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
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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
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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 <
I t is i n t e r e s t i n g t o note t h a t i f
E.
i s compact and
P
E
f o r any
y
E
IIyII <
K,
which s a t i s f y
1LiLk
-
o P(x)
l l pi
q1 o P ( y ) 1
<
K:
06
T h e hpace
- cornpeetion 1 1 I . (The
06
P
we have
(i = l , . . . , k ) ,
P = E'
f :E
- continuous +
above d e f i n i t i o n . D e f i n e
f o r some
i = 1,.
B
. . ,m
F
{Pl,...,Pkl
@ :E
IRk by
+.
s u c h t h a t f o r any (where
z
i=l
by
completely
hi(y) 5 1
C
E
P
- complete. and
> 0
X E
hl
y E IRk
( @ ( x i ) , & ) for
E
,.. .,Pk (XI), and
+
i
. There e x i s t
IR s u c h t h a t
I
m
IR
C
B,II@(x)- @ ( x i ) l l < 6 / 2
,.. . ,hm : IRk
for a l l
B
To
be selected a s i n the
$ ( X I = (P1(x)
m
s p t hi c B
0
T~
i s g i v e n t h e sup-norm)
IRk
non-negative continuous f u n c t i o n s
m
id
A 8 F is P-continuous,
functions is
be, P - c o n t i n u o u s ,
and
6 > 0
xl,...,xmE
F
F = IR 1 .
and t h a t t h e s p a c e o f P
choose
+
A i h t h e adgebfia g e n e h a t e d
I t i s e a s y t o see t h a t e v e r y e l e m e n t o f
bounded, and l e t
f :E
. t o p o l o g y i d d e d i n e d Ln a manneh
analogclub t o t h e c a d e
show d e n s i t y , l e t
is
P
w a s d i s c u s s e d in[2].
- cantinuoud dunctionh
A 8 E, tohehe
:T
for
P(B1) -P(B1)
P(y)ll < 2 ~ The . converse i m p l i c a t i o n follows because
PROPOSITION 1:
PROOF:
P
T h e r e f o r e , f o r any x,y E B1
E
t h e n t o t a l l y bounded. The case i n which
and
C
i n F' such t h a t
ql, . . . , q k
I qi(y) 1 .
+ sup
E
P("E)
> 0 , then since
i s compact i n F , t h e r e a r e u n i t v e c t o r s
P
P such that
i s P-continuous i f and o n l y i f P i s com-
P E P("E,F)
p a c t . Indeed, i f
t h e :T
B CE,
E B
-
-
.,Pk}'C
and a f i n i t e s u b c o l l e c t i o n I P 1 , . .
0
Ilf(x)
IIP(x)
and bounded set
> 0
= I ,...,m.
POLYNOMIAL APPROXIMATION A N 0 A QUESTION OF SHILOV
Choose p o l y n o m i a l s
ql,..
.,qm : IRk
Then, t h e f u n c t i o n
q :E
* F
and, f o r a l l
A 0 F
x
E
m
Z hi($(x)) = 1 i =1
such t h a t f o r
IR
d e f i n e d by
q(x) =
B,
< 2E
since
+
for
x
{ e i ) , Nemirovskir and Semenov [ 6
f : E
-f
E
B.
< 6
i
E
i =1
orthonormal
E:
0
E
E
,
T~
-
- complemapping
G(E). A
> 0 , t h e r e i s a f i n i t e s e t {A1,
IN), t h e n I I f ( x )
basis
s a i d t o b e JzeguCaJz i f
E L ( E , E ) and 6 > 0 s u c h t h a t i f x , y
(j=l,...,k,
m C qio@(x) * f ( x i )
1 have p r o v e d t h a t t h e
F , where F i s a Banach s p a c e , i s
P(’E,E)
., m r
Q.E.D.
P ( E ) c o n t a i n s t h e r e g u l a r f u n c t i o n s on
any bounded s e t B C E and
i =1,..
,
When E i s a s e p a r a b l e H i l b e r t s p a c e w i t h
t i o n of
5
E
I
.. .,Ak)
B s a t i s f y (A.7 (X
f(y)I] <
E.
for C
- Y) rei) I
bi(E,F) d e n o t e s t h e
s p a c e o f r e g u l a r mappings from E t o F, which i s a F r d c h e t s p a c e w i t h r e s p e c t t o t h e t o p o l o g y o f u n i f o r m convergence on bounded s u b s e t s Of E.
Nemirovskir and Semenov found t h a t e v e r y f
t h e :T
-
c o m p l e t i o n of
E
@(E) i s contained h
P(E)
S i n c e r e g u l a r f u n c t i o n s are bounded on bounded sets, a n a l g e b r a . I t i s n o t d i f f i c u l t t o show t h a t
a(E)
a l ( E ) C G ( E ) and
is that
6
ARON
a l l functions
m
f o f the form
f(x)
x a. i=l 1
=
m
(x,e.)
n
are i n
1
bi(~)
.Z Jail2 < and ni 1. 3 f o r a l l i . On t h e o t h e r hand, i=l a 2 ( E ) $ @ ( E ) s i n c e f ( x ) = (x,x) $’ @ ( E ) . I n d e e d , suppose f E @ ( E ) ,
provided
let
y > 0,
and l e t
g :E
+
IR be any f u n c t i o n which i s uniformlycon-
t i n u o u s on bounded s u b s e t s o f A1,-..,Ak
I(A.(x 3
E
-
and
L(E,E)
I
y ) ,ei)
Thus any such
such t h a t i f
6 > 0
< 6 , then
For a p p r o p r i a t e
E.
- yII
IIx
<
being regular, is a
g,
x,
and s o
E, 0
lb -
y
E
> 0, there
E
B1
Ilg(x)
are
satisfy
-
g(y)II <
y.
l i m i t o f polynomials,which
c o n t r a d i c t s t h e p r e v i o u s l y mentioned r e s u l t o f Nemirovskir and S e r m ~ v . I t is t r i v i a l that
@ ( E , E ) i s c l o s e d under c o m p o s i t i o n
L ( E , E ) . I t i s a l s o c l o s e d under c o m p o s i t i o n
l e f t by e l e m e n t s o f
t h e r i g h t by e l e m e n t s o f respond t o a g i v e n
f
L(E,E)
m(E,E) n g(E,E)
since, i f
and
E @(E,E)
E
f o T, for
6 ( E , E ) will c o r r e s p o n d t o
space
on t h e
> 0,
A1,
then
...,Ak
cor-
E f(E,E)
A1 o T , .
. . ,Ak
on
oT
E
T E L ( E , E ) . I n p a r t i c u l a r , the
i s a c l o s e d 2-sided i d e a l i n
L ( E , E ) which am-
t a i n s t h e f i n i t e r a n k o p e r a t o r s . Hence, e i t h e r R(E,E) nL(E,E) = fK(E,E), t h e compact o p e r a t o r s , o r lows t h a t for
E
that i f
B1
then
x,y
6
IIx - y I1 <
id
E @(E,E).
> 0 , there are
E.
Al,
I n t h e second c a s e ,
...,Ak
E L ( E , E ) and
( ( A j ( x - y ) , e i ) [ < 6 (j = I ,
satisfy
...,k ,
iEN),
m
i=l
11
such
6 > 0
But t h e n m
However, s i n c e
it fol-
11
i =1
is n o t r e g u l a r ,
i d $ dl(E,E).
F i n a l l y , w e b r i e f l y r e v i e w t h e case of d i f f e r e n t i a b l e a p p r o x i mation by p o l y n o m i a l s . I n [ 7 ] n respect t o the topologies TC
,
t h e a u t h o r s examined for
n =0,1,.
. .,
a.
Here,
a’ ( E ) n T
with is the
l o c a l l y convex t o p o l o g y g e n e r a t e d by all. seminorms of the form
POLYNOMIAL APPROXIMATION AND A QUESTION OF SHILOV
where
are compact s u b s e t s o f
E
jth - T a y l o r c o e f f i c i e n t of
f
and
K
denotes the
L
I t was f o u n d t h a t i f
(al(E)
, , =
,T:)
( a1(E),T:)-
n
T~
=
(resp.
E has
,
Cn(E)
for
( P ( E ) , . r cn) -
the
b
at
. .,
a.
x,
,
5 n,
and z J f ( x )
a n e l e m e n t of P(1E). property,
then
One n o t e s t h a t i n t h i s case,
The c o m p l e t i o n of
was s t u d i e d i n [ 2 ]
-tn)
j
j E IN,
approximation
n =1,2,.
.
,
7
to
a 1( E ) w i t h r e s p e c t
where t h e s e l o c a l l y convex t o p 0
-
l o g i e s a r e d e f i n e d by t h e gene a t i n g f a m i l y o f seminorms o f t h e form
f E al(E)
and
121,
v a r i e s among t h e compact ( r e s p . bounded) s u b s e t s o f
X C E
where
sup {
-+
j E IN, j
5 n.
(Of c o u r s e , s i n c e
E
1
i s r e a l , ( a ( E ) , T ) ~= 0
by t h e S t o n e - W e i e r s t r a s s theorem f o r
T = T~
n
Pf ( E ) : t h e c l o s u r e i n P("E) o f s p a n { Ipn: n I n t h e case o f T u' i t w a s shown t h a t i f E ' 1
proximation property, then
f :E
subsets of
-+
E.
F
such t h a t i f
( i )i j f ( x ) E
h a s t h e bounded
x, y E B
satisfy
-
ap
P
C w u ( E , F ) be t h e s p a c e o f
-+
on
funcbounded
F : f o r a l l bounded
Ivi(x
-
y)
I
sets and
< 6 (i = l , . . . , k )
-
TJZand
( i i ) a j f E C w U ( E , P ( J E ) ) 1.
I n this s e c t i o n , E
and
F are complex Banach s p a c e s .
f i r s t r e c a l l s o m e of the u s u a l t o p o l o g i e s o n pact-open
17 P("E).
f ( y ) II < € 1 . Then, i f E ' h a s t h e bounded a p p r o x i m a t i o n 1 j 5 n, ( a (E) , T ; ) ~ = i f E c"(E) : f o r a l l x E E , j E IN,
IIf ( x )
SECTION 2.
let
n 1. 1,
1 I p E E ' } = (a (E) , T : ) *
> 0 , t h e r e i s a f i n i t e set {P1,...,Pk) c E '
6 z 0
E
C(E)
= { f E c ~ ( E ): i J f ( x ) E pf ('3)
A
T h a t i s , Cwu(E,F) = (f : E
and a l l
property,
= T ~ ) .F o r
which a r e weakly u n i f o r m l y c o n t i n u o u s
B C E
then
,Tz)
x E E , j E IN, j 5 n ) . L e t
for all tions
( a (E)
0
E
t o p o l o g y on
H(E,F)
.
T~
H(E,F)
. T~
We
is the caw
i s t h e compact-open t o p o l o g y
of
8
ARON
i n f i n i t e o r d e r , g e n e r a t e d by seminorms of t h e form
where
K
i s compact and
E
C
vex t o p o l o g i e s
between
T
togolopgy a s s o c i a t e d to
T~
T~ ;
We w i l l c o n s i d e r l o c a l l y con-
j E IN. and
T
~
where ,
i s the bornological
T&
i n p a r t i c u l a r , o u r r e s u l t s are v a l i d f o r
t h e Nachbin p o r t e d t o p o l o g y T ~ .I n t h i s s e c t i o n , w e s t u d y t h e 1 p l e t i o n o f a ( E ) and a m ( E ) w i t h r e s p e c t t o t h e s e t o p o l o g i e s . [ 11
,
t h i s study i s continued f o r t h e topology Of course, H(E) = P(E)-
for
T ~ , and
T
b' h e n c e f o r a l l weaker to-
p r o x i m a t i o n p r o p e r t y , t h e n g i v e n a compact s e t
,
sup { If(x)
w e can s e l e c t
-
such
T E E 63 E
f o T(x) I : x
K } <
E
E.
'If
iT(E)
-
K)
< E . T h u s , /If
1 w e h a v e shown t h a t ( a ( E )
,.
1
( a (E) , T ~ ) = H(E)
then
-
if
K
C
that
Then, s i n c e
m e n s i o n a l , w e can f i n d a complex p o l y n o m i a l
In
0
p o l o g i e s , v i a t h e Taylor series expansion. A l s o ,
f E H(E)
com-
E has the
ap-
E,
E
> 0,
and
/If
-
f o TI!K
!
T ( E ) i s f i n i t e di-
P :T(E)
P o TllK < 2 ~ S . ince
+
C
such
that 1
P o T E a (E),
, T ~ = ) H ~ ( E ) . The c o n v e r s e i m p l i c a t i o n ,
E h a s the a p p r o x i m a t i o n p r o p e r t y ,
if ap-
is
p a r e n t l y unknown.
1
To study ( a (E)
{f
E
f i r s t remark
: Z n f ( x ) E P ~ ( ~ Ef o) r a l l
H(E)
and a l l
, T ) ~ ,w e
n E IN}
.
x
1
that
( a ( E ) , T ) ~=
E (equivalently, for
Now w e c o n s i d e r holomorphic mappings
x =o)
f : E
-+ F
Cwu(E,F) d e f i n e d a t t h e end of S e c t i o n n Pwu(nE,F) t h e i n t e r s e c t i o n of P ( E,F) and CwU(E,F);
which a r e a l s o i n t h e s p a c e 1. W e d e n o t e by
1 P ( E,F) i s a l s o d e n o t e d Ewu(E,F). I n [ 2 1 , t h e f o l l o w i n g properties wu of s p a c e s of f u n c t i o n s which are weakly u n i f o r m l y continuous on bcimmded
sets were proved.
PROPOSITION
2: ( a ) 1 6
f E C",(E,F),
aLL bounded s e t s
B C E.
then
fo
i~ c o m p a c t i n F 6o'r
POLYNOMIAL APPROXIMATION AND A QUESTION OF SHILOV
(b) Pf ("El 8 F (C)
Pwu(nE,F) i
(d) L e t
Pw,(nE,F)
C
b a
lohehe
n
hT
E
.
P(nE,F).
w i t h a b b o c i a t e d bymmettic n-lineah mapping
4 and
Gu(%,F)
6
all
c l o b e d nubnpace 0 6
P E P("E,F)
A. Then P
doh
only id t h e rnapphzg CE$7U(E,P(11-hfF))f
C(x) ( y ) :Axyn-l. I n t h i b c u b e , t h e h a n g e
d u c t contained i n
06
C .in
in
PWU("-lE,F).
hub t h e appkoximation p h o p e k t y i6 and o n l y i6 6 o t evehy
(el E '
n
Banach Apace F and ewehy n , Pf ( E) 8 F Let
9
Px(nE,F) denote the closed subspace of
of all those polynomials P such that
=
n Pwu( E,F)
.
P("E,F) consisting
P(B1) is compact in F
.
The
space of compact holomorphic mappings from E to F , HK(E,F), is the set of holomorphic mappings (equivalently, for
f (U) is
euety
PROOF: B
=
B1
x
(a) I
= 0)
Let
E
F
such that for each
, 'the following
H(E,F). Then
f
E
(b):
and choose
Let
E
6 > 0
> 0
and
E
M
, anf(x)
,
E
E
.id
PK(%,F).
and (without loss of generality) let {pl,...,pk}
C
E'
as in the definiE
B1
(i =1,.. .,k) and if A is the symmetric
mapping corresponding to P
E
x such that
HK(E,F) id and oney
tion of CwU(E). By the polarization formula, if x, y (cPi(x - y) I < 6
x
is proved.
( e q u i v a f e n t L y , d o t x = O ) and n
E
E
f
+
there is a neighborhood U of
compact in F. In [ 3 1
PROPOSITION 3 :
60%
x
f : E
satisfy n -linear
ARON
10
z
Since
- = --(E x 1
are i n IIAx"-l ZP
E B~ , E
= f 1, 1 5 i
i
1
x
B1 and
-
<
E
- y)I < 6 n" . Since n!
. ..
andy=-(Ely 1 + + E ~ Y +- E ~nZ ) n n (i =1,. , k ) , w e c o n c l u d e that
..
n-1 a P ( x ) = nAx
,
we
conclude
that
Pw,(n-lE,E').
E
E
+
E ~ - ~E xZ )
[qi(;(
Ayn-'ll
(b) * ( c ) :
x
+ ... +
5 nl.
T h i s i m p l i c a t i o n f o l l o w s from P r o p o s i t i o n 2 ( a ) .
( c ) * ( a ): A s s u m e E + 1 a P ( x ) = Axn-'
ZP E p K ( n - l E , E 1 ) , E E'
mula, i t f o l l o w s t h a t t h e
(XlI..
.I X I l - l
some
i =l,...,k.
(i=l,...,k).
E
>O,
,...,x
that
the
mapping
i s compact. By t h e p o l a r i z a t i o n f o r
-1
) E En-1
is compact. Hence, g i v e n s u c h t h a t f o r any (xl
n
so
+
-
l i n e a r mapping
A ( X ~ , . . . , X ~ - ~E) E '
t h e r e is a f i n i t e s e t
-
~ - ~ E )BY-',
A s s u m e now t h a t
x,y
Then by t h e symmetry o f
C E'
EV lr...,pkPk)
~ ~ A ( x ~ , . . . , x ~ - ~ )(pill
<
E
-
y)
I
E
A,
B1
satisfy
IVI(x
for
<
E
POLYNOMlAL APPROXIMATION AND A QUESTION OF SHILOV
11
POT a n a r b i t r a r y t e r m a b o v e ,
so t h a t
- P(y) I
IP(x)
< 3~n.
Q.E.D.
Note t h a t i n ( c ) * ( b ) a b o v e , i t i s e s s e n t i a l t h a t t h e p o l y n o -
m i d l be a d e r i v a t i v e s i n c e , i n g e n e r a l , Pw,( tained i n e r t y and
PK(nE,F). F o r example, i f F = C,
n Pwu( E )
then
=
E'
n
E,F) is p r o p e r l y
con-
h a s t h e a p p r o x i m a t i o n prop-
11
Pf ( E )
PK(nE) = P(nE)
,
and t h e
inclusion is proper i n general. Combining t h e above r e s u l t s , w e g e t t h e f o l l o w i n g . THEOREM 5:
all
n
E Dl
~ e tf
hen
E H(E).
and a l l
&
x E E, a n f ( x ) E Pwu(nE).
,the a p p h o x i m u t i o n p a a p e h t q , t h e n
df
E HK ( E , E ' )
PROOF:
.
(al(E)
f E
Fuathexmohe, id
,T)
,.
.id
und
W e o n l y prove t h e second p a r t o f t h e theorem. f E
i f a n d o n l y i f for any
n E IN a n d
x
E
by P r o p o s i t i o n 2 ( e ) . By P r o p o s i t i o n 4 , if
id and o n l y .id
E H~ ( E , E ' )
a(anf(x))
E
PK
(
n- 1 E,E').
f o l l o w s by P r o p o s i t i o n 3 .
Since
E , given
E'
hub
ontg
.id
1 ( a (E)
,TI
A
E , d n f ( x ) E P f ( n E ) = Pwu(nE)
anf (x)
Af =
E
n=l
Pwu(nE) i f a n d
,
the
only result
Q.E.D.
F i n a l l y , w e remark t h a t i n [ l ] Banach s p a c e
dot
, we
f E H(E), i f E
l o c a l l y weakly u n i f o r m l y c o n t i n u o u s .
H
show t h a t f o r a n y (E,E')
complex
i f and o n l y i f f i s
12
ARON
REFERENCES
R.
M . ARON,
Weakly u n i f o r m l y c o n t i n u o u s and weakly s e q u e n t i a l l y
c o n t i n u o u s e n t i r e f u n c t i o n s , t o a p p e a r i n P r o c . 1nf.Di.m. Holomorphy 1 9 7 7 , e d . J . A. Barroso, N o r t h H o l l a n d . R . M. ARON and J . B . PROLLA, P o l y n o m i a l a p p r o x i m a t i o n o f
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 , t o a p p e a r . R . M. ARON a n d R. M.
SCHOTTENLOHER, Compact h o l o m o r p h i c m p i n g s
o n Banach s p a c e s and t h e a p p r o x i m a t i o n p r o p e r t y , Journal Functional Anal. 2 1 (1976)
,
7
- 30.
KURZWEIL, On a p p r o x i m a t i o n i n r e a l Banach s p a c e s ,
Math. 1 4 ( 1 9 5 4 ) , 214
Studia
- 231.
KURZWEIL, On a p p r o x i m a t i o n i n r e a l Banach s p a c e s b y a n a l y t i c
o p e r a t i o n s , S t u d i a Math. 1 6 ( 1 9 5 7 ) , 1 2 4
- 129.
-
A. S. NEMIROVSKI? and S. 14. SEMENOV, On p o l y n o m i a l approxima t i o n of f u n c t i o n s o n H i l b e r t s p a c e , Math. USSR S b o r n i k 2 1 ( 1 9 7 3 ) , 255
J. B. PROLLA and C .
- 277.
S.
GUERREIRO, An e x t e n s i o n
of
Nachbin's
t h e o r e m 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 o n Banach spces with t h e a p p r o x i m a t i o n p r o p e r t y , A r k i v f o r Math. 14(1976), 251
- 258.
G . E. SHILOV, C e r t a i n s o l v e d a n d u n s o l v e d p r o b l e m s i n the theory
o f f u n c t i o n s i n H i l b e r t s p a c e , V e s t n i k Moscow U n i v . S e r . I , 25(1970) , 66
87
- 89.
- 68;
Moscov Univ. Math. B u l l .
25(1972) ,
Approximation Theory and FunctionaZ Analysis J.B. Prolla ( e d . ) 0 North-Holland Publishing Company, 1979
ANALYTIC HYPOELLIPTICITY O F OPERATORS OF P R I N C I P A L TYPE
J. BARROS NET0 Ma thema t i c s Department Rutgers University New Brunswick, N e w J e r s e y 0 8 9 0 3 , USA
Let
P ( x , D ) = Pm(x,D) + P , - l ( ~ , D )
+
...
be a d i f f e r e n t i a l o p e r a t o r w i t h a n a l y t i c c o e f f i c i e n t s i n an open set C2 of I R N . Suppose t h a t
P
i s o f p r i n c i p a l t y p e and, i n a d d i t i o n , sat-
i s f i e s t h e h y p o e l l i p t i c i t y c o n d i t i o n : a l o n g t h e null b i c h a r a c t e r i s t i c s t r i p of
Re(Pm)
,
I t f o l l o w s from 1 2
the function
h a s o n l y z e r o s o f even o r d e r .
Im(P,)
1 t h a t the d i f f e r e n t i a l operator
h y p o e l l i p t i c . Indeed, i n h i s paper [ 2 1
,
P
is
analytic
T r e v e s p r o v e s t h a t , f o r dif-
f e r e n t i a l o p e r a t o r s of p r i n c i p a l type, t h e following p r o p e r t i e s
are
e q u i v a l e n t : hypoel l i p t i c i t y : a n a l y t i c - hypoel l i p t i c i t y ; s u b - e l l i p t i c i t y and t h e above c o n d i t i o n on t h e z e r o s of t h e imaginary p a r t o f
pm'
Our aim i s t o p r e s e n t a n o t h e r proof of t h e f a c t t h a t , f o r
op-
e r a t o r s o f p r i n c i p a l t y p e , t h e h y p o e l l i p t i c i t y c o n d i t i o n abve implies analytic-hypoellipticity.
By u s i n g t h e f a c t o r i z a t i o n
formula
for
p s e u d o d i f f e r e n t i a l o p e r a t o r s , w e can r e p l a c e , modulo a n a l y t i c r e g u l a r i z i n g operators, the d i f f e r e n t i a l operator pseudodifferential operator
L = Dt
-
A(x,tlDX)
by
P
of
-
an
analytic
order
1, where
A ( x , t , D x ) i s an a n a l y t i c p s e u d o d i f f e r e n t i a l o p e r a t o r o f o r d e r 1, with r e s p e c t t o t h e v a r i a b l e x, only, and a n a l y t i c c o e f f i c i e n t s depending 13
on
14
( X It )
BARROS NET0
. L e t , then ,
be a f i r s t o r d e r a n a l y t i c p s e u d o d i f f e r e n t i a l o p e r a t o r d e f i n e d i n an open s e t i n
Rn+'
0
which w e can assume, w i t h o u t loss of g e n e r a l i t y ,
t o contain the o r i g i n . L e t
be t h e symbol o f
L, where t h e p r i n c i p a l symbol
a n a l y t i c f u n c t i o n of a l l i t s v a r i a b l e s on
Sl x
T
- A(x,t,€,)
Rn+l \ { 0
neous of d e g r e e 1 w i t h r e s p e c t t o ( ~ , r ) w, h i l e e a c h an a n a l y t i c f u n c t i o n i n respect t o
X(x,t,€,)
=
T
0
=
X ( O , O , ~0 ) w i t h
+
a(x,t,€,)
is
X .(x,t,E)
-j w i t h
6. W e s h a l l r e a s o n i n a c o n i c neighborhood of t h e
( O , O , ~ o , ~ o )such t h a t
write
homoge-
1,
-3 Bn\ { 0 1 , homogeneous of d e g r e e
61 x
i s an
point If
we
i b ( x , t , E ) o u r b a s i c assumption w i l l
be
( S o , r o ) # (0,O).
t h e following one:
contained i n
w
x
rl,
n u i t a b l e c o n e i n IRn+l
THEOREM:
Undek
the above
whetre
I" i n
con2aining
abbumptionb,
t h e phojection 0
(5
,T
t h e openat on
pakamethix. Mote p k e c i b e e y , t h e h e i n a C o n t i M u O U A
0
ad
a
),
L
han
LinCclh
a local opehatoh
ANALYTIC HYPOELLIPTICITY OF OPERATORS OF PRINCIPAL TYPE
w i t h t h e doelowing a d d i t i o n a d p h o p e h t i e h a)
Ix
i b
16
:
a h e g u l a h kehned w i t h h e b p e c t -to t h e v a h i a b d e b ( x , t )
and ( y , s ) , i.e . , t h e m a p p i n g 6
t h a n n &ohm b)
c)
(YlS)
d)
PROOF:
w,
bubbet
then
IR ( x l t l y I s )
butbe:
de2
t o assume t h a t
x
,
u 0 ady.tic
W x
r'.
w;
U x U.
r'
.is
then condition ( 4 ) implies t h a t
connected b(x,t,E)
Moreover, i n w h a t f o l l o w s w e a r e going
b 2 0 . The case
b
5
0
is t r e a t e d i n a s i m i l a r way.
Define t h e F o u r i e r i n t e g r a l o p e r a t o r
with
andg-tic-pbeudo
be a n y h e k a t i u e l g corn-
an a n a l y t i c &unction i n
(which i s always p o s s i b l e ) W
w
ahe
t~~ahe a t s o a n a d y t i c i n
Assuming t h a t t h e c o n i c neighborhood
never changes s i g n i n
tK
u E E'(U); i d
U and b e t
and
KU
i b
06
thanbpobe
i t 6
Locak i n t h e 6oLdowing
in
wheneuea
:
t h e O p e h a t o h K and
pact open
i n a n analyZic hunction
IK ( x , t , y , s )
t h e ketnel
(x,t) #
ern (u):
into
C: (u)
16
BARROS NET0
T i s a s m a l l number g r e a t e r t h a n
where
0 t o be chosen
later
where t h e phane 6 u n c t i a n $ and t h e a m p L i t u d e d u n c t i a n k are
and
t o be
d e t e r m i n e d i n such a way t h a t
(8)
with
+
LKU = u
for ail
RU,
cz(u),
E
R an a n a l y t i c regularizing operator.
W e choose t h e p h a s e f u n c t i o n
with
u
t and
@ ( x , t , t ' , < ) as t h e s o l u t i o n o f
t' belonging t o the i n t e r v a l
[-TIT]. S i n c e
h(x,t,E)
i s a n a n a l y t i c f u n c t i o n of all i t s v a r i a b l e s , t h e r e i s a unique lution
so-
( 9 ) , a n a l y t i c w i t h r e s p e c t t o a l l i t s v a r i a b l e s a n d ham-
$ of
geneous of d e g r e e 1 w i t h r e s p e c t t o
5.
As f o r t h e a m p l i t u d e f u n c t i o n , w r i t e
as a f o r m a l sum where e a c h t e r m k V i s homogeneous of d e g r e e -v w i t h respect to
5
. The
functions
kv,
v = 0,1,2,.
. ., are
o b t a i n e d as so-
l u t i o n s of t h e f o l l o w i n g t r a n s p o r t e q u a t i o n s :
I
f
(11)
DtkO
-
n
B
AS j(xrtl€, +
j =l
$,I
D x j ko + C k o = 0
ANALYTIC HYPOELLIPTICITY OF OPERATORS OF PRINCIPAL TYPE
n
v = 1/2,
17
v- 1
... .
Setting
i t c a n be p r o v e d , u s i n g s u i t a b l e e s t i m a t e s f o r
i s a n a n a l y t i c symbol 11
1
and t h a t
K
$r
and
d e f i n e d by
k
j'
that
is
(6)
p s e u d o d i f f e r e n t i a l o p e r a t o r . F i n a l l y , one can show t h a t i t s d i s t r i bution kernel
IK s a t i s f i e s ( 5 ) and p r o p e r t i e s a ) , b) , c ) and d )
a
of
t h e theorem. The e x i s t e n c e o f s u c h a k e r n e l i m p l i e s t h e n t h e a n a l y t i c p o e l l i p t i c i t y of
hy-
P.
REFERENCES
[1
I
L. BOUTET DE MONmL, O p e r a t e u r s p s e u d o d i f f e r e n t i e l s a n a l y t i q u e s e t o p e r a t e u r s d ' o r d r e i n f i n i , Ann. I n s t . Fourier 22(1972),
229 [2]
- 268.
F. TREVES, A n a l y t i c - h y p o e l l i p t i c p a r t i a l d i f f e r e n t i a l eqUati.0of p r i n c i p a l t y p e , Corn. P u r e and Appl. Math. 537
- 570.
24(1971)/
This Page Intentionally Left Blank
Approximation Theory and Functional AnaZysis J.B. Prolla ( e d . ) 0 North-Holland Publishing Company, 1979
KOROVKIN APPROXIMATION IN FUNCTION SPACES
HEINZ BAUER Mathematisches Institut der Universitat Erlangen-Nurnberg D-8520 Erlangen, Bismarckstr. 1 1/2 Federal Republic of Germany
INTRODUCTION The starting point of this survey lecture is Korovkin approximation for a linear space JE of continuous real-valued functions on a compact metrizable space X where the approximating operators defined on the total space
C(X)
are
of continuous real-valued functions
on X. This type of setting is called here absolute Korovkin appmkmation. Chapter I recalls the main results, in particular the characterization of the Korovkin closure of the given function
space
X.
Motivations, details and references to the relevant literature canbe found in the author's survey article [ 3 1 . Chapter I1 is devoted to the problem of determining theKorovkin closure in cases where it is not all of
C(X). The main tools arethe
introduction of the state space of X and the use of convexity arguments. The results of this Chapter arose from discussions Leha. Details will be published
with
G.
elsewhere.
Chapter I11 studies problems of the so-called theory of
rela-
tive Korovkin approximation. Here the approximating operators are no longer defined on all of subspace d: of
C(X)
C(X)
but rather on a fixed closed
containing
JC.
ter are due to Leha I7 1 . 19
linear
Most of the results of t h i s C h a p
20
BAUER
I. ABSOLUTE KOROVKIN APPROXIMATION W e s h a l l t r e a t h e r e a b s o l u t e Korovkin a p p r o x i m a t i o n
only f o r
s p a c e s o f c o n t i n u o u s f u n c t i o n s on a compact, even metrizable though t h e main r e s u l t s e s s e n t i a l l y remain t r u e f o r l o c a l l y
space compact
spaces [ 4 1 . Consequently, l e t C(X)
X b e a compact m e t h i z a b l e space, d e n o t e
t h e l i n e a r space of a l l c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s o n
and by
Jc
i.e. a point separating,
a 6 u n c t i a n d p a c e (on X I ,
s u b s p a c e of
*
II
of uniform c o n v e r g e n c e A sequence
( sup-norm)
w i l l be c a l l e d ( J C - )
E
c(X)
.
usual
norm
-+
C(X)
admissible i f l i m I1 Tnh n+m
f
linear
of p o s i t i v e l i n e a r operators
(Tn)n
Tn : c ( X )
A function
X
c o n t a i n i n g t h e c o n s t a n t f u n c t i o n 1. T h e s p a c e C ( X )
C(X)
w i l l b e c o n s i d e r e d a s a normed s p a c e e q u i p p e d w i t h t h e
II
by
- h 11
= 0
- Ill
= 0
for all
h E X.
satisfying
l i m IITnf
n-tw
f o r a l l a d m i s s i b l e s e q u e n c e s w i l l b e c a l l e d a K o h o u k i n d u n e t i o n (with
respect t o Kor(3C)
3C
of
Jc
1. The set of a l l t h e s e f u n c t i o n s is t h e Kohovkin d o b u h e
JC.
O b v i o u s l y , it i s a l i n e a r s p a c e s a t i s f y i n g
i s c a l l e d a K o h o u h i n Apace i f
Kor(JC) = c ( X ) .
KOROVKIN APPROXIMATION IN FUNCTION SPACES
21
The Korovkin c l o s u r e c a n b e c h a r a c t e r i z e d by means o f t h e f o l lowing envelope technique: For a n a r b i t r a r y
f E c ( X ) tclnenvelopesare
defined:
and f = s u p {h
V
Functions The s e t
A
JC
E 3C :
f o r which
f E C(X)
h 5 f}.
A
f = f ( = f ) are called V
of t h e s e f u n c t i o n s i s a l i n e a r s u b s p a c e of
JC-abbine.
c ( X ) contain-
i n g J f . I t t u r n s o u t t o c o i n c i d e w i t h t h e Korovkin c l o s u r e : A
Kor
THEOREM 1:
= JC
.
Another c h a r a c t e r i z a t i o n o f t h e Korovkin c l o s u r e i s o b t a i n e d b y means o f t h e r e p r e s e n t i n g measures. A p o s i t i v e Radon measure i s a lrepheoenting meaoulre f o r a p o i n t
I
x
E
X
(with r e s p e c t to
hdp = h ( x )
for a l l
The s e t o f t h e s e m e a s u r e s w i l l be d e n o t e d by M x ( J f ) .
t a i n s t h e Dirac measure The s e t
LEMMA 1:
FOR.
Mx(X)
f E
E~
I t always
d e f i n e d by t h e u n i t mass i n
x
E
x,
JC)
if
h E JC.
con-
x.
i s t h e n d e s c r i b e d by t h e f o l l o w i n g key
C ( x ) and
on X
lemma:
we have
n
T h i s leads t o a new d e s c r i p t i o n o f t h e f u n c t i o n s i n J C , and hence o f
22
BAUER
t h e Korovkin c l o s u r e : A f u n c t i o n
f
E
is
c(X)
JC-affine i f and only
if
for a l l
x E X
and a l l r e p r e s e n t i n g measures
p E Mx(JC).
A s a consequence o f t h i s and Theorem 1, w e o b t a i n a c h a r a c t e r -
i z a t i o n of Korovkin s p a c e s . I t u s e s t h e n o t i o n of t h e Ch0que.X boundaty aJCX
x
E
of X
X with respect t o
permitting only
which by d e f i n i t i o n i s t h e s e t o f p i n t s
a s a r e p r e s e n t i n g measure:
E~
T h e g i v e n 6 u n c t i a n .&pace JE
THEOREM 2 :
id
JC
LA
a K a a o u k i n Apace.
4
and o n l y
aJCx= x. I t i s t h i s r e s u l t which a l l o w s i n many c o n c r e t e examplesaquick
proof o f a Korovkin-type theorem. I n p a r t i c u l a r , K o r o v k i n ' s c l a s s i c a l r e s u l t follows almost i m m e d i a t e l y . I t states t h a t , for a compact i n X = [ a r b ] on t h e real l i n e
terval
IR, t h e l i n e a r h u l l of t h e t h r e e
f u n c t i o n s 1, i d , i d Z ( i d = i d e n t i t y map
x
+
x ) i s a Korovkin s p a c e .
11. DETERMINATION AND GEOMETRICAL INTERPRETATION
OF THE KOROVKIN CLOSURE
I n t h e e x i s t i n g l i t e r a t u r e few a t t e n t i o n h a s been g i v e n t o t h e d e t e r m i n a t i o n of s p a c e , hence where
f o r t h e case where
Kor(SC)
aJCX
+
X.
n o t a Korovkin
is
W e s h a l l have a c l o s e r l o o k a t this prob-
l e m i n p a r t i c u l a r f o r t h e case of 1, i d
JC
and a t h i r d f u n c t i o n
X = [ a , b l where
Jc
i s t h e l i n e a r hull
u E C ( X ) . A direct application
Theorem 1 i n c o n n e c t i o n w i t h t h e c h a r a c t e r i z a t i o n o f
of
JC-affine func-
t i o n s by means o f r e p r e s e n t i n g measures t u r n s o u t t o b e d i f f i c u l t , i n
KOROVKIN APPROXIMATION IN FUNCTION SPACES
23
g e n e r a l . However, i d e a s from t h e t h e o r y o f i n t e g r a l r e p r e s e n t a t i o n i n convex compact s e t s l e a d t o a s a t i s f a c t o r y method. C o n t i n u i n g i n t h e g e n e r a l s i t u a t i o n of C h a p t e r I , w e d e n o t e by t h e A t a t e Apace o f
S = S(x)
9 : JC
+
IR
which a r e n o r m a l i z e d , i . e .
s e t of t h e t o p o l o g i c a l d u a l E =
JC c o n s i s t i n g
c a l c o n t i n u o u s embedding
XI; i t i s c o m p a c t rne-tfiizabLe i n the space
j : X
([ 1I
pp. 79
-
j(a,x)
82;
=
+
x
the evaluation functional for well-known
i s a convexsub-
p ( 1 ) = 1. S
e q u i p p e d w i t h t h e weak t o p o l o g y
X'
of a l l p o s i t i v e linear f o m
a(JC',X). There is a canoni-
S, namely
E X.
j (x) = 8x
where
6 x is
are
The f o l l o w i n g p r o p e r t i e s
1 , pp. 1 2 1 - 1 2 5 ) .
[2
( t h e set of e x t r e m e p o i n t s ) ;
ex S
i n particular,
-
S = conv j ( X )
,
and h = lo j
+
1 IS
i s a n o r d e r and norm p r e s e r v i n g b i j e c t i o n o f of
A(S)
, t h e space of a l l continuous a f f i n e functions
F o r t h e case
dim JC
t h i s is a bijection of
+
m
o r f more g e n e r a l l y , for
JC o n t o
Y = j ( X ) ) and l e t u s d e f i n e f o r
envelopes
JC
a : S
+
IR.
closed i n c ( X )
A(S).
L e t u s c o n s i d e r now a compact s e t
(like
X onto adense subspace
Y such t h a t g E C(Y)
the
ex
S C Y C S
"geometrical"
24
BAUER
and = s u p {a E A ( S )
2Y
By
:
a 5 g
on
Yl.
we denote t h e space
A(Y,S)
As a consequence of t h e above p r o p e r t i e s of
j
we obtain
: X + S(X)
t h e c a n o n i c a l isomorphism
of
A(j
the function space
C([a,bl1.
E
-
y
j(X)
= { 1 ) x GU
g%
- -gGu
,
given
a
€
IR
GU
such t h a t
GU i s t h e graph o f u,
where
S i s t h e dace
g E A(GU,S)
and
function
h u l l of
u i s n e i t h e h c o n v e x noh c o n c a v e . Then t h e r e
y E S. Consequently, f o r e v e r y
-
with
S can be i d e n t i f i e d w i t h t h e c l o s e d convex
Suppose t h a t
ists a point
X = [ a r b ] a compact i n t e r v a l i n
Xu = l i n (1, i d , u}
Since
t h e s t a t e space GU
JC.
Consider
E x a m p .L e :
u
A
onto
(X),S)
F
of
Y
S
generatedby
concave
the
ex-
function
d e f i n e d on S , v a n i s h e s a t y and hence a t e v e r y p o i n t o f
Y = S. This proves t h a t s t r i c t i o n of f u n c t i o n s o f
F
phism between
A(S)
and
Gu
=
A(S) A(GU,S).
ZGu
to
i s a f f i n e on GU
Hence t h e
re-
d e f i n e s a c a n o n i c a l isomor-
Consequently,
T h e r e f o r e , w e can o n l y e x p e c t t o have i s concave o r convex.
S.
A
JCu
. = Kor(JCU) * A
X u = Xu
I f follows from Theorem 2 t h a t
n
Xu =
Xu CCX)
if u
is
KOROVKIN APPROXIMATION I N FUNCTION SPACES
25
e q u i v a l e n t t o t h e s t r i c t c o n c a v i t y o r c o n v e x i t y o f u. S i n c e o n e o n l y h a s t o o b s e r v e t h a t e v e r y r e p r e s e n t i n g measure has
i d E Xu E Mx(Jcu)
x as barycenter. So l e t u s assume t h a t
u i s an element of t h e s e t
C(la,b] 1.
concave f u n c t i o n s i n
of
K
I n what f o l l o w s w e s h a l l
get
all addi-
t i o n a l i n f o r m a t i o n a b o u t t h e b e h a v i o u r o f t h e map
d e f i n e d on ing
u.(
K . W e can i n t r o d u c e a pre-order
v (u,v E K )
Then t h e r e l a t i o n
u
if
<
v
Mx(Xv)
C
r e l a t i o n on
Mx(Xu) h o l d s f o r a l l
expresses t h a t
As c o n s e q u e n c e o f t h e c h a r a c t e r i z a t i o n o f
K by d e f i n -
x
[a,b].
E
v i s more concave t h a n
u.
JC-affine f u n c t i o n s by means
o f r e p r e s e n t i n g m e a s u r e s a n d o f Theorem 1 w e o b t a i n t h e i m p l i c a t i o n
T h e r e a r e two e x t r e m e cases: t h e a f f i n e f u n c t i o n s on
[a,bl
are the
minimal, t h e s t r i c t l y convex c o n t i n u o u s f u n c t i o n s on
[a,b]
are t h e
maximal e l e m e n t s o f
K.
l i n { l , i d ) = A([a,b]) Much b e t t e r
The and
corresponding
Korovkin
closures
are
C([a,b]), respectively.
r e s u l t s c a n b e o b t a i n e d by making u s e o f A l f s e n s ' s
n o t i o n o f boundary ( a f f i n e ) d e p e n d e n c i e s [ 1 ]
.
e r a l framework of t h i s C h a p t e r . F o r a p o i n t
y E S
W e r e t u r n t o t h e gen-
t h e set
B
Y
of
a l l b o a n d a h y d e p e n d e n c i c h i s , by d e f i n i t i o n , t h e l i n e a r s p a c e o f a l l v on
s i g n e d Radon m e a s u r e s
S which a r e s u p p o r t e d by t h e G 6 - s e t ex S
a n d which a n n i h i l a t e a l l a f f i n e c o n t i n u o u s f u n c t i o n s o n
I
adv = 0
for
all
S :
a
A(S).
BAUER
26
As a consequence of t h e minimum p r i n c i p l e f o r lower semicontinuous concave f u n c t i o n s , a f u n c t i o n by i t s r e s t r i c t i o n t o
g
+
g I ex S
died: q
S.
E
-
C(ex S). T h i s subspace and,conse-
can be d e s c r i b e d as f o l l o w s :
A(Y,S)
A(Y,S)
(a)
ex Therefore,
d e f i n e s a n order and norm p r e s e r v i n g iscmoq%sm of A(Y,S)
PROPOSITION 1:
tion i n
i s u n i q u e l y determined
e x S , hence i n p a r t i c u l a r t o
onto a c e r t a i n l i n e a r subspace of quently,
g E A(Y,S)
A dunction
-
q E C(ex S) i b ,the f i e b t h i c t i o n
06
a
id and o n l y id t h e d o l l a w i n g trnro c o n d i t i o n s m e
-
A(ex S,
Y
PROPOSITION 2 :
Foh
A&-
S);
C o n d i t i o n (b) i s s t i l l redundant. B y u s i n g t h e f a c e F g e n e r a t e d by a p o i n t
bunt-
o f S,
y E S , i t can be improved:
euehy d u n c t i u n
-
q E C ( e x S) and euwy point
y
E
S
t h e SoLlowing t w o c o n d i z i o n d ahe e q u i v a l e n t : (a)
1
qdv = 0
doh
alL
V E B
*
Y '
W e r e t u r n now t o t h e d i s c u s s i o n of t h e
E x a m p l e n
:
W e choose f o r u a concave polygon
p r o p e r v e r t i c e s . T h i s means t h a t
on
[a,b]
with
u i s of t h e form
u = i n € ( a l , . . . , an+l) where
all...,an+l
a r e a f f i n e f u n c t i o n s on
[a,b 1
such t h a t a . 3
5%
KOROVKIN APPROXIMATION IN FUNCTION SPACES
holds only i n t h e t r i v i a l case point
y i n t h e i n t e r i o r of
Furthermore
ex S
.. ) .
j = k (n = 1 , 2 , .
conv GU
S =
i s t h e set of t h e
n
27
For an
w e have
= conv GU
+2
Xu
q E A(ex S , S )
F
A(S) =
A(S,S)
i s c a n o n i c a l l y isomorphic t o t h e l i n e a r s p a c e of
all
satisfying
i
qdv = 0
A
Furthermore w e know t h a t phic t o
= S. Y S. T h e r e f o r e
vertices of
i t f o l l o w s from t h e t w o p r e c e d i n g p r o p o s i t i o n s t h a t
and hence
arbitrary
A ( G ~ ,S )
Xu
for a l l
i s (by means o f
v E B .
Y
0) canonically isomor-
and hence t o t h e l i n e a r s p a c e o f a l l
q E A(ex S , S )
satisfying
for all
jqdv = 0
v
The l a t t e r c o n d i t i o n i s empty s i n c e Indeed, e v e r y
z E GU
ex S
an extreme p o i n t . Consequently,
.
ex
S.
E G
all
U
\ e x S.
z E GU.
of
s i n c e FZ i s a segment o r r e d u c e s t o A
i s canonically
Xu
I t i s e a s y t o check t h a t
with v e r t i c e s i n
for
0
BZ =
z
h a s a unique r e p r e s e n t a t i o n a s b a r y c e n t e r
a p r o b a b i l i t y measure on
A(ex S,S).
and a l l
BZ
E
isomorphic
to
y lies i n exactly n triangles
These produce
n -1
l i n e a r i n d e p e n d e n t vec-
Since B i s d e t e r m i n e d by a system o f 3 l i n e a r equaY Y t i o n s i n n + 2 v a r i a b l e s , w e o b t a i n dim B = n -1. This p r o v e s o u r Y f i n a l r e s u l t , namely tors of
B
A
dim 3Cu = dim Xu
+
n-1 = n+2.
Formally, t h i s e q u a l i t y a l s o h o l d s f o r The c a n o n i c a l isomorphism between t h e same t i m e c l e a r t h a t t h e e l e m e n t s o f
n =O. A
Xu h
Xu
and
A(ex S , S ) makes a t
a r e piecewise
affine.
BAUER
28
More p r e c i s e l y : L e t
...
x1
u h a s p r o p e r v e r t i c e s . Then
< xn
be t h o s e p o i n t s i n
su
is
t h e space o f
C ( [ a , b ] ) which are a f f i n e on e v e r y i n t e r v a l
where
x0 = a
x
and
n +2
where
functions
in
[ X ~ , X ~ + ~i = ] O , ,...,n,
~ =+ b . ~ T h i s can be s e e n a l s o d i r e c t l y by means
of t h e r e p r e s e n t i n g measures. T h i s d e s c r i p t i o n of ber
[a,b]
A
Xu
makes thenun-
of i t s dimension e v i d e n t .
111. RELATIVE KOROVKIN APPROXIMATION
W e r e t u r n now t o t h e s i t u a t i o n s t u d i e d i n C h a p t e r I. Hence i s a f u n c t i o n s p a c e on a compact m e t r i z a b l e s p a c e X.
of h e l a t i v e Korovkin approximation i f t h e r o l e o f
X
W e s h a l l speak
C ( X ) in
absolute
Korovkin approximation i s t a k e n o v e r by a c l o n e d dunctian n p a c e
d:
c o n t a i n i n g JC as l i n e a r subspace:
J € c d:
Consequently, a sequence (JC, f )
-
= d:
c C(X).
( T n ) n E IN
of p o s i t i v e l i n e a r maps i s called
-admissible i f
l i m IITnh nA function
f E
spect t o
and
JC
h o l d s for a l l
-
hII = 0
i s c a l l e d a kek?ative K o h o v k i n d u n c t i o n (with
re-
L) if
(X,X)-admissible sequences. The set of t h e s e functions
i s t h e r e l a t i v e Korovkin c l o s u r e
Kor(JC,L)
for a l l
Kor(JC,E)
i s a function space s a t i s f y i n g
of
JC
w i t h r e s p e c t t o I.
KOROVKIN APPROXIMATION IN FUNCTION SPACES
JC i s
29
c a l l e d a K o t r o v k i n space w i t h hb6pec.t t o d: i f
K o r ( X , E ) = L. As
i n t h e a b s o l u t e case t h e main p r o b l e m s are t o c h a r a c t e r i z e and t o d e c i d e whether
i s a Korovkin s p a c e w i t h r e s p e c t t o
JC
Kor(W,I)
E.
L e t us c o n s i d e r f i r s t t h r e e
E x a m p t e
1)
b :
Je = l i n
X = 1-1, + 1 1 ,
{l, i d , i d 3 )
and
E = l i n 11, i d , i d 2 , i d 3 1 . I t f o l l o w s from t h e c o n s i d e r a t i o n s i n C h a p t e r I1 t h a t
aJC x
= [ -1,
-
u
1
W e s h a l l see t h a t
[ 1
,
and
11
K o r ( J C , L ) = d:
a,x
= X.
.
that
aJCx = aEx W e s h a l l see t h a t
3)
Let
=
io,ii.
Kor(Je,e) = W
.
X be t h e c l o s e d u n i t d i s k i n
of a l l a f f i n e f u n c t i o n s on
IR2
,
X , and l e t
JC
E be t h e s p a c e
C(X)
Then
is t h e u n i t circle, i.e.
boundary o f
Sagkin 1 9
X
. We
s h a l l see t h a t
] announced a r e s u l t t h a t - a t
f i n i t e dimensional s p a c e and s u f f i c i e n t f o r
JC
- t h e condition
Kor(JC,e) =
A(X1
of
which are harmonic i n t h e open disk.
a l l functions i n
ajtX = a E X
t h e space
.
t h e topological
Kor (JC,,)
=
r
.
l e a s t f o r t h e case o f a afcX = a E X
is necessary
The f i r s t two examples show
that
BAUER
30
t h i s i s n o t t r u e . However, w e s h a l l see t h a t Choquet b o u n d a r i e s
and
ascX
s p a c e w i t h respect t o
the
equality
the
of
i s s u f f i c i e n t f o r Jc to be a Xorovkin
a,X
E i f i n a d d i t i o n t h e common boundary aJcX = a E X
is closed. This a d d i t i o n a l condition i s f u l f i l l e d i n t h e t h i r d
ex-
ample. Crucial f o r t h e r e l a t i v e theory is t h e notion of the 'c aJcX
Choquet boundaay
which by d e f i n i t i o n i s t h e set
aEs p For
1: = C ( X )
asc x ,
hence
&&Latiwe
X
= IX E
= Mx(J)
: Mx(3C)
1.
t h i s i s e x a c t l y t h e d e f i n i t i o n of t h e Choquet boundary
axx
=
aJcc ( x ) X .
Immediate consequences o f
the
definition
a r e t h e f o l l o w i n g two remarks:
aJc x
=
ad:x
1: aJcx c aLx
-
E aEx c a,x;
a3cE x
=
aJcx.
A c c o r d i n g l y , w e have i n t h e above Examples:
1)
axe x
2)
E aKx = a3c x
=
a 3c x
(since 3)
aJcE x
=
=
a,x
+
Mx(3C)
a3c x
(since
= [-it
=
aE x
+
Mx(JC)
Also the notion of
a function
f E C(X)
-
=
1
io,ii
Mx(E) =
u i T1 , 1 1 ;
for the origin
x = 0);
topological boundary
Mx(L)
f o r a l l i n t e r i o r points of
X).
X-affine functions w i l l b e g e n e r a l i z e d . For h
t h e d e f i n i t i o n of t h e e n v e l o p e s f
t h e one g i v e n i n C h a p t e r I. A f u n c t i o n
and
f
v
is
f w i l l be c a l l e d (JC, 6 ) -addine
31
KOAOVKIN APPROXIMATION IN FUNCTION SPACES
if
f E E
and i f A
for a l l
f ( x ) = f (x) v
The s e t
gE
Obviously,
x E
of t h e s e f u n c t i o n s i s a l i n e a r s u b s p a c e o f
d: = C ( X )
for
w e have
$C(X)
=
rx.
d: :
x, h
The (JC,E)-affine f u n c t i o n s do n o t p l a y t h e same r o l e a s i n t h e a b s o l u t e t h e o r y . The f o l l o w i n g r e s u l t g e n e r a l i z e s o n l y o n e
part
of
Theorem 1.
PROPOSITION 3:
(3C,E) - a d d i n e d u n c t i o n in a & & z t i v e
Euehy
Kotlouhin
6un c t i o n :
W e sketch t h e proof:
Z ( x ) = c(x)
on t h e set
S =
E
>
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)
<
E
for all
T h i s i m p l i e s f o r an a r b i t r a r y (K,E)-admissible sequence
(Tn)
x
E S.
that
32
BAUER
(Tnf) c o n v e r g e s u n i f o r m l y on
<
g
E
satisfies
d:
h o l d s for a l l
E
f.
From t h i s
and
(Tnf) converges uniformly t o
p r i n c i p l e follows t h a t Indeed, a f u n c t i o n
to
S
I1 gII <
f
if
E
t h e maximum even on and
only
X.
if
aEx.
x E
W e o b t a i n two c o r o l l a r i e s :
ARY 1:
Kor(JC,E)
id
= d:
-
E aLx c aJex .
T h i s f o l l o w s by o b s e r v i n g Lemma 1 which i m p l i e s a c h a r a c t e r i z a t i o n of t h e r e l a t i v e Choquet boundary, namely
where
n
{ f = f 1 s t a n d s f o r t h e set V
COROLLARY 2 :
Kor(X,E) = d:
A
{x E X : f ( x ) = f ( x )} . V
id t h e t w o C h o q u e t b o u n d a h i e d a X X
and
ad:x a h e d o s e d a n d c o i n c i d e . T h i s follows f r o m t h e f i r s t remark f o l l o w i n g t h e d e f i n i t i o n of t h e r e l a t i v e boundary. C o r o l l a r y 2 s e t t l e s Example 3 . I t can b e s e e n from Example 1 t h a t o n e c a n n o t e x p e c t t o
the equality
G'
= Kor(JC,d:) i n P r o p o s i t i o n 3 w i t h o u t a d d i t i o n a l as-
sumptions. I n d e e d , s i n c e
aLX = X
w e have
ample. However, w e know from C h a p t e r I1 t h a t tion
id3
have
i s n e i t h e r convex n o r concave on
P r o p o s i t i o n 4 w i l l make clear why
GE II
=
sn
3C = K
s i n c e t h e func-
[-1, + 11.
K o r ( X , E ) = x and hence
i n t h i s ex-
Furthermore
2' * Kor(X,C).
The p r o o f o f P r o p o s i t i o n 3 u s e s a p r o p e r t y of t h e c l o s u r e S of t h e Choquet boundary S
aEX
which h o l d s f o r much s m a l l e r c l o s e d
sets
i n c e r t a i n c a s e s . I t i s t h i s o b s e r v a t i o n which l e a d s f r o m p r o p o s i -
t i o n 3 t o Theorem 3. A set
S C X
i f a function i n
w i l l be c a l l e d L - d e t e h m i n i n g i f i t i s closed and vanishes i d e n t i c a l l y provided t h a t i t v a n i s h e s a t
KOROVKIN APPROXIMATION IN FUNCTION SPACES
all points of S . A closed set t e h m i n i n g if for every
> 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)) <
E
>
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 ) ]<
The d e s c r i p t i o n of
E
on
that,
E
K.
i f and o n l y i f , g i v e n any
f
el
C ( K , E ) a s above, there
E
i n the unit b a l l
go : Uo
+
E
Ei
of
continuous with
such t h a t
for a l l E
on
x E K
and a l l
e ' E E;
.
HF (K) as t h e r i g h t s i d e of t h e equa-
t i o n i n 4 . (2) i s o f c o u r s e L n d e p e n d e n t o f t h e h y p o t h e s i s on F i n 4
l
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
a n d so i s t h e i n c l u s i o n
H F ( K , E ) C E E H ~ ( K ) which f o l l o w s from
d e s c r i p t i o n . Hence, a s o b v i o u s l y l i n e a r subspace of
whenever
46
this
E mE H F ( K ) i s U k m q b a t o p o l o g i c a l
w e have
HF(K,E),
i s c o m p l e t e . So, by S c h w a r t z ' s t h e o r e m , t h e a . p . o f HF(K)
E
clearly implies t h e equality V
= E 8E H F ( K )
HF(K,E)
f o r complete l.c. Let
[or let
spaces E,
even w i t h o u t t h e h y p o t h e s i s of 4 .
b e c o m p l e t e and l e t t h e a s s u m p t i o n o f 4.(1) b e s a t i s f i e d
E
HF(K)
Then t h e p r e c e d i n g t w o t h e o r e m s imply:
have t h e a.p.1.
E
E
HF (K) C E E AF (K)
I1 C
E
6E
AF(K) C AF(K,E).
So w e o b t a i n from S c h w a r t z ' s theorem:
5. COROLLARY: bouhhoodb
let
F ( U ) had t h e
K,
06
a g a i n abbume t h a t , doh
ub
a.p.
boa each
bOme
babib
VL
06
neigk-
t e , t t ( K ) =HF(K)
L1 E ul,and
be valid. Then
AF(K) = HF(K)
h o l d b doh a11 c o m p e e t e 1 . c .
hub t h e a.p.
i6 and o n l y id
( o h , e q u i v a t e n t e y , doh
AF(K,E) =HF(K,E)
Scwtuchl bpaceA E.
I f , i n concrete e x a m p l e s , one examines t h e methods t o a proof of
A (K)
F
methods a l s o p r o v e s p a c e s E. AF(K,E)
= HF(K),
A (K,E)
F
it turns out very aften
= HF(K,E)
that
lead these
f o r , a t least, a r b i t r a r y B a n a h
C o r o l l a r y 5 shows t h a t i t s u f f i c e s t o p r o v e
= HF(K,E)
that
the equality
f o r a l l Banach s p a c e s E t o o b t a i n b o t h t h e a . p .
of
46
BlERSTEDT
AF(K) = HF(K)
and
even f o r a r b i t r a r y c o m p 1 e t e t . c .
AF(KiE) = HF(K,E)
s p a c e s E . On t h e o t h e r hand, sometimes t h e methods used
in
proving
A F ( K ) = H F ( K ) may a l s o b e a d a p t e d t o y i e l d
a d i a e c t proof of t h e a.p.
o f t h i s s p a c e , and t h e n
h o l d s f o r a l l c o m p l e t e Rc.
AF(K,E) = HF(K,E)
s p a c e s by C o r o l l a r y 5 , t o o . I n f a c t , C o r o l l a r y 5 d e m o n s t r a t e s
that
t h e two a p p r o a c h e s which w e have j u s t o u t l i n e d are e q u i u a L e n t .
S i m i l a r l y , if E i s a complete 1.c. s p a c e and i f AF(K) =HF(K)
REMARK:
t h e n t h e a . p . of
E o r of
AF(K) = H (K) a l s o implies
F
I
AF(K,E) =HF(K,E)
.in g e n e h u e .
2. APPROXIMATION ON PRODUCT SETS L e t us now t u r n t o a d e s c r i p t i o n of t h e € - p r o d u c t
resp.
com-
p l e t e € - t e n s o r p r o d u c t o f t w o ( o r m o r e ) spaocs of type AF(K) resp. H F ( K ) . Such a d e s c r i p t i o n f o l l o w s e a s i l y from t h e (well-known) general"6fice phoduct t h e o h e m " f o r s u b s p a c e s o f , s a y , C ( K 1
x K2).
( T h i s s l i c e prod-
u c t theorem w a s f i r s t s t a t e d i n E i f l e r 1171, b u t h e p o i n t s o u t
that
t h e r e s u l t is a l r e a d y i m p l i c i t l y c o n t a i n e d i n G r o t h e n d i e c k [ 2 1 ] . F o r more g e n e r a l s l i c e p r o d u c t t h e o r e m s , f o r some i d e a s c o n n e c t e d
with
t h e u n d e r l y i n g method, and f o r more a p p l i c a t i o n s compare [ 4 1 a n d [ 5 ] . )
So l e t X1 such t h a t
X1
km-spaces,
x
a n d X2 X2
in a
b e t w o 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 ) spaces km-bpace.
Then b o t h
a n d , on t h e o t h e r hand, X1
x
X2
X2 a r e and i f a t l e a s t one o f t h e s p a c e s
p a c t (or i f both resp. F2
X1 and X2 are hemicompact
d e n o t e c l o s e d L.c.
subsheaves of
know ( b y a p p l y i n g B l a s c o ' s r e s u l t on t h e
sets o f c o m p l e t e l y r e g u l a r k m - s p a c e s , p r o p o s i t i o n on k - s p a c e s i n t h e p r o o f o f sheaf
F1
E
F2
"
is
X1,
on
X1
x
X2
exists:
X2 must
be
k m , i f b o t h X1
and
Xl
X2
and
i s even locally ccnr
klR-spaces). C
resp. C
Let
F1
Then we
X2' X1 k m - p r o p e r t y o f open
sub-
cf .[12 ], instead of Arhangel'skir's [ l o ] , 1 . 1 0 ) t h a t t h e "product
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
F1
E
i s u n i q u e L y d e t e h m i n e d by t h e f o l l o w i n g r e q u i r e m e n t s :
F2
For a l l open subsets
U.
Ui
a n d , f o r a l l open
r
41
F1
F2
u1
U2'V1
C
on
3 Vi
v2
( i= 1 , 2 ) ,
Xi
Xi
= rF 1
ulvl
(F1~F2)(Ll1xU2)
F 1 (U1) € F 2 ( U 2 ) ,
=
( i=1,2),
~r
F2 u2 v2
F rUVd e n o t e s t h e c a n o n i c a l r e s t r i c t i o n mapping
where
F and where t h e € - p r o d u c t
with r e s p e c t t o t h e sheaf
l i n e a r mappings i s d e f i n e d i n , s a y , [ 7 ]
.
F(U)
of
L e t us now i n t r o d u c e t h e f o l l o w i n g n o t a t i o n : ri
c a l p r o j e c t i o n of open s u b s e t s
U
X1x
of
X1
onto
X2 x X2
( i = l , 2 ) , and,
Xi
F(V)
+
continuous
i s thecanonifor
arbitrary
,
Then w e g e t a g e n e h a t d e s c r i p t i o n of
F1
E
F2
on open s e t s
uc
%"x2
as f o l l o w s :
co of uniform convergence on compact sub-
endowed w i t h t h e t o p o l o g y sets of
U , and t h e c a n o n i c a l r e s t r i c t i o n mappings o f t h e s h e a f F1€F2
are j u s t t h e o r d i n a r y r e s t r i c t i o n s of f u n c t i o n s . F1€ F2
is a ctobed
L.c.
and
subsheaf of
i n h e r i t e d by
6. THEOREM:
have:
F1
Cxlxx2 E
F2
= Cxl E C x 2
([lo I ,
.
N u c l e a r i t y of
F1
F2 i s
1.2 c 1 .
Let Ki be a compact
d u b d e t 06
Xi
( i= 1 , 2 ) .
Then
We
4a
B I E RSTE DT
= {f E C ( K 1 x K 2 ) ; doh
f(t,.)
EF2(g2)
( t , x ) E K1xK2},
all
= If E C(K1 x K 2 ) ;
f ( t , . ) may be a p p h o x i m a t e d u n i ~ o m l y on
F2
K2 b y dunctionn belonging t o K2,
604
each
again w i t h t h e
PROOF:
on o p e n n e t n containing on
f ( * , x ) may be a p p h o x i m a t e d u n i 6 o h m l y
and
by 6unctionn beeonging t o K1
0
and f ( - , x )
F1
on open h e t b
containing
( t , x ) E K1 x K 2 } ,
& u p - nohm 0 6
C (K1 x K 2 ) ,
and:
P a r t s (1) a n d ( 2 ) f o l l o w i m m e d i a t e l y from t h e s l i c e
theorem f o r s u b s p a c e s o f
C(K1
x K2)
it s u f f i c e s t o v e r i f y
(K1)
8 HF
H
F1 i m m e d i a t e , t o o . The i n c l u s i o n
lows readily
K1
2
q u o t e d above.
(K2)
C
F~ (K1
H F~
product
To p r o v e
x K2)
,
which i s
H F (K1) E H F ( K ) fol1 2 and from t h e p r e v i o u s d e s c r i p t i o n o f t h e s h e a f F1 E F2 H F ~ F ~ ( xK K ~2 )
C
from t h e d e s c r i p t i o n o f t h e € - p r o d u c t on t h e r i g h t hand s i d e , c f . ( 2 ) .
VECTOR-VALUED APPROXIMATION ON COMPACT SETS
Finally and h e n c e
49
-
A F ~ F (K1 X K 2 ) , b e c a u s e 2 (by t h e d e s c r i p t i o n o f F1 E F,) :
A F ~ ( K ~E A) F ( K 2 ) 2
0
A F1
A s 6 . (1) ( t o g e t h e r w i t h t h e d e s c r i p t i o n o f
end o f t h e p r e c e d i n g p r o o f ) shows, A F (K1) 1 b e n t h i c t l y c o n t a i n e d i n AF1 E F2 ( K 1 x K 2 ) ,
E
@
0
0
K1 x K2 = K 1 x K 2
C
0
0
0
a t the
F2(K1 X K 2 )
AF (K2) w i l l i n
2
general
a n d i t i s e a s y to construct
examplen f o r t h i s phenomenon. However, a s i m p l e t o p o l o g i c a l
assump-
t i o n f o r c e s e q u a l i t y h e r e , a s p a r t ( 2 ) o f o u r n e x t result dermnstrates.
7 . THEOREM:
(1) L e t , d o h dome b a n i n
have t h e a . p . 06
doh each
n e i g h b o u h h o o d n oQ
U2 E U 2 . [ lnntead oh
H
F2
(K2)
06
K~
Ull
06
neighbowrhooh
let,
U1 E inl
oh
,
have t h e
F2(U2)
Qoh
06
some b a n i n U 2
a.p.
I
.
Then
each
604
t h i h , we c o u l d a l n o 4equLte
t o have t h e a . p .
Kl, Fl(Ul)
H
F1
(K1)
( 2 ) 16
K1
and
K2
a4e
"Qat", i . e . Aatibay
0
Ki=Ki
(i = 1,2),
we g e t :
PROOF:
(1) The remark i n b r a c k e t s i s o b v i o u s from 6 , ( 3 ) and Schwartz's
theorem. For the proof of (1) u n d e r the a s s u m p t i o n on
F1 r e s p .
F2
BlE ASTEDT
52
i t s u f f i c e s t o show d e n s i t y of So l e t
set
U
and
f E H F ~ F , ( K ~ x K2)
containing
K1
H
(K2) in H F1 E F2 (K1 K 2 ) * F2 b e g i v e n a n d f i n d a n open
8 H
(K1)
F1
> 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
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t h e Summeh
6hom
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lokalkonvexer Funktionenraume,
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VECTOR-VALUED APPROXIMATION
[ 9
1
K.-D.
BIERSTEDT, B .
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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-
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.
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
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[12]
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(19771, t o
preprint
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[13]
J . BLIEDTNER and W .
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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
[20]
BIERSTEDT
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 .
[22]
HO-VAN-THI-SI,
1231
I . NAMIOKA a n d R. R. PHELPS, T e n s o r p r o d u c t s of compact c o n v e x
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
N.
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.
[28 1
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.
VITUSHXIN, U n i f o r m a p p r o x i m a t i o n b y h o l o m o r p h i c functions, J. F u n c t i o n a l Anal. 20 ( 1 9 7 5 1 , 1 4 9
(30 1
24
- 822.
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
e q u a t i o n s , P r o c . Amer. Math.
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 <= g i f f
f(t)
2
degree
g ( t ) for all
5
t E [a,b
2 i s Dedekind-dense
1"
.
Then t h e set of a l l p o l y n o m i a l s o f
i n C [ a,b]. Since the set E : = { E * E ~ C [a,b] ! E > 0)
g e n e r a t e s c o n v e r g e n c e i n t h e sup-norm w e c a n c o n c l u d e t h e
classical
Korovkin- t h e o r e m from theorem 1.
REFERENCES
[1
1
B.
BANASCHEWSKI, Uber d i e V e r v o l l s t a n d i g u n g g e o r d n e t e r Gruppen
,
Math. Nachr. 1 6 (19571, 5 1 - 7 1 . 12
1
W.
A.
LUXEMBURG a n d A. C. ZAANEN, RiebZ Apace&. Vol. I., North- H o l l a n d , Amsterdam-London, 1 9 7 1 .
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. Prolla ( e d . ) 0 North-Holland Publishing Company, 1979
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
P . L.
BUTZER,
L. STENS and M.
R.
WEHRENS+)
L e h r s t u h l A f u r Mathematik Aachen U n i v e r s i t y
of
Technology
Aachen, Western Germany D e d i c a t e d t o P r o f e s s o r E i n a r H i l l e on t h e o c c a s i o n of h i s e i g h t y f i f t h b i r t h d a y on 28 June 1 9 7 9 , i n f r i e n d s h i p a n d h i g h esteem.
1. INTRODUCTION AND PRELIMINARY RESULTS I n 1963 one of t h e a u t h o r s [ 7
1 r a i s e d t h e question*)
whether
i t i s p o s s i b l e t o c o n s t r u c t a s i n g u l a r c o n v o l u t i o n i n t e g r a l of a funtion
f
d e f i n e d on
[
- 1,1] , s a y , which
d e g r e e n and which a p p r o x i m a t e s O(n-"),
where C
!C
5 1, p r o v i d e d
a
0
w (rl:f:C)
1
[
- 1 ,1 1
functions
f
f
i s an a l g e b r a i c polynomial
u n i f o r m l y on [
-
o?
1 , 11 w i t h o r d e r
belongs to
f
i s t h e modulus o f c o n t i n u i t y o f
f
(see ( 7 . 3 ) )
and
is t h e s e t o f a l l real (or complex)-valued c o n t i n u o u s
on
[
- 1 , 11
endowed w i t h t h e u s u a l sup-norm I1 * I I c
.
The
p o i n t is t h a t t h i s i n t e g r a l s h o u l d n o t b e derived under som3 (triqonomtric)
+) T h i s a u t h o r
w a s s u p p o r t e d by DFG r e s e a r c h g r a n t Bu 166 / 2 7
which
i s g r a t e f u l l y acknowledged. *)
T h i s problem h a s s i n c e b e e n r e c a l l e d i n some l a n g u a g e o r o t h e r by,
f o r example, G. F r e u d
[ 28
1 and 71
R.
DeVore
[ 2 1 , 23
1
.
BUTZER.STENS and WEHRENS
12
s u b s t i t u t i o n from a c o r r e s p o n d i n g i n t e g r a l which i s a
( s u c h as t h e i n t e g r a l o f F e j s r - K o r o v k i n ;
polynomial o f degree n e.g.
(141
, [9,
trigonometric
801).
p.
T h i s p r o b l e m was s o l v e d i n some f o r m o r o t h e r by F r e u d (1964), S a l l a y Vsrtesi-Kis
[ 49 ]
[ 60 ]
[ 46 ]
Rodina
(1964)
,
Saxena
(1967), DeVore
(1969) , B o j a n i c [ 5
1
[ 14
[ 50
[20
( 1 9 6 9 ) , Chawla
( 1 9 7 3 ) , Freud-Sharma
and Butzer-Stens
see
I ( 1 9 6 7 ) , V G r t e s i [ 591 (1967) ,
1 (19681, Bojanic-DeVore [ 18 I
[ 29
1 ( 1 9 7 6 ) . Most
[ 271
(1970)
1
411 (19711,
Mathur
1 (1974) , Szabados
[ 6
[ 561
(1976)
of t h e p o l y n o m i a l s c o n s t r u c t e d by
t h e s e a u t h o r s are of d e g r e e 4 n - k w i t h k = 2 o r 4 , some a p p r o x i m a t e f u n i f o r m l y o n l y on
[-1
+
E
,1 -
E]
for each
E
> 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
,
Lupa$
[ 38, 39
1 and StenS-
1 5 5 1 considered t h e i n t e g r a l
(1.2) X2n(X;U)
:=
n
2
3
+ 3n + 3
2"
k=o
+Pk(X)Pk(U)
-1
P("*) n
b e i n g t h e Jacobi p o l y n o m i a l . Note t h a t x,,(x;u))
0,
1-i
xZn(x;u)du
= 2 and, as w i l l be s e e n below, t h e k e r n e l c a n more s i m p l y be rewrit-
t e n as
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
73
Lupaq, for example, showed that
However, (Jhf) (X) is a polynomial of degree
.
2n, and not n
One purpose of this paper is to present a systematic
approach
to these two problems, thus to study direct approximation theorems for algebraic approximation processes that are built up analogously well-known trigonometric convolution integrals. Normally
to
one would
expect to examine the convergence of convolution integrals like
to f(x) for
n+-. The best known example of such an integral is that
of Landau-Stieltjes (see e.g. 140
I),
[37, p. 91, [ 4 8 , p. 1471,[22, p. 261 ,
the kernel of which is given by
Here it is known (see e.g. [ 22, p. 221I)that f E Lipl(a;c[-l+E, 1 0 < a
5 1,
El),
0 < E < 1, implies that
The integral is again an algebraic polynomial of degree 2n. however, difficulties occur at the end points
f
1 of the
[-1, 11 since the classical translation operator (-r;g) (x) := g(x
-
T;
used,
Here, interval namely
u) , leads one outside of the interval [-1,1 1
.
74
BUTZER,STENS and WEHRENS
The q u e s t i o n now i s w h e t h e r i t i s p o s s i b l e t o employ a n a l g e
-
b r a i c c o n v o l u t i o n c o n c e p t (which depends o n an a s s o c i a t e d t r a n s l a t i o n c o n c e p t ) f o r which t h e s e d i f f i c u l t i e s do n o t o c c u r a n d f o r which there h o l d s some " c o n v o l u t i o n theorem" i n t h e form t h a t
if
is a s u i t -
T
a b l e t r a n s f o r m such a s
f
for suitable functions
(u)Qk(u)w(u)du
ak
(k
€
P = { 0 , 1 , 2 , . ..])
to
,orthogonal with respect
the
weight
w (x), t h e n
f
*
g
being t h e s u i t a b l e convolution of
f
and g . T h i s would enable
o n e t o u s e i n t e g r a l t r a n s f o r m methods a n d , as i s w e l l - k n o w n ,
such
methods u s u a l l y e n a b l e one t o s o l v e a v a r i e t y o f problems b y a r e d u c t i o n t o a s t a n d a r d p r o c e d u r e ( r e c a l l t h e F o u r i e r t r a n s f o r m method i n t h e s o l u t i o n o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s ; see e . g . [ 9 ,
Chap.VII]).
Hereby t h e a i m , is t o employ p u r e l y a l g e b r a i c means i n t h e proofs,the only connection w i t h t h e p e r i o d i c t h e o r y being of s t r u c t u r a l n a t u r e , namely a n a p p r o a c h v i a c o n v o l u t i o n i n t e g r a l s t o g e t h e r w i t h t r a n s f o r m methods. T h e r e f o r e i n none of t h e p r o o f s r e s u l t s of F o u r i e r a n a l y s i s
w i l l be u s e d , a s was t h e case i n a f e w i n s t a n c e s i n the Chebyshev transform a p p r o a c h of B u t z e r - S t e n s [ 1 2 , 1 3 , 1 4 , 1 5 , 1 6 1 , S t e n s [ 5 4
1
.
The t r a n s f o r m w e s h a l l a p p l y i s t h e q u i t e w e l l developd Legendre t r a n s f o r m . Although F o u r i e r - L e g e n d r e series have b e e n known
for
at
least a c e n t u r y , t h e product formula l e a d i n g t o t h e t r a n s l a t i o n o p e r
ator b e i n g a l r e a d y known t o Gegenbauer
1301 i n 1 8 7 5 ,
the
Legendre
t r a n s f o r m p o i n t o f view s e e m s t o have b e e n f i r s t e m p h a s i z e d b y W a t e r [ 5 8 I o n l y i n 1950 ( s e e a l s o [ 51, p. 4 2 3 , 454 ]
and
c i t e d t h e r e ) . The main r e s u l t s needed h e r e are b u i l t
the up
literature in
Stens-
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
Wehrens
[551
, but
Letting 15 p <
m,
f i n e d on
l e t u s r e c a l l some of t h e b a s i c c o n c e p t s .
s t a n d e i t h e r f o r t h e s p a c e C [-1,11 o r L p ( - l , l )
X
5
L ~ ,
o f a l l r e a l ( o r complex)-valued measurable f u n c t i o n s f de[-1,1]
f o r which t h e norm
i s f i n i t e , t h e Legendre t r a n s f o r m o f
Here
76
f E X
i s d e f i n e d by
P k ( x ) i s t h e Legendre polynomial o f d e g r e e k , namely 2 k
=
Pk(X)
I n view o f t h e f a c t t h a t
one h a s
s o t h a t ( 1 . 8 ) d e f i n e s a bounded l i n e a r o p e r a t o r mapping X i n t o (co)I t h e s p a c e o f all r e a l ( o r complex)-valued sequences that
Urn,-
m
CakIkzO
such
a
= 0. k The c l a s s i c a l t r a n s l a t i o n o p e r a t o r
In contrast t o
T;
,rh
d e f i n e s f o r each
e a r o p e r a t o r from X i n t o i t s e l f w i t h
r;
is h e r e r e p l a c e d by
h E [-1,11
I h h 11 [ x , x ] = 1
a positive linand t h e usual
76
BUTZER, STENS and WEHRENS
IIrhf
limh+l-
-
f
Itx
f
= 0 . The a s s o c i a t e d c o n v o l u t i o n p r o d u c t
*
g is
d e f i n e d as (1.12)
If
f
(f
E
x,
*
g ) (XI
:=
-
(x
1 ( x )g (u)du
(T Uf
g E L l , t h e convolution
*
f
E
[-1,11).
g e x i s t s a s a n e l e m e n t of
X
together with
IIf
(1.13)
*
gllx <_ II f IIx 114 Ill
which i s t h e form t a k e n on by t h e c o n v o l u t i o n t h e o r e m ( 1 . 7 )
i n the
Legendre case. The d e r i v a t i v e a l s o b e i n g d e f i n e d v i a t r a n s l a t i o n , it i s t o be e x p e c t e d t h a t t h e d e r i v a t i v e i n t h e Legendre frame w i l l be The s t r o n g ( L e g e n d r e ) d e r i v a t i v e o f
f E X
unusual.
i s the function
g E X
f o r which
hl i+ml -
(1
-
f
Thf
1- h
1 D f = g. D e r i v a t i v e s
we w r i t e
defined iteratively. d e n o t e d by
WG
.
-
91Ix = 0;
Dr of h i g h e r o r d e r
The set o f a l l
1:
= 2,3,...
are
f o r which D r f e x i s t s i s 1 Note t h a t t h e s t r o n g d e r i v a t i v e D f , f E W i , cof E X
incides with t h e pointwise d e r i v a t i v e
The c o u n t e r p a r t s o f t h e modulus o f c o n t i n u i t y
and
Lipschitz
class h e r e t a k e on t h e form (1.15)
w
L
L
(6;f;X) Eul(6;f) 1
:=
s u p IIThf 65hLl
- fllx
(-1 < 6 < 1)
APPROXIMATION BY ALGEBRAIC M N V O l U f l O N INTEGRALS
77
The main purpose of this paper is to give a unified
treatment
of algebraic approximation theory via the Legendre transform method, in particular, to study conditions upon the sequence of functions 1 {Xn’n EP C L ( - 1, 1) such that (1.17)
1imIIf n-
* xn - f l l X
=
0
and to investigate the rate of convergence in (1.171, expressing in terms of the modulus of continuity (1.15). In addition
to
it
such
direct approximation theorems, the matching inverse theorems willalso be considered, emphasis being placed upon the so-called case
of
non-optimal approximation. The case of optimal or saturated approximation is dealt with briefly. The aim will be to employ
elementary
means in establishing the direct theorems (thus without appealing to the general theorems based upon intermediate space methods of Berens [3
1 and Butzer-Scherer
[ 10, 11 1
,
as was carried out
by
Bavinck
11, 2 1 1. Concerning the inverse theorems, they will either be dedwed
via the associated theorems of best algebraic approximation (as developed in Stens-Wehrens [ 5 5 1 ) or from a general result based upon a Bernstein-type inequality. This material is considered in Sec. 2 . One major aspect is to study various examples of suitable kernels that can be classified under the Legendre transform
approach.
These are given by various summability methds of the Fourier-Legendre series of
f
E
X:
(1.18)
namely by the Fej6r means of means (Sec. 3 )
,
f
E
X (Sec. 5), by the
Fejgr-Korovkin
the Rogosinski means (Sec. 6), certain de La
vall6e
78
BUTZER,STENS and WEHRENS
Poussin sums (Sec. 5) , by the Weierstrass and integral means, as w l l as bythe Landau-Stieltjes integral in the Legendre frame, all
three
treated in Sec. 4 . The Fej&
means are defined by
which may be rewritten in the form of an algebraic convolution integral (1.20) where (1.21)
Fn(x) := Z i z 0 (1 - n + 1 (2k+1)Pk(x)
(x
E
1-1,lI;
n El!?).
A particular case of the results to be established asserts that
This solves the stated problem in its original form [ -1, 11 for
0 <
on the whole
a < 1, onf being an algebraic polynomial precisely
of degree n. For the more difficult case
a =
1 we proceed as follows.
In
Fourier analysis the Fejsr-Korovkin kernel may be defined asthat even non-negative trigonometric polynomial tn of the form
for which the coefficient al takes on its largest possible (given by
cos(n/(n+2))).
In the corresponding algebraic case
amounts to finding that algebraic polynomial
value this
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
pn(x) = 1 + Ek n = l (2k which i s non-negative on [-1, 1
+ l)bkPk(x)
79
(x E [-1,lI;
1 and f o r which bl a t t a i n s
n E P),
i t s maxi-
mum. The s o l u t i o n o f t h i s e x t r e m a l problem i s e x a c t l y t h e
Fejgr-
Korovkin k e r n e l f o r t h e Legendre c a s e , d e f i n e d by
being t h e l a r g e s t r o o t of notes the l a r g e s t integer
5
P,(x)
r
and
(n X i
N = En/2 1 + 1
de
-
x).
Again a s p e c i a l case o f o u r r e s u l t s s t a t e s t h a t
T h i s shows t h a t t h e a s s o c i a t e d Fejgr-Korovkin means
f
Knsolveour
problem even i n i t s e x t e n d e d form on t h e whole [-1, 11
f
Kn
ac-
t u a l l y b e i n g a ( p u r e ) a l g e b r a i c polynomial of d e g r e e n . Most of t h e above r e s u l t s are n o t o n l y v a l i d i n
C [ -1, 11 b u t also i n Lp (-1,l).
A s mentioned, two of t h e a u t h o r s s e t up a Chebyshev
transform
method w i t h e s s e n t i a l l y t h e same aim i n mind, namely t o g i v e aunified approach t o a s many problems a s p o s s i b l e on t h e approximationof functions
f belonging t o
C [-1,
1 1 or
LE(-l,l),
15 p <
OD,
polynomials. The Chebyshev method h a s t h e a d v a n t a g e t h a t v a r i e t y of problems can be c o n s i d e r e d , such a s a l l w i t h moduli o f c o n t i n u i t y
of
those
bydgebkdc a
greater connected
higher order, including t h e f r a c t i o n a l
c a s e . The d i s a d v a n t a g e , however, i s t h a t it i s n o t as " p u r e l y " a l g e b r a i c as i s t h e Legendre t r a n s f o r m approach.
Although t h e l a t t e r is
BUTZER, STENS and WEHRENS
80
more i n t r i c a t e as it i s n o t c o n n e c t e d w i t h t h e p e r i o d i c F o u r i e r t h e o r y , i t h a s t h e advantage t h a t no "bad-looking" weight f a c t o r s e n t e r i n t o 2 -1/2 in t h e p i c t u r e as is t h e c a s e w i t h t h e weight w(x) = (1 - x ) t h e Chebyshev t h e o r y . The q u e s t i o n o f c o u r s e a r i s e s why n o t t r e a t t h e m a t t e r by t h e more g e n e r a l J a c o b i t r a n s f o r m approach. T h e r e a s o n i s t h a t
we
first
wanted t o p r e s e n t a n approach t h a t i s n o t o n l y as uncomplicated
but
a l s o as complete as p o s s i b l e . However,much of t h e material p r e s e n t e d can r e a d i l y be c a r r i e d o v e r t o t h e J a c o b i frame. As i n d i c a t e d
Bavinck
[ 1, 2 ] c o n s i d e r e d more or less some of o u r r e s u l t s i n t h e latter frame.
But it c a n p e r h a p s be s a i d t h a t h i s a i m w a s t o g e n e r a l i z e t t i g o n o n i e t h i c approximation t h e o r y t o t h e J a c o b i frame w i t h o u t b e i n g concerned
w i t h t h e c o n n e c t i o n s t o t h e problems of a t g e b h a i c a p p r o x i m a t i o n
in
t h e c l a s s i c a l sense. F o r a b a s i c u n s o l v e d problem i n t h e Legendre approach see 1171.
2.
GENERAL THEOREMS ON CONVOLUTION INTEGRALS T h i s s e c t i o n i s concerned w i t h theorems on t h e convergence
of
g ene ra l convolution i n t e g r a l s
where
{
xp Ips
A
i s a k e r n e l , i.e.
xP
E
L1 (-1, 1) w i t h
(2.2)
-
p b e i n g a p a r a m e t e r r a n g i n g o v e r some set A which
t e r v a l ( a , b ) with
0 5 a < b
s
,o r
the set
is e i t h e r an in-
P. L e t
p,
be
one
of
m
APPROXlMATlON BY ALGEBRAIC CONVOLUTION INTEGRALS
the points
a,b
or
+ a .
I n t h e following M denotes a p o s i t i v e constant, t h e value
of
which may b e d i f f e r e n t a t e a c h o c c u r r e n c e . M i s always independentof t h e q u a n t i t i e s a t t h e r i g h t margin.
PROPOSITION 1:
be a izehneL s u c h t h a t
{xpjpE
Let
l i m III f P
P'P
lim
P+P,
- f Itx
x''(k) P
= 0
(k E PJ = { 1 , 2 , 3 , . . . } ) .
= 1
I n t h i s p r o p o s i t i o n , t h e proof o f which f o l l o w s by t h e BanachS t e i n h a u s theorem, i t may be d i f f i c u l t t o v e r i f y c o n d i t i o n ( 2 . 3 ) t h e a p p l i c a t i o n s . I f t h e k e r n e l i s however p o s i t i v e , i . e . f o r almost a l l
M=l
in
xp(u)
0
u E (-111)1 p E A, t h e n (2.3) i s always s a t i s f i e d with
i n view o f ( 2 . 2 ) .
T h i s l e a d s t o t h e f o l l o w i n g Bohman-Korovkin-
type r e s u l t :
PROPOSITION 2:
16 t h e heanel
t o w i n g annettions in
~ Q U ~ I J L Z t o~ ~( 2R. 4~)
lim
P'P,
(2.7)
~x,),,,i n
xi(1)
pobitiue, each
and
06
t h e dot-
(2.5):
= 1,
(-1 < 6 < 1).
82
BUTZER. STENS nnd WEHRENS
PROOF:
T h a t ( 2 . 5 ) i m p l i e s ( 2 . 6 ) i s o b v i o u s . On t h e o t h e r hand,
if
( 2 . 6 ) h o l d s , t h e n by ( 2 . 2 ) a n d t h e p o s i t i v i t y o f t h e k e r n e l ,
1
< 1-6
6, 1
(l
-
T h i s gives (2.7). C o n c e r n i n g ( 2 . 7 ) for
f
* ( 2 . 4 1 , one u s e s t h e f a c t
that
E X
which i m p l i e s ( 2 . 4 )
by s t a n d a r d a r g u m e n t s .
0
When d e a l i n g w i t h p o s i t i v e k e r n e l s , t h e r e h o l d s t h e
following
g e n e r a l r e s u l t on t h e r a t e o f c o n v e r g e n c e i n ( 2 . 4 ) :
PROPOSITION 3:
LCf
C x p J p E A be
a punitive k e h n e L , a n d L e t
d t h i c t l y panitive dunction d e d i n e d an A n u c h t h a t
lim
P"Po
IP
IP(P)
be a
=o.
T h e doLLowing anbehtionb ahe e p u i v a L e n t :
PROOF:
L e t u s f i r s t mention t h e f o l l o w i n g i n e q u a l i t i e s needed
[55; S e c . 51),
(see
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
L
(2.12) w1(61;g;x) 5 12(1 +
L )w1(62'g;x) 1- 62
1-61
83
( g E X ; -1 < 61,62
< 1).
Now if (2.9) holds, then (2.10) follows as in (2.8) by (2.12)
and
(2.2) since
To prove the converse, we apply
for k = 1 and
Since
P1
E
g
= xD
to deduce
1 W x , (2.9) follows by (2.11).0
The particular case
v (p)
= 1
-
xi (1)
gives, noting that
Inequality (2.15) shows that the approximation error dependson the smoothness properties of the functions involved. Therefore
one
might expect that the rate of convergence in (2.4) could be arbitrarily good if f is sufficiently smooth. Later on it will be seen
that
84
BUTZER.STENS and WEHRENS
t h i s s i t u a t i o n r e a l l y t a k e s p l a c e f o r a t l e a s t one c o n v o l u t i o n
in
-
t e g r a l . But f o r a m a j o r i t y o f approximation p r o c e s s e s t h e r e e x i s t s a
c r i t i c a l o r d e r which cannot be t r a n s p a s s e d u n l e s s f i s
a constant.
T h i s i s t h e familiar s a t u r a t i o n phenomenon.
A handy c r i t e r i o n d e c i d i n g whether t h e s a t u r a t i o n property holds f o r a convolution i n t e g r a l
{Ip$ €A
a) I &t h e hehnee
PROPOSITION 4 :
i s g i v e n by
{ x p l p E A 0 4 t h e pJLocebA
natisdied condition
lim
(2.16)
- X " (k)
1
sup
P (PI
PV,
whehe
~p
i n a duncttion
impeieb t h a t
i n Phop. 3 , t h e n
ah
f = const.
(a.e.1 *)
b) 7 6 rnoheoveh t h e h e
.then t h e h e
EXibtA
.o
.
C ~ i A t bA O M e
ko E IN
buch t h a t
a t &ant o n c n o n - c o n b t a n t [ n o n - t h i v i a e )
that (2.19)
IIIpf
I n o-thuzeh w o t d ~ , f
*)
- fllx = ~
O(P(P))
~ ~ Autuhated i n lX w i t h ohdm O ( v (~p ) ) ,
" ( a . e . ) " means t h a t a n a s s e r t i o n h o l d s for a l l X=C[-l,ll,
a n d f o r almost a l l
x E
p
~
+
x E [-1, 11
1-1, 11 i f X = L p ( - l , l ) ,
p,
. if
1zpcm.
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
86
By (2.17) one h a s on a c c o u n t of (1.10) and ( 1 . 1 4 ) t h a t
PROOF:
If"(k)
I X i ( k ) -11
lim
T h i s means t h a t
P'P,
1
5 IIIpf
-
fllX = 0 ( 9 ( p ) )
11 -X''(k)I
(p
jf^(k)/ /p(p)
o t h e r hand, by ( 2 . 1 6 ) t h e r e e x i s t s a s u b s e q u e n c e {pj)j=l limj+=
1
x ; . ( k ) -11 / t p ( p j ) 3
u n i q u e n e s s theorem (see 1 5 5 , Sec.
=
ck > 0 , e a c h
2 ] ) t h i s gives
Concerning p a r t b), one h a s f o r
p,;
kGlN).
~-
0 , k c N. On
=
W
such t h a t
-t
,lim.
the
P j =Po,
k e lN.
By t h e
f = const.
(a.e.1.
by ( 2 . 1 3 ) and(2.18) that
f =Pk 0
which c o m p l e t e s t h e proof.0 Whereas C o r . 1 g i v e s a d i r e c t a p p r o x i m a t i o n r e s u l t , a n i n v e r s e one i s g i v e n by
Let
PROPOSITION 5:
incquaLity
16
f E
x
06
ohdek
{XnlnE a b e a k e f i n e L hatib6ying a B e f i n n t e i n - t y p e y > 0 , nameLy
c a n be apphoximated b y t h c convoLution
intcghat
f
*
Xn
nuch t h a t
doh home
PROOF:
0 < a < 1, t h e n
L e t u s follow t h e c l a s s i c a l arguments u s i n g t e l e s c o p i n g sums
( c f . [9, p. 110 1 . S e t t i n g n=3,4,.
L
f E Lipl(a;x).
.., t h e n
by ( 2 . 2 1 )
u2
= f *X22
,un
=
*
2n
-
f*X2n-l
for
BUTZER,STENS and WEHRENS
86
11 Unll
(2.22)
5 M2-nya
(n = 2 , 3 , . . . I f
S i n c e t h e c o n v o l u t i o n p r o d u c t i s commutative and a s s o c i a t i v e , o n e c a n
rewrite Uk
Uk = ( f
as
-f *
X2k-l)
* x 2 - (f - f * x
2
(k=3,4,.
k ) .X2k-l
..I.
T h i s i m p l i e s by ( 2 . 2 5 ) , ( 2 . 2 0 ) and ( 2 . 2 1 ) t h a t hhUk
-
Uk
IIx 5
M ( l
which i s a l s o v a l i d f o r
- h)
I2
ky ( 1 - k ) y a 2
+
( k - 1 ) Y2-kYa
1
k = 2 . T h i s y i e l d s by ( 2 . 2 4 ) and ( 2 . 2 2 ) t h a t
I f one now chooses m such t h a t f o r t h e n ( 2 . 2 6 ) g i v e s ( a s e.g.
-
6 > 1 2-',
i n 19, p. 1 0 1 1 ) t h a t
This i s the desired assertion.
0
Z m - l < (1-6)-'" L w1(6;f)
5
< PI
M ( 1 -8)'.
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
87
Finally a result on best approximation by algebraic polynomials will be stated: it may be used for establishing direct approximation theorems for certain approximation processes.
It also gives another
way in proving inverse theorems for polynomials kernels leading
to
optimal approximation. As usual, let
En(f:X)
=
inf IIf - pnIIx PnEPn
be the best approximation of gree
f
E
X
(n E P)
be algebraic polynomials of de-
5 n, Pn being the set of such polynomials. Concerning the rate
of convergence in (2.27)
lim En(f;X) = 0 n+-
we have the following counterpart of the classical Jackson and Bernstein theorems (see [ 55, Sec. 5 1 ) .
(2.29)
Drf
E
Lipl(a;X), L i.e.
wL(6;Drf;X) = O ( ( 1 - 6 l a ) 1
The purpose of the next sections is to apply results to several concrete approximation processes.
the
(6
-+
1-1.
foregoing
88
BUTZER. STENS and WEHRENS
3 . THE ALGEBRAIC CONVOLUTION INTEGRAL OF FEJER-KOROVKIN
Concerning Cor. 1 it is of interest to search
for
positive
m
~ ~ which l ~ =belong ~ to Pn such that the rmdulus u:(G(U
kernels {
;f;X)
is as small as possible. This amounts to determing such kernels which xG(1) is asclose to 1 as possible. Note that
for
Xi(1) < 1
in
view of (2.14). This will lead to the Fejgr-Korovkin kernel of Sec. 1. Therefore we will now discuss some extremal properties of
the
coefficients of the elements of the s e t
THEOREM 1:
a) F O R . aLL
+ t h e h e hoLdb pn E N Pn
m = 6 (n + 1)/ 2 1, and
whehe
b) FOR. e a c h euen
n E P
Xm+l thexe
i n t h e .hR.gebt h o o t exibtb
a
UniQUe
06
Pm+l
+
pnE NPn
.
ouch
that
c) F O J L n E IN o d d t h e l r e e x i b t b no d)
FOR. each
pn E NP;
afid e a c h
p, E
PROOF:
doh
(3.3) hoe&.
j E JN t h e k e hoLdb
t h e t i g h t hand i n e q u a l i t y b e i n g v a l i d doh a L l inequaLity oMtg
NPn+ buch that
n
E
P, t h e L e d t
hand
n 2 N0 = N o ( j ) .
First we need the Gauss-Jacobi mechanical quadrature formula
89
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
I1
(3.5)
J-1
where with
x ,k
I
q2k-1 (u)du = .Ekj=1 X j ,kq2k-1 ( x j,k)
15 j 'k,
d e n o t e t h e r o o t s of t h e Legendre polynomial pk
-1 < x ,.+ l , k < ~ j , k < 1 , j = l 1 2 1 . . . l k - l ,
and
X
j ,k
t h e "Legendre
abscissas"
Now l e t
+
pn E NPn
.
Choosing
1 = pn A(0) =
(3.6)
m = U (n+l)/ 21
-+ 9+l j=1
o n e h a s by ( 3 . 5 )
j ,m+lPn ( x j,m+l 1
To p r o v e p a r t s b) a n d c ) we f i r s t assume t h e e x i s t e n c e Pn
E
NP',
a
of
s a t i s f y i n g ( 3 . 3 ) . By ( 3 . 6 ) a n d (3.7) one h a s
S i n c e a l l t e r m s i n t h i s sum are nm-negative and o n e h a s by ( 3 . 6 )
-
XjIWl(xm
+
-x
)
j ,m+l
#'a,
90
BUTZER, STENS and WEHRENS
(3.9)
Pn(xl,m+l) =
"l,m+l
*
T h e r e - > r e pn i s u n i q u e l y d e t e r m i n e d a t t h e p o i n s Since t h e
xi,m+l
,i = 2 , . . . ,m + 1 a r e
t i v e , i t must have
sesses ( n
n odd, i . e .
+
1 <j cm+l.
j,ml' -
i n (-1,1) and pn i s t o be p o s i -
m d o u b l e roots. T h i s i m p l i e s t h a t f o r even n
i s t h e only polynomial For
x
in
satisfying (3.3)
N P:
m = (n
+
a c o n t r a d i c t i o n t o ( 3 . 9 ) , proving c )
P
.
1)/2, (3.8) s t a t e s t h a t pn
1 ) / 2 d o u b l e z e r o s , meaning t h a t
E
~,(XI
pos
-is
I 0 . But t h i s
.
Concerning p a r t d ) , t h e r i g h t s i d e of
( 3 . 4 ) f o l l o w s from
where ( 2 . 1 3 ) , Cor. 1, ( 2 . 1 1 ) a n d t h e e q u a l i t y
(3.11)
(D1P . ) ( x ) = 3
(x E [-1,ll : j
E
P)
( c f . 155, Sec. 3 1 1 were a p p l i e d . I n o r d e r t o v e r i f y t h e l e f t s i d e of (3.4) f i x
j E IN. F o r
n
j
+
1 o n e h a s by ( 3 . 5 ) t h a t
91
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
.
b e i n g v a l i d as I P . ( x ) I i s e v e n , a n d xlln=-xnIn 7 From [ 57, S e e . 7.21 1 it now follows t h a t j P . ( x ) a t t a i n s i t s maximum 7 v a l u e i n [0,1 - q ] a t x = l -rl i f rl > 0 i s s m a l l enough. This t h e latter e q u a l i t y
means t h a t
max 0 5 x ~ x ,
c h o s e n l a r g e enough. So
jpj(x)I =
IP.(xJ
lln 3.12) y i e l d s t h a t
f
In view of B r u n s ' i n e q u a l i t y (see e . g .
5,
we f i n d with a s u i t a b l e
E (x lln
l f n) + (1
1 = P . ( l ) = P.(x
I
3
Since plete.
-
n
= P.(X
3
if
lrn
n
is
[57.(6.21.5)1)
, 1)
that
Xlfn)Ppn)
l i m n + m P I ( 5 ) = PI (1) = j ( j + 1 ) / 2 , 3
) j
7
t h e proof
of
i s con-
d)
0
The p o l y n o m i a l
pn o f
(3.10) s a t i s f i e s f o r even
n
E
the
DJ
same e x t r e m a l p r o p e r t y a s d o e s t h e t r i g o n o m e t r i c F e j g r - K o r o v k i n k e r n e l . T h e r e f o r e o n e may j u s t l y c a l l t h e k e r n e l (Legendre-) F e j 6 r - K o r o v k i n k e r n e l . I f c i s e l y t h e polynomials
p,
of
Kn
of
n i s e v e n , t h e Kn
(3.10) ; i f
(1.22)
the
are
pre-
n i s odd t h e n Kn(x) =Dn-,(x).
Concerning t h e approximation behaviour of t h e a s s o c i a t e d F e j g r -Korovkin c o n v o l u t i o n i n t e g r a l w e h a v e THEOREM 2.
Let
{KnInEp
b e .the L e g e n d h e - Fe j e h - K o t o v k i n
/Lchii&
dcdiirzcd
BUTZER,STENS and WEHRENS
92
b y ( 1 . 2 2 ) . One hub
6otr
f
E
X
L
The proof o f p a r t a ) f o l l o w s by Cor. 1. Concerning b ) , fELipl(a;X) i m p l i e s by (3.13) t h a t
u t ( G N ; f ) = O ( (1
-zN)a )
can be d e r i v e d e i t h e r from Prop. 5 s i n c e
f
*
=
O ( n - 2 a ) . Tileconverse
Kn E
Pn
C
:W
and
(see [ 5 5 , Sec. 51 ) , o r from Prop, 6. P a r t c) f o l l o w s r e a d i l y by Prop. 4, Thm. l d ) and t h e f a c t t h a t
1
-
= 1
K''(1)
L e t us remark t h a t Bavinck [ 2
- -xN
= O(n'2),
1 employs t h i s k e r n e l t o e s t a b l i s h
a Jackson-type theorem ( i n t h e J a c o b i f r a m e ) . I n [ 1 1 he also t r e a t s t h e external p r o p e r t y b u t does n o t d i s t i n g u i s h between t h e c a s e s
n
b e i n g even o r odd; f o r odd n h i s proof i s n o t q u i t e c o r r e c t s i n c e the q u a d r a t u r e formula used i s no more e x a c t i n t h i s case.Newman-Shapiro
1431 and Feinerman-Newman [ 2 5 , p. 1031 h a n d l e t h e e x t r e m a l
problem
f o r even n i n a somewhat more g e n e r a l frame b u t do n o t d i s c u s s r e s u l t s o f t h e t y p e o f Thm. 2 b )
,
c).
As an immediate consequence o f Thm.
positive kernel
{XnlnEP,
Xn
E
Pn
,
l d ) one h a s t h a t f o r e v e r y
t h e r e holds
An a p p l i c a t i o n of Prop. 4 a ) t h e n l e a d s t o t h e f o l l o w i n g a l g e b r a i c counterpart of a r e s u l t i n
[9,
p. 8 8 1 which i s r e l a t e d t o
a
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
93
result of Korovkin [ 3 6 , p . 128 1 .
COROLLARY 2:
dome
604
f
E
16
x
Xn
+
NPn, .then
E
i m p e i e b .that
f = const. (a.e.1.
In other words, one cannot approximate a non-constant function
f
X
E
by positive polynomial convolution integrals of degree n with
O(II-~).
an order better than
4.
THREE FURTER POSITIVE ALGEBRAIC CONVOLUTION INTEGRALS
Whereas the first example below is an algebraic polynomial
of
degree 2n, the second and third are non-polynomial.
4.1. THE SINGULAR INTEGRAL OF LANDAU-STIELTJES As remarked in the introduction, the classical Landau-Stieltjes
integral approximates functions [-1 +
E,
l - -] ~ for
E
E
f E cl-1,
11 uniformly
only
on
(0,l). Using the kernel of this integral we
now construct a new singular integral by means of the Legendre con
-
volution which forms an approximation process in ‘21-1, 11 aswell as in
The
Lp(-lll). This kernel is defined by
AZn belong to NP;n
, and
for the
one has
xo
= 1,
BUTZER,STENS and WEHRENS
84
1
2n (2n - 2 ) = ( 1 / 2 ) j 0 t-li2(1- t I n d t = (2n + 1) ( 2n
A2n
L e t us l i s t t h e p r o p e r t i e s o f
.-.1) . .. ...2, 3 ,
needed f o r t h e a g p r o x i m a t i o n t h e -
orem.
LEMMA 1:
a ) F O R all
n
E
= 1
A5n(l)
(4.3)
b) F a h each
Mn-1’2
(4.4)
-<
k
E
t h e h e haldh
IP
-
l/(n
IN, X h e R e
l-A;n(k)
5
+
1)XZn.
eXihth
a conntant M > 0
72k(k+l)/(r1+l)h~~
huch t h a t
( f E X; n
E
PI.
c ) The z j a l l o ~ u i n g B e h n h t e i n - t y p e i n e q u a l i t y h o l d s :
PROOF:
P a r t a ) f o l l o w s i m m e d i a t e l y by i n t e g r a t i n g t h e i d e n t i t y
By a p p l y i n g a ) and ( 3 . 4 ) w e o b t a i n t h e r i g h t - h a n d s i d e of i n e q u a l i t y (4.4).
AZn
Concerning t h e l e f t side, o n e h a s i n view of t h e p o s i t i v i t y o f
and ( 4 . 6 ) for e a c h
1
-
6
E
A$n(k)
(0,2), k
2
E
IN,
- J‘Il - 6 (1 - Pk ( u )A 2 n
(u)d u
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
Since
P'(1) = k(k
k i n €l-62u51
I Pi(u)l
f r o m below by ( 6 / 2 )
+
1 ) / 2 , w e may choose some
> 0.
',
*
A2n
E
P2n
C
Wi
X2n
( 0 , 2 ) such t h a t
be
estimated
The d e s i r e d i n e q u a l i t y now follows by ( 4 . 2 ) .
and
T h e r e f o r e w e have o n l y t o show t h a t
First let
E
Furthermore, t h e i n t e g r a l can
To p r o v e p a r t c) w e o b t a i n from [ 5 5 , f
6
95
n 1 2 . By t h e e s t i m a t e d x2-1 d 1 dx [ 2 dx A 2 n ( x ) J !
one f i n d s t h a t
= n/(n-l)
+ 15
3,
S e c . 31 and ( 1 . 1 3 )
that
96
BUTZER,STENS and WEHRENS
which y i e l d s ( 4 . 5 ) by ( 4 . 2 ) . A s i m i l a r c a l c u l a t i o n shows t h a t
n = 1; t h e c a s e
al s o holds f o r
n = 0 i s obvious.
(4.7)
0
As an a p p l i c a t i o n of Cor. 1, Prop. 5 and Prop. 4 we o b t a i n
THEOREM 3.
Foh
t h e ninguLah i n t e g h a l
06
Landau-Stieetjen
f
*
A2n,
f E X I thetre hoLdb a ) IIf
*
AZn-
L
fllx ( 2 4 u l ( l - [
c) T h e i n t e g h a e
cieh
o (n-li2) .
06
-1 ( n + l ) X Z n l ;f;X)
(n E PI.
Landau-Stieetjeb i n n a t u m t e d i n X w i t h
oh-
4 . 2 . THE INTEGRAL OF WEIERSTRASS The k e r n e l of Gauss-Weierstrass w i t h r e s p e c t t o t h e
Legendre
system i s given by
LEMMA 2:
T h e hexnee
wt(x) i n non-negative
604
ale
XE
[ - I l l1
8
t > 0,
and n a t i n hi e n t h e Betnntein-typL i n e y u a e i t y
The proof o f t h e f i r s t p a r t can be found i n Bochner [ 4 , p. 11461 o r Hille-Phillips -Nessel-Trebels (C,l)-sununable
[ 3 3 , p. 612 1 ,
and ( 4 . 9 ) can be deduced from a r l i c h
[ 31, (3.9)] s i n c e t h e Legendre series o f
f E X
is
(see Sec. 5 below).
An a p p l i c a t i o n of Cor. 1, Prop. 5 w i t h
Xn = w
~ ,,a n~d Prop. 4
97
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
yields
Foh t h e a l g e b h u i c G a u b b - w e i e h b t R a n 6 c o n v o k ? u t i o n ..integaaL
THEOREM 4.
f
*
wt , f
E X, -theee hoed6
a) I I f
b) f
*
E
(t
w t - fll, (24ul(e L -t ;f;X)
Lipl(cx;X) L
*Ilf
c) T h e i n t e g h a k ?
f
*
wt - f llx
* wt i n
=
O(ta)
(t
-f
o+;
O
n a t u h u t e d i n X u i t h o d e & O(t), t+O+.
Concerning the infinitesimal generator of the Weierstrass well as Abel method
Of
’ 0).
as
summation of the Legendre series see e.g.[33,p.610 1.
4 . 3 . THE INTEGRAL MEANS
In 1 5 5 , Sec. 3 1 there had been introduced the (Legendre-) integral means
Ahf, f
€
X, h
€
( - l f l ) .They are defined as the convo-
lution integral with kernel
(4.10)
K(x;h):=
I o
f
otherwise,
the Legendre coefficients of which are given by k = O
k E IN.
Note that
BUTZER, STENS and WEHRENS
98
Since
K(x;h)
the A f h
1. 0 , x E [ - l , l l , h
E (-1,l) and
l i m h + l - [ ~ ( * ; h l A ( l =1, )
d e f i n e an a p p r o x i m a t i o n p r o c e s s on X f o r
THEOREM 5.
Foh t h e i n t e g h a t meand
a ) IIAhf
-
f
%f
= f
*
h
-+
1
-.
K ( * ; h ) , f E X , o n e han:
I t x i M w Ll ( h ; f ; X )
(h E ( - 1 t l ) ) *
c ) T h e i n t e g h a e meand ahe n d t u t r a t e d in X w i t h o.tdek O(l-h), h +1-. PROOF:
P a r t a) f o l l o w s by (4.121, C o r . 1 and (2.121, y i e l d i n g
also
t h e d i r e c t p a r t of b ) . Concerning t h e i n v e r s e p a r t i n b ) , w e make u s e o f t h e B e r n s t e i n i n e q u a l i t y ( c f . [ 5 5 , Sec. 3 I
and a p p l y Prop. 5, s e t t i n g
Xn ( x ) = K ( x ; l
- l/n).
Finally part c) fol-
lows from Prop. 4 by ( 4 . 1 2 1 . 0
5.
THE INTEGRALSQFFEJER AND DE LA VALLEE POUSSIN
I n analogy t o t h e t r i g o n o m e t r i c t h e o r y , t h e FejLr k e r n e l r e s p e c t t o t h e Legendre s y s t e m i s d e f i n e d by
where
Dn
i s the (Legendre-IDirichlet kernel
with
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
The a s s o c i a t e d a l g e b r a i c F e j 6 r c o n v o l u t i o n i n t e g r a l
(recall (1.19),
( 1 . 2 0 ) ) i s a c t u a l l y t h e n-th
of t h e
Fn
f E X , The L1
-
norms
are u n i f o r m l y bounded (see [ 321) , i.e. there e x i s t s 1
Fn
such t h a t
IIFnll I
5
F, n E 3P
. Since
I.
(1
FA(k) =
with
*
(C,1)-mean of the partial
series ( 1 . 1 8 ) of
sums of t h e F o u r i e r - L e g e n d r e
( a n f ) := f
l i m n + m FA(k) = 1,
l i m IIf
(5.3)
n-
*
k E IN
- ) n+ 1
,
O c k c n
0
,
k'nf
, one
h a s by Prop.
1
( f E XI.
F n - fllX = 0
Next c o n s i d e r t h e d e L a V a l l g e P o u s s i n k e r n e l
(5.4)
v
mln
(XI :=-
m
D (x) + 1 zn k=n-rn k
I n view of t h e i d e n t i t y
(F-l
= 0)
(5.5)
it follows t h a t
If
m d e p e n d s on n i n s u c h a way t h a t
( ~ ~ [ - 1 , 1 1 m, ; n
E
P, men).
BUTZER, STENS and WEHRENS
100
t h e n IIvmInll1
5 MI n
IP. I n t h i s case Prop, 1 a g a i n i m p l i e s t h a t t h e
E
associated integral
f
*
v
forms an approximation p r o c e s s mln n o t i n g t h a t t h e Legendre c o e f f i c i e n t s of v ~ are , ~g i v e n by
(5.7)
1
,
O
~
(n+l-k)/(m+l)
,
n
-m
0
,
k >n.
I n contrast t o the trigonometric (Legendre-) Fej&-Korovkin
Fejk
kernel
.
~
n
-
m
+ 1 5k 5 n
and
that
t r e a t e d i n Sec. 3 , t h e (Legendre-)
k e r n e l f a i l s t o be p o s i t i v e ( e . g . F n ( - l ) = -1/2 i f [ 261 )
k
on X I
of
Fejdr
n is odd;-
also
Using ( 5 . 5 ) t h i s e n a b l e s one t o show t h a t t h e (Legendre-) de
L a V a l l g e P o u s s i n k e r n e l i s n o t a p o s i t i v e k e r n e l , a t least f o r cer-
tain
m,n E IP.
Hence one c a n n o t a p p l y C o r . 1 i n order to obtain direct
approximation theorems. W e p r o c e e d i n a n o t h e r f a s h i o n ; i n t h e p e r t
-
o d i c case it i s due t o S t e s k i n [ 5 3 1 ,
LEMMA 3 .
F a t a&b
f E X , n,m E Ip, m
5 n
thefie hold4
(5.8)
(5.9)
PROOF:
for a l l
If
From (5.7) and t h e c o n v o l u t i o n theorem ( 1 . 1 4 ) it follows that
Pn-m
E
pn-m
p E ( f ) denotes
a
polynomial of b e s t approximation i n
Pn t o
f
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
in
X,
IIf
*
101
then (5.10) y i e l d s Vm,n-
fllX
5 Ilf
The p r o o f of
*
v
-
( 5 . 9 ) b e i n g p r a c t i c a l l y t h e same as i n [ 5 3 1
,
we
w i l l o n l y s k e t c h i t . One h a s by ( 5 . 1 ) , ( 5 . 4 ) for all k , n E m , 2 k - 1 < n + 1 < 2 k
f
-
f
*
Fn =
n+ 1 i f - f *
vo
+
,o
xk-l 2j-l ( f - f * v . 1 j=1 2~-l-l,2J-l
One d e d u c e s by ( 5 . 8 )
< - 8F n + l
Z n + l E .( f i x ) , j=O J
which i s t h e d e s i r e d i n e q u a l i t y f o r
n
2 1.
If
n = 0 i t follaws by ( 5 . 8 ) ~ .
O f c o u r s e ( 5 . 8 ) g i v e s a u s a b l e estimate o n l y i n t h e case
when
102
BUTZER,STENS and WEHRENS
limn+m(n-m(n))=
Choosing e . g . m = U@nl f o r some
+a.
O
h a s i n view of Prop. 6 and La, 3
*
v , f E X I be t h e a L g e 6 ~ ~ a idce La k.&%e Poutdin m,n c o n w o h t i a n i n t e g h a t w i t h i n t e g h a l m = [Bn] I 0 < B < 1,dedined w i a (5.4).
THEOREM 6.
Let
f
Une had d o h e a c h
r
E
P
a ) IIf * v ~ , ~ f l l X- < Mn-2rw:(1
c) The i n t e g h a l
ahbithahy dunction
~p
*
-n-2;Drf;X)
(f E
wi;
n GIN).
i b n o t s a t u h a t e d i n X , i. e . doh any mtn ab i n P h o p , 3 t h e h e E X i h d b a n f E X , f #const.
f
v
( a . e . ) ( e . g , f any p o L y n o m i a l l , s u c h t h a t
I n t h e case
n - m r e m a i n s bounded, one can p r o c e e d s i m i l a r l y a s
f o r t h e F e j k means below, n o t i n g t h a t
f
*
Fn = f
f o r t h e Gegenbauer c a s e ) . Concerning t h e F e j & means
f
*
Fn
v
ntn
(see
[341
w e have by (51.9)~Prop.
6
and Prop. 4
One has d o h
THEOREM I:
b) f
E
f
L i pLl ( a ; X ) * / I f
*
*
Fn
Fn
f
-
E X,
n
+ m
f I I x = U ( n- 2 9
I
(0 < a < 1 / 2 1
I
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
c) T h e integhal
f
*
Fn
i n natuhated in X w i t h ohdeh
Note that Kallaev [ 3 5 1 not only showed that rated in
C [ -1, 11 with order
103
f
*
Fn
U (n-l),
is satu-
I) (n-l) but actually determined
the
i. e. the class of functions f for which F n - fllC = O(n-1) . It consists of all f E C[-1, 11 for which
saturation class, IIf
*
G E Lipl(l;C), where
In other words, he studied part b) in the case
a =1/2. For a point-
wise analogue of part a) see Rafal'son [451.
6.
THE ROGOSINSKI SUMMATION METHOD It is well-known that the convolution integral generated by the
Dirichlet kernel Dn of (5.2) fails to be an approximation processin X unless X = Lp(-l,l), 4 / 3 c p < 4 (see [441, 1421). Instead of Dn we now consider the "shifted" Dirichlet kernel
(n E IP
where
{xn}
i
x E [-1,1]) ,
is a suitable sequence of reals in (-1,l). This
may by Christoffel's summation formula be rewritten in
the
kernel closed
form
The construction (6.1) corresponds to the
trigonometric
Rogosinski
BUTZER. STENS and WEHRENS
104
k e r n e l (see [ 4 7 1 and e . g .
[9,
p. 56, 1 0 6 1
which forms an a p p r o x i m a t i o n p r o c e s s i f e . g .
5,
= s / ( 2 n ) . I n view o f P r o p , 1 t h e
xn
5,
=
1 ~ / ( 2 n+ 1)
or
i n ( 6 . 1 ) have t o be c h o s e n i n
s u c h a way t h a t
l i m P (x ) = 1 n+m k n
(6.3)
(k E m ) r e s p e c t .
lim x n+m
= 1,
for i n t h i s case
l i m R"(k) = l i m P ( x ) = P (1) = 1 k n+m n n-tm k n
(6.4)
The f o l l o w i n g lemma g i v e s s u f f i c i e n t c o n d i t i o n s f o r (6.2)
and
(6.3) t o hold:
Let
LEMMA 4:
wheae
Xn
{xnIn
Ep
c (-1,l) be d u c h t h a t
i n t h e l u n g e s t zeho
06
-
P n ( x ) , xo
atrbitaahy. Then
and ( 6 . 3 ) ahe v a l i d .
PROOF:
By Bruns' i n e q u a l i t y ( r e c a l l ( 3 . 1 3 ) ) w e have
(6.2)
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
for
n+m.
Here
This implies ( 6 . 3 ) . Applying Abel’s transformation twice
-APk (x,)
2Pk+l (x,)
105
+
Pk(xn)
-
-2 Pk (xn ) is the first and A Pk (xn ) =Pk+2 (xn)the second difference of {Pk (xn ) 1k E P ’
= Pk+l (x,)
Making use of the well-known representation (see e.g.[57, p. 881) 20s [ ( k + 1/2)arc cos u 1 d u (u - x)li2 (1 - u2) 1/2
and the generalized mean value theorem, we obtain with -2 A Pk(xn)
X(u) E (0,l)
=
Since 1 (u-xn)1/2(1 - u 2 ) 1/2 du 5
1
one can estimate the second difference using (6.6) by
1
1/2
d’
106
BUTZEA.STENS
and
WEHRENS
I n a s i m i l a r way one h a s f o r t h e f i r s t d i f f e r e n c e
and u s i n g Markov’s i n e q u a l i t y ( e . g . [ 3 6 , p. 99 1 ) f o r
x
E
(O,l),
Applying t h e s e t h r e e i n e q u a l i t i e s in ( 6 . 7 ) l e a d s t o
which y i e l d s (6.2) s i n c e
IIFklll
5
F,
k
E
IP
, and
(see e.g.
[321 1
I n view o f Prop. 1, La. 4 and t h a t (6.3) i m p l i e s ( 6 . 4 1 ,
there
follows
REMARK:
Note t h a t t h e assumptions of La. 4
are
fulfilled
in
the
107
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
p a r t i c u l a r case
xn
Cor. 3 . The k e r n e l
-
= x
" s o t h a t one may u s e
n R (x;;,) n
-xn
may a l s o be r e w r i t t e n f o r
T h i s form shows t h a t t h e Fejgr-Korovkin k e r n e l a c t u a l l y t h e s q u a r e of
i n s t e a d of
Rn(x;zn)
K2n-2
x
xn
#'En
in
as
is
of (1.22)
a p a r t from a n o r m a l i z a t i o n f a c t o r :
The a n a l o g o u s r e s u l t i n t h e t r i g o n o m e t r i c case, namely
that
k e r n e l i s e s s e n t i a l l y t h e squaire of a p a r t i c u l a r R o g o s i n s k i
this
kernel,
w a s f i r s t n o t i c e d by S t a r k [ 5 2 1 . L e t u s f i n a l l y c o n s i d e r t h e rate o f a p p r o x i m a t i o n i n ( 6 . 8 ) . By t h e c o n v o l u t i o n theorem one h a s f o r e a c h
c) The i n t e g h a l
x with vtdek PROOF:
Let
f
*
Rn
06
Pn
'n
Legendhe-Rogvbinbki i d n a t u k a t e d i n
O(n'2). p:
E
P,
be
a
p o l y n o m i a l o f b e s t approximaticm t o f
E
X.
.
BUTZER STENS and WE HRE NS
108
Using Minkowski's i n e q u a l i t y and ( 6 . 1 0 1 , one o b t a i n s
which i s t h e f i r s t i n e q u a l i t y of p a r t a ) ; t h e s e c o n d f o l l o w s by Prop. 6 a ) , ( 6 . 6 ) and ( 2 . 1 2 ) . P a r t b) f o l l o w s by a ) and P r o p , 6b) or (3.14) and Prop. 5. F i n a l l y t h e s a t u r a t i o n r e s u l t can be d e r i v e d from Prop. 4 since
Note t h a t K a l l a e v
351 states w i t h o u t p r o o f t h e s a t u r a t i o n class
o f a p a r t i c u l a r Rogosinski-type i n t e g r a l .
APPROXIMATION RATES EXPRESSED I N TERMS OF CLASSICAL
7.
MODULI
OF
CONTINUITY
The r e a d e r may o b j e c t t o t h e f a c t t h a t t h e r a t e s o f c o n v e r g e n c e
of t h e v a r i o u s c o n v o l u t i o n i n t e g r a l s g i v e n by p a r t s b) a n d a) of C0r. 1 and Thms. 2-8 a r e e x p r e s s e d i n terms o f moduli of c o n t i n u i t y
L i p s c h i t z classes which are d e f i n e d v i a ( r h f )( x ) 2
-
or
Legendre difference 1 f ( x ) and n o t t h e classical differences (\f) (x) = f (x+ h) - f ( X I ,
( A h f ) ( x ) = f ( x + h)
the
+ f ( x - h ) - 2f ( x ) . However,
t h e "Legendre
"
Lipschitz
c l a s s e s a r e a c t u a l l y e q u i v a l e n t t o c e r t a i n p o i n t w i s e Lipsckitz classes
109
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
defined via
( A i f ) ( x ) , i =1,2, a t l e a s t f o r
shown i n 155, S e c . 6
X = C [-l,l
1
as
was
1.
The p o i n t w i s e moduli of c o n t i n u i t y of f with respect t o t h e differences L i p s c h i t z c l a s s e s of o r d e r
a20
E
C [-l,l1
Aif ( x ) I i =1,2, a n d t h e a r e d e f i n e d by
I t w a s i n d e e d shown t h a t
(7.2)
I
1/21
P-Lipl (a;C )
(0 < a
P-Lip2 ( a ; C )
(0 < a < l ) .
L
Lip ( a ; C ) = 1
Using t h e s e r e s u l t s w e have
for x
E
[-1,1]
associated
BUTZER, STENS and WEHRENS
110
I f one i s o n l y i n t e r e s t e d i n d i r e c t a p p r o x i m a t i o n t h e o r e m s , it
i s a l s o p o s s i b l e tQc o n n e c t o u r r e s u l t s on r a t e s o f convergence w i t h
t h e c l a s s i c a l moduli of Lp
- spaces.
where
11
c o n t i n u i t y and L i p s c h i t z c l a s s e s ,
even
in
They a r e g i v e n by
I I x [a,bl i n d i c a t e s t h e X-norm t a k e n over t h e i n t e r v a l [a,bl.
By a well known r e s u l t on b e s t a p p r o x i m a t i o n ( c f . [ 2 4 , p . 1451) r m l y
we f i n d i n view o f Prop. 6b)
111
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
The l a t t e r t h r e e i m p l i c a t i o n s will now be used t o d e r i v e t h e
direct
r e s u l t s . I n t h e c a s e of t h e i n t e g r a l s o f de La V a l l g e P o u s s i n
and
Fejgr-Korovkin one o b t a i n s by (5.81, and Thm. 2b) t o g e t h e r w i t h
the
inequality
respectively, the following
COROLLARY 4:
one had
a ) Fax -the d i n g u l a k intqd
60'1- a n y
r
E
f
*
To e x t e n d t h e l a s t i m p l i c a t i o n t o
the assertion t o the case
Fax
[BnP ,n'
fEX, O < B < l ,
IP
b) Foh -the b i n g u l a t i n t e g t a C
COROLLARY 5:
f *v
X = C [ - l , lI
f ' E Lipl(l;C)
.
Xn
,€
E X,
v n e hub
a =1 w e have
t h e h e hoLds
to
restrict
112
BUTZER,STENS and WEHRENS
W e need t h e f o l l o w i n g i m p l i c a t i o n s
PROOF:
(7.8)
f ' E L i p l ( a ; C ) ; + f E L i p 2 ( l + a ; C )* f E P - L i p 2 ( ( l + a ) / 2 ; C )
(7.8) follows by ( 7 . 7 )
and t h e d e f i n i t i o n o f
c a n b e found i n [ 5 5 , S e c 6 1 , I f (7.8)
and ( 7 . 9 )
that
wt
f'
6 ; f ; C ) = O(1
L i p 2 , P-Lipz;
L i p l 1 ; ~ ,) t h e n
€
-
(OCaLl)
6 1 , 6 +l-,
one
(7.9)
has
by
yielding t h e as-
s e r t i o n by Thm. 2 b ) . o C o r . 4b) a n d C o r .
5 solve t h e problem p o s e d i n t h e i n t r o d u c t i o n
as w e l l a s i t s e x t e n s i o n n o t o n l y i n Lp(-lll)
,
12 p <
,
C[-1,1] -space
a p a r t form t h e c a s e
but also
f ' E L i p l ( l ; L p ) . The a 1
g e b r a i c d e La V a l l i e P o u s s i n sums h a v e a much
better
-
approximation
b e h a v i o u r . According t o Cor 4a) t h e y a c t u a l l y a p p r o x i m a t e f E X
for
w i t h t h e same o r d e r as do t h e a l g e b r a i c p o l y n o m i a l s
any g i v e n of
best
approximation. L e t us r e c a l l t h a t t h e i n t e g r a l s c o n s i d e r e d i n t h i s p a p e r c o n v o l u t i o n i n t e g r a l s o f t h e form ( 2 . 1 )
are
( t h e c o n v o l u t i o n b e i n g under-
s t o o d i n t h e Legendre s e n s e ) a n d n o t o f t h e form (1.4). These i n t e
-
g r a l s may, however, r e a d i l y b e r e w r i t t e n i n t h e form
I-,
1
with
x
P
(X;
u) d u = 2 , x E -1, 11
. For example , t h e
g r a l c a n be w r i t t e n a s ( 7 . 1 0 ) w i t h
Rogosinski. i n t e -
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
113
Another example o f a s i n g u l a r i n t e g r a l w r i t t e n i n t h e form (7.10)
Jlnf
was the integral
o f ( 1 . 2 ) . Lupag' r e s u l t ( 1 . 3 ) f o r t h i s i n t e -
g r a l c a n a l s o be d e r i v e d f r o m C o r . 1. I n d e e d , s i n c e
one h a s t h a t f o r e a c h IIJZnf
(7.11)
-
f E X
f l l X ( 2 4 u lL( l
-
3
n2 If
X = C [-1, 11
r
+ 3n + 3
( n E Ip).
;f;X)
a n e a s y c a l c u l a t i o n ( c f . [ 5 5 , Sec. 6
1 ) shows t h a t
which y i e l d s
T h i s i s ( 1 . 3 ) a p a r t from a c o n s t a n t . Using o u r methods i t c a n b s shown t h a t
0 (n-l-a) rated i n holds f o r
JZnf a p p r o x i m a t e s
provided
f
'
X with order
E
f
C [ -1, 1I
in
Lipl (a;C)I 0 < a
2 1.
- space
Moreover,
even
with order
JZnf i s s a t u -
l I ( n m 2 ) , A r e s u l t o f t h e t y p e o f Thm. 2b) also
JZnf.
L e t u s f i n a l l y c o n c l u d e w i t h a n u n s o l v e d problem. Does there n e x i s t a t r i a n g u l a r m a t r i x of d i s t i n c t nodes ( x ~ , ~ , } ~ =n ~ E Ip, -1 < x
-
k,n
< + -
n {qk,n ( x ) )k=O
1
and a t r i a n g u l a r m a t r i x
n E IP
, defined
m a t o r operators ( L n f ) (XI =
on [ -1, + 1 ]
c!=*
of
,
fundamental f u n c t i o n s
s u c h t h a t t h e l i n e a r sum-
f ( % , n ) 9 k , n ( ~ )f , E
p o s i t i v e a l g e b r a i c p o l y n o m i a l s o f d e g r e e n and
c 1-1,
11
I
are
BUTZER, STENS and WEHRENS
114
provided
f E L i p 2 ( a ; C [-1, 11 1,
The p o i n t i s t h a t t h e
0
pkrn
s h o u l d be d e f i n e d a s c o n s t r u c t i v e -
ly a s t h e c o r r e s p o n d i n g f u n c t i o n s i n t h e case o f t h e B e r n s t e i n polyn o m i a l s d e f i n e d over
[ -1, 11
9k,n(x) = 2
,
namely
-n n ( k ) (1
+
k
x) (1
-
a n d t h e s e a r e known t o a p p r o x i m a t e o n l y w i t h o r d e r
-a/2
O(n
)
under
t h e same h y p o t h e s i s .
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S. SHAPIRO, J a c k s o n ' s t h e o r e m i n h i g h e r d i -
m e n s i o n s . I n : Apphaximation T h e a h y ( P r o c . C o n f . , O b e r w o l f a c h , 1963; Eds. P. L. B u t z e r a n d J. Korevaar) Vol.
5 , B i r k h a u s e r Verlag, B a s e l - S t u t t g a r t ,
-
ISNM
1964, p.208
-218.
[44]
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Amer. Math. [45]
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, 19(1975),88-91.
APPROXIMATION BY ALGEBRAIC CONVOLUTION INTEGRALS
11s
[46]
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[ 49 I
M. SALLAY , Uber ein Interpolationsverfahren (Russ.sun.). Magyar Tud. m a d . M a t . Kutat6 Int. Kozl. 9(1964) , 607-615 (1965).
[SO1
R. B. SAXENA, On a polynomial of interpolation. Studia Math. Hungar. 2(1967) , 167 - 183.
[ 511
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[ 52 ]
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[53]
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Sci.
R. L. STENS, Charakterisierungen der besten algebraischen ap proximation durch lokale Lipschitzbedingungen. In: App h o x i m a b i o n T h e o h y (Proc. Conf., Bonn. 1976; Eds. R. Schaback and K. Scherer) Lecture Notes in Math. 556, Springer Verlag, Berlin-Heidelberg-New York, 1976, p. 403 415.
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[60 1
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- 128.
Approximation Theory and Functional Analysis J.B. ProZla (ed.) 0 North-Holland Fublishing Company, 1979
WEIGHTED APPROXIMATION
NON-ARCHIMEDEAN
J O S E PAUL0 Q.
CARNEIRO
I n s t i t u t o de Matemstica U n i v e r s i d a d e F e d e r a l do Rio d e J a n e i r o R i o de J a n e i r o , R J , B r a z i l
1. INTRODUCTION
F be a non-archimedean,
Let
non-trivially
i s l o c a l l y compact f o r i t s n a t u r a l t o p o l o g y ,
Hausdorff s p a c e , E o v e r F, a n d
r
v a l u e d f i e l d , which a zero -dimensional
X
a non-archimedean l o c a l l y convex Hausdorff s p a c e
t h e d i r e c t e d f a m i l y of a l l non-archimedean c o n t i n u o u s
seminorms on E
.
11
( F o r the d e f i n i t i o n of t h e s e t e r m s , see
,
[2
1
and [ 3 1 . ) d e n o t e s t h e v e c t o r s p a c e , o v e r F , o f a l l continuous func-
C(X,E)
t i o n s from X t o E, and L i s a f i x e d v e c t o r s u b s p a c e o f An L-weight i s a f u n c t i o n w from X t o F such t h a t uppersemicontinuous f
E
L. I f
f E L
+
topology
sup [ w ( x ) f (x)], f o r
XEX
T~
,
under which
space. I n t h i s case, If
S
p
E W,
L i s a non-archimedean
E S),
f o r each x
E
x
x
E
w
E
W
and
wighted
locally
such t h a t
lwil
X,
convex
we
El
a n d we s a y t h a t
w1,w2
5 h[w[, for 121
to
X
there exists
s ( x ) # 0. W is c a l l e d dihected i f , given and
r
(L,rW) i s c a l l e d a n o n - a h c h i m e d e a n Nachbin Apace.
v a n i s h i n g on X if f o r e v e r y
h > 0
E
endow L w i t h a
i s a c o l l e c t i o n o f f u n c t i o n s from
S ( x ) = ( s ( x ) E E; s
p
t h e n t h e non-archimedeanseminom
r ,w
E
is
p o (wf)
and n u l l a t i n f i n i t y , f o r e v e r y
i s a s e t o f L-weights,
W
C(X,E).
s E S
E W,
S
set
i s non-
such t h a t there exists
i = 1 , 2 . The f o l l o w i n g
122
CARNEIRO
f a c t s a r e clear: (i) I f
W i s n o n - v a n i s h i n g on
(ii)If
W is directed,
for
r, w
p E
then ( L , T ~ ) i s Hausdorff.
XI
p [ w ( x ) f ( x ) ]< €1,
t h e n t h e s e t s i f E L;
E W,
E
> 0 , form a b a s i s o f 0-neighborhoods
for ( L , T ~ ) . (iii)I f
W
i s n o n - v a n i s h i n g a t X I t h e n , f o r e a c h x E X, 6,:
L
+
E,
6 x ( f ) = f ( x ) , i s c o n t i n u o u s and l i n e a r .
d e f i n e d by
2. EXAMPLES L = C ( X , E ) and W i s t h e s e t o f F - c h a r a c t e r i s t i c fun*
(i) I f
t i o n s o f compact s u b s e t s o f open t o p o l o g y i n ( i i )I f
then
X
compact-
is t h e
T~
C(X,E).
L = Co(XIE)I
t h e v e c t o r s p a c e of a l l c o n t i n u o u s f u n c -
t i o n s from X t o
E which v a n i s h a t i n f i n i t y , a n d W = { l } ,
then
i s t h e uniform topology i n
‘cW
In particular, i f (iii)I f
X
Co(E,X).
i s compact, w e g e t C o ( X , E ) = C ( X , E ) .
X i s l o c a l l y compact, L = C b ( X , E )
,
the vector
o f a l l bounded c o n t i n u o u s f u n c t i o n s from
,
W = Co(X,F)
Cb(X,E)
-
then
X
to
El
i s c a l l e d t h e strict topology
‘cW
W e n o t i c e t h a t i n a l l t h e s e c a s e s W i s non-vanishing and t h a t , i n t h e f i r s t t w o cases, W
at
and in
X,
is directed.
3 . THE BOUNDED CASE OF THE NON-ARCHIMEDEAN Given a s u b a l g e b r a A o f
space
BERNSTEIN-NACHBIN PROBLEM
C(X;F) a n d a Nachbin s p a c e
( L , T ~,)
t h e non-archimedean B e r n s t e i n - N a c h b i n problem c o n s i s t s i n d e s c r i b i n g t h e c l o s u r e of a v e c t o r s u b s p a c e t h a t is, s u c h t h a t For p ( x ) = V(y)
?,:
M C L
which is a module o v e r
A,
AM C M.
y E XI
we say t h a t
f o r every
~p E
A.
x I y
W e d e n o t e by
(mod A) X/A
i f and
only
t h e quotient of
if
X
1 23
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
by t h i s e q u i v a l e n c e r e l a t i o n . Each
of X , o n e e a s i l y sees t h a t
being a closed
Y E X/A
i s a s e t of
WIy
I ,T
Lly-weights,
closure
cides with (or, equivalently, contains) the set that
fly
belongs t o the
-c
wlY
c l o s u r e of
MIy
and then
i s t h e n s a i d t o be
) i s a non-archimedean Nachbin s p a c e . M WIY t o c a t i z a b t e undeh A i n (L,-rW) i f t h e
(L
subset
of
M in
= If
ZA(M) in
L coin€
L
such
Lly}.
The h e b t h i c t e d non-archimedean B e r n s t e i n - N a c h b i n p r o b l e m
s i s t s i n a s k i n g f o r c o n d i t i o n s u n d e r which l o c a l i z a b i l i t y
con-
holds.
The bounded cane of t h e non-archimedean B e r n s t e i n - N a c h b i n prob-
l e m o c c u r s when t h e f u n c t i o n s i n A a r e bounded on t h e every function i n W
.
support
of
We h a v e t h e n t h e f o l l o w i n g :
THEOREM 1:
I n t h e bounded c a h c , ewekg
l o c a l i z a b l e undet
PROOF:
in
A
A-moduee c o n t a i n e d i n
L
i h
(L,rW).
The p r o o f i s a n a d a p t a t i o n o f t h e one g i v e n by Nachbin ( [ 4 ] )
i n t h e real case, and i s b a s e d o n t h e f o l l o w i n g
LEMMA:
LeZ A
b e a u n i t a k y c t o d e d hubaegebha
w i t h t h e hupkemum naktn. 1 6 , doh eweky bet and
06
X w h i c h i n d i n j o i n t Sham
(pl,.
.. , p n
E
A
PROOF OF THE LEMMA:
and
6 :Cb(X,F)
Cb(X,F)
Y E X/A, K y
equipped
i n a compactnub-
Y , t h e n thehe e X i b t
Z1,...,Zn
E X/A
huch t h a t :
n (iii) Z pi(x) = 1, i =1
X
06
Let +
x E X. BX
b e t h e Banaschewski c o m p a c t i f i c a t i o n o f
C(BX,F) t h e i s o m e t r y s u c h t h a t
p ( f ) = Bf is the
CARNEIRO
124
o n l y c o n t i n u o u s e x t e n s i o n of of
A under
f?
,
f
BX (see [ 5 J ) . Then
to
i s a u n i t a r y c l o s e d subalgebra of
t h e q u o t i e n t space
n
and
f?X/f?A
:
fix
+
C(BX,F). I f
Y E G I w e have t h a t
t h i s case, Y n X E X/A.
is d i s j o i n t f r o m
Y n X f (I
Then
KynX
Lemma 2.1,
Y1,..
.,Yn
and
Vi
(i) hiln(K ( i i ) llhiIlm
Letting
Jli
element o f
hlr...,hn
2 1
BX/BA,
i~ E
E II ( X ) = G
By such
\T(KZ )
,
i G . By [6!,
C(GrF) such t h a t :
i = l,...,n;
= 0
= hi o
E
a.
=
i s an open c o v e r i n g of
(ViIlLiLn
w e can f i n d
Y E n ( X ) and, i n
x(Ky
121
then
) . NOW,
i s a compact s u b s e t o f X I which
Y n X. T h i s i m p l i e s t h a t
i = 1, . . . , n ,
(see [ 6 1
i f and o n l y if
the f i n i t e intersection property, there exist n that ( K ~ in x = 4. P u t t i n g zi = Yi n X for
is
G
the canonical projection,
G
t h e n G i s compact, H a u s d o r f f , a n d z e r o - d i m e n s i o n a l for
BA, the image
i = l,...,n
C(BX,F) , t h e n e a c h
Jli
i s constant
in
each
so t h a t , by t h e non-archimedean S t o n e - W e i e r s t r a s s
Theorem f o r compact Hausdorff z e r o - d i m e n s i o n a l s p a c e s (see [ 6 I ) , each
Jli
belongs t o
BA. P u t t i n g
pi = Jlilx1 we g e t t h e d e s i r e d r e s u l t .
Now w e go back t o t h e proof o f t h e Theorem. I f a l g e b r a of
B is the
C(X,F) g e n e r a t e d by A and t h e c o n s t a n t f u n c t i o n 1, t h e n
i t i s clear t h a t M i s an A-module i f and o n l y i f i t i s a
and t h a t M i s l o c a l i z a b l e under c a l i z a b l e under
and
A
11 E
such t h a t , f o r The set
r
i n L i f and o n l y i f i t
A C Cb(X,F) and
L e t then
Cb(X,F). C l e a r l y , X/A = X / i . p = 11
be g i v e n . F o r e a c h
x E Y Ky =
m U
B-module,
is
lo-
Thus, w e c a n assume A t o be u n i t a r y .
B i n L.
Suppose f i r s t t h a t in
sub-
and
j -1,
{x E X;
j =1 p a c t and d i s j o i n t from Y.
let
be t h e c l o s u r e of A
f E LA(M), € > O f wlI...,wrnE there exists
Y E X/A,
...r m ,
Iwj(x) (Ilf(x)
Iwj(x)[ Ilf(x)
-
g,(x)ll
- gy(x)ll 2
E)
By t h e Lemma, w e c a n f i n d Z 1 , . . . , Z n
W
gy E M
< E.
i s comE X/A
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
and gi
--
, ... , 9, E 5 , we g e t , for
91
gzi
I
Iw,(x)
such t h a t
lPi(x)
for
x
E
I
Ilgi(x)
E
X,
1
KZi
..
= 0,
X, i =1,. ,ri , j
x
as one c a n see by d i s t i n g u i s h i n g I t follows t h a t ,
I
9,
-
f(x)II
5
the cases
125
llPiIlm
5
. . ,m:
..
Letting
1
= 1,.
I,
E [ ~ ~ ( x )
x
E KZ
j = I,...,m,
i
and
x
B
KZi.
n
so t h a t
aigi
As
E AM C
M,
t h i s proves t h a t
f
b e l o n g s t o t h e c l o s u r e of
M
i n L. F i n a l l y , i n t h e g e n e r a l case when E E
in
A, w
> 0,
E
wl,
W,
A C C(X,F)
and
for
each
i s bounded on t h e s u p p o r t of w, g i v e n f E =CA(M), m ,wm E W, p E r , w e t a k e Y = U supp w j , which is closed j =l
...
X I and g e t
A l y C Cb(Y,F). Replacing
X , W, L, A , M I
T~
by
126
CARNEIRO
y, w l y r L l y ' A I Y f M I y ,
s u l t , obtaining t h a t in
Lly
.
such t h a t
But s i n c e
w . (x) = 0
3 proves t h a t f
AlY
i s c o n t a i n e d i n t h e c l o s u r e of
(MIy)
COROLLARY:
if
in
A
(i) Given
€
-
E,
f o r x EY, j=l,...,m.
t h i s holds f o r every
.rW-closure of
in
M
L.
which
X E X,
So M i s
lo-
A t p a h a t i n g and we u h e i n t h e bounded cane, t h e n : f
E
TW-dUA6uhe a d M .in
t o the
belongb
f ( x ) b e t o n g n t o .the c t o s u k e
x
we c a n f i n d
fly E xAly(Mly),
g(x))] <
MIy
(L,T~).
M(x) i n
06
doh
E,
x.
i n denoe i n
( i i )M
x E X \ Y,
f E L, t h e n
.id
evehy
x
3
A .id
76
L
p[w.(x)( f ( x )
belongs t o t h e
c a l i z a b l e under
PROOF:
d:
Since i t i s e a s i l y seen t h a t
MIy
gly E
r e s p e c t i v e l y , w e a p p l y t h e p r e v i o u s re-
TW l Y
L,
M(x) in d e n n e i n
id
L(x),
doh evehy
x.
(i) Since
t o n s . Given t h e n
A i s s e p a r a t i n g , t h e e l e m e n t s of > 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 <
€
g(x))] <
X/A a r e s i n g l e there exists,
X,
A k ' E,
where
for j = l , - . . , m ,
The r e s u l t f o l l o w s by Theorem 1.
( i i )F o l l o w s from ( i ) .
REMARK:
I t W i s non-vanishing
on
X,
then t h e converses of
( i ) and
( i i ) i n t h e C o r o l l a r y h o l d , even w i t h o u t t h e h y p o t h e s e s on A .
4 . THE SCALAR CASE
In the scalar case, E = F a b s o l u t e value. I f t h e subspace subalgebra A of
i s (non-archimedean) normed by L of
C(X,F) c o n t a i n e d i n
t h e n , as p a r t i c u l a r c a s e o f Theorem 1:
C(X,F) i s f i x e d , t h e n L i s a n A-module.
We
the every have
127
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
l n t h e b o u n d e d c a s e , e u e h y subaLgebha
THEOREM 2:
A
06
con-
C(X,F)
t a i n e d i n L in LocaLizabCe undeh i t n e L d i n (L,rW).
THEOREM 3 ( S t o n e - W e i e r s t r a s s ) :
Let A b e a s u b a l g e b ~ a
i n
C(X,F)
C ( X , F ) , and aAdUme t h e bounded cade.
contained i n t h e subspace L ad
f E L
Then, i n ohdeh t h a t a g i v e n
06
~ ~ - c L o ~ u0 6h e A
belongs t o t h e
L, it is su,jdicien-t - t h a t b o t h (i) q ( x ) = 0,
d o h euehy
g
( i i )g ( x ) = g ( y ) , d o h e u e h y
16 W flee e h b ah y
i b
.
E A,
Y
E
X/A,
impeies
( i i )i m p l i e s t h a t
h be t h i s c o n s t a n t v a l u e . Fixed
that there exists constant i n
Y,
g
f(x)
= f(y). ah&
aLso
x
E
Y,
then
j =1,.
such t h a t
is constant i n
xo E Y , i f
h
Y.
# 0 , ( i )i m p l i e s
..,m.
E
9(Xo)
> 0.
So t h a t
wl,...,w
i,t h e
m
]w.(x)j I h ( x ) - f ( x ) I = O < E , I
W is non-vanishing on X I and
i W - c l o s u r e of
q ( x ) = 0 , for every
E W,
f E -fA(A).
As t o t h e n e c e s s i t y , assume t h a t
f belongs t o
f
such t h a t g ( x o ) # 0 . Since q i s also h and h i y = f l y . ( I f X = O , h=O h = g E A
E A,
w i l l do). Then, g i v e n
that
f ( x ) = 0 , and
nun-vaninhing on X , t h e n t h e s e c o n d i t i o n s
f E - f A ( A ) . Given
for
g
impLies
For t h e s u f f i c i e n c y , i t i s enough, by Theorem 2, t o show t h a t
PROOF:
Let
E A,
A
g E A, t a k e
in
L.
Then i f x
Ax
as i n
E
5 1 , and
X is
we
get:
s i n c e E i s H a u s d o r f f , so t h a t f o r every f(x)
f ( x ) = 0. S i m i l a r l y , i f
q E A, t h e same r e a s o n i n g w i t h
= f(y).
6,
-
6y
g(x) =g(Y), shows
that
128
CARNEIRO
then f bdongn t o t h e c o m p a c t - o p e n (i) g(x) = 0,
4 u h euetiy
g
( b ) Id
a nubalgebha
A
06
f E Cb(X,F), t h e n
and
Co(X,F)
i n L o c a l l y compact,
X
f(x) = 0
g E A, impeieb
Longb t o t h e uni6vhm cLonuhe
(c) 16
id and o n d y id
A
impeien
A,
E
( i i )g ( x ) = g ( y ) , 6oh e u e h y
06
CL#bUhe
f E CO(X,F), t h e n
06
A i n a nubalgebha
be-
and
Cb(X,F)
b e l u n g b -to t h e A t h i c t c l o b u h e
f
f
id and v n d y id
A
06
f(x) = f(y).
06
i d and
A
o n l y id (i) g(x) = 0,
( i i )g ( x ) = g ( y ! ,
g E A, i m p e i e b
euehy
doh
f ( x ) = 0.
g E A, i m p l i e d
d o h euehy
f(x) = f(y).
5. DENSITY I N TENSOR PRODUCTS S and T are, r e s p e c t i v e l y , v e c t o r subspaces of C(X,F) and
If E,
then
S 8 T
t h e form
x
+
d e n o t e s t h e set of a l l f i n i t e sums o f f u n c t i o n s
s ( x ) t , with
s
t
E S,
E
T.
a r e z e r o - d i m e n s i o n a l Hausdorff s p a c e s , and t i v e l y , vector subspaces of
C(X1,F)
denotes the s e t of a l l f i n i t e (x1,x2)
+
THEOREM 4 :
sl(xl)s,(x,),
with
sums
s1
E
and
of
S1 and S2
are,
C(X2,F), the
is an A-module,
s i n c e A i s non-vanishing
at
functions
A 8 E
i n
and ( A 8 E) ( x ) = E ,
X.
and
s1 of
X2
respec-
then
1 6 A i n bepaha-ting and n o n - v a n i b h i n g on X ,
A 8 E
Corollary.
X1
s2
t h e form
S1, s2 E S 2 .
and i6 we a t e i n t h e bounded c a n e , t h e n
PROOF:
Similarly, i f
of
A 8 E C
L,
T w - d e n n e i n L.
f o r every
x E X,
I t s u f f i c e s t h e n t o a p p l y Theorem 1,
129
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
COROLLARY 1:
( i ) C(X,F) 0 E
i b
C ( X , F ) , d o t t h e compact
dens e i n
open
-
t o p a e o g y. (ii) I d doh-
X
is l o c a l C g compact,
(K(X,F)
t h e unidoam t o p o l o g y .
tinuoun (iii) I d
K[X,F) 8 E
X
is d e n b e i n Co(X,E), con-
i n the s e t od abl
6unct i onn w i t h compact b u p p o h t ) .
bCaeah
is eocaley compact, Cb(X,F) 8
in d e n b e i n %(X,E),
E
topology.
d o t the ntaict
COROLLARY 2 (Dieudonng) : (i) (C(X1,F)
Q C(XZ,F))
8 E
id
dense i n
C(Xl
x X2,E),
do&
t h e compact-open t o p o e o g y . ( i i )C(X1,F) @ C ( X 2 , F )
in denbe i n
C(X1
X2)
x
@ F.
6 . EXTENSION THEOREMS
THEOREM 5:
Id E i d a non-aachimedean Fhzhchet bpaCe a u e h F , and
i n a non-empty compact s u b n e t X,
t h e n eueay
04
t h e z e k o - d i m e n s i o n a l Haundoh6d o p c e
E - uat ued COntinUOUb d u n c t i o n o n Y can b e ext ended t o
a bounded co n ti nuous d u n c t i o n o n PROOF:
X.
.
W e w i l l employ a t e c h n i q u e due t o D e La F u e n t e [ 7 I
l i n e a r mapping
Ty : C ( X , E )
* C ( Y , E ) , d e f i n e d by
c l e a r l y c o n t i n u o u s f o r t h e compact-open For
Y
S C C(X,E),
denote
u n i t a r y s u b a l g e b r a of
Ty(S)
by
C ( Y , F ) , and
S i n c e t h e c o n s t a n t f u n c t i o n s belong By Theorem 1, C o r o l l a r y ,
Assume f i r s t t h a t
Cb(X,E) X
Iy
Sly.
topologies i n both Then
M = Cb(X,E)
to
Ty(f) = f l y
i s an
,
is
spaces.
ly
A = Cb(X,F)
ly
The
is
a
A-module.
M , M ( x ) = E l f o r each X E Y .
is dense i n
i s compact. Then C ( X , E )
s p a c e , a n d so i s i t s q u o t i e n t by t h e c l o s e d s u b s p a c e
C(Y,E).
is
a
Fr6chet
K = T-l(O) Y
.
Now
CARNEIRO
130
we c l a m t h a t
,
C (X,E)
C ( X , E /K
i s l i n e a r l y and t o p o l o g i c a l l y i s o m o r p h i c t o
f o r which i t i s enough t o p r o v e t h a t
homomorphism. I n d e e d , g i v e n
U,
a b a s i c neighborhood of 0 i n
then
U = {g E C ( X , E ) ; p [ g ( x ) ] <
E;
x
Then
V = {h
E;
x E Y}
of
E
C(Y,E); p[h(x)] <
E X}
C ( Y , E ) . S i n c e it i s e v i d e n t t h a t
0 in
is atopological
Ty
f o r some
i s an open Ty(U)
C
g E C(X,E).
j o i n t from that
9
is
Y.
G = { t E X;
0 on G , 1 on Y , and f E U
Therefore, C(X,E)
V n [C(X,E)
ly
and
NOW,
1
9
= Cb(X,E)
jy
g = hix
THEOREM 6:
16
there exists
9 E C ( X , F ) such
< 1 on X . Then f = 9 g E C(X,E)
i s c o m p l e t e , and Cb(X,E)ly
BFX
thus
h E Ty(U). closed
in
= C(Y,E).
t h e Banaschewski compact-
h E C(BFX,E)
such t h a t
ly.
Then,
f =hly.The
i s the required extension.
E i n a nun-ahchimedean Fhzchet npace
i n a cloned n u b n e t
ly 1,
i s compact a n d d i s -
X. B y t h e p r e v i o u s r e s u l t , C ( Y , E ) = C ( B F X , E )
f E C(Y,E),
function
E)
T y ( f ) = h , which p r o v e s t h a t
i n the general case, take
i f i c a t i o n of
06
oveh
F , and
Y
t h e zeho-dimennianaL LocaLLy compact HaUAdOh66
s p a c e X , t h e n evehy 6unction in
PROOF:
I
2
> 0.
h = g l y E V,
there exists
X,
Since i t i s a l s o dense, we g e t
C(Y,E).
tion i n
p[g(t)]
By u l t r a - n o r m a l i t y o f
i s such t h a t
given
Then
E
neighborhood
i t i s enough t o p r o v e t h e r e v e r s e i n c l u s i o n . L e t t h e n with
r,
p E
C(X,E),
Co(Y,E)
can be extended to a d u n c -
Co(X,E).
W e o m i t t h e p r o o f , which i s s i m i l a r t o t h a t o f Theorem 5 .
REFERENCES
[ 1]
A. F. MONNA, Analyne nun-ahchimzdienne, E r g e b n i s s e 6er MathemWr und i h r e G r e n z g e b i e t e , Band 5 6 , S p r i n g e r - V e r l a g , B e r l i n , 1970.
131
NON-ARCHIMEDEAN WEIGHTED APPROXIMATION
[21
L . N A R I C I , E.
BECKENSTEIN a n d G. BACHMAN, F u n c t i o n a l A n a l y n i n and V a l u a t i o n T h e u h y , P u r e a n d A p p l i e d M a t h e m a t i c s , v o l . 5 , Marcel D e k k e r , I n c . ,
[ 31
J. P . Q.
New York, 1971.
CARNEIRO, Aphoximacaa Pondehada naa-ahquimediana,(Doc-
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 d a d e F e d e r a l d o Rio d e J a n e i r o , 1 9 7 6 ; An. A c a d . B r a s . C i . [ 4
1
W e i g h t e d A p p h o x i m a t i a n d o h ALgebhan and MaduLed 0 6 C a n t i n u o u n F u n c t i o n o : R e a l and S e l d - A d j o i n t Complex Caben,
L. N A C H B I N ,
A n n a l s of M a t h .
[51
50 ( 1 9 7 8 ) , 1 - 3 4 .
G.
BACHMAN, E.
81 ( 1 9 6 5 1 , 289
- 302.
BECKENSTEIN, L. N A R I C I a n d S . WARNER,
R i n g s of
c o n t i n u o u s f u n c t i o n s w i t h v a l u e s i n a topological f i e l d , T r a n s . Amer. M a t h . [6
1
J. B.
SOC. 2 0 4 ( 1 9 7 5 ) , 9 1 - 1 1 2 .
PROLLA, N o n a r c h i m e d e a n f u n c t i o n s p a c e s . To a p p e a r
in:
L i n e a h Spacen and A p p h o x i m a t i a n ( P r o c . C o n f . , O b e m l f a c h , 1 9 7 7 ; E d s . P . L . B u t z e r a n d B . SZ. - N a g y ) , ISNM vol. 40, B i r k h a u s e r Verlag, B a s e l - S t u t t g a r t , [ 7
1
A.
1978.
DE LA FUENTE, Algunon h e d u l t a d a n n a b h e a p h a x i m a c i o n d e d u n -
c i o n e o v e c t o h i a l e n t i p a t e o h e m a W e i e h n t h a b n - S t o n e . , Doct o r a l D i s s e r t a t i o n , Madrid, 1 9 7 3 . [ 8
1
L. NACHBIN, Elementn Co.
[9
1
Inc.,
06
Apphoximatian Theahy, D .
1 9 6 7 . R e p r i n t e d by R . Krieger C o .
Van N o s t r a n d
Inc., 1976.
J. B . PROLLA, A p p h o x i m a t i o n a d V e c t o h V a l u e d F u n c t i o n n , NorthH o l l a n d P u b l i s h i n g Co.,
Amsterdam,
1977.
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Appro&mation Theory and Functional Analysis J.B. Prolla ( e d . ) 0 North-Holland Publishing Company, 1979
T H E O R I E SPECTRALE EN UNE INFINITE DE VARIABLES
JEAN- P I E RRE FE R R I E R I n s t i t u t d e Mathsmatiques P u r e s
U n i v e r s i t 6 d e Nancy 1 5 4 0 3 7 Nancy Cedex, F r a n c e
1. L ' u t i l i t g d ' u n e t h 6 o r i e s p e c t r a l e e t d ' u n c a l c u l f o n c t i o n n e l h o l o -
morphe e n une i n f i n i t 6 d e v a r i a b l e s a 6 t 6 m i s e e n l u m i s r e p a r l a rec h e r c h e d e c o n d i t i o n s d ' u n i c i t 6 p o u r l e c a l c u l f o n c t i o n n e l holomorphe d ' u n nombre f i n i d e v a r i a b l e s e t d e s a l g s b r e s d s p e c t r e s noncompacts (cf 1 2 1 1.
Disons, de
schgmatique,
f-1
que l ' u n i c i t 6 e s t
6tablie
pour undomaine s p e c t r a l pseudoconvexe e t e n p a r t i c u l i e r p o l y n o m i a l e ment convexe e t q u e , d ' a u t r e p a r t , t o u t domaine d e
an
peut s ' i n t e r -
p r 6 t e r comme l a p r o j e c t i o n d ' u n domaine p o l y n o m i a l e n e n t convexe, mais d ' u n nombre i n f i n i d e v a r i a b l e s . D e faCon c l a s s i q u e , 6 t a n t donn6e une a l g s b r e
A , commutativeet
2 6lGment u n i t 6 ( t o u t e s l e s a l g s b r e s s e r o n t suppos6es &sorimis telles), on se donne d e s 616ments
a = ( al , . . . , a n )
o ( a ) de
de
Cn
al,...'a
A e t on d 6 f i n i t
de
n
comme l ' e n s e m b l e d e s p o i n t s
t e l s que l ' i d 6 a l engendrg p a r
al
-slI..
le spectre
s = ( s l , . . . , sn)
.,an - sn
s o i t pro
-
p r e , p l u s p r 6 c i s 6 m e n t comme l e f i l t r e d e s c o m p l 6 m e n t a i r e s d e s p a r t i e s S, d i t e s spectrales,
s u r l e s q u e l l e s on p e u t t r o u v e r
s +. u . ( s ) b o r n 6 e s v g r i f i a n t 1
2.
Z(ai
-
si)ui(s)
des
fonctions
= 1.
Pour d 6 c r i r e une s i t u a t i o n s e m b l a b l e e n d i m e n s i o n i n f i n i e : il e s t
n a t u r e 1 de remplacer donn6e d e
a l l . . . ,an
Cn
p a r un e s p a c e l o c a l e m e n t convexe
E
et la
p a r celle d'une a p p l i c a t i o n l i n 6 a i r e born6e 133
a
134
FERRIER
du dual E' de E dans
A.
La notion de spectre correspond alors 5 ce qui suit: systsme fini
cp =
(cpll...,9n) d'616ments de E'
plication lingaire continue a
9
,.. . ,a (cp n 1 )
= (a (ql)
semble
Scp
de E dans
9
et son spectre a
spectral pour
s
un
dire uneap-
Cn, on peut considgrer
o(a9)
v6rifiant des conditions d'uniformitg: n
de c p .
c'est
, plus
prgcisgment un en-
On s'intgresse 5 des familles
9.
fonctions ui exprimant que
,
pour
(SV)
est fixe (ou majorg),et les
E o ( a ) sont born6es
indgpendamment P Ainsi les ensembles spectraux sont-ils remplacgs par des fa-
milles d'ouverts
-1
~2 =
cp
9
S
(S
9
)
cp
qui s'ordonnent en un systsme projec-
tif. Avec les notations qui prgcgdent le calcul fonctionnel classique est un morphisme
f
+
f[ a 1 de l'algsbre
O(6s)
des fonctions ho-
lomorphes 1 croissance polynomiale sur le domaine spectral S dans A , c'est 5 dire telles que f 6:
soit bornge pour un certain entier
ofi
CN
est la distance dans
fiS
L'algibre
A
N,
au complgmentaire de S .
qui intervient en dimension infinie ades 618ments
de la forme
f = Z X f 9
9
correspondant 5 une famille spectrale (S9) I f
9
E
O(S
s9
tes de 9 .
(2)
)
oii
I
hcpI <
ml
ofi
et vgrifie dans cette algsbre des majorations indgpendanLe calcul fonctionnel s'obtient en posant
f[a]
=
CX
f [ a 1.
v 9
v
P l u s pr6cissment on le dgfinit d'abord pour des sommes telles que (1)
n'ayant qu'un nombre fini de termes, et on le prolonge
au complGt6, l'alggbre
&t
par
passage
ayant 6tG dgfinie elle-mGme de cette %on.
THcORlE SPECTRALE EN UNE INFINIT$ DE VARIABLES
135
3 . Un problsme, clef pour l'unicit6 du calcul fonctionnel en un non-
bre fini de variables, se pose: peut-on consid6rer l'algsbre & c m une algsbre de fonctions sur un domaine z de E ? De faqon 6vidente
aq dgfinie par
si Z est la partie de la limite projective des
on a un morphisme de
(R sur une algsbre
dz de
fonctions sur Z,dont
l'injectivit6 n'est malheureusement pas Claire. S'il n'y
a
probleme dans le cas d'un produit, la situation n'est pas dans le cas d'un produit fibr6 sur un domaine de nier est pseudoconvexe (cf [ 1 1 I
[ 2
1
pas
de
6lucidge
Cn, sauf si ceder-
).
4 . Dglaissant ici le probldme de savoir si les fonctions holomorphes
du calcul fonctionnel sont des fonctions, concentrons-nous
sur
spectre et cherchons si on peut remplacer dans certains cas le &me
projectif des
sys-
aq par un domaine 52 de E. Pour cela il faut
pouvoir connaitre des familles (S
Ip
)
2 partir de la seule donnge
I1 est nature1 de considgrer, pour n donngI les familles continues d'applications lingaires 9 de E dans des parties
Sp
de
le
Cn telles que
Sq
den.
*-
Cn etles familles
contient l'image par 9 de
n,
ce qui se traduit plus exactement par le fait que
(3)
I1 faut noter 5 ce sujet que la dernisre condition g6nGral impossible pour
L'ouvert
A
90
,
avec A parcourant
1 0 , 11
rend
,
en
le choix
sera spectral si pour tout choix (S9 ) conforme
ce qui pr6csde on a
S9 E o ( a ) avec uniformit6 par rapport 5
v
9.
5
136
FERRIER
Un c a s p a r t i c u l i s r e m e n t s i m p l e e s t c e l u i d ' u n e s u i t e born&
e t d ' u n e s u i t e bornde ( S n ) t e l l e q u e
de A
(an)
S n E a ( a n ) a v e c uniformi&
E e s t l ' e s p a c e L"(c) e t ( a n ) s ' i i i e n t i f i e 1 L (CC) d a n s A. Peut-on a l o r s a f f i r m e r quela
par rapport 5 n ; l'espace
5 une a p p l i c a t i o n a d e partie
C du p r o d u i t d e s
Sn
,
d6finie par
i n f 6s ( s ) > 0 , e s t s p e c n n
t r a l e pour a ?
If f a u d r a i t p o u r cela q u e p o u r un dl6ment IP d e l a s p h d r e u n i s
d e E', c ' e s t ait
s
d i r e une s u i t e ( A n )
p(f'2) E a ( a @ ) , c ' e s t
de
L1(C)
A
n
1
= 1o n
2 dire
e t avec uniformit6 p a r r a p p o r t B (A 1 . n En e f f e t , s ' i l e x i s t e E > 0 t e l q u e c o n t i e n t l a boule ouverte boule o u v e r t e
z1
t e l l e que
B (zn,€) e t
6
~ ( 6 1 =)
B ( X A n z n , ~ )d e s o r t e que
'n
(zn)
2
ZA S
n n
61P(a)( Z A n
, alors
E
'n
contient
Zn)
2
la
E.
5. On p e u t donc se p o s e r de f a q o n g d n d r a l e l e probldme s u i v a n t x s t a n t
donn6e une s u i t e b o r n g e (a,) d e que
sn
N A e t une s u i t e (Sn) d e
E a ( a ) avec uniformits par rapport
n
l a r e l a t i o n ( 4 ) pour t o u t e s u i t e (1,) de
avec u n i f o r m i t 6 p a r r a p p o r t
S (A,)
CN
telle
B n , est-ce que l ' o n
L1(CC) telle que
a
X /Anl
= 1,
Banach.
On
?
Consid6rons l e cas p a r t i c u l i e r d ' u n e a l g s b r e
de
v d r i f i e t o u t d ' a b o r d , e n p r e n a n t d e s caractzres, l ' i n c l u s i o n suivante, dans l a q u e l l e Sn e s t remplac6 p a r l ' e n s e m b l e t e r s e c t i o n du f i l t r e
s p ( an ) ( q u i e s t l ' i n -
u (a,))
C e t t e mgme i n c l u s i o n montre donc q u e pour t o u t choix de SnE5(aJ,
on a l a r e l a t i o n ( 4 ) . Cependant il r e s t e r a i t 5 d t a b l i r
l'uniformit6
THEORIE SPECTRALE EN UNE INFINITE DE VARIABLES
par rapport au choix d'une s u i t e (An)
137
de l a s p h g r e u n i t e d e
L1(c),
I1 n ' y a p a s d e d i f f i c u l t 6 s i on remplace l a b o r n e sur les coef-
ficients avec
u
i
E > 0
p a r l e f a i t q u e S c o n t i e n n e un E-voisinage d u f i x e . En e f f e t s i
AE
spectre
d 6 s i g n e l ' e n s e m b l e des p i n t s dont
la distance 5 4 est strictement infgrieure
on a
E
( 2 x n s p (an)
.
On e s t a i n s i c o n d u i t 5 L t u d i e r la c r o i s s a n c e des
coefficients
s p e c t r a u x e n f o n c t i o n de l a d i s t a n c e a u s p e c t r e . Dans un s e n s on a l ' i n L g a l i t 6 :
q u i s ' d t a b l i t facilement en prenant que
Ix(ui)
I 5
llui
II
t = x(a)
E
s p ( a ) e t en
sachant
.
La q u e s t i o n fondamentale c o n c e r n e l ' a u t r e s e n s : p e u t - o n
tout
E
> 0
pour
t r o u v e r une b o r n e d e s c o e f f i c i e n t s u i ( s ) avec d(s,sp(a)),c
q u i s o i t i n d g p e n d a n t e de
a , I1 a II 5 1 ?
B I B L I OGRAPHIE
[ 1]
J.-P.
FERRIER, T h g o r i e s p e c t r a l e e t a p p r o x i m a t i o n p a r des f o x t i o n s d ' u n e i n f i n i t 6 de variables, C o l l . An. H a r m . Comp l e x e , La Garde - F r e i n e t 1977.
[ 2
1
K . NISHIZAWA, A propos de l ' u n i c i t 6 du c a l c u l f o n c t i o n n e l h o l o -
morphe d e s b - a l g s b r e s , [ 3
1
t h s s e , U n i v e r s i t d de Nancy, 1977.
L . WAELBROEK, Etude s p e c t r a l e d e s a l g s b r e s compl&es,Acad.
Belg. C1. S c i . M6m., 1 9 6 0 .
Roy.
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Approxhation Theory and Functional Am Zysis J.B. Prolla ( e d . ) Oh'orth-rrlot land Zishtng Cornparry, 1979
MEROMORPHIC UNIFORM APPROXIMATION ON CLOSED SUBSETS OF OPEN R I E M A " SURFACES
P. M.
GAUTHIER*
Dgpartement de Mathgmatiques e t de S t a t i s t i q u e U n i v e r s i t g de M o n t r g a l , Canada D e d i c a t e d i n memory o f A l i c e Roth
1. INTRODUCTION
Let
F be a ( r e l a t i v e l y ) c l o s e d s u b s e t o f an open Riemann s u r -
f a c e R. Denote by
H(F) and
M(F) r e s p e c t i v e l y
the
holornorphic
and
m e r o m r p h i c f u n c t i o n s on ( a neighbourhood o f ) F. L e t A(F) d e n o t e t h a f u n c t i o n s c o n t i n u o u s o n F a n d h o l o m o r p h i c on t h e i n t e r i o r
Fo o f F.
R e c e n t l y , t h e problem of a p p r o x i m a t i n g f u n c t i o n s i n A(F) u n i f o r m l y b y f u n c t i o n s i n H ( R ) h a s been c o n s i d e r e d by S c h e i n b e r g [ 1 7 ] . I n t h e p r e s e n t p a p e r , w e c o n s i d e r t h e problem of a p p r o x i m a t i n g a g i v e n f u n c t i o n on
F u n i f o r m l y by f u n c t i o n s i n t 4 ( R ) a n d o b t a i n , as
a corollary,
a
r e s u l t r e l a t e d t o S c h e i n b e r g ' s . Our method o f a p p r o x i m a t i o n i s b a s e d o n t h e t e c h n i q u e o f t h e l a t e A l i c e Roth I15
1.
W e s h a l l r e l y on S c h e i n b e r g 1171 for s o m e r e s u l t s
to-
on t h e
pology o f s u r f a c e s . W i t h o u t loss of g e n e r a l i t y , w e s h a l l assume t h a t e v e r y Riemann s u r f a c e
its closure i n of
*
R if
R i s connected. A s u b s e t i s bounded i n
R i s compact. A Riemann s u r f a c e
R'
is an
if
extenhion
R i s ( c o n f o r m a l l y e q u i v a l e n t t o ) an open s u b s e t of
Research supp0rtedbyN.R.C.
R
R ' , If
of Canada a n d M i n i s t s r e de 1 ' E d u c a t i o n
d u Qu6bec. 139
140
GAUTHIER
furthermore
ii #
R',
i s an e o b e n t i d e x t e n s i o n of R. W e s h a l l say
R'
t h a t a c l o s e d s u b s e t F of
a
R i s ebbentiaLLy
0 6 d i n i t e genUb i f F has
c o v e r i n g by a f a m i l y of p a i r w i s e d i s j o i n t open sets, e a c h
n i t e genus. Denote by morphic on
i t s on
r?(F) t h e uniform l i m i t s on F of functions
R w i t h p o l e s o u t s i d e of
F and by
F of f u n c t i o n s holomorphic on
c o m p a c t i f i c a t i o n of
R
of f i -
R
. R*
wro-
G ( F ) t h e uniform l i m -
w i l l d e n o t e t h e o n e point
.
The c e n t r a l problem i n t h e q u a l i t a t i v e t h e o r y of a p p r o x i m a t i o n
i s t h a t o f a p p r o x i m a t i n g a g i v e n f u n c t i o n on a g i v e n s e t . I n t h i s d i r e c t i o n w e s t a t e our p r i n c i p a l theorem.
THEOREM 1: ( L o c a L i z a t i o n ) :
Mite genuo i n an o p e n Riemann o u k d a c e R . T h e n , a d u n c t i o n f i(F)
6.i-
Let F be c l o s e d and k ? b b E n t i a l l y 0 6 i b
in
id and o n l y i d
doh eVehy
compact
bet
K i n
R.
I f we drop t h e c o n d i t i o n t h a t
F be e s s e n t i a l l y o f f i n i t e genus,
t h e n t h e theorem i s no l o n g e r t r u e I 9 d i t i o n , for
I
.
However, w e may drop the con-
R p l a n a r , s i n c e it is t r i v i a l l y v e r i f i e d by a l l F
.
In
t h i s s i t u a t i o n , Theorem 1 i s due t o A l i c e Roth [ 1 5 ] . An immediate consequence o f Theorem 1 i s t h e f o l l o w i n g
Walsh-
t y p e theorem, which w a s f i r s t o b t a i n e d for p l a n a r R by N e r s e s i a n I 1 4 1.
THEOREM 2:
Let
F b e c l o b e d and e b b e n , t i a l L y
o p e n Riemann b u t 6 a c e that
0 6 d i n i t e genub i n
R. A b u d d i c i e n t c o n d i t i o n d o h
an
A(F) = G;I(F) i A
141
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
dolr
e v e h y bounded o p e n b e t
V in R.
By t h e Bishop-Kodama L o c a l i z a t i o n Theorem [ 1 2 1 ,
w e may r e p l a c e
t h e open s e t s V by p a r a m e t r i c d i s c s . The f o l l o w i n g i s a Runqe-type theorem.
THEOREM 3:
Let F be c l o b e d and ebAenLiai%'y
open Riernann bu4dace R
.
Then
H(F)
C
06
d i n i t e genub
G ( F ) . MO4eOVe4, H ( F )
C
in
an
H(F)
and o n l y i 6 R*\ F 0 c o n n e c t e d and l o c a l l q c o n n e c t e d . R e c e n t l y , w e p r o v e d Theorem 3 f o r more r e s t r i c t e d p a i r s (F , R ) [ 7 1 . From Theorem 2 , w e h a v e a c o r o l l a r y o n Walsh-type approximation by h o l o m o r p h i c f u n c t i o n s .
THEOREM A:
( S c h e i n b e h g I171 1 :
L e t F be c t o n e d and
enben,tiatXy
06
d i n i t e genun i n a o p e n Riemann h u t d a c e R . A b u d d i c i e n t c o n d i t i o n doh A(F) =
i ( i~ n t h) at
R* \ F
b e c o n n e c t e d and toca&?y
connected.
S c h e i n b e r g a c t u a l l y o b t a i n e d t h i s r e s u l t for somewhat mre gene r a l p a i r s ( F I R ) . F o r a r b i t r a r y p a i r s ( F I R ) , t h e c o n d i t i o n t h a t R*\ F b e c o n n e c t e d and l o c a l l y c o n n e c t e d i s a l s o n e c e s s a r y b u t
no
longer
s u f f i c i e n t [ 9 ] . I n f a c t , S c h e i n b e r g h a s shown t h a t t h e r e i s no t o p o l o g i c a l c h a r a c t e r i z a t i o n o f p a i r s ( F I R ) f o r which A ( F ) = H ( F ) [17].
R*\ F
i s c o n n e c t e d , i t f o l l o w s from t h e
Bishop-Mergelyan Theorem 1 2 1 t h a t
F s a t i s f i e s t h e h y p o t h e s e s of The-
PROOF OF THEOREM A:
Since
orem 2, when t h e sets V are p a r a m e t r i c d i s c s . Thus, i f E
> 0,
there is a
g1
E M(R) with
Now by Theorem 3, t h e r e i s a
q E H(R)
such t h a t
f e A ( F ) and
GAUTHlE R
I42
g(z)l < ~
z2 E F. ~
/
T h i s c o m p l e t e s t h e p r o o f of t h e c o r o l l a r y .
A c l o s e d set F i n R i s c a l l e d a s e t of Carleman t i o n by meromorphic f u n c t i o n s , i f f o r e a c h t i v e and c o n t i n u o u s on F
,
there is a
g
€
approxima-
f E A ( F ) and e a c h M(R)
E
psi-
with
The n e x t r e s u l t c h a r a c t e r i z e s such sets c o m p l e t e l y when
Fo =
8. This
r e s u l t i s known € o r R p l a n a r 1 1 4 1 .
THEOREM 4 :
L e t F b e cloned w i t h emp.ty i n t e h i o h i n arz open Riemaw
dace R . Then F i n a 6 e t
06
CUhJ?emaM apphoximation
by
5Wr-
mehomohphic
dunctionn id and o n l y id
doh
each compact
bet
K.
2. FUSION LEMMA
Using Behnke-Stein t e c h n i q u e s , Gunning and Narasimhan I l l ] have shown t h a t e v e r y open Riemann s u r f a c e R can b e v i s u a l i z e d i n a v e r y c o n c r e t e way. I n d e e d , t h e y showed t h a t f i c a t i o n ) above t h e f i n i t e p l a n e
(11.
R c a n be s p r e a d ( w i t h o u t rami-
R a d m i t s a l o c a l l y i n j e c t i v e holomornhic f u n c t i o n
is t h e spread.
-
To be p r e c i s e , t h e y p r o v e d t h a t p . Thus
R
P
(11
W e w i s h t o r e c o n s t r u c t t h e Cauchy k e r n e l of Behnke-Stein on R , -1 something r e s e m b l i n g ( q - p ) C o n c e p t u a l l y w e p r e f e r t o t h i n k of p
.
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
and q as b o t h l y i n g o n R
ble t o t h i n k of two c o p i e s the
z and
,
143
however, f o r p r o o f s , it may b e p r e f e r a R
P
and
R
9
of
-
R s p r e a d respectively above
5 planes:
We c o n s t r u c t a n open c o v e r o f
R x R.
If
(w,q)
E
R x R,
let
and D b e d i s c s a b o u t p and q r e s p e c t i v e l y w h i c h l i e s c h l i c h t P 9 is o v e r B . S e t U(p,q) = Dp x D . C o n s i d e r t h e Cousin data which q (5 z ) ” on U ( p , q ) . S i n c e R x R is S t e i n , t h e f i r s t Cousin probD
-
l e m c a n be s o l v e d . Hence t h e r e i s a meromorphic f u n c t i o n whose s i n g u l a r i t i e s are o n
@
on R x R
t h e d i a g o n a l . I n t h e neighbourhood o f
a
d i a g o n a l p o i n t , w e h a v e , i n local c o o r d i n a t e s (forever more g i v e n by P x
PI, that
i s holomorphic. O ( c
I
z ) means
O ( p , q ) , where
p ( p ) = 5 and p ( q ) = z .
We s h a l l p e r s i s t i n t h i s a b u s i v e n o t a t i o n , s i n c e i t is i n v a r i a n t under l o c a l change of c h a r t s w i t h i n t h e a t l a s g i v e n by
the function
O
a Cauchy k e r n e l on
ResZ@(
p x p. W e c a l l
R since
,z) = 1
.
We s h a l l now e x t e n d t o s u r f a c e s t h e p o w e r f u l F u s i o n
Lemma
of
A l i c e Roth [15].
FUSION LEMMA:
Let
K1, KZ, and K be c o m p a c t b u b b e t b o $ a n o p e n R i e -
mann buhdacc? R , w i t h K1 and K2 d i n j o . i n t . T h e h e i n a ponh%~e numbs
144
GAUTHIER
a Auch t h a t id ml
Aatihdying,
doh
and m2 aae any t w o meaomaaphic i u n c t i a n h
home
Iml
then theae
i.4
In W e may assume
PROOF:
bourhoods and
K
U1 and U2 of
LK <
E
,
2
mjIKuK
\ K
R huch t h a t
on
U.
I
60.
j =1,2,
.
aE
j
# @. Thus, w e can c o n s t r u c t oDen n e i g h -
K1 and K2 r e s p e c t i v e l y s u c h t h a t
i s precompact. Moreover, w e may assume t h a t
R \ U2
aries of
- m2
a d u n c t i o n m , meaamoaphic
(2)
R
> 0,
E
(1)
on
O f i =@ 1 2 t h e bound-
and U,L c o n s i s t of f i n i t e l y many d i s j o i n t smooth J o r d a n
c u r v e s . L e t E be t h e compliment o f compact neighbourhood o f
( R \ U2)
i s u n i f o r m l y bounded f o r
z E G , where
U
i n R . L e t G be a p r e -
U1 U U2
K2
U
K.
then
i s a Cauchy k e r n e l f o r
0
W e i n t r o d u c e now a n a u x i l i a r y f u n c t i o n
i n [0,1] such t h a t
0 i s 1 o n U1 and
@ is
( 3 ) i s u n i f o r m l y bounded, t h e r e i s a c o n s t a n t
with values
$8 E C1(R) 0 on
U2
a > 2
R.
. Then,
since
such t h a t
(4)
for
z E G.
R e t u r n i n g now t o our rneromorphic ml
and m2
By (1) w e c a n f i n d a precompact neighbourhood /q(Z)
I
<
E
, z
E
v.
follows. F i r s t , set
W e replace
U
, we of
put q=m,.-~. K
such
that
q by a f u n c t i o n q1 c o n s t r u c t e d
as
ME ROMORPHICAPPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
q l = q
(5)
Now set q1 = 0
146
V,UU,UE.
on
elsewhere. Thus,
(6)
Iql(z)i <
E
,
z E E.
Set
Since g is a Cauchy integral, g is holomorphic outside of E. Consequently I
is holomorphic in U2 Z E U
(for ql(z) =
m
,
set
pl(z)ql(z) = 0)
.
For
1'
is meromorphic and has the same poles as ql. To see that f is also holomorphic on U , we invoke the formula
Hence
GAUTHIER
146
For
z E U , q1 = q
i s holomorphic. Thus morphic on
U1
U
and
f
1J2 U U
i s holomorphic i n
and hence
U ,
w i t h t h e same p o l e s a s
q,
By
i s mero
-
t h e Runge
-
f
Behnke-Stein Theorem [ l ] t h e r e i s a meromorphic f u n c t i o n m3
on
R
f o r which
Finally we put
m = m 2 +m3
,
and w e have t h e f o l l o w i n g e s t i m a t e s :
K1 U K
on
K2 u K
Im-m21 5
If1
+ im3
+ Im3-fi
-
5
f / < E + (a
j@ll
-
2
This c o m p l e t e s t h e p r o o f o f t h e f u s i o n lemma.
)
q l + 141
+~
E
= a€.
+
on
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
147
I n t h e f u s i o n lemma, i t i s c l e a r t h a t i f
m
w e may t a k e
j
E A(K
j
U
K)
,
j =1,2.
Another consequence o f t h e f u s i o n lemma i s t h e f o l l m i n g BishopKodama
L o c a l i z a t i o n Lemma.
LOCALIZATION LEMMA
06
K
net
(Kodama [12]):
Let
a n o p e n Riemann h u k d a c e , and hUppOhe t h a t
t h e h e e X i b t h a c l u b e d pahame.thiC d i h c
Then
f E
f be g i v e n u n a c o m p a c t h u b 604
DZ w i t h centek
each
E K
z
z buch .that
k(K).
3 . PROOFS OF THEOREMS
C o n s i d e r f i r s t Theorem 1. The n e c e s s i t y i s t r i v i a l . Toprove the s u f f i c i e n c y , suppose f i r s t t h a t which t h e and t h a t
R' R'
- closure F i s open. L e t
of
R h a s a n e s s e n t i a l e x t e n s i o n R' i n
#F
F i s compact. W e may assume t h a t
I G n 1 be a n e x h a u s t i o n o f R
by
domains
with
' I n t h e F u s i o n Lemma, l e t
F\ R'.
ber
Gn+l,
and
Gn+l K1,
K , and
U Gn = R .
K2
be t h e s e t s
-
Gn,
-
F n Gn+l,
and c o n s i d e r t h e s e as compact s u b s e t s of t h e Riemann surface
For each
n = l t 2 , 3 , ...#
t h e F u s i o n Lemma g i v e s u s a p o s i t i v e n u m
a n , and w e may assume t h a t
GAUTHIE R
148
1 < an < an+l
If
E
.
i s a g i v e n p o s i t i v e number, w e s e l e c t t h e p o s i t i v e numbers
E ~ , E ~ , E ~ , . , .so
that m
E
n+l
<
E
n
and
By t h e h y p o t h e s e s t h e r e e x i s t f u n c t i o n s
E
c
E n < Y .
qn
E
n=l
M(R)
such t h a t
and t h e r e f o r e
(12)
n=1,2,3,
By t h e Behnke-Stein
Theorem [ 1 1 , w e may assume t h a t
by t h e Fusion Lemma, f o r e a c h
r
n
E M ( R 1 ) such t h a t , f o r
r
r
n =1,2,3,.
n=1,2,3,
...
n
n
The i n e q u a l i t i e s ( 1 3 ) y i e l d m
T h e r e €0re
m
I
.. , t h e r e
... . q
n
E M(R').Thus,
exists a function
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
149
is meromorphic in m
R =
U
n=l Gn
.
From (ll), (13) and (lo), there f o l l o w s for
< -
2al
03
+ C E V < E .
1
+ c IrV(z) n
for
z E F1 '
n- 1
qv(z)I <
z E F n \ Fn-l
c
1
,
E,+
En -+
CO
C
2an
n=2,3,.
E
~
<
n
,E
.. .
Thus f can be approximated uniformly on F by functions meromorphic on R , and the proof is complete for the special case that R has an essential extension in which F is bounded. To prove Theorem 1 in general, we shall invoke the specialcase just proved to inductively construct a sequence of meromorphic functions which converqes to an approximating function. Suppose R is an arbitrary open Riemann surface and closed subset. of R for which F has a covering
{ Vj j
F
is
a
by pairwise
GAUTH IE R
150
d i s j o i n t open s e t s , e a c h o f f i n i t e genus. We may assume t h a t e a c h V
meets F , from which i t f o l l o w s t h a t t h e f a m i l y n i t e . For each
we triangulate V
j,
j
is locally fi-
{Vjl
and s e t
j
where T r e p r e s e n t s a n a r b i t r a r y 2-dimensional closed t r i a n g l e o f t h e W e c a l l { P j l a polygonal coverand j' P . p o l y g o n a l . I t i s c l e a r t h a t t h e segments which make up u a P
t r i a n g u l a t i o n , and
F
j
= F
in V
3 j a r e l o c a l l y f i n i t e . R e p e a t i n g t h e same argument, w e can f i n d , f o r each
j
r
a polygon
Qj
with
P
C
j
Qj C Qj
C
W e may c o n s t r u c t an e x h a u s t i o n
Vj
.
of
{G.}
1
R bypolygonalbound-
e d domains i n s u c h a way t h a t
G
3
.
I
~
W e may a l s o assume t h a t each
to each
aQk.
That is, aG.
I
v
,~
=k >~ 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