n d t h e d e f i n i t i o n o f t h e c o r r e s p o n d i n g topologies.

Now w e want t o p r o v e t h a t A i s s u r j e c t i v e i f w e impose

some

f u r t h e r conditions.

4 . LEMMA:

subset

06

Fon Ei.

i =1,2

El

Asbume t h a t

'Let g b e a a n y 6unc;tion i n a)

Fon a n y ( j, k )

b e a L . c . bpace and l e t

L e t Ei

E

x m

E2 k

i b

604

be an open

any ( j , k )

E

E!

k E2

IN 2

.

cco(~l,cco(~2)). m

IN2 t h e mapping f ( j r k :Ql )

dehilzed b y f ( j t k ) (x1,xi,y1,y2)

is c o n t i n u o u s .

a km-6pace

Ri

.

x

Q2

X

x

+

IR

SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY

331

a ) Observe t h a t f o r any open s u b s e t R of a 1.c. s p a c e E l a n y

PROOF:

1 . c . s p a c e F and any

i s c o n t i n u o u s on

K

x

f

QJ

E

m

Cco(QIF)

the function

f o r any compact s u b s e t

K

R and

of

any

compact s e t Q i n E and has v a l u e s i n F . Hence f o r any compact s u b s e t K1

of

Rl

and any compact s u b s e t Q1

of

Ell

the function

i s l i n e a r and c o n t i n u o u s f o r any compact s e t K 2 i n p a c t s e t Q 2 i n E2 , w e g e t from lemma 1.b) t h a t b e l o n g s t o C(K1 x QllC(K2 x Q,k)) = C(K1 x Q! x K2 x

b)

x

-

) [

x K2 x

Q:'

x

Q,). k

is a kR-space

The s e c o n d a s s e r t i o n i s a consequence o f t h e f o l l o w i n g con-

siderations:

=: A ( t )

pk((g(j)(

d;) =C(K1

T h i s p r o v e s t h e c o n t i n u i t y o f f ( j r k )I s i n c e El1' f o r any ( j , k ) E IN 2

.

R 2 and any com-

+

B(t).

302

MElSE

uniformly i n

y1 E Qi

By lemma 2.2

=

k y 2 E Q,

and

.

we g e t

lo

1

f ( j r k + ' )(x,

+ t h l r x 2 + T t h Z r y l r ( h 2 , y 2 ) )dT

f ( J r k + l ) i s u n i f o r m l y c o n t i n u o u s i n a neighbourhood of t h e compact set {xl} x {x,) x Qi* x Q2k hence w e also have I t f o l l o w s from a ) t h a t

uniformly i n

5. THEOREM:

equal6

Foh

(Ei)AA

Then t h e mapping ib

i =1,2

a topological

l e t Ei

E!

x E:

m

A : Cco(Ql

ment o f

m

C c o ( a l f C ~ o ( ~ 2 ))

any ( j , k ) A(f) : x l + f

E

.

2 D l

(Xlf*)r

Q1

A i s s u r j e c t i v e . L e t g be any e l e

-

1 . By lemma 4 t h e f u n c t i o n f : (x,,x2) +g(xl) (x,)

Cco(Ql,C~,(Q,)

is obvious t h a t

+

doh

Ei.

06

idomohphibm.

m

i s c o n t i n u o u s on

be an open bubbet

i b a km-hpace

x "2)

F i r s t l e t us show t h a t

PROOF:

be a q u a b i - c o m p l e t e l.c.bpace w h i c h

t o p o l o g i c a l t y and l e t Qi

buathekmoke t h a t

Addume

k y2 E Q 2 .

and

y1 E Q:

x Q,.

W e s h a l l prove

A ( f ) = g, hence

I n order t o prove

m

m

f E CCo(al

x

a,).

Then it

A is surjective.

f E Cc0(R1 x

a,)

let usremark t h e following:

SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY

and a s u r j e c t i o n u

Let (j,k) E

:

{I,...f j j

U

303

{l'f...fk'}+.{lf,..,j+k~

b e g i v e n . Then w e d e f i n e a c o n t i n u o u s l i n e a r map ru :(El x E 2 ) j + k

+

E

by

= ( ( e l , u (1)'

-

r e l (, j~ )

-(j,k)

By lemma 4.a) t h e f u n c t i o n Ql

X

R2

x

(El

any (x1,x2)

€

X

R1

:= f ( I , k ) o r

fU

-f a(j,k) ( x l , x 2 , * )

E2)j+k and x

' (e20 (1')'. . - 'e2u ( k ' )

X

E2,1R),

is c o n t i n u o u s . Using t h e mappings duction t h a t

f

m

belongs t o

Cco(R1

for

is ( j + k ) - l i n e a r

-f a(I 'k)

R2. Because of t h e c o n t i n u i t y o f c]+'(El o

c o n t i n u o u s on

is

U

t h e map-

d e f i n e d by

f u( J ' k ) I t is e a s y t o p r o v e by i n x

Q,).

Let

us

show

that

f

is

Gsteaux-differentiable:

u

Define

u2

:

0

U (1')

+

{l}

*

1 '

11)

0

U

{l}

by

ul(l)

= 1

and

define

~ ~ ( 1 =' 1. ) Thenwe g e t from 4 . b ) t h a t f o r

by

x = (x1,x2) E Q1 x R2

+

and

h = ( h l , h 2 ) E El

E2

x

1

+

f ; O r 1 ) ( x ) ) [ h ] , and f E Cco(R1 x R,) ul 2 by lemma 2.3. From t h i s and l e m a 4 . b ) w e g e t by i n d u c t i o n t h a t f o r

Hence

f'(x)[h] = (f(l'')(x)

any 1 E W t h e f u n c t i o n

f

be r e p r e s e n t e d as a sum of

k in

INo w i t h

j +k =

L

is i n

L

Cco(Rl

x Q,)

and t h a t

f ('1

can

f, ( J r k ) where t h e sum r u n s o v e r a l l j and and o v e r c e r t a i n

(J

.

This proves

that

MEISE

f E

cco (a, m

x

Q,).

Hence w e have shown t h a t

A : Cm c o ( ~ xl o 2 )

b i j e c t i v e . From t h e r e p r e s e n t a t i o n o f (A'l ( g ) )

follows t h a t A - l

m

+ c ~ ~ ( Q ~ , ci s~ ~ ( ~ ~ i n d i c a t e d above i t

i s c o n t i n u o u s . Then A i s a t o p o l o g i c a l isomorphism

by p r o p o s i t i o n 3 .

REMARK:

R e s u l t s of t h e same t y p e as i n theorem 5 a r e also g i v e n

in

t h e l e c t u r e n o t e s of F r o h l i c h e r a n d Bucher [151 ( w i t h a d i f f e r e n t defin i t i o n o f d i f f e r e n t i a b i l i t y ) and i n Colombeau [111I [ 1 2 ] . Itseems t o b e i m p o s s i b l e t o g e t t h e r e s u l t on (DFM)-spaces g i v e n below by

bor-

n o l o g i c a l methods. Concluding t h i s s e c t i o n , l e t us combine theorem 5 and some

of

t h e r e s u l t s i n s e c t i o n 3 . Then w e g e t

6 . THEOREM:

Let El

and E 2 b e e i t h e k (F)-Apactd o h (DFM)-bpacesand

L e t Oi be a n o p e n d u b n e t a d Ei

doh

i =1,2.

Then we h a v e t h e

dot-

bowing t o pob a g i c a l 16 a ma h p hid m d

7 . THEOREM: Ei

.

Foh

i =1,2, b e t

be an open s u b s e t

Cli

06

t h e L . c . space

Assume t h a t e i t h e n . 1)

El

2)

El

and E 2 and

E2

ahe ( F ) - d p a c e n , o n e

06

a h e (DFM) - s p a c e s , o n e

w h i c h had a . p . ,

05

ah

wkich A a ( D F S A ) -npace.

T h e n t h e d o l l o w i n g hold4

8. REMARK:

The d u a l of

CEo(Sa)

forms a n a t u r a l g e n e r a l i z a t i o n of the

s p a c e of d i s t r i b u t i o n s w i t h compact s u p p o r t t o i n f i n i t e

dimensions.

SPACES OF DIFFERENTIABLE FUNCTIONS ANDTHE APPROXIMATION PROPERTY

306

It is obvious that many of the results of this article can regarded as results on the dual of

also

be

m

Cco(Q). E.g. theorem 3.10 is of

importance in connection with the theorem of Paley -Wiener -Schwartz (in order to see this one has to extend several results

to

complex

valued functions on R , then (for certain 1.c. spaces E) one can define the Fourier-Laplace transform of any

m

T E Cco(Q,fl!)'

morphic function on the complexification of EA growth condition). Theorem 6 can be used

* : Cmco (E)'

x

CZo(E) '

-+

to

,

as a

holo-

satisfying acertain

define

a

convolution

Czo(E) I . The precise formulation of the results

just mentioned will be contained in a subsequent paper.

REFERENCES

[

11

A. ARHANGEL'SKII, Bicompact sets and the topology Soviet Math. (Doklady) 4 (1963),, 561 - 564.

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spaces,

[ 21

R. ARON, Compact polynomials and compact differentiable mappings between Banach spaces, in "Si?minaihe P i t h h e L e h f l g ( A n a l y b e ) Annee 1974/75", Springer Lecture Notes Math. 524 (1976), p. 213-222.

[ 31

R. ARON, Approximation of differentiable functions on a Banach space, in 'I 1 n d i n i t e d i m e n d i o n a l hoComo/rphq a n d appfic&ovl~': North-Holland Mathematics Studies (19771, p. 1-17.

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K.-D. BIERSTEDT and R. MEISE, Lokalkonvexe Unterraume in topologischen Vektorramen und das c-Produkt,manuscripta math. 8 (1973)I 143 -172.

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BIERSTEDT and R. MEISE, N u c l e a r i t y and t h e Schwartz prope r t y i n t h e t h e o r y of holomorphic f u n c t i o n s on

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t o appear

in

Acta

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F. BOMBAL GORDON and J. L. GoNZaEZ UAVONA, La p r o p i e d a d

de

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t a Acad. C i . Madrid 70 ( 1 9 7 6 1 , 7 2 7 - 7 4 1 . [ l l ] J. F. COLOMBEAU, Uiddekentiation

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bpaced,

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[191

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307

Grundlehren

der Math. 159 (1969). [20] J. B. PROLLA and C. S. GUERREIRO, An extension of Nachbin's theorem to differentiable functions on Banach spaces with the approximation property, Ark. Mat. 14 (19761, 251 - 258. [21] H. H. SCHAEFER, T o p o L o g i c a L v e c t o h dpaces, Springer 1970. [221

L. SCHWARTZ, Theorie des distributions 5 valeurs I, Ann. Inst. Fourier 7 (19571, 1-142.

[ 231

M. DE WILDE, R6seaux dans les espaces lin6aires 2 semi-normes, Mgmoires SOC. Royale Sc. Lisge, 5e sGrie,l8, 2 (1969).

[24I

S. Y A W U R O , Uia6ekentiaL

CdCU&Uh

vectorielles

i n t o p o t a g i c a L fitzeah

Springer Lecture Notes in Math. 374 (1974).

hpaceb,

This Page Intentionally Left Blank

Approximation Theory and Functional A n a l y s i s J.B. Prol2a ( e d . ) 0North-HoZland Publishing Compmzy, 1979

A LOOK AT APPROXIMATION THEORY

LEOPOLDO NACHBIN I n s t i t u t o de Matemgtica U n i v e r s i d a d e F e d e r a l do Rio de J a n e i r o 20.000 R i o de J a n e i r o RJ ZC-32 Brazil Department of Mathematics U n i v e r s i t y of R o c h e s t e r R o c h e s t e r NY 14627 USA

1, INTRODUCTION I would l i k e t o d e s c r i b e v e r y b r i e f l y how I w a s l e d t o

become

s e r i o u s l y i n t e r e s t e d i n Approximation Theory, t h a t i s , t o i n d i c a t e t h e m o t i v a t i o n t h a t I had i n my mind. T h i s f i e l d h a s d e v e l o p e d i n B r a z i l i n t h e p a s t t e n y e a r s or so, t h a n k s a l s o t o t h e work of S i l v i o Machado,

Joao Bosco

Prolla

and

Guido Z a p a t a , as w e l l as t h e r e s e a r c h s c h o o l t h a t t h e y formed. I f I had t o r e d u c e b i b l i o g r a p h i c a l r e f e r e n c e s t o a b a r e

mini-

mum, i n what c o n c e r n s t h e work of t h e B r a z i l i a n s c h o o l i n Approximat i o n Theory and i t s r e l a t i o n s h i p t o t h e r e s e a r c h o f o t h e r g r o u p s ,

I

would q u o t e my monograph Element6 ad A p p t o x i m a t i a n T h e o h y ( 1 9 6 7 ) , as w e l l as P r o l l a ' s monograph Apphoximation (1977) (see [ 3 4 1 up-to-date

,

0 4 Vectoh Vatued

[ 5 4 1 ) . However, t h e b i b l i o g r a p h y

at

the

Funciionb end

is

and complete w i t h r e s p e c t t o t h e work by Machado, P r o l l a ,

Zapata and m y s e l f .

I t i s extremely incomplete o th er wis e.

emphasize t h e f o l l o w i n g aspects:

Let

me

310

NACHBIN

1)

I s h a l l r e s t r i c t myself

h e r e t o t h e r e a l v a l u e d c a s e . The

v e c t o r v a l u e d c a s e was t r e a t e d i n a d e s i r a b l e d e g r e e

of

generality

(see also

through v e c t o r f i b r a t i o n s by Machado [ 1 6 ] and P r o l l a [ 4 0 ] [35 1

I361 1 * 2)

I n t h e complex c a s e ,

Bishop and W e i e r s t r a s s - S t o n e 3)

I p o i n t o u t t h e work by Machado on the

theorems [ 181

.

W e c a l l a t t e n t i o n t o t h e work by Zapaka on Mergelyan's the-

orem and q u a s i - a n a l y t i c classes [ 65 ] (see a l s o [ 541 ) 4)

.

(See a l s o [ 541 )

.

Weighted approximation i n t h e c o n t i n u o u s l y

differentiable

c a s e was s t u d i e d by Zapata [631 , [ 6 4 1 . 5)

A d e n s i t y theorem f o r polynomial a l g e b r a s of

continuously

d i f f e r e n t i a b l e mappings i n i n f i n i t e dimensions and i t s

relationship

t o t h e Banach-Grothendieck

approximation p r o p e r t y was i n v e s t i g a t e d by

P r o l l a and G u e r r e i r o I 5 3 I (see a l s o [ 38 1 ) . 6)

Nonarchimedean Approximation Theory h a s

P r o l l a [ 561,

and C a r n e i r o [ 7 1

,

[ 8

1

.

been

sthdied

by

2 . APPROXIMATION OF CONTINUOUSLY DIFFERENTIABLE MAPPINGS

I n 1 9 4 7 , M a r s h a l l S t o n e came from t h e U n i v e r s i t y o f Chicago t o l e c t u r e a t t h e U n i v e r s i d a d e F e d e r a l do R i o de J a n e i r o (known t h e n as U n i v e r s i d a d e do B r a s i l ) f o r t h r e e months.

He

offered

a

beautiful

c o u r s e on "Rings of Continuous F u n c t i o n s " . Among o t h e r t h i n g s ,

he

t a l k e d a b o u t h i s c e l e b r a t e d p a p e r A GenehaLized W C i e & A t h U A A A p p h o x i -

m a t i o n Theohem which he had j u s t w r i t t e n . I t was p u b l i s h e d n e x t y e a r i n volume 21 (1948) of Mathematics Magazine. T h i s is a good

example

of an a r t i c l e t h a t became famous i n s p i t e of t h e f a c t

is

that

was

p u b l i s h e d i n an o b s c u r e j o u r n a l . S t o n e ' s c o u r s e d e a l t w i t h c o n t i n u o u s f u n c t i o n s , and was

going

t o have a l a s t i n g i n f l u e n c e on m e . I t was d u r i n g a n d shortly a f t e r i t t h a t , i n 1948, I t h o u g h t of and proved, b u t d i d n o t

gublish

then,

A LOOK AT APPROXIMATION THEORY

31 1

I will

w h a t I c a l l e d the W e i e r s t r a s s - S t o n e theorem f o r modules [ 3 4 ] .

come back t o t h i s a s p e c t i n a b r i e f w h i l e . The r e a s o n I d i d n o t publ i s h r i g h t aw'ay t h a t r e s u l t f o r modules w a s t h i s . I t took u n t i l 1960

to r e a l i z e

- 1961,

years

me

w h i l e I v i s i t e d B r a n d e i s U n i v e r s i t y f o r four months,

th!, i n t e r e s t f o r Approximation Theory o f

modules i n p l a c e

o f a l g e b r a s , and t o g e t s t a r t e d i n w e i g h t e d a p p r o x i m a t i o n p r o p e r f o r continuous functions. I n 1948, I went t o t h e U n i v e r s i t y of Chicago v i s i t during 1948-1950,

for

a two

a t t h e i n v i t a t i o n o f S t o n e . While t h e r e ,

had an a p p o r t u n i t y , i n 1 9 4 9 , o f p r e s e n t i n g a t And& Weil's

I

seminar

the t h e n r e c e n t a r t i c l e "On i d e a l s of d i f f e r e n t i a b l e f u n c t i o n s "

Hassler

year

by

Whitney, j u s t p u b l i s h e d i n volume 70 (1948) of t h e American

J o u r n a l of Mathematics. A f t e r my l e c t u r e ,

I r v i n g Segal

asked

me:

how a b o u t a s i m i l a r r e s u l t f o r a l g e b r a s of c o n t i n u o u s l y differentiable f u n c t i o n s , a l o n g the l i n e s o f t h e W e i e r s t r a s s - S t o n e t h e o r e m ? I n o t h e r

words, t h e problem w a s t o describe t h e c l o s u r e of

a subalgebra

continuously d i f f e r e n t i a b l e functions , or e q u i v a l e n t l y , to

of

describe

t h e closed subalgebras of c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s ,

in

t h e s p i r i t of t h e W e i e r s t r a s s - S t o n e theorem. To t h e b e s t of my knowl e d g e , t h i s problem h a s remained open so f a r : see below f o r t h e con-

jecture t h a t I h a v e i n mind i n t h i s r e s p e c t . Pressed by S e g a l ' s q u e s t i o n , I s t u d i e d i m m e d i a t e l y i n 1949 [ 231

t h e noteworthy case of d e n s e s u b a l g e b r a s , t o o b t a i n t h e f o l l o w i n g r e s u l t . L e t E be a r e a l m - d i f f e r e n t i a b l e ( m = 0 , 1 , f i n i t e dimension. Denote by

...,-)

Cm(E) t h e a l g e b r a o f

m - d i f f e r e n t i a b l e real f u n c t i o n s o n E ,

all

manifold

continuously

endowed w i t h t h e t o p o l o g y

o f uniform convergence on t h e compact subsets of and a l l t h e i r d i f f e r e n t i a l s up t o o r d e r m .

of

T~

E of suchfunctions

312

NACHBIN

(Nl)

F o h euehg

x

E

E, thehe i d

(N2)

F a t euehg

x

E

E,y

E

E

E

f E A

duck t h a t f ( x ) #O.

# y , ,?%thehe0

E, x

f E A nuch t h a t

f(x) # f(y).

F a h evehy

(N3)

x

x , thehe

at

ib

t # 0

and evehy t a n g e n t u e c t o h f

E A

t o E

buck t h a t

These c o n d i t i o n s do n o t depend on m . The case

excluded

m=O

by t h e above r e s u l t i s c o v e r e d by t h e W e i e r s t r a s s - S t o n e

theorem.

Coming back t o t h e q u e s t i o n S e g a l a s k e d m e i n 1 9 4 9 , b i t by b i t I was l e d t o f o r m u l a t e t h e f o l l o w i n g c o n j e c t u r e . I f i t i s t r u e ,

the

Whitney i d e a l theorem a n d t h e above d e n s i t y theorem are subsumed

by

i t . For t h e s a k e o f s i m p l i c i t y o f terminology and n o t a t i o n o n l y , l e t

15 m

us assume t h a t of

f o r some

lRn

n = 1,2,.

16

belongb t o t h e

CeObUhe

modueo

jD"g(x)

06

-

and evehy

U/A

Daf(x) 1 c

oadeh a t

mobt

Notice t h a t

E

doh

A

any

f

K ad

E

U

doh

T~

x E K

and

QO

equivalence are

Cm(U) t h e n f

i d (and ahJay4

contained i n

> 0, theke i b

E

m=

A.

LA a d u b a l g e b k a a d

Cm(U)

in

doh e u e h g compact h u b b e t

ceabb

D"

A

consider the

U, a c c o r d i n g t o which x , y E U

f E C m ( U ) and 06

subset

t o a r b i t r a r y E and t o

C"(U),

f(x) = f(y) for a l l

CONJECTURE 2 :

i s a nonvoid open

E=U

.., e x t e n s i o n

d e f i n e d by A on

U/A

e q u i v a l e n t when

id),

and t h a t

A is a subalgebra of

being easy. I f relation

m

g

any

dome

E

A

pahtiae

OMLy

equivalence buch

that

dehiuadiue

epual t o m.

f belongs to t h e c l o s u r e of

A in

Cm(U)

for

T~

when t h e above c o n d i t i o n h o l d s t r u e f o r e v e r y compact s u b s e t K o f U, by d e f i n i t i o n , n o t j u s t f o r t h o s e K c o n t a i n e d i n

some

equivalence

313

A LOOK AT APPROXIMATION THEORY

class modulo

The above c o n j e c t u r e i s a n a s p e c t of what I called

U /A.

LacaLizabiLity (see below too). If t r u e , t h e above c o n j e c t u r e h a s a n a t u r a l e x t e n s i o n t o modules i n p l a c e of a l g e b r a s . T h e r e i s a more n a i v e c o n j e c t u r e , which i s e a s i l y s e e n

to

f a l s e . W e m i g h t i n d e e d c o n j e c t u r e t h a t e v e r y s u b a l g e b r a A of which i s c l o s e d f o r

p l e convergence a t p o i n t s o f

t i a l s up t o order m. F o r

E of f u n c t i o n s and a l l t h e i r d i f f e r e n -

m = 0, t h i s i s i n d e e d t h e case; as a matter

of f a c t , t h e s t a t e m e n t t h a t g e b r a s of

Co(E)

Cm(E)

of s i m -

i s a l s o c l o s e d f o r t h e t o p o l o g y rms

T~

be

and

T~

= C(E) i s easily

have t o same c l o s e d s u b a l -

T~~

seen

to

be

to

equivalent

the

W e i e r s t r a s s - S t o n e theorem.

Lef: A be f:he n u b a t g e b f i a 0 6

EXAMPLE 3:

f(l/k) = f(0)

n u c h bha-t

Then A i d c t o b e d

60fi

60k

- c ~

a&!

k=1,2,

C1(n) a 6 a t e

... and

f

E

m

tnofieaweh Z,=,f'(l/n)/n

but it in n o 2 c t o n e d

doh

C1 ( R ) 2

=O.

71s.

A few y e a r s a g o , I a s k e d J a i m e Lesmesthe q u e s t i o n o f e x t e n d i n g

t h e above Theorem 1 t o i n f i n i t e d i m e n s i o n s . I a l s o d i d

raise

q u e s t i o n d u r i n g a l e c t u r e I gave a t Madrid, where Jos6 L l a v o n a

that got

i n t e r e s t e d i n i t . Recent work a l o n g t h i s l i n e w a s done by L e s m e s [13] and P r o l l a [ 4 9 ]

,

[ 531 i n B r a z i l , and by Llavona [14 I

,

[15]

i n Spain.

W e now summarize t h a t a s p e c t v e r y s u c c i n t l y , a l o n g t h e l i n e s o f 1381.

L e t E l F be Hausdorff r e a l l o c a l l y convex s p a c e s , E # 0 , U a nonvoid open s u b s e t o f

m

and

E

t h e v e c t o r s p a c e of a l l mappings

=

1,2,.

f :U

+

F

..

,m.

W e d e n o t e by

FfO,

?(U;F)

t h a t a r e c o n t i n u o u s l y m-

d i f f e r e n t i a b l e i n t h e following sense: 1)

f

is f i n i t e l y m-differentiable;

dimensional v e c t o r subspace

S

w e assume t h a t the r e s t r i c t i o n

of

E with

f ] (U

13 S )

t h a t is, f o r every f i n i t e

S # 0

+

fa,(

k

E;F)

U n S nonvoid,

is m - d i f f e r e n t i a b l e i n t h e

c l a s s i c a l s e n s e . Thus we h a v e the d i f f e r e n t i a l s dkf : U

and

NACHBIN

314

for

k =0,1,.

. ., k

5 n,

k

w i t h v a l u e s i n t h e v e c t o r s p a c e gas ( E;F) o f

Ek t o

a l l symmetric k - l i n e a r mappings of 2)

F.

The mapping

i s c o n t i n u o u s of e v e r y

longs t o t h e v e c t o r subspace k - l i n e a r mappings of

We endow

.,

k = 0,1,.

Ek

to

k 5 m . I n p a r t i c u l a r , d k f ( x ) be-

k ~ ; of ~ ) all Is

c o n t i n u o u s symmetric

F.

Cm(U;F) w i t h t h e t o p o l o g y

T~

of seninorms depending on t h e p a r a m e t e r s k ,

fc

d e f i n e d by t h e

4, K, L

a c o n t i n u o u s seminorm on

b e i n g nonvoid compact subsets of

family

F and K , L

respectively.

U, E

W e s h a l l u s e t h e n o t i o n of polynomial a l g e b r a ; see t h e convent i o n on page 6 3 , [ 5 4 1 .

THEOREM 4 :

Let

m 2 1 and A be a p o L y n a m i a l b u b a t g e b t a

SUppObe t h a t t h e t e i n a b u b n e t G c o n t i n u o u n L i n e a z endomohphibmh duch

06

06

t h e v e c t a h Apace

E'

?(U;F).

06

06 all

@ E

E w i t h d i n i t e dimenbionaL h a g u ,

that: 1)

T h e i d e n t i t y mapping

IE

belong4 t o t h e clonute 0 6 G doh

t h e compact-open t o p o l o g y an t h e v e c t o h npace a L L c o n t i n u o u n L i n e a h endomohphibmb 2)

Fon evetry that

06

and e v e h y

f

E

h t h i c t i o n ( f o J) IV = f o (J I V)

A,

; E)

06

U

06

E.

J E G, e v e t y n o n v a i d open nubbet V

J(V) c U

E(E

it 6 o l l o w h t h a t t h e

belongn t a t h e

nuch ze-

ctonuhe i n

316

A LOOK A T APPROXIMATION THEORY

Cm(U;F)

Then A i n d e n s e i n (Nl)

Foh e v e h y

x

(N2)

Fah evehy

x E U, y

that (N3)

6

in

f

E

A

nuch

thehe i b

f

E A

buch

x # y, t h e h e

U,

E

h u c h t h a t f ( x ) # 0.

f E A

U, t h e h e in

f ( x ) # f(y)

Foh e V e h y

don .rm i d and ondy id:

.

x E U, t E E , t # 0,

that

aa ft ( X I If

E

= d f ( x ) (t)

# 0.

i s f i n i t e d i m e n s i o n a l , c o n d i t i o n s 1) and 2 ) of Theorem 4

are s a t i s f i e d by G r e d u c e d t o

IE. Hence Theorem 4 i m p l i e s

Theorem

1. C o n d i t i o n 1) o f Theorem 4 i m p l i e s t h a t Grothendieck approximation p r o p e r t y , t h a t i s , closure of

E' 8 E

in

L(E;E)

E

has

the

belongs

IE

f o r t h e compact-open

Banachto

topology.

the Thus

Theorem 4 leads t o t h e f o l l o w i n g c o n j e c t u r e :

CONJECTURE

5:

F a h evetry

given

E,

t h e 60tLaiuing c o n d i t i o n n a h e equi-

vatent: Foh a h b i t h a h y

U, F , m 1. 1, t h e n e v e h y poLynamiaL

adgebha A i n d e n h e i n

o n l y id) A E

hatihdieh

Cm(U;F) d o h

(Nl)

,

-rm id ( a n d

nubaLwayn

(N2) , ( N 3 ) .

hab t h e Ranach - G h o t h e n d i e c k a p p h o x i m a t i o n p h o p e h t y .

I t i s known t h a t ( C 1 ) i m p l i e s ( C 2 ) . The c o n j e c t u r e i m p l i e s ( C l ) i s an a t t e m p t t o improve Theorem 4 .

that

(C2)

316

NACHBIN

I n t h e d i r e c t i o n of r e s e a r c h t h a t I j u s t mentioned,

there

is

more g e n e r a l l y t h e q u e s t i o n of s t u d y i n g Approximation Theory f o r a l g e b r a s o r modules of c o n t i n u o u s l y d i f f e r e n t i a b l e v e c t o r v a l u e d mappi n g s by u s i n g w e i g h t s . T h i s q u e s t i o n however i s s t i l l wide open,

in

s p i t e o f t h e a v a i l a b l e r e s u l t s . See t h e n e x t s e c t i o n f o r t h e c o n t i n uous c a s e .

3 . WEIGHTED APPROXIMATION FOR MODULES AND ALGEBRAS OF CONTINUOUS F"C-

TIONS L e t m e t a l k now a b o u t t h e W e i e s t r a s s - S t o n e theorem f o r m o d u l e s , how i t l e d m e t o t h e B e r n s t e i n a p p r o x i m a t i o n problem and what I t h e n c a l l e d t h e w e i g h t e d a p p r o x i m a t i o n problem ( o r t h e B e r n s t e i n - N a c h b i n a p p r o x i m a t i o n problem, a c c o r d i n g t o a more r e c e n t terminology by other authors). Let

E be a c o m p l e t e l y r e g u l a r t o p o l o g i c a l s p a c e , and C ( E ) de-

n o t e t h e a l g e b r a o f a l l c o n t i n u o u s r e a l f u n c t i o n s on E endowed w i t h t h e compact-open t o p o l o g y . T h e r e i s t h e i d e a l theorem f o r I f I i s an i d e a l i n

C ( E ) and

b e l o n g s t o t h e c l o s u r e of

-1

1

I

I-'(O)

in

as

C(E) which reads =

nf

,

I f-'(O)

C ( E ) i f and only i f

then

follows. f

E

f vanishes

C (E) on

(0). More g e n e r a l l y , t h e r e i s t h e W e i e s t r a s s - S t o n e

a l g e b r a A of

C ( E ) which r e a d s as f o l l o w s . L e t

l e n c e r e l a t i o n on E d e f i n e d by for every

f E A. C o n s i d e r

x1

A in

E / A

be t h e e q u i v a -

i f xl, x2 E E and f ( 5 ) = f ( x 2 )

n f € A f"(0)

A-'(O)=

o f e q u i v a l e n c e c l a s s e s modulo E / A belongs to t h e c l o s u r e of

- x2

theorem f o r a sub-

which e i t h e r is one

o r e l s e is v o i d . Then

C ( E ) i f and only i f

on e v e r y e q u i v a l e n c e class modulo E / A -1 A (0) i s nonvoid.

and f

f

f E C(E) i s constant

v a n i s h e s on A-'(O)

if

I n t h e i d e a l theorem, we have a module I over the algebra A = C ( E ) .

A LOOK AT APPROXIMATION THEORY

31 7

I n t h e W e i e r s t r a s s - S t o n e theorem, w e have a module bra

A

+

A over t h e alge-

IR g e n e r a t e d b y A and a l l c o n s t a n t r e a l f u n c t i o n s on

w e c o n s i d e r e d j u s t a v e c t o r subspace W of

C(E)

,

w e would

If

E.

have

a

module W o v e r t h e a l g e b r a A of a l l c o n s t a n t r e a l f u n c t i o n s on E . I n t h e succession of these t h r e e c a s e s , t h e algebra of m u l t i p l i e r s vari e s from t h e l a r g e s t t o t h e s m a l l e s t p o s s i b i l i t y c o n t a i n i n g t h e unit. More g e n e r a l l y , l e t A be a s u b a l g e b r a o f

C ( E ) whichwe mayrylw

w

assume t o c o n t a i n t h e u n i t w i t h o u t loss o f g e n e r a l i t y , and l e t a v e c t o r subspace o f

C ( E ) which i s a module o v e r

be

A so t h a t A W C W .

The W e i e r s t r a s s - S t o n e theorem f o r modules r e a d s a s f o l l o w s . 1 n t r o d u c e as b e f o r e t h e e q u i v a l e n c e r e l a t i o n E/A to t h e closure of

s e t K of E

E

W in

on E .

Then

f E C(E) belongs

C ( E ) i f , a n d o n l y i f , f o r e v e r y compact sub-

c o n t a i n e d i n some e q u i v a l e n c e c l a s s modulo E/A and every

> 0, there is

g E W

such t h a t

Ig(x)

-

f ( x )1
0

T(V) C X V , form a b a s i s o f neighborhoods a t 0 ; i n equiva-

such t h a t

l e n t t e r m s , when c o r r e s p o n d i n g t o e v e r y neighborhood U of

0 in

W

w e may f i n d a n o t h e r neighborhood V o f 0 i n W and E > 0 such t h a t k k Urn T ( E V) C U. More g e n e r a l l y , t h e members o f a c o l l e c t i o n C o f k =O l i n e a r o p e r a t o r s on W are s a i d t o be " s i m i l a r l y d i r e c t e d " i f the neighborhoods such t h a t

a t 0.

V of

T(V) C

0 in

W

,

f o r e a c h o f which there i s X = A ( V , T ) > 0 T E C , form a b a s i s of neighborhoods

X V f o r every

D i r e c t e d n e s s o f a l i n e a r o p e r a t o r i m p l i e s i t s c o n t i n u i t y . Both

d i r e c t e d n e s s and s i m i l a r d i r e c t e d n e s s r e d u c e t o continuity when a normed s p a c e . These c o n c e p t s a r i s e o n l y i n t r e a t i n g

more

t o p o l o g i c a l v e c t o r s p a c e s . Thus t h e h y p o t h e s i s i n Theorem t h a t the operators i n i s f i e d when

THEOREM 6:

W is

general 6

below

A be s i m i l a r l y d i r e c t e d is a u t o m a t i c a l l y sat-

W i s a normed s p a c e .

The p a i h A , W ha6

b0Me

h e p h e s e n t a t i o n b y continuous A e a L

a Haubdoadd s p a c e lukich 4~ lady convex

dunc.tionb id and o n L y i6 W

i6

undeh A ,

i n A a t e bimieahdy d i t e c t e d .

and the

0pehU.tOth

A WOK AT APPROXIMATION THEORY

76 the pait

THEOREM 7: h u e

6uncXionn and

undex A ,

S

i 4

A , W han 40me h e p h e n e n t a t i o n b y

06

a wectoh oubnpuce

t h e n t h e q u o t i e n t paih

A / S , WIS

16 t h e p a i h

areal d u n c t i o n b , t h e n 16

dea W

A, W

bpeC.tkae

cont i nuoun

which in

inuahiant

aeptenentation

S i 4 cloaed i n

had n u m e h e p h e d e n t a t i o n

W.

by cant i nuoun

nynt hebi d hoedn i n t h e doLl!owing

S i n a cLoned p h o p e h v e c t o h bubnpuce

A,

W

hab dome

b y Cona%tUOUb heal! 6unct i onn i6 a n d o n l y id

THEOREM 8:

319

t h e n S in t h e intehnection

06

06

W which

in

inwahiant un-

a l l ! C t 0 4 e d w e c t o h n u b 4 p a c ~0 6

w h ich a x e i n v a a i a n t undeh A , have cadimenhion o n e i n W and c o n - .

tain S. The p a s s i n g t o a q u o t i e n t s t a t e m e n t of Theorem 7 i m p l i e s

spec;

t r a l s y n t h e s i s i n Theorem 8 , which may be viewed a s an a b s t r a c t v e r s i o n o f the W e i e r s t r a s s - S t o n e theorem f o r modules. L e t u s a l s o p o i n t

,

then

Theorem 8 becomes t h e f o l l o w i n g s t a t e m e n t . Every c l o s e d p r o p e r

vec-

o u t t h a t , when

tor subspace

A i s reduced t o t h e scalar o p e r a t o r s

S o f a l o c a l l y convex s p a c e

a l l c l o s e d vector s u b s p a c e s o f and c o n t a i n

W

of

W

is t h e i n t e r s e c t i o n

of

W which have codimension one i n

S . As i t i s c l a s s i c a l , such a s t a t e m e n t

is

W

equivalent

t o t h e Hahn-Banach theorem. Thus Theorem 8 may be looked upon

as

a

g e n e r a l i z a t i o n of b o t h t h e W e i e r s t r a s s - S t o n e theorem f o r modules a n d t h e Hahn-Banach theorem f o r l o c a l l y convex s p a c e s . We may t h e n ask t h e f o l l o w i n g n a t u r a l q u e s t i o n . To what e x t e n t

the c o n d i t i o n o f t h e o p e r a t o r s i n

A b e i n g s i m i l a r l y d i r e c t e d i s mu-

c i a l f o r the v a l i d i t y o f Theorem 6, o r Theorem 7 , or Theorem 8 ? Lo-

c a l c o n v e x i t y under

A

i s n o t superfluous.

In fact,

r e d u c e d t o t h e scalars o p e r a t o r s o f

W , t h e n i t may

e v e r y c l o s e d p r o p e r v e c t o r subspace

S of

s l l closed vector subspaces of

and c o n t a i n

S,

letting be

A

false

be that

is the intersection

of

W which have condimension one i n

W

W

i n case W i s n o t assumed t o be l o c a l l y convex.

The

NACHBIN

320

answer t o t h e above n a t u r a l q u e s t i o n i s no. The example t h a t I found i n 1957 l e d m e t o t h e c l a s s i c a l B e r n s t e i n a p p r o x i m a t i o n problem, a s 1 s h a l l describe next.

EXAMPLE 9 :

t i o n s on

Let

R

W be t h e F r g c h e t s p a c e o f a l l c o n t i n u o u s r e a l f u n c -

t h a t are r a p i d l y d e c r e a s i n g a t i n f i n i t y . C a l l

t h e a l g e b r a o f a l l r e a l p o l y n o m i a l s on

a

R . Every

E

A = P (33)

is

that

C(lR)

s l o w l y i n c r e a s i n g a t i n f i n i t y g i v e s r i s e t o t h e c o n t i n u o u s l i n e a r opTa : f E W

erator

+

af

E W

which i s d i r e c t e d i f and only a is bounded.

Thus A may be v i e w e d . a s a commutative a l g e b r a operators of

of

continuous l i n e a r

W c o n t a i n i n g the i d e n t i t y o p e r a t o r o f

W , b u t each such

o p e r a t o r i s d i r e c t e d i f and o n l y i f t h e c o r r e s p o n d i n g p o l y n o m i a l

is

c o n s t a n t . I t i s c l e a r t h a t W i s l o c a l l y convex u n d e r A .

is

w

some

E W

v a n i s h i n g nowhere

i n lR s u c h t h a t

W ( t h i s i s e a s i l y seen t o be e q u i v a l e n t

v a n i s h i n g nowhere i n of

BAP

-2

or BA P

t o e x i s t e n c e o f some

W which i s i n v a r i a n t u n d e r

lR, i t can be shown t h a t A w

any c l o s e d v e c t o r s u b s p a c e o f condimension o n e i n W .

in

b e l o w ) . Then t h e c l o s u r e

p r o p e r v e c t o r subspace o f never vanishes i n

i s n o t dense i n

Aw

w

E

W

t h a t i s n o t a f u n d a m e n t a l w e i g h t i n the sense

R

-1

There

W is a closed

Since

w

i s n o t contained

in

A.

W which i s i n v a r i a n t under

A, having

Thus Theorem 8 d o e s n o t h o l d i n t h i s case due

t o l a c k o f d i r e c t e d n e s s . A f o r t i o r i Theorem 7 a n d Theorem 6

do n o t

h o l d i n t h i s c a s e f o r t h e same r e a s o n . This counterexample l e a d s us t o t h e

CLUbbiCae

&MnAZeh a p p o x i -

m a t i o n p t o b L e m , u s u a l l y f o r m u l a t e d i n t h e f o l l o w i n g t w o forms, where P(lRn)

i s t h e a l g e b r a o f a l l r e a l p o l y n o m i a l s on IRn B AP

and

- 1.

Let

v : IRn

+

IR,

b e an upper s e m i c o n t i n u o u s " w e i g h t "

Cvm(lRn) be t h e v e c t o r s p a c e o f a l l

tends to

... .

f o r n = 1,2 ,

f E C(IEln)

such

that

0 a t i n f i n i t y , seminormed by II f Ilv = s u p { v ( x ) If ( x ) ; x EW

Assume t h a t

vf n

v i s r a p i d l y d e c r e a s i n g a t i n f i n i t y , t h a t is P(Rn) CCv,(*).

1.

A LOOK AT APPROXIMATION THEORY

When i s

dense i n

P(IRn)

321

Cvw(lRn) ? W e t h e n s a y t h a t

mentaL w e i g h t . W e s h a l l d e n o t e by R n

v is a

dunda-

t h e s e t o f a l l s u c h fundamental

w e i g h t s i n t h e s e n s e of B e r n s t e i n . F o r t e c h n i c a l r e a s o n s w e a l s o i n -

rn

troduce the set

rn

Clearly

BAP

i n g to

0

C

f o r a l l k > 0.

Rn

E

This inclusion i s proper.

Rn.

- 2.

vk

o f a l l such v such t h a t

Let

Cw(lRn)

be t h e Banach s p a c e o f a l l

a t i n f i n i t y , normed by

the s p e c i a l case of

E

C(#)tend-

Ilfll= s u p { i f ( x ) I ; x E lRn 1 ;

Cvm(lRn) when

w

v = l . Assume t h a t

rapidly decreasing a t i n f i n i t y , t h a t i s

w a w e i g h t . When i s

f

P(IRn) w dense i n

P(lRn) w

it

is

E C(IRn)

is

and c a l l

Cm(IRn),

C

Cw(lRn) ? W e t h e n s a y

that

w

is a 6undarnentaL w e i g h t . If

w

E

C(IR")

is rapidly decreasing a t i n f i n i t y , then w i s a

f u n d a m e n t a l w e i g h t i n the s e n s e of v a n i s h e s on B AP

- 1. H o w e v e r

v a n i s h on that

and

IRn

IRn

B A P -1

I wI

BAP- 2

i f and o n l y i f

is a fundamental weight i n t h e

a fundamental w e i g h t v i n t h e s e n s e of

a n d may f a i l t o be c o n t i n u o u s .

is

It

B AP

sense

of

-1

my

€3 A P

i n t h i s sense

i s a b e t t e r way o f l o o k i n g a t t h e c o n c e p t

m e n t a l w e i g h t s i n t h e s e n s e of B e r n s t e i n t h a n

never

w

of

funda-

- 2.

The f o l l o w i n g a r e t h e s i m p l e s t c r i t e r i a f o r a n upper s e m i c o n tinuous function

v : IR

+

IR+

t o belong to

rl ,

thus t o

R1 ,

by

i n c r e a s i n g d e g r e e of g e n e r a l i t y : BOUNDED CASE: ANALYTIC CASE:

v

hub a b o u n d e d buppoht.

Thehe ahe

C > 0

and

c > 0 dvh w h i c h , doh any x E IR,

we have

QUASI-ANALYTIC CASE:

We h a v e

1 z;=l -

VM,

=

+

-

whehe,

{oh

NACHBIN

322

m = O,l,...,

In

we b e t

B A P - 1, t h e s u b a l g e b r a

Cvm(IRn), and we have t h e weight BAP

- 2,

i s contained i n

C(IRn)

Cvm(IRn). I n

v i n the definition of

t h e submodule P ( I R n ) w o v e r t h e s u b a l g e b r a

is contained i n of

of

P(IRn)

of

P(IRn)

C (IR")

c,(IRn), and w e have t h e w e i g h t w i n t h e d e f i n i t i o n

P(EP)W. Thus

was l e d

I

t o t h e following general

formulation

of t h e

weighted a p p h o x i m a t i o n phobLem. The v i e w p o i n t t h u s adopted embraces the

Weierstrass

- Stone

theorem f o r modules, t h u s f o r a l g e b r a s ,

B e r n s t e i n approximation problem. A c t u a l l y , it i s guided by

and t h e the

idea

of e x t e n d i n g t h e c l a s s i c a l B e r n s t e i n approximation problem i n t h e same s t y l e t h a t the Weierstrass

- Stone

theorem g e n e r a l i z e s

W e i e r s t r a s s theorem (see [ 3 4 ] f o r d e t a i l s )

.

t h e classical

L e t V be a s e t of upper semicontinuous p o s i t i v e r e a l f u n c t i o n s

on a completely r e g u l a r t o p o l o g i c a l s p a c e E.

d i m c t e d i n t h e s e n s e t h a t , i f vl, v 2 v1 5 X v and

such t h a t

v2

E V,

and any

v E V

E

f

+

is

V

i s called

f E C ( E ) such t h a t ,

a for

> 0 , t h e c l o s e d s u b s e t CxEE; v ( x ) - i f ( x ) l L E I

i s compact, w i l l be denoted by seminorm

V

t h e r e a r e h > 0 and v E V

5 X v. Each element of

w e i g h t . The v e c t o r subspace of C ( E ) o f a l l any

W e assume t h a t

CVm(E).

It f l l v = sup I v ( x )

0

Each

If ( x ) 1 ; x E E

n a t u r a l topology on t h e w e i g h t e d d p a c e

CV,(E)

v

determines a

E V

on

the

CVm(E).

is defined

by

the

f a m i l y of a l l such seminorms. Let

A

C

C ( E ) be a s u b a l g e b r a c o n t a i n i n g t h e u n i t , and W

be a v e c t o r subspace. A s s u m e t h a t W i s a module o v e r A W C W.

A

,

C

CVm(E)

that

is

The w e i g h t e d a p p h u x i m a t i o n pAObeem c o n s i s t s of a s k i n g f o r a

d e s c r i p t i o n of t h e c l o s u r e of

W in

CVm(E) under such c i r c u m s t a n c e s

We s a y t h a t W i s LocaLizabLe undefi A i n Wm(E)when the following

A LOOK AT APPROXIMATION THEORY

condition holds true: i f of

W

in

CVm(E) i f

f(x)1


0

such t h a t

The n t h i c t w e i g h t e d appaoxi-

E X.

mation phab-tern c o n s i s t s of a s k i n g f o r n e c e s s a r y and s u f f i c i e n t c o n d i tions i n order t h a t W e d e n o t e by

W b e l o c a l i z a b l e under G ( A ) a s u b s e t of

A as a n a l g e b r a w i t h u n i t ,

W e a l s o introduce a subset W a s a module o f

t h a t i s , such t h a t t h e s u b a l g e b r a

G(W) of

f o r t h e t o p o l o g y of

A

of

A

C(E)

.

W which t o p o l o g i c a l l y g e n e r a t e s

t h a t i s , the submodule over A of

A,

G(W) i s dense i n W

by

CV,(E).

A which t o p o l o g i c a l l y g e n e r a t e s

G ( A ) and one i s d e n s e i n

g e n e r a t e d by

A in

f o r t h e topology of

W

generated

CVm(E).

A b a s i c r e s u l t i s t h e n t h e f o l l o w i n g one.

THEOREM 10:

w

E

doh

G(w),

any

Addume

thehe id

x

E E.

t h a t , 604

Y

E

rl

v

eUChg

E

V,

euehy

a

E G(A)

and e u e h q

nuch t h a t

Then W i n locaLiza6Le undeh A i n

CVm(E).

W e may combine Theorem 10 w i t h t h e i n d i c a t e d c r i t e r i a f o r memb e r s h i p of

rl.

COROLLARY 11: evekg

L e t u s c o n s i d e r e x p l i c i t l y the a n a l y t i c case.

Anbume t h a t , d o h e v e h y

w E G(W), t h e t e a t e

6 o h any

x

E

E.

Then W

i b

C > 0

and

v

E

V,

c > 0

evehy

a E G(A)

and

nuch t h a t

LocaLizabLe undeh A i n

CV=(E).

A s a p a r t i c u l a r c a s e o f t h e above r e s u l t s f o r modules,

w e have

324

NACHBIN

t h e f o l l o w i n g o n e s f o r a l g e b r a s . For s i m p l i c i t y s a k e , assume t h a t

i s s t r i c t l y p o s i t i v e , t h a t is, f o r every that

v ( x ) > 0 . L e t A be c o n t a i n e d i n

,

there is v E V

su&

We s a y t h a t A i s

lo-

E E

CV,(E).

C V m ( E ) when t h e f o l l o w i n g c o n d i t i o n

calizabte i n f E CV,(E)

x

then

always o n l y i f )

belongs t o t h e c l o s u r e o f

f f

holds

A in

true:

CV-(E)

if

is c o n s t a n t on e v e r y e q u i v a l e n c e class mdulo

W e d e n o t e by

G ( A ) a s u b s e t of

V

A which t o p o l o g i c a l l y

if (and E/A.

generates

A as an a l g e b r a w i t h u n i t , t h a t i s such t h a t t h e s u b a l g e b r a o f A g e n -

e r a t e d by

G ( A ) and one i s d e n s e i n

A

f o r t h e topology of

CVm(E).

The p a r t i c u l a r c a s e i s t h e n t h e f o l l o w i n g one.

W e may combine Theorem 12 w i t h t h e i n d i c a t e d c r i t e r i a f o r membership of

rl.

COROLLARY 1 3 : ahe

C > 0

d o h any

x

and

E E.

W e quote

L e t us c o n s i d e r e x p l i c t l y t h e a n a l y t i c case.

Andume t h a t , 6 0 4 e v e h y

c > 0

buch

Then A [34]

,

i b

v

E

V and evehy a E G ( A ) , t h e t r e

that

localizable i n C V m ( E ) .

[37] for additional details.

A LOOK AT APPROXIMATION THEORY

325

FEFE RENCES

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ARON, Approximation of d i f f e r e n t i a b l e

R. M. ARON, Polynomial approximation and a q u e s t i o n of G.E. Shilov, i n A p p h o x i m a t i o n T h e o h y a n d FunctionaL A n a L y n i A J . B.

P r o l l a ) , Notas de Matemztica ( 1 9 7 9 )

,

(Editor:

North-Holland,

t o appear. R. M. ARON and J. B. PROLLA, Polynomial approximation of

dif-

f e r e n t i a b l e f u n c t i o n s on Banach s p a c e s , J o u r n a l f i i r d i e Reine und Angewandte Mathematik, t o appear.

K.-D. BIERSTEDT, Verallegemeinerungen das S a t z e s von StoneWeierstrass , Jahrbuch ijberblicke Mathematik(1975),109-135.

K.-D. BIERSTEDT, Some g e n e r a l i z a t i o n s of t h e W e i e r s t r a s s

and

Stone-Weierstrass theorems, Anais da Academia B r a s i l e i -

r a de C i k c i a s 4 9 ( 1 9 7 7 1 , 507 K.-D.

- 523.

BIERSTEDT, A remark on vector-valued approximation onam-r p a c t s e t s , approximation on product s e t s , and t h e

ap-

proximation p r o p e r t y , i n A p p h o x i m a t i o n T h e o h y a n d Func-

t i a n a l A n a L y d ( E d i t o r : J . B. P r o l l a ) , Notas de Matemzt i c a ( 1 9 7 9 ) , North-Holland,

J.

Q.

t o appear.

CARNEIRO, AproximaGao ponderada nao-arquimediana, A n d s

da Academia B r a s i l e i r a de C i s n c i a s 50 (1978) , 1 - 3 4 .

J . P. Q. CARNEIRO, Non-archimedean weighted approximation, i n A p p h o x i m a t i o n T h e o a y a n d FunctionaL A n a e y d i n (Editor: J. B.

P r o l l a ) , Notas de Matemgtica ( 1 9 7 9 ) , North-Holland,

t o appear. J.-P.

FERRIER, Suk l ' a p p h o x i r n a t i o n p o n d z h e e , P u b l i c a t i o n s

S6minaire d'Analyse Moderne, U n i v e r s i t 6

( 1 9 7 2 ) , Canada.

de

du

Sherbrooke

NACHBIN

326

(10 ]

G.

GLAESER, A l g s b r e s e t s o u s - a l g s b r e s de f o n c t i o n s d i f f g r e n t i a -

37(1965),

b l e s , Anais da Academia B r a s i l e i r a de C i i n c i a s

395

- 406.

[ll]

C . S. GUERREIRO, I d e a i s

de

fun@es d i f e r e n c i g v e t g ,

[12]

C. S. GUERREIRO, Whitney's s p e c t r a l s y n t h e s i s theorem

Academia B r a s i l e i r a de C i g n c i a s 49 (1977), 41

Anais

- 70.

in

da

in-

f i n i t e dimensions, i n A p p h a x i m a t i o n T h e o k y and F u n c t i o n a l

Analydin (Editor: J. B. P r o l l a ) , (1979), North-Holland,

Notas

de

Matemdtica

t o appear.

[ 131

J. LESMES, On t h e approximation of c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s i n H i l b e r t s p a c e s , R e v i s t a Colombiana de Mat e m s t i c a s 8 (1974), 217 - 223.

[14]

J. G. LLAVONA,

A p t u x i m a c i o n de d u n c i o n e h d i d e h c n c i a b t e n , Uni-

v e r s i d a d Complutense de Madrid (19751, Spain. [ 15 ]

J . G. LLAVONA, Approximation o f d i f f e r e n t i a b l e f u n c t i o n s ,

to

appear.

[16 ]

S. MACHADO, Aproximaqgo ponderada e m f i b r a d o s vetoriais, Anais da Academia B r a s i l e i r a de C i d n c i a s 4 3 (19711, 1 - 2 1 .

1171

S. MACHADO and J . B. PROLLA, An i n t r o d u c t i o n t o N a c h b i n s p a c e s , R e n d i c o n t i d e l C i r c o l o Matematico d i Palermo

119

- 139.

21 (1972),

[18]

S . MACHADO, On B i s h o p ' s g e n e r a l i z a t i o n of t h e Weierstrass-Stone theorem, I n d a g a t i o n e s Mathematicae 39 (1977), 218 224.

[19]

S. MACHAW and J . B. PROLLA, Concerning t h e bounded c a s e o f t h e Bernstein-Nachbin approximation problem, J o u r n a l of t h e Mathematical S o c i e t y o f J a p a n 29 (19771, 451 - 458.

[20]

S. MACHADO and J . B. PROLLA, The g e n e r a l complex

-

Bernstein-Nachbin approximation problem, 1 ' I n s t i t u t F o u r i e r 28 (1978), 193

- 206.

case of

the

Annales

de

A LOOK AT APPROXIMATION THEORY

327

[Zl] P. MALLIAVIN, Approximation p o l y n o m i a l e p o n d s r z e e t p r o d u i t s c a n o n i q u e s , i n A p p h o x i m a t i o n Theohy and F u n d a n d A n d y b A ( E d i t o r : J. B. P r o l l a )

,

N o t a s de Matemhtica (1979)

, North-

Holland, t o appear. [22]

R . MEISE, S p a c e s o f d i f f e r e n t i a b l e f u n c t i o n s a n d t h e approxima-

t i o n property, i n Apphoximation Theohy

A n a t y b i n ( E d i t o r : J . B. P r o l l a ) , ( 1 9 79) , North-Holland,

[231

and

Functionat

de

Platerngtica

Notas

t o appear.

L . NACHBIN, S u r l e s a l g s b r e s d e n s e s de f o n c t i o n s

diffgrentia-

b l e s s u r une v a r i b t z , Comptes Rendus d e l'Acad6mie S c i e n c e s de P a r i s 228 ( 1 9 4 9 ) , 1549 [241

- 1551.

L. NACHBIN, A l g e b r a s o f f i n i t e d i f f e r e n t i a l order a n d t h e e r a t i o n a l c a l c u l u s , Annals o f Mathematics 413

[251

des

- 437.

70

op-

(1959),

lo-

L. NACHBIN, On t h e w e i g h t e d p o l y n o m i a l a p p r o x i m a t i o n i n a c a l l y compact s p a c e , P r o c e e d i n g s o f t h e N a t i o n a l Acao f S c i e n c e s o f t h e USA 4 7 (1961), 1055 -1057.

[26 1

L. NACHBIN, S u r 1 ' a p p r o x i m a t i o n p o l y n o m i a l e pond6rs.e d e s

fonc-

t i o n s r6elles c o n t i n u e s , A t t i d e t L a 11

Riunione det Ghoupement d e M a t h z m a t i c i e n b d ' EXpm5.4.kn Latine, FirenzeBologna 1 9 6 1 ( 1 9 6 3 ) , 4 2 - 5 8 , E d i z i o n i Cremonese, I t a l y . [271

L. NACHBIN,

R g s u l t a t s & c e n t s e t probldmes d e n a t u r e a l q a r i q u ?

e n t h g o r i e d e l ' a p p r o x i m a t i o n , P t o c e e d i n g n 06 t h e

t e t n a t i o n a t COnghe6b o 6 M a t h e m a t i c i a n s , Stockholm (19631, 379

[281

- 384,

ln1962

A l m q v i s t a n d W i k s e l l s , Sweden.

L. NACHBIN, S u r l e thsorzrne de Denjoy-Carleman p o u r les a p p l i c a t i o n s v e c t o r i e l l e s i n d s f i n i m e n t d i f f g r e n t i a b l e s quasia n a l y t i q u e s , Comptes Rendus d e 1'Acadgmie

des Sciences

de P a r i s 256 (19631, 8 6 2 - 8 6 3 . [29 I

-

L. NACHBIN, F o n c t i o n s a n a l y t i q u e s e t q u a s i - a n a l y t i q u e s v e c t o r i e l l e s e t l e problgme d ' a p p r o x i m a t i o n d e B e r n s t e i n , S z m i n a i h e P i e h h e Letong ( A n a l y s e ) , I n s t i t u t H e n r i P o i n C ~ (1963) , F r a n c e .

328

NACHBIN

[30 ]

L.

NACHBIN, Weighted a p p r o x i m a t i o n o v e r t o p o l o g i c a l s p a c e s and

t h e B e r n s t e i n problem

over f i n i t e dimensional

vector

s p a c e s , Topology 3 ( 1 9 6 4 ) , s u p p l . 1, 1 2 5 - 1 3 0 . [31 ]

L . N A C H B I N , Weighted a p p r o x i m a t i o n f o r a l g e b r a s and modules of

c o n t i n u o u s f u n c t i o n s : real and s e l f - a d j o i n t Annals o f Mathematics 8 1 (19651, 289 [32]

-

complex cases,

302.

L . NACHBIN, Aproximaqao p o n d e r a d a d e f u n q o e s c o n t i n u a s p o r po-

lin6mios, A t a d do T e h c e i h a Coloquio B h a d i L e i h a d e Mate-

m z t i c a , F o r t a l e z a 1 9 6 1 (1965), 1 4 6 - 189, I n s t i t u t o

de

Matem6tica P u r a e A p l i c a d a , B r a s i l . 133 ]

L . NACHBIN, Weighted a p p r o x i m a t i o n f o r f u n c t i o n

F. T . B i r t e l ( 1 9 6 6 ) , 330

134 ]

algebras

q u a s i - a n a l y t i c mappings , i n F u n c t i o n A l g e b h a d

L . NACHBIN,

- 333,

and

(Editor:

S c o t t a n d Foresman, USA.

E l e m e n t 4 a 6 apphoximatian Rheahy (1967) , Van N o s t r a n d .

R e p r i n t e d ( 1 9 7 6 ) , K r i e g e r , USA. 135 ]

L . NACHBIN, J . B . PROLLA a n d S. MACHADO, Weighted a p p r o x i m a t i o n ,

v e c t o r f i b r a t i o n s and a l g e b r a s o f o p e r a t o r s , J o u r n a l de

Mathgmatiques P u r e s e t A p p l i q u g e s 5 0 ( 1 9 7 1 ) , 2 9 9 [36 ]

L . NACHBIN, J . B . PROLLA and S . MACHADO,

- 323.

Concerning weighted

approximation, v e c t o r f i b r a t i o n s and a l g e b r a s of

opera-

t o r s , J o u r n a l o f Approximation Theory 6 ( 1 9 7 2 1 , 80 - 8 9 . [371

L.

N A C H B I N , On t h e p r i o r i t y o f a l g e b r a s o f c o n t i n u o u s f u n c t i o n s

i n w e i g h t e d a p p r o x i m a t i o n , Symposia Mathematica 17(1976),

169 [ 38

I

- 183.

L. NACHBIN , S u r l a d e n s i t 6 d e s s o u s - a l g g b r e s p o l y n o m i a l e s

d'ap-

p l i c a t i o n s c o n t i n h e n t d i f fgrentiables ,Seminaihe Piehhe

LeLung e t Henhi S k a d a (Andyde) , 1976/77,

Springer Verlag

L e c t u r e Notes i n Mathematics, t o appear. [39]

J . B.

PROLLA,

Vectah

dibhatiann

and

aLgebhan a d o p e h a t o f i b ,

P u b l i c a t i o n s du S g m i n a i r e d ' A n a l y s e Moderne, U n i v e r s i t g de S h e r b r o o k e (1968/69)

,

Canada.

A LOOK AT APPROXIMATION THEORY

[40]

J. B.

329

PROLLA, Aproximaqiio p o n d e r a d a e S l g e b r a s

de operadores,

A n a i s d a Academia B r a s i l e i r a d e C i g n c i a s 43(1971), 23

[41]

L , B.

PROLLA, The w e i g h t e d Dieudonn6 t h e o r e m

for

- 36.

density

in

t e n s o r p r o d u c t s , I n d a g a t i o n e s Ebthermticae 33(1971), 170-175. 142

I

J. €3. PROLLA, Weighted s p a c e s o f v e c t o r - v a l u e d

c o n t i n u o u s func-

t i o n s , A n n a l i d i Matematica P u r a e d A p p l i c a t a 145 [43 ]

J . B.

- 158.

PROLLA, B i s h o p ' s g e n e r a l i z e d S t o n e - W e i e r s t r a s s f o r weighted s p a c e s , Mathematische 283

[44 1

89 (1971),

- 289.

theorem

Annalen 1 9 1 (1971) ,

J . B . PROLLA, Weighted a p p r o x i m a t i o n o f c o n t i n u o u s

functions,

B u l l e t i n o f t h e American M a t h e m a t i c a l S o c i e t y 7 7 ( 1 9 7 1 ) , 1021-1024.

[45 I

J . B.

PROLLA, Weighted a p p r o x i m a t i o n a n d s l i c e p r o d u c t s of iscdu-

l e s o f c o n t i n u o u s f u n c t i o n s , A n n a l i d e l l a S c u o l a Nomle S u p e r i o r e d i P i s a 2 6 ( 1 9 7 2 ) , 5 6 3 571.

-

[46 1

J . B . PROLLA a n d S . MACHADO, W e i g h t e d G r o t h e n d i e c k

subspaces,

T r a n s a c t i o n s o f t h e American M a t h e m a t i c a l S o c i e t y (1973) [471

J. B.

,

247

- 258.

186

PROLLA, Modules od c o n t i n u o u s f u n c t i o n s , i n

Functional A n a e y b i n and A p p l i c a t i a n n ( E d i t o r : L. N a c h b i n ) ,S p r i n g e r

V e r l a g L e c t u r e N o t e s i n M a t h e m a t i c s 384 (19741, 123- 128. [48]

J . B.

PROLLA, Then c o n d e 4 e n c i a b n a b h e t e o h i a

de aphoximacion,

P u b l i c a c i o n e s d e l D e p a r t a m e n t o de E c u a c i o n e s F u n c i o n a

-

lest U n i v e r s i d a d d e S e v i l l a ( 1 9 7 4 1 , S p a i n . (491

J. B.

PROLLA, On p o l y n o m i a l algebras o f c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s , R e n d i c o n t i d e l l a Accademia N a z i o n a l e d e i L i n c e i 57 ( 1 9 7 4 ) , 481

[501

J . B.

- 486.

PROLLA, On o p e r a t o r i n d u c e d t o p o l o g i e s , i n A n a l y n e

tionnelee e t Applications 225 - 2 3 2 ,

Hermann, P a r i s .

( E d i t e u r : L. N a c h b i n )

Fonc-

(1975) ,

NACHEIN

330

[51]

J.

1521

J . B. PROLLA,

B. PROLLA,

Dense approximation f o r polynomial algebras, Bonner

Mathematische S c h r i f t e n 8 1 (1975) , 115

- 123.

Aphaximacizn en litgebhab p a l i n o m i c a b de duncianu dibehenciabled,Publicaciones d e l Departamento de An6l i s i s Matemstico, U n i v e r s i d a d de S a n t i a g o de Compostela (19751, S p a i n .

1531

[54

1

J . B. PROLLA and C . S . GUERREIRO, An e x t e n s i o n

of Nachbin's theorem t o d i f f e r e n t i a b l e f u n c t i o n s on Banach s p a c e s w i t h t h e approximation p r o p e r t y , Arkiv f o r Matematik 1 4 (19761, 251 - 258.

J. B . PROLLA, Apphoximation o d u e c t o t - v a t u e d d u n c t i o n n ,

de Matemitica 6 1 ( 1 9 7 7 ) [55]

J. B.

,

Notas

North-Holland.

PROUA, The approximation p r o p e r t y f o r Nachbin s p a c e s , i n Appaaximation Theohy and F u n c t i o n a l A n a l y b i b ( E d i t o r : J. B. P r o l l a )

, Notas

de Matem6tica (19791, North - H o l l a n d ,

t o appear. [56 1

J. B.

PROLLA, Non-archimedean f u n c t i o n s p a c e s , i n Lineah Spaces

a n d Appkoximation ( E d i t o r s : P. L. B u t z e r and B.Sz-Nagy)

,

I n t e r n a t i o n a l S e r i e s i n Numerical Mathematics 40 (1978) , 1 0 1 - 1 1 7 , B i r k h a u s e r Verlag B a s e l , S w i t z e r l a n d . [57]

J. B. PROLLA and S. MACHADO, S u r l ' a p p r o x i m a t i o n polynomialeen

dimension i n f i n i e , Acted d e l a VT Rzunion du Ghoupernent d e Mathematiciend d ' Exphehbion L a t h e , Palrna de M a l l o r c a 1977, S p a i n , t o a p p e a r . [581

W.

H.

SUMMERS, Weighted bpaceb and w e i g h t e d a p p h o x i m a t i o n , PUb l i c a t i o n s du S i m i n a i r e d'Analyse Moderne , U n i v e r s i t S de Sherbrooke ( 1 9 7 0 ) , Canada.

[59 1

W.

H.

SUMMERS, The bounded case of t h e weighted

approximation problem, i n FunctionaL Analydid and A p p t i c a t i o n b (Editor: L. Nachbin) , S p r i n g e r V e r l a g L e c t u r e Notes i n Mathematics

384 ( 1 9 7 4 ) , 1 7 7 - 183.

331

A LOOK AT APPROXIMATION THEORY

160 ]

W.

H. SUMMERS, Weighted a p p r o x i m a t i o n f o r modules o f c o n t i n u o u s f u n c t i o n s 11, i n Anak?yne F a n c t i o n n e L l e ( E d i t e u r : L . Nachbin) ( 1 9 7 5 ) , 2 7 7

[61]

G.

I . ZAPATA, A p t o x i m a C Z a

- 283,

et

Appticationd

Hermann, P a r i s .

p o n d e a a d a paka d u n ~ o e b &&?hen&&&,

M o n o g r a f i a s do C e n t r o B r a s i l e i r o de P e s q u i s a s F i s i c a s 30

(1971) , B r a s i l .

[62]

G.

I . ZAPATA, S u r le problsme d e B e r n s t e i n e t l e s a l g s b r e s

de

f o n c t i o n s c o n t i n h e n t d i f f g r e n t i a b l e s , Comptes Rendusde

1'Acadgmie des S c i e n c e s de P a r i s 274 (1972) , 70 [631

G.

- 72.

I . ZAPATA, B e r n s t e i n a p p r o x i m a t i o n problem f o r differentiable

f u n c t i o n s and q u a s i - a n a l y t i c weights, T r a n s a c t i o n s

of

t h e American M a t h e m a t i c a l S o c i e t y 182 (19731, 503-509. [64]

G.

I . ZAPATA, Approximation f o r w e i g h t e d a l g e b r a s o f d i f f e r e n -

t i a b l e s f u n c t i o n s , B o l l e t t i n o d e l l a Unione I t a l i a n a 9 ( 1 9 7 4 ) , 32 [651

G. I . ZAPATA, Weighted a p p r o x i m a t i o n , Mergelyan t h e o r e m and q u a s i - a n a l y t i c w e i g h t s , A r k i v f o r Matematik 1 3 ( 1 9 7 5 ) ,

255 [66 ]

Matematica

- 43.

G.

- 262.

I . ZAPATA, Fundamental seminorms, i n A p p a o x i m a t i o n Theoty and

F u n c t i o n a l Anadgdid

( E d i t o r : J. B . P r o l l a ) ,

Matemztica (1979) , N o r t h - H o l l a n d ,

t o appear.

Notas

de

This Page Intentionally Left Blank

Approximation Theory and Functional Analysis J.B. ProZla led. I @North-HoZZand PubZishing Company, 1979

BANACH ALGEBRAS OVER VALUED FIELDS

LAWRENCE N A R I C I

St. John's University

Jamaica, N e w York, 11439, USA and EDWARD BECKENSTEIN

S t . John ' s U n i v e r s i t y S t a t e n I s l a n d , N e w York 1 0 3 0 1 , USA

ABSTRACT By " G e l f a n d t h e o r y " h e r e is meant t h e s t u d y o f t h e c o n s e q u e n c e s o f t o p o l o g i z i n g t h e maximal i d e a l s of a Banach algebra.

The

i s most r i c h when t h e u n d e r l y i n g f i e l d i s t h a t of t h e complex

R o r some o t h e r v a l u e d

b e r s . I f t h e u n d e r l y i n g f i e l d is

theory num-

field,

a

t h e o r y c a n s t i l l b e d e v e l o p e d however and t h a t i s d i s c u s s e d here. F i r s t t h e G e l f a n d t h e o r y for complex Banach a l g e b r a s i s reviewed

briefly;

t h e n t h e a n a l o g o u s t h e o r y f o r t h e case when t h e f i e l d c a r r i e s a nonarchimedean r e a l - v a l u e d v a l u a t i o n i s p r e s e n t e d . I n t h e c o u r s e o f t h e

l a t t e r d i s c u s s i o n , a S t o n e - W e i e r s t r a s s t h e o r e m is needed. I n t h e l a s t p a r t of t h e p a p e r some versions of t h e S t o n e - W e i e r s t r a s s

theoremwhich

h o l d i n a l g e b r a s of c o n t i n u o u s f u n c t i o n s over f i e l d s w i t h n o n a r c h i medean v a l u a t i o n are d i s c u s s e d .

1. CLASSICAL GELFAND THEORY.

If

I,

G i s an open s u b s e t o f

a t o p o l o g i c a l vector s p a c e , a map

[9

1

C , t h e complex numbers,

x:G + X

333

and

X

is

is a n a l y X h i n G i f t h e

NARlCl and BECKENSTEIN

334

d i f f e r e n c e q u o t i e n t has a l i m i t a t each p o i n t i n

G.

For t h e v e c t o r - v a l u e d v e r s i o n o f L i o u v i l l e ' s theorem t o

hold,

t h e v e c t o r s p a c e must have a good s u p p l y o f c o n t i n u o u s l i n e a r f u n c t i o n a l s . The d u a l s p a c e X' must be t o t a l i n t h e s e n s e t h a t i f e v e r y vanishes a t

f E X'

x , then x must be

1.1. LIOWILLE'S THEOREM:

0.

16 X LA a TVS

and

X'

i A

totae then

, p.211).

([l1

x : & + X in e n t i h e and b o u n d e d , t h e n x mu4.t be c o n h t a n t .

id

F o r t h e remainder o f t h e r e s u l t s i n t h i s s e c t i o n w e a s s u m e t h a t X is a complex commutative Banach a l g e b r a w i t h i d e n t i t y e

A complex number

A

v e r t i b l e . The s e t

06

i s a hegueah p a i n t

if

x E X

x

-

(11 e 11 =1). X e is i n

-

p ( x ) of r e g u l a r p o i n t s o f

over t h e r e s o l v e n t map

rx : p ( x )

X,X

+

x i s an open set.More-1 ( x -Xe) i s a n a l y t i c , "11,

+

p. 2 0 8 ) . An i m p o r t a n t consequence o f t h e s e r e s u l t s i s :

1.2.

GELFAND - M A Z U R THEOREM ( [ i ] , p.

bpecthum

u(x)

06

212):

x, t h o b e compgex numbetrn

( a ) F o t evehy A

doh

which

x E

x,

the

-

he

ib

x

n o t i n v m t i b e e , i b n o t empzy. ( b ) 16 X i n

a d i v i n i a n a l g e b t a [ a l l nanzeto elementn have i n -

v e t b e d ) t h e n X i6 i n o m o t p h i c and i n o m e t h i c t o

PROOF:

(b) Since

a(x) #

@, x

Since X i s a d i v i s i o n algebra,

-

Xe x

-

&.

i s n o t i n v e r t i b l e for sore A Xe

must be

E

Q.

0 , i . e . x = Xe.

The proof of p a r t ( a ) depends h e a v i l y on t h e L i o u v i l l e theorem. Consequently one would s u s p e c t t h a t t h i s r e s u l t would

not

transfer

e a s i l y t o Banach a l g e b r a s o v e r o t h e r f i e l d s , and i n d e e d t h i s i s c a s e . Even i n r e a l Banach a l g e b r a s t h e r e may be e l e m e n t s w i t h

the empty

spectrum. A s l o n g as t h e u n d e r l y i n g f i e l d i s Q , however, wecan obtain v e r s i o n s o f t h e above r e s u l t f o r l o c a l l y convex Hausdorff and l o c a l l y m-convex a l g e b r a s ( [ lI

,

p . 212

- 3).

algebras

The o n l y change that

336

BANACHALGEBRASOVERVALUEDFIELDS

o c c u r s i s t h a t t h e " i s o m e t r y " of p a r t ( b ) i s r e p l a c e d by "homeomor phism". For a t i m e i t was wondered ( [ 6 1 ) l o g i c a l d i v i s i o n a l g e b r a s o t h e r t h a n Q.

i f t h e r e w e r e complex top.

1 ,[ 1, p. 2141)

Williamson "12

showed t h a t t h e r e were by p r o v i d i n g an a l g e b r a i c a l l y c o m p a t i b l e

t with

Q(t) of r a t i o n a l f u n c t i o n s i n

pology f o r t h e f i e l d

-

to-

complex

coefficients. ( b ) above i s t h a t

An i m p o r t a n t consequence o f maximal i d e a l M of

X

. We

(I:

for any

d e n o t e t h e c o s e t (complex number) x + M by

x(M). I t now becomes p o s s i b l e t o view on t h e s p a c e M o f maximal i d e a l s of function

is

X/M

2 which s e n d s M i n t o

X as a collectionoffunctions X

.

We a s s o c i a t e x

x(M).Once M h a s been

X as a c o l l e c t i o n

it becomes p o s s i b l e t o view

of

X with

the

topologized,

c o n t i n u o u s func-

t i o n s mapping M i n t o Q. Among o t h e r t h i n g s , even w i t h o u t

endowing

M w i t h a t o p o l o g y , i t now f o l l o w s t h a t

a(x)

1.3

= %(MI.

I n a l g e b r a s o f c o n t i n u o u s o r a n a l y t i c f u n c t i o n s ( [ 1 1 , p.202-3) c h a r a c t e r i z a t i o n s such as 1 . 3 are t h e r u l e f o r d e s c r i b i n g

spectra,

i . e . , t h e spectrum of a f u n c t i o n x i s i t s r a n g e . We endow M w i t h t h e weakest t o p o l o g y which w i l l make each t h e maps

Z

c o n t i n u o u s and c a l l t h i s t h e Gelband t o p o l o g y .

of then

M

becomes a compact Hausdorff s p a c e .

i s a Banach a l g e b r a w i t h i n v o l u t i o n s a t i s f y i n g t h e

A B*-algebra

condition

II x* x 11 = I1 x 11 2 . The c e l e b r a t e d r e p r e s e n t a t i o n theorem

of

Gelfand and Naimark states:

1.4. REPRESENTATION OF B*-ALGEBRAS ( 1 1

algebta, t h e n X

06

CVfltiflUVUb

maximal i d e a &

i b

1

, p.

259f.

:

iboaethically i ~ o m a ~ p h ti oc t h e algebha

c o m p l e x - v a l u e d dunctiond on t h e compact 06

16 X i b a

X with

bUp

nahm ( a n d p V i n t w i b e

bpaCC

VpehUtiVnb).

C(M

U

B*I

Q)

04

NARlCl and BECKENSTEIN

336

2.

GELFAND THEORY OVER VALUED F I E L D S

Here w e assume t h a t

X i s a commutative Banach

algebra

with

i d e n t i t y o v e r a f i e l d F where t h e norm on X and t h e v a l u a t i o n on F each s a t i s f y t h e s t r o n g IIx + y 11 5 max (Ilx 11 t y a r e t h a t if

,

("nonarchimedean")

II y I1 )

. Among

IIxII ZIlyll , t h e n

triangle

inequality:

t h e consequences of t h i s i n e q u a l i IIx+ yII =max (Ilxll

, IIy 1 1 )

and t h a t

e v e r y p o i n t i n a s p h e r e i s a c e n t e r . A l l norms and v a l u a t i o n s areassumed r e a l - v a l u e d .

A d e t a i l e d d i s c u s s i o n of such normed s p a c e s

a l g e b r a s can be found i n

[lo ] ,

and

such s p a c e s b e i n g c a l l e d n o m c k i m e d u n

hpaces, The c r i t i c a l r e s u l t ( ( 1 . 2 ) ) t h a t

each

e l e m e n t have

nonempty

spectrum f a i l s t o h o l d f o r nonarchimedean a l g e b r a s . There may b e e l e ments w i t h empty spectrum ( [ l o ] , p. 1 0 5 ) . The w o r s t consequence t h i s is t h a t w e c a n n o t s a y t h a t

X.

X/M

X/M

i s merely a s u p e r f i e l d of

F.

of

i s F f o r e a c h maximal i d e a l of If we hypothesize

separately

t h a t e a c h element have nonempty spectrum t h e n , e x a c t l y a s i n p r o o f o f (1,2) ( b ) , d i v i s i o n a l g e b r a s are i s o m e t r i c a l l y i s o m o r p h i c t o t h e

d e r l y i n g f i e l d . We d e f i n e a Geldand atgebha t o be

a

commutative Banach a l g e b r a X w i t h i d e n t i t y such t h a t each maximal i d e a l

M of

un-

nonarchimedean X/M = F

for

X.

Although w e c a n n o t show t h a t each e l e m e n t h a s n o n e m p t y s p e c t r u m i n an a r b i t r a r y nonarchimedean Banach a l g e b r a , w e c a n show f o r any x that

u ( x ) i s c l o s e d and bounded, t h e proof b e i n g a b o u t t h e same

f o r t h e complex c a s e

([lo] ,

p . 114). Thus i f

e a c h e l e m e n t h a s compact spectrum. A l s o ( c f . true that

u ( x ) = G(M)

F

is locally

(1.3) 1 i t i s

as

compact, generally

fl F.

I n an a t t e m p t t o d u p l i c a t e t h e complex Gelfand t h e o r y , w e wish t o i n t r o d u c e a t o p o l o g y t o t h e maximal i d e a l s . Two main c h o i c e s

are

a v a i l a b l e : R e s t r i c t c o n s i d e r a t i o n of what e l e m e n t s x are t o b e chosen

or c o n s i d e r o n l y c e r t a i n maximal i d e a l s . More s p e c i f i c a l l y we consider

([lo] , p . 1 1 7 f . l :

337

BANACH ALGEBRASOVER VALUED FIELDS

2.1.

THE GELFAND SUBALGEBRA

maximal i d e a l M I x(M)

2.2.

9: or

F

E

THE GELFAN'D IDEALS

f o r every

X

x

Those

E

such t h a t f o r

X

every

Those maximal i d e a l s M such that x(M) E F

:

Mg

x.

I n t h e f i r s t c a s e w e r e t a i n a l l t h e M ' s ; i n t h e s e c o n d , a l l the x's.

I t now f o l l o w s t h a t ( a ) f o r each

M E M, M 0 X

9 = F); ( b ) X = X

(i.e.l X / M n X

i s a Gelfand

iff M = M 4 9 g (X i s a Gelfand a l g e b r a i f f e a c h maximal i d e a l i s a Gelfand i d e a l or

(maximal) i d e a l i n X

9

9

X c o i n c i d e s w i t h i t s Gelfand s u b a l g e b r a ) ;

gebra o f

(c) X

g

is a closed subal

-

X.

W e may now c o n s i d e r t h e f o l l o w i n g t o p o l o g i e s .

Define t h e w c a k G e L d a n d Z o p o L o g y

2.3. THE WEAK TOPOLOGY: weakest topology f o r

M such t h a t each

i n d u c e s t h e weak Gelfand t o p o l o g y on

2.4.

x

E

X

4

t o be t h e

i s continuous.

Mg'

Define t h e s t t o n g GcLdand t o p o l o g y

THE STRONG TOPOLOGY:

t h e weakest t o p o l o g y f o r M

g

This

t o be

such t h a t e v e r y X E X i s c o n t i n u o u s . T h i s is

c l e a r l y s t r o n g e r t h a n t h e weak Gelfand t o p o l o g y .

REMARKS: M

9

( a ) S t r o n g t o p o l o g i e s y i e l d s p a c e s w i t h more s t r u c t u r e . (b)

i s g e n e r a l l y n o t b i g enough t o y i e l d i n f o r m a t i o n a b o u t

t h e Gelfand i d e a l s

M

of

X

X whereas

are r i c h enough t o h e l p d e s c r i b e

X

g' a ( x ) = B ( M ) . ( c ) These t o p o l o g i e s a r e unigg 9' a r e complete. Thus M o r M i s compact f o r m i t i e s and M and M g 99 9 gg i f f t h e y are t o t a l l y bounded.

e.g.

if

x

E

X

99

g

then

The l a s t remark h e l p s t o o b t a i n t h e f o l l o w i n g compactness sult.

re-

NARlCl and BECKENSTEIN

338

2.5.

COMPACTNESS

and

Mgg

([lo],

7 6 F in eocaU?y c o m p a c t t h e n

p. 124):

ahe n t h u n g e y c o m p a c t . Convehnek?y i d

Mgg

11X

A4

oh

h t h o n g l y compact, t h e n e i t h e t F in L o c a l L y compact o h t h e 06

any element i n X

2.6.

9

i b

in 0 - d i m e n n i a n a L and each

ad t h e npaeen

Each

06

npecthum

t h e Geldand t o p d O g i 5 5

Mg' Mgg'

M

1 3

t o p o C o g y i b t o t a L C y d i b c o n n e c t e d and Haundoh66.

([lo],

in

9

n o n e m p t y , compact, and nowhehe d e n b e .

DISCONNECTEDNESS ([lo], p . 1 2 5 ) :

2.7. SEPARATION

Mg

p. 1 2 6 ) :

X

4

i n fie nfivng

T h e d o L l o ~ i n 9n t a t e m e n . t n a t e e q u h a -

Cent. (a) The (weak1 Geldand t a p o l o g y on M ( b ) T h e dunctionn dhom

X

(c) The dunctiono dhom X ( d ) The map M + M n X

g

in

g g

i n Haundohdd.

nepahate p a i n t n .

nepahate pointb

hth0Mgly.

1-1.

Maximal i d e a l s must a l w a y s be o f codimension 1. C o n v e r s e l y , i n

I , Gleason p r o v e d t h a t a l i n e a r s u b s p a c e o f codimension

1

in a

complex commutative Banach a l g e b r a w i t h i d e n t i t y i s a maximal

ideal

I 5

i f f i t c o n s i s t s o f s i n g u l a r e l e m e n t s . Hence,in a nonarchimedean Banach a l g e b r a , one might c o n s i d e r t h e q u e s t i o n : I f

M i s a l i n e a r subspace

o f codimension 1 c o n s i s t i n g s o l e l y o f s i n g u l a r e l e m e n t s , must M b e a G e l f a n d i d e a l ? The f a c t t h a t G l e a s o n ' s

argument uses d e e p theorems

from complex v a r i a b l e t h e o r y g i v e s warning t h a t

the

nonarchimedean

q u e s t i o n c o u l d be d i f f i c u l t . In [ 2 ]

t h e a u t h o r s c o n s i d e r e d G l e a s o n ' s q u e s t i o n i n t h e topo-

l o g i c a l a l g e b r a (endowed w i t h t h e compact-open t o p o l o g y ) C (T,F) c o n t i n u o u s f u n c t i o n from a t o p o l o g i c a l s p a c e T i n t o

a

of

topological

f i e l d F . I t i s shown t h e r e t h a t G l e a s o n ' s r e s u l t i s t r u e i f F i s t h e f i e l d of complex numbers, f a l s e i f

F i s t h e reals, a n d t r u e i f F i s

a n u l t r a r e g u l a r f i e l d c o n t a i n i n g a t l e a s t t h r e e p o i n t s under a n y t h e following conditions.

of

BANACH ALGEBRAS OVER VALUED FIELDS

339

1. F

i s n o t a l g e b r a i c a l l y closed.

2. F

p o s s e s s e s a s e q u e n c e o f d i s t i n c t e l e m e n t s converging to 0.

3. F

i s d i s c r e t e l y valued.

4 . The t o p o l o g y of

i s g i v e n by a v a l u a t i o n .

F

is ultranormal.

5. T

2

W e s a y t h a t a Gelfand a l g e b r a i s fiegulah i f t h e f u n c t i o n s s e p a r a t e p o i n t s and closed subsets of

2.8.

REGULAR:

X

i d

M strongly.

f i e g u l a f i i d 6 t h e I w t a k l Geldand t o p o l o g y

c o i n c i d e n w i t h t h e h u l l - k e f i n e l t o p o l o g y on M .

( I10

1

,

on

M

p. 1 3 5 ) .

I n t h e complex case, X i s r e g u l a r i f f t h e h u l l - k a r n e l t o p o l o g y

i s Hausdorff and t h e p r o o f r e l i e s h e a v i l y on t h e compactness o f M i n t h e Gelfand t o p o l o g y . By c!ioosing nonarchimedean a l g e b r a s i n which M

i s n o t compact, one o b t a i n s c o u n t e r e x a m p l e s t o ' i f t h e X is regular".

topology is Hausdorff, t h e n

U be the u n i t b a l l i n

Let

each maximal i d e a l M 1 . that

U C W. If

hull - k e r n e l

X and l e t

S i n c e II x(M) 11'

U = W, w e c a l l

II x I1

W = { x I Ilx(M)II

f o r every M

I

5

it i s clear

X a v*-aLgebaa.As w i l l ba seen shortly,

t h e V*-algebras are t h e nonarchimedean a n a l o g s o f B*-algebras (2.10)).

for

1

I t i s e a s y t o v e r i f y ([lo 1

I

(

see

p . 1 4 8 ) t h a t V*-algebras m u s t be

semisimple.

2.9.

16

T i n a 0 - d i m e n n i o n a e compact Haubdofid6 Apace and F in com-

p l e t e t h e n T in homeomofiphic t o t h e n p a c e M a d maximal C(T,F) undeh t h e map

t

+

Mt

S

i d

06

= { x E C(T,F) I x ( t ) = 0 ) urhefl \#i c~hhieA

t h e GeLdand t o p o l o g y . A C A ~ ,C ( T , F ) L A a V * - a t g e b k a

a d d i t i o n , id

idcaln

( [ l o ] , 9. 1 5 4 ) . I n

0 - d i m e n h i a n a l , compact and Haubdok.d6 t h e n S 0 ho-

meomohphic t o T id6 C(S,F)

i b

ibomofiphic t o

C(T,F).

As a f i r s t r e p r e s e n t a t i o n t h e o r e m w e have 2 . 1 0 . ( [ 1 0 ] , ~ . 164)

16

Xg i b a V*-Gd6and

aegebfia and

Mg

in compact

NARlCl and BECKENSTEIN

340

then X

9

in i n o m e t h i c a L L y i n o m o h p h i c t o

t o w n 2ha.t id X

C(Mg

I

dhom w h i c h it

p),

a V*-Gebaand a t g e b t a i n w h i c h

X id

in idorne-th.icaley i n o m o h p h i c t o

bl

601-

i d compact t h e n

C(M,F).

F o r t h e p r o o f of ( 2 . 1 0 ) one n e e d s a version of

a Stone-Weierstarss

t h e o r e m f o r a l g e b r a s of c o n t i n u o u s f u n c t i o n s which t a k e v a l u e s i n

a

nonarchimedean v a l u e d f i e l d . Such t h e o r e m s a r e t h e s u b j e c t o f t h e n e x t and l a s t p a r t o f t h e p a p e r .

3 . STONE-WEIERSTRASS THEOREMS

F d e n o t e s a f i e l d w i t h nonarchimedean v a l u a t i o n . G e n e r a l i z i n g a r e s u l t of Dieudonn6 ([ 4 ] ) , K a p l a n s k y ( [ 7 1 1

ob-

t a i n e d t h e f o l l o w i n g a n a l o g of t h e c l a s s i c a l S t o n e - W e i e r s t r a s s t h e o -

rem.

3.1.

KAPLANSKY-STONE-WEIERSTRASS

THEOREM:

([ 7

1,

i n a compact Haundohdd n p a c e and Y a nubaegebha & a t e 4 p o i n t s and COntainb C O n b . t U M f b t h e n

[

10, p . 162 ]

06

'Id T

:

C ( T , F ) w h i c h nepa-

Y i n dende i n

C(T,F).

An immediate c o n s e q u e n c e o f t h i s is

3.2. and

([

71,

[ 1 0 , p.

1631 1:

Y a bubatgebha

05

16 T i n a LocaLLy c o m p a c t Haundohbd n p a c e

C-(T,F)-continuoud

buncfionb which vanidh a t

i n d i n i t y - w h i c h n e p a t a t e n p a i n t d and c o n t a i n d conn.tan;tA then Y i n denbe in

Ca(T,F). A s h a s b e e n o b s e r v e d b y Nachbin

( [ a 1 ) , i t is

n o t r e a l l y neces-

s a r y t o c o n s i d e r s u b a l g e b r a s Y f o r S t o n e - W e i e r s t r a s s t y p e theorems: sub-modules s u f f i c e . T o q u o t e just one o f many p o s s i b l e i l l u s t r a t i o n s of t h i s

viewpoint ([ 3 ]

I

f o r example) w e h a v e t h e f o l l o w i n g r e s u l t of

P r o l l a ' s. 3.3.

( [ 111

,

Cor. 2.5):

Le-t T be a compact Haubdohd6

dpace,

X

a

341

EANACH ALGEBRASOVER VALUED FIELDS

n o n a h c h i m e d e a n nohmed b p a c e o v e h

F

whehLe A i n a n e p a h a t i n g n u b a e g e b h a Then i d

denne i n

and

Id a n A-bubmoduLe

06

W i d denne i n C(T,X)

06

C(T,x),

C(T,F). d o h eaclz

t i n T, V 7 ( t ) ={w(t)lwEW}

X.

REFERENCES

[ 11

E. BECKENSTEIN, L. N A R I C I a n d C .

SUFFEL, T o p o L o g i c a L A l g e b h a b ,

North-Holland P u b l i s h i n g Co., [ 21

E. BECKENSTEIN, L. NARICI,

C.

Amsterdam, 1977.

SUFFEL and S . WARNER,

Maximal

ideals i n algebras of c o n t i n u o u s f u n c t i o n s , J. Anal. Math. 31(1977) , 293

[

31

[ 41

- 297.

R. C . BUCK, A p p r o x i m a t i o n p r o p e r t i e s of vector - v a l u e d

t i o n s , P a c i f i c J. Math. 5 3 ( 1 9 7 4 ) , 85

J. DIEUDONNE, S u r l e s f o n c t i o n s c o n t i n u e s p - a d i q u e s , Math. 6 8 ( 1 9 4 4 ) , 79

51

[ 61

A.

- 95.

func-

Bull.Sci.

GLEASON, A c h a r a c t e r i z a t i o n of maximal i d e a l s , J.Anal. Math.,

vol. 1 9 ( 1 9 6 7 ) , 1 7 1

- 172.

I . KAPLANSKY, T o p o l o g i c a l r i n g s , B u l l . Amer. M a t h . SOC. 45(1948) 809

[ 71

- 94.

- 826.

I. KAPLANSKY, T h e Weierstrass t h e o r e m i n f i e l d s w i t h valuations, P r o c . Amer. Math. SOC. 1 ( 1 9 5 0 ) , 356 - 3 5 7 .

[ 81

L. NACHBIN, A p p h o x i m a t i o n T h e o h y ,

van Nostrand, P r i n c e t o n , l 9 6 7 .

R e p r i n t e d by Krieger P u b l i s h i n g C o . , n u e , H u n t i n g t o n , N. 91

6 4 5 New Y o r k

Ave-

Y., 1 9 7 6 .

M. NAIMARK,.Nohmk?d Ringd, N o r d h o f f , G r o n i n g e n , T h e N e t h e r l a n d s , 1964.

[lo 1

L. N A R I C I , E . BECKENSTEIN a n d G . BACHIVYW, FuncfhnrLt Aaa.tydi.6 and V a L u a t i o n T h e o h g , Marcel D e k k e r , N e w Y o r k , 1 9 7 1 .

NARlCl and BECKENSTEIN

342

[ll]

J . B.

PROLLA, Nonarchimedean f u n c t i o n spaces. T o a p p e a r

Birkhauser Verlag, Basel-Stuttgart,

[12]

J. H.

in:

Lineah S p a c u and A p p h a x i m a t i o n ( P r o c . Conf .,Oberwolfach, 1 9 7 7 ; E d s . P . L. B u t z e r a n d €3. S z . - N a g y ) , I S N M v o l . 4 0 , 1978.

WILLIAMSON, On t o p o l o g i s i n g t h e f i e l d C ( t ) Math. SOC. 5 ( 1 9 5 4 ) , 729 - 734.

,

Proc.

Amer.

Approdmation Theorg and Functional AnaZysds J.B. Prolla l e d . ) 0iVort.h-Holland PA Zishing Company, 1979

APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS

PHILIPPE NOVERRAZ U n i v e r s i t g d e Nancy I Ma t h g m a t i q u e s 5 4 0 3 7 NANCY CEDEX, F r a n c e

If

U

i s a n open and c o n n e c t e d s u b s e t o f

a , an a p p l i c a t i o n

s i o n a l l o c a l l y convex v e c t o r s p a c e E o n (resp. [ -

a,

+ 1

an i n f i n i t e

dimen

-

f :U+a

i s s a i d t o b e h o t o m o h p h i c ( r e s p . p.&.U&ubhahmonLc) i f

)

a)

f

b)

t h e r e s t r i c t i o n of

i s continuous ( r e s p . upper semicontinuous)

f

t o any f i n i t e d i m e n s i o n a l

subspace

i s holomorphic ( r e s p . plurisubharmonic). L e t us d e n o t e by

(resp. P ( U )

H(U)

,

P,(U))

t h e s e t o f holomor-

p h i c ( r e s p . p l u r i s u b h a r m o n i c , p l u r i s u b h a r m o n i c a n d c o n t i n u o u s ) funct i o n s on

U.

If

K i s a compact , s u b s e t of

= Ix E

(U)

In

an,

n

2

2,

1) Any v i n

u,

U

,

l e t u s d e n o t e by

v ( x ) 5 s u p v , wv E P ( U ) ) . K

t h e f o l l o w i n g r e s u l t s are w e l l known ( 3 ) : P(U) i s t h e p o i n t w i s e d e c r e a s i n g l i m i t

of

p l u r i s u b h a r m o n i c f u n c t i o n s i n a s t r i c t l y smaller o p e n

2)

(ie

U'

of

If

U i s pseudo-convex

compact

U

K of

Cm

set

d(U', C U ) > 0).

U)

then

(ie Kp(u)

343

Kp(U)

-

i s compact i n

Kpc(U)

.

U

f o r any

344

NOVERRAZ

If

U i s pseudo-convex,

compact s u b s e t of al,

..., a j

Iv

If

then f o r v i n

U there e x i s t

fll..

Pc(U)

. , f 7.

,

in

E >

0 and K

and

H(U)

p o s i t i v e numbers such t h a t

-

K = KH(U)

sup ai l o g I f i

i

i s compact i--a pseudo-convex open set

U,

then any holomorphic f u n c t i o n i n a neighborhoodof Kcan be a p p r o x i m a t d u n i f o r m l y on K by elements If

u

H(U).

-

A

and U' are pseudo-convex, U C U' t h e n K H ( U ) , = K H ( " , )

f o r any compact s u b s e t of in

of

U i f and o n l y i f

H(U') i s d e n s e

H(U) f o r t h e compact open topology.

P r o p e r t i e s 31, 4 ) and 5 ) have been g e n e r a l i s e d t o l a r g e r c l a s ses of l o c a l l y convex s p a c e s w i t h Schauder b a s i s i n c l u d i n g

Banach

spaces ( 6 ) .

8, c o n d i t i o n

W e s h a l l i n v e s t i g a t e c o n d i t i o n s 1 and 2 . I n

i s o b t a i n e d by r e g u l a r i s a t i o n ( i e c o n v o l u t i o n ) of

se-

v by a D i r a c

quencer so it i s n a t u r a l t o c o n s i d e r s o m e measure.

1)

For t h e sake

of

s i m p l i c i t y w e s h a l l c o n s i d e r h e r e only ( i n f i n i t e dimensional) Banach spaces and Gaussian measures f o l l o w i n g Gross ( 5 ) . I t i s w e l l known t h a t i n a Banach s p a c e E there are no

s t i t u t e t o t h e Lebesgue measure t h a t means t h e r e does n o t e x i s t

sub-

a

measure i n v a r i a n t by t r a n s l a t i o n s o r r o t a t i o n s . A Gaussian measure l.~ on E can be c h a r a c t e r i z e d as follows: there e x i s t s an H i l b e r t space H

v

d e n s e l y and c o n t i n u o u s l y imbeded i n E such t h a t

u

t h e c y l i n d r i c a l Gauss measure on t h e c y l i n d r i c a l s e t s of

arises H

1-I

.

from The

t r i p l e t ( H p , i , E ) is c a l l e d an a b s t r a c t Wiener space. The f o l l o w i n g p r o p e r t i e s hold:

1)

L e t be

T in

P(E,E), i f

and i s u n i t a r y t h e n

p

T restricted t o H

i s i n v a r i a n t by

T

i s i n P(H H ) I-r P I !J ( i e pT-' = 11).

APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS

L e t be

2)

clx(A)

= p(x

+

,A

A)

346

u

Bore1 i n E l t h e n

and p x are

e i t h e r e q u i v a l e n t o r o r t h o g o n a l , t h e y a r e e q u i v a l e n t if and only i f

x belongs t o

H

P

.

W e have t h e f o l l o w i n g Lemma:

16

LEMMA 1:

i n a Gaubbian meabutre on E and

p

hatrmonic 6uncLion i n an open n u b s e t U

doh

E we h a v e

Suppose t h a t v i s bounded from above i n t h e b a l l B ( x , r ) , t h e

mapping

x

+

eiex

- invariant,

but

a pLuhisub-

r bmale enough.

PROOF:

Te

06

i d v in

V(X)

induce& a

u n i t a r y mapping T e on H

P

I

so

u

is

and w e have

5

v(x

+

y e i e )do

.

The r e s u l t f o l l o w s from

Fubini

t h e o r em. L e t us note

P R O P O S I T I O N 1: 1)

A(v,x

A(x,v r )

p(r)A

2)

v(x) = l i m r =O

3)

A(x,v,r)

LA apLutribubhatrmonic d u n c t i o n 06

a c nwex and inctreasing dunc-tion

i n in6initely

any x i n E t h e f u n c t i o n y = 0.

and

0 6 Log r .

A(x,V,r).

L e t us r e c a l l t h a t a f u n c t i o n

entiable a t

x

y

+

cp

H

u - di66etrenZiable. 9 is H -differentiable

u

( x + y ) , d e f i n e d on

Hcl

I

if

for

is differ

-

NOVERRAZ

346

PROOF:

1) i s a consequence of t h e f a c t t h a t p l u r i s u b h a r m o n i c

func-

t i o n s depending o n l y from I1 x I) are l o g a r i t h m i c a l l y convex. S i n c e v i s upper s e m i c o n t i n u o u s , f o r any

2)

5 v(x) +

v ( x + y)

w e have

II y II 5 r X f E hence

for

E

> 0

E

Is a consequence of a r e s u l t of Gross (5).

3)

L e t us n o t i c e t h a t , u n l e s s

v i s continuous, A(v,x,r)

i n g e n e r a l a continuous f u n c t i o n of

is

not

x.

A s a consequence of 2 ) and 3 ) w e have:

A p l u t i b u b h a h m a n i c dunc-tion v

PROPOSITION 2:

wine l i m i t

a nequence

06

06

i-6

L a c a l l y ,the p o i n t -

i n , 5 i n i t d y H - d i d , 5 e h e n t i a b l e @~L5ubha/unonic

iuncztio nA . T h e r e i s a n o t h e r way t o a p p r o x i m a t e bounded f u n c t i o n s : l e t p be a Gaussian measure o f p a r a m e t e r

vt{ll x 11 2

c1

> 01

+

f u n c t i o n Ptf ( x ) = f Ptf

0

i,

if f (x

t

+

+

t > 0 , then

t h e n Gross ( 5 ) h a s proved t h a t

0

1.1

/f(x)

-

f

is uniformly continuous

f u n i f o r m l y on E .

tends t o

For

the

y)ut(dy) is i n f i n i t e l y H -differentiable if

i s bounded and m e a s u r a b l e . Moreover i f

PROOF:

t and

= 1

vt(E)

E

f(y)I 5

< 0, t h e r e i s E

.


0, t h e r e i s

,

i f g i v e n a s t r i c t compact sub-

such t h a t T E E* 8 E ,

K

C

EB

a n d i s compact i n

such t h a t

%(T(x)- x)

0,

E

K

and i s compact i n

T E E* Q E

f E JCS(U;F)

,E

> 0

such t h a t

EB

,

361

so t h a t

pB(T(x)

-

x)
0 , 6

i s t h e complement o f

d i s t ( K , C E B ( U 87 EB) ) (where C (U n EB) EB EB U n EB i n E B ) , s u c h t h a t B ( f ( x ) f(y)) < E,

whenever

pB(x

x E K

-

and

-

y) < 6. Since

( P r o p o s i t i o n 1.15), t h e n f o r e a c h

f l U n ~ B is

x E K,

continuous

is

there

6x

>

0,

A x 5 d i S t E ( K , C E (U n E B ) ) , s u c h t h a t B ( f ( X I - f (y)) < E/ 2 , for B B n pB(x y ) < 6 x . S i n c e K C U 17 EB i s compact i s EB , K C . L J B(xi,GXi), 1=1 f o r some s e t {x, xn} C K . ( B ( a , r ) = { x E EB; p B ( x a ) C r , when

-

,...,

and

a E EB

-

Define

r > 0)).

y(x) = sup { 6

Then

y :K

+

Now f o r any B(x,6)

C

B(xi,6

‘i

Since E has the

for a l l

x E K.

for all

x

Let

Uo =

-

pB(x

-

...,n }

xi);

i=l,

i s c o n t i n u o u s a n d y > 0. L e t

R

x E K

E

xi

),

and

y E B(x, 6 )

I

for

x E K.

6 = i n f { y ( x ) ; x E K).

there

is

some

i

with

thus

S.a.p.,

there is

T E E* Q E

such that ~ f , ( T ( x- )x ) c 6 ,

By the a b o v e , w e g e t t h a t

K. L e t

{gl

, .. . , gn}

be a b a s i s i n

T ( E ) and l e t

U n EB n T ( E ) . S i n c e f i s S i l v a - h o l o m o r p h i c ,

f

can

be

c o n s i d e r e d as a h o l o m o r p h i c mapping from t h e f i n i t e d i m e n s i o n a l balanced s e t

Uo

into

F,

PAQUES

n f(z) = f ( B

i=l

where (z,,

..., z n )

f

ECn,

subsets of Uo. S i n c e is

E

P

F

5

zigi)

=

z

IPl= 0

ZPf

P '

and c o n v e r g e n c e i s uniform on compact

T(K) C U

(1

EB

and i s compact i n

there

Uo,

M E IN, such t h a t

Thus, i f

x E K,

Since

t h e proof i s complete. NOW, w e g i v e a n e x t e n s i o n of t h e p r e v i o u s theorem

class of s u b s e t s of

2 . 2 DEFINITION:

t o be

Let

s a i d t o be d i n i t e d y S

2 . 3 REMARK:

E ,

If

U be a non-void open s u b s e t of

Pb(E;C)

- Runge

(Paques [ 111)

2 . 4 THEOREM:

E,

said

T ~ ) .U

is

i n E i f for e a c h f i n i t e dimensional sub-

Eo

*

i s a Banach s p a c e , t h i s d e f i n i t i o n c o i n c i d e s w i t h

t h e D e f i n i t i o n 2 . 1 of A r o n - S c h o t t e n l o h e r [ 2 open s u b s e t o f

is

E. U

is d e n s e i n ( J C s ( U ; ( c ) ,

i s S-Runge i n

U n Eo

E

another

E.

S-Runge in E i f

s p a c e Eo of

to

then

U

1

.

If

U

is

a

i s f i n i t e l y S-Runge and S-Runge

balanced in

E.

. L e t E have t h e

S.a.p.

and L e t

U

b e an o p e n

nubbet

THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS

06

Jfs(U;C)

E which i h 4 i n i t e d y S-Runge. Then

JCs(U; F)

604

Q

F

i b

383

T

S

-denhe i n

F.

e v e h y l k ~ c a e e y convex hpace

F o r t h e p r o o f o f Theorem 2 . 4 i t w i l l b e n e e d e d

the

following

p r o p o s i t i o n , which h a s i m p o r t a n t c o r o l l a r i e s .

2 . 5 PROPOSITION:

L e t F b e a dpace s a t i d d y i n g t h e doLeowing

tion: I d K i n a compact bubded

v e x huLL

06

F , t h e n t h e ceohed

06

(Pb(nE;F),

( a na lo g o u s 6ohmuLaA hold a b o h a t

f o r all

to

T : 3CS(U;F)

Let

+

f E XS(U;F),

I$

f o r each

JCs(U;C).

E

JCs(U;C) F ' and

E

U

Lo a nun-void

x

f E Jcs(U;F)

ous. Indeed, l e t

U.

and

Clearly,

n E IN).

doh

+

Xs(U;C)

seminorm on

K

x E K}, where

(Tf) ($)

belongs

$ E F'.

Tf : FA

p be a rS-continuous

p ( g ) = sup { Ig(x) 1 ;

-tS),

b e d e f i n e d by (Tf) ($1 ( x ) = ( $ of) (x),

F

W e now show t h a t t h e l i n e a r map

by

F. 7 6

abno.ecl*eLy con-

E , then

o p e n 6 u b d e t 06

PROOF:

06

r ( K ) , i n a compact n u b s e t

K,

C

U

is continu-

JCs(U;C)

defined

is a strict

compact

s e t . By h y p o t h e s i s , t h e closed a b s o l u t e l y convex h u l l o f compact s u b s e t o f f i n e d by

for all

fine

F.

C a l l it

$ E F'.

Hence

Now

E

f(K)

q b e t h e seminorm on

Let

F'

is a de-

L). I t f o l l o w s t h a t

Tf E Z ( F & ; 3 C s ( U ; C ) ) .

A E Xs(U;C)

g ( x ) E (FA)' = F

$ E F'.

L.

q ( $ ) = s u p { II$(t) I; t

L e t now

condi-

E F = L(F;,

JC,(U;C)).

by t h e formula

g is weakly S-holomorphic,

For each

x E U, de-

g ( x ) ( @ ) = ( A @ )( x ) ,

hence

S

for

- holomorphic

all by

PAOUES

364

C l e a r l y , Tq = A, a n d t h e r e f o r e T is onto Xs(U;C) E F .

P r o p o s i t i o n 1.20.

On t h e o t h e r hand, T

i s i n j e c t i v e by t h e Hahn-Banach Theorem.

r e m a i n s t o show t h a t

T i s a homeomorphism.

Let T(g) = s u p

6

I

E c s ( F ) and

Ig(x) I;

x

E

K C U

KI,

b e a s t r i c t compact

g E Jcs(U;Cl.

It

subset.

Let

t h e n , f o r every f EJCS(U;F),

we have by t h e Hahn-Banach Theorem, t h a t

This completes t h e p r o o f .

2.6 COROLLARIES OF THE PROPOSITION 2 . 5 :

nubnet a 6

a)

16 U i d

U

- void

nun

Open

E , we h a v e :

16

F i b a c o m p k t e bpace a n d

F oh (X,(U;C),

T

~

)hub

the

a p p t o ximatio n p ~ ~ o p e t t yt ,h e n

I n p a t t i c u l a t id E had d i n i t e d i m e n n i o n a n d F

i d

a com-

pLete d p a c e , t h e n

b)

16 F had t h e a p p t o x i m a t i o n p t o p e t t y a n d

condition

06

Pmpodition 2.5, then

JCs(U;C)

dadiddied

B F

the

d Ts-deue

i n XS(U;F). c)

(X,(U;C),

ill

T,)

Jc,(U;C)

npuced

F.

had 8 F

t h e a p p t o x i m a t i o n p k o p e t b y id a n d i d

Ts-denAe i n

JcS(U;F),

doh

only

a l e am&

THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS

366

The proof of a) f o l l o w s from P r o p o s i t i o n 2.5 and

( a ) . The

1.33

Proposition

proof of b ) f o l l o w s from P r o p o s i t i o n 2 . 5 a n d

Proposition

1 . 3 3 (b); and c ) f o l l o w s from P r o p o s i t i o n 2.5 and P r o p o s i t i o n 1 . 3 3 ( c ) and P r o p o s i t i o n 1 . 2 2 .

PROOF OF THEOREM 2 . 4 :

Let

be a s t r i c t compact s e t , B E c s ( F )

K C U

f E JCs(U;F). By h y p o t h e s i s , t h e r e i s

and

EB

,

so t h a t g i v e n

pB(T(x)

-

x)
0, there

E

for a l l

E,

B E BE

x

E

K,

U be a n o n

- void

apphoximatiofl phopehty,

( E * , T ~ ) i s a complemented

subspace

of

386

PAQUES

(Xs(U;C),T

:) hence E$ h a s t h e a p p r o x i m a t i o n p r o p e r t y . From

~

w e have t h a t , i f a

for

f E XS(U;C),

t h e mapping

U,

clear t h a t

Da : ( J c s ( U ; C ) ,

To show c o n t i n u i t y , l e t

a

Then t h e r e i s

E.

B E BE

6 > 0 , be s u c h t h a t

Let

-1

defined by Da(f) = 6 f ( a ) ,

T ~ +E;, )

i s a c o n t i n u o u s p r o j e c t i o n onto

D2 = Da.

s u b s e t of i n EB.

i s a q u a s i - c o m p l e t e s p a c e , t h e n E h a s t h e S.a.p..

E

For

this,

+

compact

K C EB and i s c o m p a c t

From Cauchy

C U n EB.

SK

Indeed, it is

K be a s t r i c t

such t h a t

a

E;.

in-

e q u a l i t i e s , ( C o r o l l a r y 1.18) i t f o l l o w s t h a t

for all

f

E Jcs(U;C).

Then

i s continuous.

Da

w e show t h a t E h a s t h e S . a . p .

NOW,

.

Since

Ei

has the

p r o x i m a t i o n p r o p e r t y , t h e n f o r e v e r y b a l a n c e d convex compact of

1 E

,

EZ

f o r e v e r y s t r i c t compact subset K o f

> 0, t h e r e i s

p E 1. S i n c e

g

E

g E (E;)

m g =

Since, f o r each

for of

Bi

B E

m

E

BE

U

Fo r

so t h a t

1=1

D E

BE

vi

.., m ,

vi

(EZ)', xi

E

E, a n d f o r

-

pIIK
0). Hence

PE

= Voo=V.

Sine

ED,

C

g =

I

f o r T E E*. ED J , ( I ) = I D i s a b a l a n c e d convex

$(T) = TI

V", where V i s a c l o s e d a b s o l u t e l y 0-neighborhood in ED.

( V = {v

where

(ED);

367

Z pi i=l

@ xi.

Hence,

Therefore

and t h e n

that is,

-

S i n c e , g E E* 8 E the

6

i s i n d e p e n d e n t of

i t f o l l o w s t h a t E has

S.a.p..

2 . 8 DEFINITION:

L e t E be a l o c a l l y convex complex Hausdorff s p a c e .

i s s a i d t o have t h e S - h o l a m o h p h i c a p p h o x i m a t i o n p h o p e h t y (S.H.a.p.1

E

K C E, a s t r i c t compact set, t h e r e i s

i f given K

and

C

EB

and i s compact i n EB and g i v e n

such t h a t

pB(g(x)

-

x)
0, there is g

x E K.

such t h a t E

JCs(E;&) B E

PAQUES

368

I t is clear t h a t i f

E has t h e

S.a.p.,

t h e n E h a s t h e S.H.a.p..

For t h e converse i t i s needed t h a t E be a quasi-complete space, t h a t

i s , w e have t h e f o l l o w i n g theorem, which c o n t a i n s t h e p r e v i o u s t h e o -

E, which i s f i n i t e l y S-Runge.

rem f o r an open s u b s e t U o f

2 . 9 THEOREM:

U b e an open

which i d h i n i t e l y S-Runge. Then t h e 6 o & l o w i n g conditionh

E,

d u b b e d 06

b e a q U a d i - C O m p l E t e d p a c e and l e t

Let E

ahe e q u i v a l e n t : a)

E

S.H.a.p..

b)

Foh eweny l o c a l l y convex d p a c e

had t h e

in c)

(xs(u;C),T

d)

E

)had

~

only i n

c)

+

+

E t o be a quasi-complete space i s

needed

d).

c) i s p a r t (c) o f C o r o l l a r y 2.6, which i s t r u e f o r

open s u b s e t of

E.

c)

+

d) i s Theorem 2.7.

remains o n l y t o show t h a t proof o f Theorem 2 . 1 , ( c f . D e f i n i t i o n 2.8)

2.10 COROLLARY:

S.a.p.

t h e a p p k o x i m a t i o n ptopekty.

S.a.p..

had t h e

The assumption of

b)

63 F i d -rs-dende

3ES(U;F).

REMARK:

PROOF:

F, JCs(U;C)

a)

+

Let

E

i 6 and o n l y id,

+

a ) i s obvious.

b ) . T h i s proof i s analogous

substituting

.

d)

g

E

for

HS(E;C) C3 E

be a q u a d i - c o m p l e t e d p a c e . Then 60k

each

n E IN,

(Pb(%;C),

-rs)

any It

t o the

T E E* Q E

E

had

had t h e

the ap-

pho ximation phopehty.

PROOF:

If

E has t h e

any open s u b s e t U of

S.a.p.,

i t f o l l o w s by Theorem 2 . 9 ,

E , which i s f i n i t e l y S-Runge,

h a s t h e approximation p r o p e r t y . S i n c e f o r each

n

E

that

(X,(U;C),

for T ~ )

1N, ( P b ( n E ; C ) ,rS

THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS

i s a complemented subspace of

( X S (U;(c), ' I ~ )

, we

have t h a t

369

(Pb(%;O),

'I~)

h a s t h e approximation p r o p e r t y . Conversely, i n p a r t i c u l a r , E* h a v i n g t h e a p p r o x i m a t i o n property, has the S.a.p.

E

i n t h e proof of Theorem 2 . 7 )

.

By t h e p r e v i o u s C o r o l l a r y , w e have t h a t

2.11 REMARK: quasi-complete S-Runge,

(as

s p a c e and U i s an open s u b s e t o f

El

if

E

a

is

which is f i n i t e l y

h a s t h e approximation p r o p e r t y , i f and only t h e n (Ws(U;C), T ~ ) n E IN,

i f , f o r each

(Pb("E;C), ' I ~ ) h a s t h e approximation p r o p e r t y .

REFERENCES

I1 1

R. ARON,

Tensor p r o d u c t s o f holomorphic f u n c t i o n s , Indag. Math.

35, (1973) I 1 9 2 [ 21

- 202.

R. ARON and M. SCHOTTENLOHER, Compact holomorphic mappings Banach s p a c e s and t h e Approximation p r o p e r t y , J. t i o n a l Analysis 21,

[ 31

[ 4

I

1

51

(1976) , 7

- 30.

P . ENFLO, A counterexample t o t h e approximation p r o p e r t y Banach s p a c e , A c t a Math. 130 (1973) , 309 317.

-

A.

on

Func-

in

Phoduitd ten6o&ie& t a p o e o g i q u e d e t eApace6 n u c . t e a i h e 6 , Memoirs Amer. Math. SOC., 1 6 ( 1 9 5 5 ) .

GROTHENDIECK,

C. P. GUPTA, Malgrange theorem f o r n u c l e a r l y e n t i r e f u n c t i o n s o f bounded t y p e on Banach s p a c e . D o c t o r a l D i s s e r t a t i o n , U n i v e r s i t y of R o c h e s t e r , 1 9 6 6 . Reproduced by I n s t i t u t o de Matemgtica Pura e A p l i c a d a , Rio de J a n e i r o , B r a s i l , Notas de Matemgtica, N Q 37 ( 1 9 6 8 ) .

[ 61

M. C. MATOS, Holomorphically b o r n o l o g i c a l s p a c e s and

infinite d i m e n s i o n a l v e r s i o n s o f H a r t o g s theorem, J . London Ma*. SOC. ( 2 ) 17 (19781, t o a p p e a r .

370

I 71

PAQUES

L. NACHBIN, Recent developments i n i n f i n i t e dimensional

holo-

morphy, B u l l . Amer. Math. SOC. 79 ( 1 9 7 3 1 , 6 2 5 - 6 4 0 . [ 81

In:

L. NACHBIN, A glimpse a t i n f i n i t e d i m e n s i o n a l h o l o m o r p h y ,

P h a c c e d i n g h o n ' I n , 3 i n i t e D i m e n d i o n a L Holomokphy, U n L v m i t y

0 6 Kentucky

1 9 7 3 , ( E d i t e d by T. L. Hayden and

T.

J.

S u f f r i d g e ) . L e c t u r e Notes i n Mathematics 3 6 4 , S p r i n g e r Verlag B e r l i n - H e i d e l b e r g - N e w York 1 9 7 4 , p p . 69 - 79.

I91

L . NACHBIN, TopoLogy o n S p a c e d 0 6 Holomo/rpkic M a p p i n g h , . E r g e b ~ s s e der M a t h e m a t i k und ihrer Grenzgebtete, B a n d 47, Springer

-Verlag New York I n c . 1 9 6 9 .

[lo ]

Ph. NOVERRAZ, P d e u d a - c v n v e x i t e , c a n v e x i t i i p o l y n o m i d e eA d o m d n u d ' h o L o m o h p h i e en d i m e n h i o n indinie, ca 4 8 , North-Holland,

[111

0. T. W.

Notas de M a t e m s t i -

Amsterdam, 1 9 7 3 .

PAQUES, P h o d u t o h t e n d o h i a i d d e dunqoe.4 Silva-hvlomok-

6ah

e a

p h o p h i e d a d e d e a p h o x i m a ~ i i a , Doctoral Dissertation,

Universidade E s t a d u a l de C a m p i n a s , C a m p i n a s ,

Brasil,

1977. [12 1

In: Analyhe , 3 v n c t i a n e l l e e t a p p l i c a t b n h (L. N a c h b i n , e d i t o r ) . Hermann, Paris,

D. PISANELLI, S u r l a L F - a n a l i t y c i t g . 1 9 7 5 , pp. 2 1 5 - 2 2 4 .

I131

J . B. PROLLA, A p p k o x i m a t i o n

06

Vectak Valued F u n c t i o n h ,

d e Maternztica 6 1 , N o r t h - H o l l a n d , [14]

L . SCHWARTZ, T h d o r i e des d i s t r i b u t i o n s

Notas

Amsterdam, 1 9 7 7 . valeurs

vectorielles

I , Ann. I n s t . F o u r i e r 7 ( 1 9 5 7 1 , 1 - 1 4 1 .

[151

M.

SCHOTTENLOHER, €-product a n d c o n t i n u a t i o n o f a n a l y t i c

map-

pings, I n : Anaeybe F o n c t i o n e l l e e t AppRicationn, (L. N a c h b i n , e d i t o r ) Hermann, P a r i s , 1 9 7 5 , p p . 2 6 1 - 2 7 0 . [161

J. S. SILVA, C o n c e i t o h

calmente

d e dunciio diddenenci&~eL em

COnULXVh,

L i s b o a , 1957.

C e n t ro de E s t u d o s

ebpacob

Matemsticos

lade

Approximation T h e o q and Functional AnaZyaie J . B . ProZZa (ed.) QNor th-Hc Z land Pub t i s h i n g Company, 19 79

THE APPROXIMATION PROPERTY FOR NACHBIN SPACES

JOAO B . PROLLA Depar tamento d e M a temstica U n i v e r s i d a d e E s t a d u a l de Campinas Campinas, S P , B r a z i l

1. INTRODUCTION Throughout t h i s p a p e r X i s a Hausdorff s p a c e s u c h t h a t C&(X;X)

(IK = I R o r

C)

s e p a r a t e s t h e p o i n t s of

X,

and

E

i s a non-zero locally

convex s p a c e . Our aim i s t o p r o v e t h a t c e r t a i n function spaces L C C(X;E) have t h e approximation p r o p e r t y as soon as E h a s t h e

approximation

p r o p e r t y . W e show t h i s f o r t h e c l a s s of a l l Nachbin s p a c e s C V m ( X ; E ) . Such s p a c e s i n c l u d e

C ( X ; E ) w i t h t h e compact-open t o p o l o g y ;

w i t h t h e s t r i c t topology:

, Bierstedt

that

CVm(X;IK)

that

X i s a completely r e g u l a r

[ 11

,

w i t h t h e uniform t o p o l o g y .

Co(X;E)

E = IK

v E V

When

u s i n g t h e t e c h n i q u e of E-products, had proved

h a s t h e approximation p r o p e r t y , under t h e h y p o t h e s i s k m - s p a c e , and t h a t t h e f a m i l y V o f

w e i g h t s i s such t h a t g i v e n a compact subset weight

Cb (X;E)

such t h a t

v(x)

1

for a l l

K C X, one c a n f i n d

x

a

E K.

The t e c h n i q u e w e u s e h e r e was s u g g e s t e d by t h e p a p e r

151

G i e r z , who proved t h e analogue o f Theorem 1 below f o r t h e c a s e of

of X

compact and b u n d l e s o f Banach s p a c e s . T h i s t e c h n i q u e of " l o c a l i z a t i o n " of t h e approximation p r o p e r t y was used by B i e r s t e d t , i n t h e c a s e t h e p a r t i t i o n by a n t i s y m m e t r i c s e t s ( B i e r s t e d t [ 2 1 1 , b u t

the

of

main

i d e a of r e p r e s e n t i n g t h e s p a c e o f o p e r a t o r s of L as a n o t h e r Nachbin s p a c e o f cross s e c t i o n s i s due t o G i e r z . However o u r p r e s e n t a t i o n is 371

372

PROLLA

much s i m p l e r , i n p a r t i c u l a r w e do n o t u s e t h e concept of a C (X)-convex

C ( X ) -module.

locally

I n t h e I n t r o d u c t i o n t o h i s paper, Gierz said

t h a t h i s method could be a p p l i e d t o t h e v e c t o r f i b r a t i o n s i n t h e sense of [ 8]

,

and t h i s l e d t o o u r e f f o r t a t s i m p l i f y i n g

his

proof

and

adapting it t o our context.

2. THE APPROXIMATION PROPERTY FOR NACHBIN SPACES A v e c t o h d i b h a t i o n o v e r a Hausdorff t o p o l o g i c a l space

p a i r ( X , ( F x ) x E X ) ,where each F,

i s a v e c t o r space over

X

the

is a field

IK (where K = IR or a ) . A c k o d b - ~ e c t i o nis then any element f o f t h e C a r t e s i a n product o f t h e s p a c e s A w e i g h t an

Fxl i . e .

X i s a f u n c t i o n v on

norm o v e r Fx f o r each L of c r o s s - s e c t i o n s

f

.

f = ( f (x)I x

I

X such t h a t

v ( x ) is a semi-

LVm i s a v e c t o r space

x E X. A Nachbin b p a c e

such t h a t t h e mapping

is upper semicontinuous and n u l l a t i n f i n i t y on X f o r each weight v

be onging t o a d i h e c t e d b e t V of weights ( d i r e c t e d means t h a t , given v1

, vz

E

v

V , t h e r e is some

( i = 1,2) f o r a l l

x

f

E

V

and

X > 0 such t h a t v i ( x ) 5 Av(x)

X); t h e space L is then equipped

with

the

topology d e f i n e d by t h e d i r e c t e d s e t of seminorms

and i t i s denoted by

LVa

.

S i n c e only t h e subspace w e may assume t h a t

L(x) = F,

L(x) = { f ( x ) ; f f o r each

x

C(X;IK)

L} C Fx i s relevant,

E X.

The C a r t e s i a n p r o d u c t of t h e s p a c e s F, C ( X ; M ) -module, where

E

h a s t h e s t r u c t u r e of a

denotes t h e r i n g of

all

continuous

THE APPROXIMATION PROPERTY FOR NACHBIN SPACES

i f we d e f i n e t h e p r o d u c t

IK-valued f u n c t i o n s on X I

Q E C(X;IK)

for a l l

and e a c h c r o s s - s e c t i o n

x E X. I f

W

C

373

f

Of

each

by

B C C(X;IK) is a

is a v e c t o r subspace and

L

for

s u b a l g e b r a , w e s a y t h a t W i s a B-module,

i f BW = { $ f ; $ EB, f

W ) CW.

E

W e recall t h a t a l o c a l l y convex s p a c e E h a s t h e a p p h o x i m a t i o n

p h o p e h t y i f t h e i d e n t i t y map e on E can be approximated,

uniformly

on e v e r y t o t a l l y bounded s e t i n E, by c o n t i n u o u s l i n e a r maps of f i n i t e rank. T h i s i s e q u i v a l e n t t o s a y t h a t t h e space

E ' @ E i s dense i n

L(E) w i t h t h e topology o f uniform convergence on

bounded s e t s of

E.

Let

Ec(E),

totally

c s ( E ) b e t h e s e t of a l l c o n t i n u o u s seminom

,

d e n o t e t h e spacz E s e m i P normed by p. I f , f o r e a c h p E c s ( E ) , t h e s p a c e E h a s t h e a p p r o x i P mation p r o p e r t y , t h e n E h a s t h e a p p r o x i m a t i o n p r o p e r t y . on E .

For each seminorm

THEOREM 1:

p

E cs(E)

Suppabe t h a t , d o h each

Fx equipped w&h

x E X, t h e bpace

{v(x); v E V l

hab

B c C b ( X ; I K ) be a b e l d - a d j o i n t

and

t h e t o p o l o g y dedined by t h e damily

t h e apphoximation p h o p e h t y . L e t

let E

06

beminohnb

b e p a h a t i n g b u b a l g e b h a . Then any Nachbin d p a c e

which

LVm

id

a

B-modute hab t h e apphoximation p h o p e h t y . The i d e a o f t h e p r o o f i s t o r e p r e s e n t t h e s p a c e W = LV,

being

,

a s a Nachbin s p a c e of c r o s s - s e c t , i o n s o v e r

XI

e(W),

where

each

fiber

L(W;Fx), and t h e n a p p l y t h e s o l u t i o n o f t h e Bernstein-Nachbin

a p p r o x i m a t i o n problem i n t h e s e p a r a t i n g and s e l f - a d j o i n t bounded case. B e f o r e p r o v i n g theorem 1 l e t us s t a t e some c o r o l l a r i e s .

COROLLARY 1: Fx

L e t X be a Hauddohdd b p a c e , and

604

each

be a nohmed b p a c e w i t h t h e apphoximation p h o p e h t y .

Cb(X;IK)

be a b e t i - a d j o . i n t and b e p a h a t i n g b u b a l g e b h a .

let

x E X

Let

B

C

374

PROLLA

L e t L be a v e c t o t s p a c e

x)

(X; (F,)

chodb

06

-Aectiand

pehtaining

to

nuch t h a t

x

f E L , t h e map

(1) doe evetry

+

Ilf(x)II 0 u p p a demicontinuoirn

and nuLL a t i n d i n i t y ;

i n a B-rnoduLe;

(2)

L

(3)

L(X) = F,

60%

x E

each

x.

Then L equipped w i t h nohm IIf 1 I = sup fIlf(x)lI; x E X I

had t h e

apphoximation p t o p e h t y . PROOF:

Consider t h e w e i g h t v on X d e f i n e d by

f o r each

II

f

x E X.

II = sup

REMARK:

Then

{ IIf ( x ) II ; x E

LVm

is

just

L

v ( x ) = norm of

equipped

with

FX’

norm

x).

From C o r o l l a r y 1 i t f o l l o w s t h a t a l l “ c o n t i n u o u s sums”,

t h e s e n s e of Godement [ 6

1 or

[7

in

1 , of Banach s p a c e s w i t h the approxi-

mation p r o p e r t y have t h e approximation p r o p e r t y , i f t h e X

the

i s compact and i f such a “ c o n t i n u o u s sum” i s a

“ b a s e space“

Cb(X;IK)

-module.

I n p a r t i c u l a r , a l l “ c o n t i n u o u s sums“ o f H i l b e r t s p a c e s and of C*-alg e b r a s , i n t h e sense of D i x m i e r and Douady [ 3 tion property, i f

1

have t h e approxima

X i s compact. Indeed, a ” c o n t i n u o u s sum“

sense of [ 3 1

is a

COROLLARY 2 :

Let X b e a Hauddohdd dpace buch t h a t

-

i n the

C ( X ; I I o -module.

k a t i n g ; L e t V b e a dikected b e t demicontknuoub dunctiand o n

X;

04

C b ( X ; x ) 0 bepa-

&eat-vaLued, n o n - n e g a t i v e , uppek

and l e t E be a lacuLLy convex pace

w i t h t h e apphoximation p h o p e h t y . Then C V m ( X ; E ) had t h e apphaxha.tLun pto pehty

.

PROOF:

By d e f i n i t i o n , CVm(X;E) = { f E C ( X ; E ) ;

finity, for a l l

v

€

vf

vanishes

at

in-

V), equipped w i t h t h e topology d e f i n e d

by

the

THE APPROXIMATION PROPERTY FOR NACHBIN SPACES

376

f a m i l y o f seminorms

where

v E V Let

and

p

E

denote

Lv

cs(E).

C V m ( X ; E ) equipped w i t h t h e topology d e f i n e d by

t h e above seminorms when either

or

Lv(x) = 0

by t h e seminorms

v

E

V

Lv(x) = E

{v(x)p; p

i s k e p t f i x e d . Then, for e a c h x E X , equipped with t h e topology defined

E CS(E)

1 . Hence i n b o t h c a s e s , L v ( x ) h a s

t h e approximation p r o p e r t y . I t remains t o n o t i c e spaces a r e

Cb(X;JK)-modules. T h e r e f o r e

property. Since

v E V

Lv

that

has

was a r b i t r a r y , C V m ( X ; E )

the has

all

Nachbin

approximation t h e approxima-

t i o n property.

COROLLARY 3:

(a)

Let X and E b e an i n CoaoLLaay 2 . T h e n

C(X;E)

w i t h t h e compact-open t o p o L o g y h a d t h e a p p h o x i m a -

t i o n phopehty. (b) C o ( X ; E )

N i t h t h e uni6oam t o p o L o g g had

the

appkoximation

pkopehtg.

REMARK:

I n ( a ) above, i t i s s u f f i c i e n t t o assume t h a t

C(X;IK)

is

separating.

COROLLARY 4 :

(Fontenot [ 4 1 )

A p a c e , and Let E

Let

X

b e a LocaLLy compact

be a L o c a L L y convex Apace w i t h

p a o p e h t y . T h e n c ~ ( x ; E )w i t h t h e n t a i c t t o p o e o g y

the

Haundoa66

appaoximation

B had t h e a p p k o x i -

m a tio n p h o p e h t y .

PROOF:

Apply C o r o l l a r y 2 , w i t h

COROLLARY 5:

Ale Nachbin spaced

V = {v E Co(X;JR);

06

v

0).

continuoun ncaLak-vaLu&d duncfiond

376

PROLLA

h a v e t h e apphoximation p h o p e k t y .

I n Corollary 2, take

PROOF:

E = IK.

3 . PROOF OF THEOREM 1

Let

W = LV,

Let

vo

E

For e a c h

and l e t and

V T

w

be a t o t a l l y bounded s e t .

be g i v e n .

> 0

E

J(W)

E

A C

c o n s i d e r t h e map

E ~ O T : W + F ,

for

x

E

X I where

for all

f

STEP 1:

sX o T

E

E~

:W

+

W.

E L(W;Fx).

Just notice t h a t

PROOF:

is t h e e v a l u a t i o n map, i.e., ~ ~ (= ff( x) ) ,

F,

E,

E C ( W ; F ~ ) #s i n c e

v ( x ) [ ~ ( x5 ) ~1 I f

f o r every

f o r any

v

U(x)

T E

f o r any

v E V.

T E C (W), c o n s i d e r t h e c r o s s - s e c t i o n

F o r each

and f o r each

IIv ,

E V

E

E o ~T)

c o n s i d e r t h e weight ? on X d e f i n e d by

C(W;Fx).

e(W).

?=(

Then

377

THE APPROXIMATION PROPERTY FOR NACHBIN SPACES

STEP 2:

x * ~ ( x ) I ? ( x ) ]i b uppetc . b e m i c o n t i n u o u b and vanishes

T h e map

at i n d i n i . t g o n X , d o & e a c h

PROOF:

Let

Choose

h"

xo E X

and

there exist

such t h a t

{ 1 , 2 , ...,rn}

x

x E Vi

Let

X. L e t

h')

. Then

6 > 0. Since

A

such t h a t , given

E

u

=

T(A) i s totally bounded, f E A,

there

is

such t h a t

+

v ( x )[

( T f i ) ( x )]

neighborhoods of

V2,...,Vm

for all

-

fl,f 2,...,fm

Since

in

h'

6 = 2(h"

Let

V1,

a n d assume

~ ( x o ) [ ? ( x o ) 0. L e t

-V

which i s t h e normal f r e q u e n c y f u n c t i o n

2m

cos vx d v ,

SCHOENBERG

388

if

m =1, o t h e r w i s e (m = 2,3,.

.

.)

Gm(x) i s an e n t i r e f u n c t i o n

having

i n f i n i t e l y many zeros, a l l r e a l . T h e caeddicients a d (1) satis6q t h e a s y m p t o t i c h e L a t i o n h

--

--

1 1 --1 a ( n ) = ( i n ) 2m Gm(v(hn) 2m ) + , o ( n 2m)

(4)

a4

V

where t h e " l i t t e e

n +

m ,

v.

or' dymbok? hoed4 unidaamly d o h n l L i n t e g e h d

For a proof see ( 5 , P a r t I ] , where i t i s a l s o shown byexamples (1.10),

t h a t ( 4 ) no l o n g e r h o l d s i f t h e e q u a l i t y s i g n i s a l l o w e d i n and t h a t t h e c o e f f i c i e n t n = 2k

aAn) d i v e r g e s e x p o n e n t i a l l y t o

t e n d s t o i n f i n i t y t h r o u g h even v a l u e s , i f

( 1 . 1 0 ) are r e v e r s e d anywhere i n t h e i n t e r v a l

the

0 < u < 2.rr

+

m

,

as

inequalities

.

The f o l l o w i n g d i s c u s s i o n , w h i l e n o t d i r e c t l y r e l a t e d

to

our

s u b j e c t of smoothing, w i l l show t h e c o n n e c t i o n of t h e a s y m p t o t i c rel a t i o n ( 4 ) w i t h t h e w i d e r f i e l d of p a r a b o l i c d i f f e r e n t i a l e q u a t i o n s . Observe t h a t ( 2 ) i m p l i e s t h a t

(5)

--1

--1

U(x,t) = t 2m G m ( x t 2m) =

-

-tv

2m

+ ixv

d v , ( t > 0).

The f u n c t i o n under t h e i n t e g r a l s i g n i s immediately s e e n t o s a t i s f y for all v

, the

d i f f e r e n t i a l equation

which r e d u c e s t o t h e f a m i l i a r h e a t e q u a t i o n i f

also -plane

m = l . I t follows t h a t

U ( x , t ) , d e f i n e d by ( 5 ) , is a s o l u t i o n of (6) i n t h e upper h a l f t > 0 . On t h e o t h e r hand, a p p l y i n g t o ( 2 ) F o u r i e r ' s inversion

formula and s e t t i n g

v = O r we f i n d that

ON CARDINALSPLINE SMOOTHING

These r e m a r k s imply t h e f o l l o w i n g : 'I 6

1x 1

say, a s

+

LA a b o t u , t i o n

a,

06

f (x)

389

cantinuow and a ( I X I - * )

,

then

t h e d i , j d e ~ e n . t i a l e q u a t i o n ( 6 ) Aattin6qLng t h e boundmy

condition

This p a r t i c u l a r s o l u t i o n

u ( x , t ) may now a l s o be

approximated

by t h e f o l l o w i n g n u m e r i c a l p r o c e d u r e : Draw i n t h e ( x , t ) - p l a n e

the

rectangular lattice of p o i n t s

(WAX,

n At)

(w

= 0, k 1

, ...

;

D e f i n e on it a l a t t i c e f u n c t i o n

n = 0,1,2

u

v

, ...) .

by s t a r t i n g w i t h

uw ,o = f ( v Ax) ,

and computing t h e v a l u e s a l o n g e a c h h o r i z o n t a l l i n e from those on the l i n e below i t , by means o f t h e t r a n s f o r m a t i o n (1.2). T h i s

evidently

amounts t o i t e r a t i n g (1.21, a n d a f t e r n s t e p s w e o b t a i n

(10)

For any g i v e n x a n d

t > 0,

( 1 0 ) w i l l go o v e r i n t o ( 8 ) i f w e

f o l l o w i n g : We 6 h A t c o n n e c t the. m e o h - n i z e b

Ax and

A t

do the

by ,the h d a t i a n

SCHOENBERG

380

A t = X (Ax) 2m.

(11)

Id t h e i n t e g e k b

n ahe buch t h a t

v and

VAX

+

x,

and

n A t

.+

t

an

Ax

0.

+

then

U

v,n

T h i s follows r e a d i l y from r e l . a t i o n (4): ( 1 0 ) d i f f e r s

U(X,t).

+

1 0 ) and ( 8 1 , i n view o f t h e

asymptotic

from a Cauchy-Riemann sum f o r tk integral

( 8 1 , by a q u a n t i t y t h a t t e n d s t o z e r o due t o t h e u n i f o r m i t y i n

v of

t h e error t e r m o f ( 4 ) . I t i s i n t e r e s t i n g t o n o t e t h a t it d o e s n o t matter which

for-

mula ( 1 . 2 ) w e u s e i n t h i s c o n s t r u c t i o n , as l o n g as it i s o f the degree of exactness

2m-1,

i.e.,

i t s a t i s f i e s (1.71, and above a l l t h a t i t

s a t i s f i e s t h e s t a b i l i t y c o n d i t i o n (1.10)

,

t h e t e r m "stabi1ity"meaning

h e r e s t a b i l i t y on i t e r a t i o n . F o r t h e g e n e r a l t h e o r y of F. J o h n , which t h e e q u a t i o n ( 6 ) i s a s p e c i a l example, see [ 3 1

.

of

I n t h i s s e c t i o n w e d e a l t e x c l u s i v e l y w i t h f o r m u l a e ( 1 . 2 ) which s a t i s f y t h e symmetry r e l a t i o n . I n [ 2 ]

T.

N.

E. G r e v i l l e d e a l t

with

t h e more d i f f i c u l t c a s e o f unsymmetric f o r m u l a e .

3 . CARDINAL SPLINE INTERPOLATION (see [ 9 , L e c t u r e s 1

l e m o f caadinal intehpolation i s t o f i n d s o l u t i o n s

-

4 1 ) . T h e prob-

f ( x ) of t h e i n -

t e r p o l a t i o n problem

(1)

f ( v ) = Y"

,

for all i n t e g e r s

v

,

where ( y v ) are t h e d a t a . A f o r m a l s o l u t i o n i s f u r n i s h e d b y t h e series

391

ON CARDINAL SPLINE SMOOTHING

i n v e s t i g a t e d i n 1 9 0 8 by de l a V a l l G e P o u s s i n , also l a t e r

by

E. T.

W h i t t a k e r , who c a l l e d i t t h e cahdinad b e h i e b . The d i f f i c u l t y w i t h ( 2 )

“i: :y

i s t h e s l o w decay o f t h e f u n c t i o n

as

x

-. A

+

s o l u t i o n of (1) i s t h e p i e c e w i b e l i n e a h i n t e h p o t h z t

much s i m p l e r g i v e n by

S1(x)

m

(3)

where

M2(x) i s t h e roof f u n c t i o n d e f i n e d by

in

M2(x) = x + l

,

[-1,01

M (x) = 1

2

-x

i n [ O , l l and%(x)

=o

The p u r p o s e o f cahdinad b p l i n e i n t e h p o & z t i o n i s t o b r i d g e between t h e p i e c e w i s e l i n e a r

if 1x1 ’1.

the

gap

S1(x) d e f i n e d by (31, a n d t h e c a r d i n a l

series ( 2 ) . I t a i m s a t r e t a i n i n g s o m e of t h e s t u r d i n e s s a n d s i n p l i c i t y of ( 3 ) , a t t h e same t i m e c a p t u r i n g some of t h e s m o o t h n e s s a n d s o p h i s t i c a t i o n of

(2).

Le-t m be a natuhad numbeh, and d e b

(4)

S2rn-l

b e t h e cLadb

06

= {S(X)3

cahdinad b p d i n e d

S(x)

0 6 deghee

2m-1

dedined

by

the two conditionb:

(5)

The h e s t h i c t i o n whete

v

i d

04

S ( x ) -to e u e h y u n i t i n t e n v a l

a n i n X e g e h , i b apolynomia!.

(v ,v

0 6 deghee 2

2m

+11,

-

1.

392

SCHOENBERG

For

m =1

we f i n d

S1

t o be i d e n t i c a l w i t h t h e c l a s s ( 3 )

c o n t i n u o u s p i e c e w i s e l i n e a r f u n c t i o n s . Observe t h a t t h e c l a s s o f p o l y n o m i a l s of d e g r e e s n o t e x c e e d i n g The r o l e o f t h e r o o f - f u n c t i o n t h e s o - c a l l e d centha.! B-npLine

M 2 m ( ~ ) : Waiting

SZmml c o n t a i n s 2m-1.

of (3)

M2(x)

of

x+

i s t a k e n o v e r by = max ( x , O ) ,

it

may be d e d i n e d b y

Clearly port

M2m(~)

€

S2m-l; w e also f i n d t h a t

M2m(~) > 0

i n its

sup-

- m < x < m. The B - s p l i n e s h o u l d be f a m i l i a r i n view of the fun-

damental i d e n t i t y

which a l s o shows t h a t

IM2,(x)dx

= 1 if

w e choose

f (x) = x

The r e p r e s e n t a t i o n ( 3 ) a l s o g e n e r a l i z e s , and eueAny S ( x )

2m E

. SZm-l

admitb a unique hepheoentation m

S(x) = c c

~

M*m(X

--m

whehe t h e

-

v)

I

c v ahe c o n n t a n t n . T h i s i s t h e s o - c a l l e d ntandahd heptebefl-

t a t i o n . The c o n v e r s e i s clear: Every series (8) f u r n i s h e s an e l e m e n t of

SZm-1

ments o f

.

W e now t r y t o s o l v e t h e i n t e r p o l a t i o n problem (1) b y e l e -

S2m-l.

I n t h i s d i r e c t i o n t h e r e are t w o d i f f e r e n t k i n d s o f

results.

A. T h e d a t a (y,) s e q u e n c e (y,)

ahe

06 poweh

g h o w t h (See [ 8

i s of p o w e h g n o w t h , and w r i t e

1). W e s a y t h a t t h e

ON CARDINAL SPLINE SMOOTHING

(y,)

(9)

393

E PG,

provided t h a t

y,

(10)

v

+

E

PG,

= ~ ( l v l y ) as

f

m,

f o r some

y

2

0.

y1

2

Similarly, we w r i t e

f(x)

(11)

provided t h a t

f ( x ) = O ( l x l y l ) as

x

+

f o r some

f m,

Below w e e x c l u d e t h e t r i v i a l c a s e when

m=l,

l e m i s s o l v e d by ( 3 ) w i t h o u t any r e s t r i c t i o n on t h e

THEOREM 1:

16 t h e heqUenCt

(y,)

i h

0.

s i n c e o u r prob

-

(y,,).

a d pawet g t u w t h , t h e n t h e i n t e h -

palation p t a b l e m

huh a u n i q u e h o l u t i o n

S(x)

huch t h a t

The a s s u m p t i o n ( 9 ) o f Theorem 1 i s a rough one; i t admits,e.g., a l l bounded s e q u e n c e s ( y V ) , w i t h

y = 0 i n ( 1 0 ) . The s e c o n d assump-

t i o n t o which w e now p a s s , i s much more s e l e c t i v e , and

takes

a c c o u n t t h e f i n e r s t r u c t u r e of t h e sequence: i n f a c t i t a d m i t s a narrow subclass of t h e s e q u e n c e s of

PG.

into only

As u s u a l , w i t h strongeras-

sumptions, s t r o n g e r c o n c l u s i o n s are p o s s i b l e : The i n t e r p o l a n t w i l l e x h i b i t a n i m p o r t a n t extremum p r o p e r t y .

S(x)

SCHOENBERG

394

m

B. T h e c a n e when

IAmyv12
r on H, gl(0) = O .

such t h a t

hl = 1 on

f l = hl o g l ,

f = h(gl,.

., g L )

. .,gL)

A andalso

h E C"(lR) such that h,O,

Then from Nachbin's Lenuna,there exist gl,...,gt

h(gl,..

all

for

H

i s a subalgebra of

s i n c e t h i s i s a c l o s e d s u b a l g e b r a . A l s o , fl

such t h a t

con-

h(gl,.

let

and

I n p a r t i c u l a r , f o r any

C m ( I R ) b e such t h a t

bourhood o f

IRn

i s bounded

approximates

i s p o s i t i v e on a neighbourhood of

hog

n e s s , there e x i s t

K.

without

h ( 0 ) = 0 . From the above remark, it follows t h a t

and

Let

gi

functions,

be t h e s e t o f r e a l p a r t s o f f u n c t i o n s i n A.

i s a s e l f - a d j o i n t a l g e b r a , t h e n A1

g E A1

k

1.

e R ,using

T.

f E CE(IRn)

s u p p o r t . L e t A1 A

Hence

p E P(lR )

a

h ( 0 ) = 0. Furthermore,

. . ,e.

E Corn

IRn } i s bounded i n

w e c a n approximate h on G by p o l y n o m i a l s s t a n t t e r m , since

h(0) = 0, m n

i s such t h a t

Cm(lR )

t h e W e i e r s t r a s s a p p r o x i m a t i o n theorem f o r d i f f e r e n t i a b l e

k

topology

A s s u m e a l s o t h a t C i s a s u b a l g e b r a . I n t h i s case,

7.

E C

S i n c e the set

on

c1

f

., g L )

on

A,

since

E

N o w t h e p r o o f i s complete.

z E C\lR

, let

gz

b e t h e complex f u n c t i o n on

W

FUNDAMENTAL SEMINORMS

g,(x) I t i s clear t h a t

PROOF:

g,

E

E = P(DUa. W e

In fact, for

m=O

claim t h a t

gp(IR) C E

for all

m E IN.

t h i s i s e v i d e n t . Assume t h a t t h e p r o p o s i t i o n

m E IN. L e t

t r u e f o r some

x E IR.

ci(IR).

-

Let

1 x - 2

=

437

p E P(IR). Since

-

q = gz(p

is

p ( z ) ) EP(IR),

t h e n from t h e a s s u m p t i o n i t f o l l o w s t h a t

Now t h e mapping

f E Ba

+

g f:

i s continuous hence

E Ba

S i n c e E i s a complex v e c t o r s p a c e w e h a v e

So t h e claim i s p r o v e d . F u r t h e r , t h e mapping

continuous, hence E is s e l f - a d j o i n t s i n c e for all

g:(;,)'

1

E

f

E

P(IR)

Ba

+

is.

'f

SO

E

(4,)

is

Ba

-.7E E -g,

I N : whence

E E:g

c g:

P(B)

a C

E

for all

m , n E IN.

From t h i s i t f o l l o w s t h a t t h e complex a l g e b r a A g e n e r a t e d by g, a n d

-g2

i s c o n t a i n e d i n E.

t i o n s (N) s i n c e

{gz}

Also

A

i s s e l f - a d j o i n t and s a t i s f i e s condi-

s a t i s f i e s c o n d i t i o n s (N).

From

Lemma

2

it

438

ZAPATA

P(IR) is d e n s e i n B a r t h a t i s ,

follows t h a t

LEMMA 4 :

1e.t

con6.tan.t

CzIZl

PROOF:

Let

p

,

a E SPC(IR)

z, z'

c

\ IR.

is fundamental.

T h e n t h e 4 e e x i s t 6 a pa4,itive

such ,tha.t

E

~ ( m ) .Since

gzp

i t follows

From t h e d e f i n i t i o n of

If

E

CI

r = ~ y r n z . t~h,e n

=

42 42

a, there exists

11 gzIlm =

Z

k-0

k! k+l

g Z I p = (1 + ( z

C > 0

=

cz

and m E IN s u c h t h a t

and

T o f i n i s h , i t i s enough t o o b s e r v e t h a t t h e number Cz

d o e s n o t depend on

PROOF:

Assume t h a t

- z')g2)gzlp

12'

= l + Iz-z'I CC,

p.

P a ( z ) i s unbounded. L e t

p E P(IR) be such t h a t

FUNDAMENTAL SEMINORMS

then

q

E P

(IR) and

q

-

gz =

.

g ZP

p(z)

By c h o o s i n g a c o n s t a n t C Z r i > 0

a s i n Lemma 4 i t follows t h a t

Since

P ( z ) i s unbounded, t h e n a

gz

and from Lemma 3

E P")-

is

c1

f undamen t a 1.

C o n v e r s e l y assume t h a t n

E

IN* be g i v e n . S i n c e

that

a(gZ

-

p)

a(gzq) = n a ( g Z

5

-

3.

pn

E

E

P(IR)",

q = n(l

Let

a(giq)

~ ( n, cl(gipn) ) 5

5

CirZ

1 and

-

there exists (x

-

.

pn(z) =

THEOREM 2 ( q u a s i - a n a l y t i c c r i t e r i o n ) :

a

PROOF:

on

Q: \

and

IR

p E P(IR)

Then

To f i n i s h w e l e t

unbounded.

then

ZIP).

z E

q

E

such

P ( I R ) and

i s a p o s i t i v e c o n s t a n t as is Lsrr

P) 5 1. If Ci,z

m a 4 it follows t h a t Then

gz

i s fundamental. L e t

c1

n . Hence 'i,z

Let

a

pn =

'i,z

P,(z)

is

E SPC(lR), Id

in 6undamenZaL.

Let

P(sX).

T b e a c o n t i n u o u s l i n e a r form on Let

D d e n o t e t h e s e t o f complex

Ba

s u c h t h a t T vanishes

numbers

such

that

on D. I n f a c t assuming t h i s ,

from

z

Imz < 1. D e f i n e h ( z ) = T ( g Z ) , z E D.

I t i s enough t o p r o v e t h a t

h =O

440

ZAPATA

Hahn-Banach a

-

g2 E P ( l R J a for a l l

theorem i t f o l l o w s t h a t

i s fundamental from Lemma 3 . Let

z E D, n E IN.

S i n c e T v a n i s h e s on is also t r u e f o r

If

n =O.

z, zo

E D, z

#

zo.

h(z) =

Hence

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

S i n c e II g 2 I l m 5 ( m + l ) !

Let

n 2 1 then

P(lRR) i t follows t h a t

From t h e d e f i n i t i o n of

for a l l

Then

gz

=

4,

From t h i s i t follows t h a t

h

i s holomorphic on D and

n=l

m E IN such t h a t

(2

-

zO)4,g2

0

.

i s holomorphic on D . Since

m

z

~ ( g ~ x " ?his ). zn

we have t h a t

z E D

-

and

C > 0

0

h

z E D. Then

n-

4

1

=

+

(*)

is t r u e ,

m,

a(x")

t h e n Denjoy c o n d i t i o n s i n Watson's problem are s a t i s f i e d , v a n i s h e s on

Hence

D ( [ 6 1 ) . N o w t h e proof i s complete.

hence

h

441

FUNDAMENTAL SEMINORMS

COROLLARY 1: t h e h e ahe

Let A be t h e

aLl neminohmb

d e t 06

p o n i t i v e conntantd

C

I

N , m E IN

and

c

a E SPC(IR) doh

which

(ddepending o n a)

A U C ~t h a t

... - log,

a ( x n ) 5 c(c n log n whehe log,

doh

log, n = n and

dedined b y

i d

n) n

aLL log,"

n 2 N = l o g ( 1 o gm - l n)

.id

m 2 1. Then A

PROOF:

s e t 06

id a dihected

6undamentaL neminohmd.

T h i s i s a d i r e c t consequence o f Theorem 2 o b s e r v i n g t h a t t h e

"moments" o f any t w o such seminorms have a common e s t i m a t e of the sirme type

a

Let

THEOREM 3:

SPC(IRn). 1 6 t h e k c e x i d t 6undamentaL

E

JemiMohmd

~ 1 ~ , . . .E~ SPC(IR) a ~ duch t h a t

a(fl then

for

... B

f n ) 5 a 1( f 1

...

*

an(fn)

aee

doh

flI . . . I f n E B t

6undamenXaL.

a i d

PROOF:

(9

Let

n :B x . . . x Ban a1 f l l . . . l f n E B. Then

-+

Bnta

be defined by m(fl

,...,f n) =f,@ ...@ f n ,

i i s fundamental and .rr i s c o n t i n u o u s . Hence i f t h e complex s u b a l g e b r a g e n e r a t e d by

s i n c e each

a

T(C;(IR)

then

x

-a

A C P(IRn)

...

x

C;(IR))

. Since

A

viz

A = C ~ ( I R Io R)

A

... o C ~ ( I R )

is a s e l f - a d j o i n t s u b a l g e b r a o f

I

C:(IRn)

is

442

ZAPATA

and a l s o s a t i s f i e s c o n d i t i o n s (N) , from Lemma 2 i t f o l l o w s is dense i n

Hence

Bn,cl.

a

that

A

i s fundamental.

4 . OPEN PROBLEMS

1.

2.

Give i n t e g r a l c r i t e r i a l i k e t h o s e i n [ 7 ]

for characteriz-

i n g fundamental seminorms on

IR.

Under what c o n d i t i o n s on

SPC(IR) i s i t t r u e t h a t a i s

fundamental i f and o n l y i f

a

E m

L:

i=ly a ( x n )

= + a ?

3.

If

CY

E SPC(lR) i s n o t f u n d a m e n t a l , d e s c r i b e

4.

If

a E SPC(IR) i s n o t f u n d a m e n t a l , a r e t h e r e p o s i t i v e con-

s t a n t s c , C such t h a t f o r a l l

z 5.

E C

we have

p E P(IR) , a ( p )

a

SPC(IR) i s i t t r u e t h a t

fundamental i f and o n l y i f t h e s e t { p

6.

the

E

a is

P ( I R ) , a ( p ) (1) i s

s p a c e o f e n t i r e f u n c t i o n s on

Q:?

Give a c h a r a c t e r i z a t i o n o f fundamental seminorms o n

n 7.

in

and

Ip(z) 1 5 C e C I Z I ?

Under what c o n d i t i o n s on

unbounded

5 1

IRn

,

2 2.

I s t h e set of a l l fundamental seminorms on

Same on

R

directed?

R”?

REFERENCES

[ 11

N.

AKIEZER,

On t h e w e i g h t e d a p p r o x i m a t i o n o f c o n t i n u o u s

func-

Amer. t i o n s by p o l y n o m i a l s on t h e e n t i r e number a x i s , Math. SOC. T r a n s l a t i o n s , S e r i e s 2 , v o l . 22 (1962) , 95 - 138. [ 21

S . BERNSTEIN, Le problgme d e l ‘ a p p r o x i m a t i o n d e s f o n c t i o n s con-

t i n u e s s u r t o u t l ’ a x e r 6 e l e t l ‘ u n e de ses a p p l i c a t i o n s , B u l l . SOC. Math. F r a n c e 52 ( 1 9 2 4 ) , 399 -410.

443

FUNDAMENTAL SEMINORMS

[ 31

J. P . FERRIER, Suk k ? ' a p p k o x i m a t i o n pond&ce, moderne, Univ. de Sherbrooke, 1972.

[ 4]

P. GEETHA, On Bernstein approximation problem, J. Math. and Appl. 25 (1969), 450 - 469.

[ 51

P . MALLIAVIN, L'approximation polynomiale pondGr6e sur un

[ 6]

S. MANDELBROJT, S & i e n a d h h e n t n , k z g u l a h i z a t i o n den nuiteA,app t i c a t i o n d , Gauthier-Villars, 1952.

[ 71

S. MERGELYAN, Weighted approximation by polynomials, Arwr. Math. SOC. Translations, Series 2, vol. 10 (19581, 59 -106.

[ 81

L. NACHBIN, Sur les algzbres denses de fonctions diffgrentia-

Sem.

Analysis

espace localement compact,Amx.Journal Math. 81(1959), 605-612.

bles sur une variatg, Comptes Rendus Acad. t. 228 (1949), 1549 -1551.

1 91

d'Analyse

Sc.

Paris,

05 a p p h o x i m a t i o n t h e o h y , D. Van Nostrand, 1967. Reprinted by R. Krieger Co., 1976.

L. NACHBIN, Elementd

[lo] N. SIBONY, Problsme de Bernstein pour les fonctions contintment diffgrentiables, Comptes Rendus Acad. Sc. Paris, t. 270 (19701, 1683-1685. [ll] F. TReVES, T o p o l o g i c a l v e c t o l r n p a c e d , d i n t k i b u t i o n n and KehneRs, Academic Press, 1967. [121

K. UNNI, Lectuked o n B e k n b t e i n a p p k o x i m a t i o n phob.tem, in Analysis, Madras, 1967.

[131

G. ZAPATA, Bernstein approximation problem for differentiable functions and quasi-analytic weights.Transactions Amer. Math. SOC. 182 (19731, 503-509.

[141

G. ZAPATA, Weighted approximation, Mergelyan theorem and quasianalytic weights, Arkiv for Matematik 13 (1975), 255-262.

Seminar

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INDEX

A

a l g e b r a i c convolution i n t e g r a l s

71

almost simple

214

a p p r o x i m a t i o n , non-archimedean

121

a p p r o x i m a t i o n on p r o d u c t

46

sets

37,

approximation p r o p e r t y approximation, r a t i o n a l

4 21

approximation, r e s t r i c t e d range

226

approximation, simultaneous

227

B

- differentiable

161

B e r n s t e i n problem

433

B e r n s t e i n seminorm

4 33

Be rnstein space

4 31

Birkhoff c on d i t i o n

192

B i r k h o f f i n t e r p o l a t i o n problem

189

Birkhoff' s kernel

222

b

C

c a r d i n a l series

391

cardinal s p l i n e i n t e r p o l a t i o n

39 0

c o a l e s c e n c e of matrices

1 98

c o e f f i c i e n t of c o l l i s i o n

200

compactly

-

291

regular

446

280,

373

446

INDEX

c o n d i t i o n (L)

167

cross -section

372

D Dedekind c o m p l e t i o n

64

degree o f e x a c t n e s s

385

differentiability type

164

d i f f e r e n t i a h i l i t y t y p e , compact

165

E e c h e l o n Kothe-Schwartz s p a c e s E

409

- product

Fe j&

- Korovkin

37, 269

F

kernel

78,

f o r m a l power series

354

fundamental seminorm

4 32

f undamen t a 1 w e i g h t

4 33

f u s i o n lemma

143

G Gaussian m a t r i x

2 31

G e 1f and t h e o r y

3 36

generating function

396

I i n c r e a s i n g seminorm

4 31

i n t e r c h a n g e number

202

i n t e r p o l a t i o n matrix

189

i n t e r p o l a t i o n matrix, p o i s e d

189

interpolation matrix, regular

189

79,

88

INDEX

447

K

Korovkin a p p r o x i m a t i o n

19

Korovkin c l o s u r e

20

Korovkin s p a c e

20

Korovkin' s theorem

63

L level functions

199

M meromorphic uniform a p p r o x i m a t i o n

139

N Nachbin s p a c e non-archimedean

3 72 spaces

121

0

order regularity

189

P

plurisubharmonic f u n c t i o n

34 3

p o i d s de B e r n s t e i n

237

point r6gulier

238

Pdlya c o n d i t i o n

192

P6lya f u n c t i o n s

191

p o l y n o m i a l l y c o m p a t i b l e seminorm

4 31

power growth

392

p r o p e r t y (B)

168

pseudodifferential operator

13

INDEX

446

q

Q

- regular

quasi

- analytic

229

4 39

criterion

R

r a t i o n a l approximation

421

regular interpolation matrix

189

r e l a t i v e Korovkin a p p r o x i m a t i o n

28

r e l a t i v e Korovkin c l o s u r e

28

r e s t r i c t e d range approximation

226

Rogosinski summation method

103

Rolle set

209

S S-approximation p r o p e r t y ( S . a . P . 1

359

seminorm, B e r n s t e i n

433

seminorm , fundamental

4 32

seminorm, i n c r e a s i n g

4 31

seminorm, p o l y n o m i a l l y c o m p a t i b l e

431

s h e a f o f F-morphic f u n c t i o n s

40

shift

203

S-holomorphic a p p r o x i m a t i o n p r o p e r t y (S.H.a.p.1

367

Silva-bounded n-homogeneous polynomial

353

Silva-bounded n - l i n e a r map

352

S i 1va- bounded po 1ynomi a 1

35 4

Silva-holomorphic

35 5

S i l v a - h o l o m o r p h i c , weakly

356

simple

21 3

s i n g u l a r i n t e g r a l of de l a v a l l d e P o u s s i n

99

singular Integral of Fej6r

98

s i n g u l a r i n t e g r a l of Landau-Stieltjes

93

449

INDEX

s i n g u l a r i n t e g r a l of Weierstrass

96

smoothing f o r m u l a S

386

- Runge

36 2

s t r i c t compact

3s 7

supported sequence

194

V V*- a l g e b r a

339

vector fibration

372

v e r y compact

275

w weakly S i l v a - h o l o m o r p h i c

355

weight

372,

w e i g h t , fundamental

4 33

433

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