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 of Theorem 1, t h e r e i s a f u n c t i o n g2EM(G U Q U Q ) 3 1 2
such t h a t
.
Set
m2 = g2 + p1
Set
f l = f . Then, w e may p r o c e e d i n d u c t i v e l y t o c o n s t r u c t a s e q u e n c e
m' j
s a t i s f y i n g for
Then,
E
M(Gj+l
j =2,3,..
.
U
...
Q1 u
U Qj)
, J
E
c
Imj(z)
- f(7.11
Im.(z)
- mj-l(z) I
<
n=l
J
,
2
E
u
n=l
Fn
and
3
I t i s clear t h a t
m
j
<
E -
2"
-
z E Gj-1
I
converges t o a f u n c t i o n
Im(z) - f ( z ) l <
E
,
z E
F.
m
E M(R)
*
and
152
GAUTHIER
T h i s c o m p l e t e s t h e proof o f Theorem 1. Theorem 3 was p r o v e d i n I 7 1 f o r t h e s p e c i a l c a s e t h a t R h a s an F isbounded.Theorem 3 has t w o
e s s e n t i a l e x t e n s i o n i n which
parts,
o n e o n meromorphic a p p r o x i m a t i o n a n d o n e on holomorphic approximtion. The meromorphic a p p r o x i m a t i o n f o l l o w s from t h e s p e c i a l c a s e i n exactly t h e same way a s t h e g e n e r a l form of Theorem 1 f o l l o w e d from t h e spe-
c i a l case of Theorem 1. The p r o o f o f t h e holomorphic p a r t of Theorem
3
also
follows
from t h e holomorphic s p e c i a l c a s e , b u t w e must d e f i n e t h e sets Pj l Q j and G
j
more c a r e f u l l y s o t h a t
(Gj+l
U
Q1
U
...
U
Qj
i s connected and l o c a l l y connected. F i r s t of a l l t h e exhaustion s t r u c t e d i n such a way t h a t For each R*\?
j '
R*\G
j
, let
K
j
G;+l
(G. 7
\ Gj
b e t h e set
1
can be (and u s u a l l y i s ) am-
i s connected, f o r each j of
bounded
components
.
of
j '
and
These a r e f i n i t e i n number. Connect e a c h s u c h component t o t h e i d e a l boundary o f
R by a s i m p l e p a t h which misses
F . W e may r e p l a c e t h i s
p a t h by a c o n n e c t e d p o l y g o n a l neighbourhood w i t h t h e same p r o p e r t y . C l e a r l y w e may assume t h a t t h e f a m i l y of a l l s u c h p a t h n e i g h b o u r h o o d s over a l l
j
is locally finite
and t r a n s v e r s a l t o e v e r y t h i n g w e have
153
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
constructed. L e t
P b e t h e u n i o n of a l l t h e s e p a t h neighbourhoods.Set
,
G' = G . \ P j i Then by c o n s t r u c t i o n ,
R* \
R*\
,
P! = P . \ ? 3
F!
3
7 '
(GI U
7
1
=Q.\?. 3
6;
R* \
P' u
Q! 3
... u
P!)3
and
are a l l c o n n e c t e d . I t i s e a s y t o see t h a t t h e s e s e t s are a l s o l o c a l l y c o n n e c t e d s i n c e t h e b o u n d a r i e s are l o c a l l y f i n i t e and p o l y g o n a l .
follows t h a t t h e c o v e r s
G!,
, P; , Q;
have t h e r e q u i r e d
It
properties,
T h i s c o m p l e t e s t h e p r o o f of Theorem 3 .
W e now p r o v e Theorem 4 . Suppose t h e n , t h a t
f o r e a c h compact s e t tinuous function on
K.
Let
f E C(F)
and l e t
Fo = $, and t h a t
E
be a p o s i t i v e m n -
F.
L e t {Gn} b e a n e x h a u s t i o n o f
E~
R
by p o l y g o n a l domains. S e t
= inf {E(z) : z E F n
By h y p o t h e s i s , t h e r e i s a
g1 E M ( R )
.
G ~ I
such t h a t
GAUTHIER
164
Set
go = g1 , G o =
41Ig2'".'gn-1
a,
and s u p p o s e ,
to
have been found i n
obtain
M(R)
an i n d u c t i o n ,
with t h e following
that three
properties:
L e t us c o n s t r u c t
continuously to
g n . F i r s t set
-Gn-l
U
(F n
En)
fn - gn-l
on
-
Gn-l
.
Now e x t e n d f
i n s u c h a way t h a t f n = f on F
n
17aGn
and
S i n c e , by a s s u m p t i o n ,
and s i n c e Gn i s a Lyapunov domain, it follows from Lemma 3 i n and from t h e Bishop-Kodama L o c a l i z a t i o n Theorem [12 1 t h a t
Hence t h e r e i s a f u n c t i o n
hn E M ( R )
such t h a t
[lo]
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
a n d so
gn s a t i s f i e s (15). I t i s e a s i l y v e r i f i e d t h a t
155
gn a l s o s a t i s -
f i e s ( 1 6 ) a n d ( 1 7 ) . Thus, w e have c o n s t r u c t e d i n d u c t i v e l y a s e q u e n c e n
E M(R) having t h e p r o p e r t i e s
From (16), w e see t h a t
(15), (16), a n d (17).
gn c o n v e r g e s t o a f u n c t i o n
From ( 1 5 ) a n d (16), it f o l l o w s t h a t i f all
E
M(R).
then f o r
m > n,
< - 'n 2n
As
z E F n (Gn\ Gn-l)r
g
m tends t o
m
E
n+l 2n+1
+
+
c
% < E n .
j = n + l 2'
w e have
m ,
T h is completes t h e p r o o f .
If
Fo = @
and
R*\F
i s connected
and
locally
cannected,
S c h e i n b e r g [ 1 7 ] h a s shown t h a t
F i s a s e t o f Carleman a p p r o x i m a t i o n
by f u n c t i o n s h o l o m o r p h i c on
(see a l s o [ 8 I ) . T h i s i s i m p l i c i t i n the
p r o o f of Theorem 4 .
R
I n d e e d , w e may c o n s t r u c t a n exhaustion c o m p a t i b l e
with F , t h a t i s , such t h a t
i s connected and l o c a l l y connected f o r each
j . Thus
w e may
choose
GAUTHIER
156
t h e functions
gn from
H(R).
4 . OPEN PROBLEMS
a) If
R i s p l a n a r and
f
i s g i v e n on
Theorem 3 t h a t approximation of
F
,
t h e n i t f o l l o w s fEom
f by f u n c t i o n s holomorphic
on F o r by f u n c t i o n s meromorphic
on
e q u i v a l e n t . However, t h e example i n [ 9
R
are
essentially
1 shows t h a t for
s
a
c l o s e d sets i n some Riemann s u r f a c e s , t h e r e a r e f u n c t i o n s i n
H(F) which c a n n o t b e approximated by f u n c t i o n s from The problem of a p p r o x i m a t i o n by f u n c t i o n i n
M(R).
becomes,
H(F)
t h e n , a s e p a r a t e q u e s t i o n which h a s n o t been t r e a t e d on a r b i t r a r y open Riemann s u r f a c e s . b) I f
R i s p l a n a r , t h e c o n d i t i o n i n Theorem 2 i s n o t o n l y s u f -
f i c i e n t b u t a l s o n e c e s s a r y [ 1 4 ] , I t would b e o f i n t e r e s t t o know whether it i s a l s o n e c e s s a r y on an open R
c ) There remains t h e problem (F,R)
.
of
i
m surface.
pairs
considering a r b i t r a r y
Only Theorem 4 i s complete i n t h i s r e s p e c t . Scheinberg
[ 1 7 ] 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 , R ) f o r which
A(F) = i ( F ) . This is not
at
all
o b v i o u s , b u t it i s e a s y t o see t h a t t h e r e i s a l s o no t o p 0 l o g i c a l c h a r a c t e r i z a t i o n of p a i r s
(see
6
-
(F,R) f o r which A(F) =R(F)
I).
d ) S c h e i n b e r g [17] h a s solved t h e problem of Carleman a p p r o x i mation by holomorphic f u n c t i o n s f o r t h e c a s e t h a t (see also [ 8 1 ) .
I n t h e c a s e where
R is p l a n a r ,
Fo =
necessary
and s u f f i c i e n t c o n d i t i o n s are known ( n e c e s s i t y [ 5 I c i e n c y [ 1 3 ] ) f o r Carleman a p p r o x i m a t i o n , even when What a b o u t Riemann s u r f a c e s ?
0
,
suffi-
Fo # '@.
~
MEROMORPHIC APPROXIMATION ON CLOSED SUBSETS OF RIEMANN SURFACES
167
e) T h e r e i s a l s o t h e q u e s t i o n of u n i f o r m a p p r o x i m a t i o n on u n bounded s e t s i n s e v e r a l complex v a r i a b l e s . T h i s i s p r a c t i c a l l y v i r g i n t e r r i t o r y . S e e , however, [ 4
I
.
and [16]
REFERENCES
H . BEHNKE and K.
STEIN, Entwecklung A n a l y t i s c h e r F u n k t i o n e n a u f
Riemannschen F l g c h e n , Math. Ann, 1 2 0 ( 1 9 4 9 ) , 430 - 4 6 1 . E. BISHOP, S u b a l g e b r a s o f F u n c t i o n s on a Riemann S u r f a c e ,
c i f i c J. Math. 8 ( 1 9 5 8 ) , 29 S . BOCHNER,
F o r t s e t z u n g Riemannscher F l a c h e n , Math,
(1928) , 4 0 6
J . E.
- 421.
Ann.
98
FORNAESS a n d E. L. STOUT, S p r e a d i n g P o l y d i s c s o n Complex
M a n i f o l d s , Amer. J. Math. P . M.
- 50.
Pa-
(to appear).
GAUTHIER, T a n g e n t i a l Approximation by E n t i r e
a n d F u n c t i o n s Holomorphic i n a D i s c , Izv. -ad. SSR 4 ( 1 9 6 9 ) , 319
- 326.
Functions Nauk.
Arm.
P. M. GAUTHIER, On t h e P o s s i b i l i t y of R a t i o n a l Approximation , i n Pad; and R a t i o n a l Appaoximation, 1 9 7 7 , Academic Press, N e w York, 261 - 2 6 4 .
P. M. GAUTHIER, A n a l y t i c Approximation on C l o s e d Subsets of Open Riemann S u r f a c e s , Paoc. C O M B . o n C o n n t h u c t i v e F u n c t i o n T h e ohy, B l a g o e v g r a d , Sofia ( i n p r i n t ) . P. M. GAUTHIER a n d W. HENGARTNER, Approximation s u r l e s fermds p a r des f o n c t i o n s a n a l y t i q u e s s u r une s u r f a c e d e Riemann, Comptes Rendus d e 1'Acad. B u l g a r e d e s Sciences(Dok1ady Bdgar.
Akad. Nauk) 2 6 ( 1 9 7 3 ) , 731.
P. M. GAUTHIER a n d W. HENGARTNER, Uniform Approximation on closed S e t s by F u n c t i o n s A n a l y t i c o n a Riemann S u r f a c e , Apptoximdon Theoky(Z.Ciesielski and J.Musielak, eds. 1, kidel, lblland, 1975, 63-70.
I58
GAUTHIER
[lo 1
P. M. GAUTHIER and W. HENGARTNER, Complex Approximation and Simultaneous Interpolation on Closed Sets, Can. J. Math. 29 (1977)I 701 - 706.
111 1
R. C. GUNNING and R. NARASIMPAN, Immersion of Open Riemann Surfaces, Math. Ann. 174 (1967), 103 108.
1121
L. K. KODAMA, Boundary Measures of Analytic Differentials and Uniform Approximation on a Riemann Surface, Pacific J.Math. 15 (1965) 1261 - 1277.
1131
A. H. NERSESIAN, On the Carleman Sets (Russian), Izv. Akad.Nauk Arm. SSR 6(1971), 465 - 471.
-
[141 A. H. NERSESIAN, On the Uniform and Tangential Approximation by Meromorphic Functions (Russian), Izv. Akad. Nauk Arm.SSR 7 (1972), 405 - 412. [15 1
ALICE ROTH, Uniform and Tangential Approximations by Meromor phic Functions on Closed Sets, Can. J. Math.28(1976) I 104-111.
1161
S. SCHEINBERG, Uniform Approximation by Entire Functions, d'Analyse Math. 29(1976) , 16 - 19.
J.
[I71
S. SCHEINBERG, Uniform Approximation by Functions Analytic a Riemann Surface, Ann. Math. (to appear).
on
Approximation Theory and Functional Analyeis J.B.
Prolla (ed.)
0 North-Holland Publishing Company, 1979
WHITNEY'S SPECTRAL SYNTHESIS THEOREM I N INFINITE DIMENSIONS
CLAUDIA S. GUERREIRO(*)
I n s t i t u t o d e Matemdtica U n i v e r s i d a d e F e d e r a l do R i o d e J a n e i r o Rio
0.
de J a n e i r o , B r a z i l
IJTRODUCTION
I n 1 9 4 8 H. Whitney [131, b a s e d o n a c o n j e c t u r e of
L.
Schwartz,
p r o v e d t h a t , g i v e n a non-empty open s u b s e t U C IR", t h e c l o s u r e , r e s p e c t t o t h e compact-open t o p o l o g y o f o r d e r m , o f an ideal
with
I C Rm(U)
i s d e t e r m i n e d b y i t s s e t o f local i d e a l s . The o r i g i n a l p r o o f w a s s i m p l i f i e d i n 1 9 6 6 by B. Malgrange [ 5
1.
The main c o n c e r n o f t h i s p a p e r
i s t o e x t e n d W h i t n e y ' s theorem t o open s u b s e t s o f i n f i n i t e
dimen-
s i o n a l s p a c e s . I n f i n i t e d i m e n s i o n s t h e r e are t w o e q u i v a l e n t formu
-
l a t i o n s of this theorem:
(*)
T h i s r e s e a r c h was p a r t i a l l y s u p p o r t e d by
FINEP ( B r a s i l ) t h r o u g h
-
U n i v e r s i d a d e F e d e r a l do
a g r a n t t o t h e I n s t i t u t o d e Matemstica R i o de J a n e i r o . 1 59
160
GUERREIRO
n {I
=
+
I ( a , k , ~ ) ;a
E U,
k E N, k 5 m ,
E
> 01
and
I n i n f i n i t e d i m e n s i o n s , W h i t n e y ' s theorem i s f a l s e i n formulat i o n 1, even i n t h e c a s e
U = H , a real s e p a r a b l e H i l b e r t s p a c e , and
m = l . We p r e s e n t an example of t h i s i n s e c t i o n 2 . I n f o r m u l a t i o n i t i s t r u e , w i t h r e s p e c t t o t h e u s u a l compact-open
case
m =1 w i t h some r e s t r i c t i o n s . The case
2
topology, f o r the
m 1. 2
r e m a i n s a n open
problem and o u r g u e s s i s t h a t t h e theorem i s f a l s e i n t h i s c o n t e x t . Two o t h e r d i r e c t i o n s a r i s e n a t u r a l l y i n i n f i n i t e dimensions:the f i r s t one i s t o c o n s i d e r subspaces o f d i m e n s i o n s , w i t h t h e whole s p a c e new t o p o l o g y i n
am(U)
g r n ( U ) which c o i n c i d e , i n f i n i t e
am(U);
t h e second i s t o l o o k f o r a
which c o i n c i d e s , i n f i n i t e d i m e n s i o n s ,
with
t h e u s u a l one. I n s e c t i o n 2 w e c o n s i d e r t h e c o n c e p t o f d i f f e r e n t i a b i l i t y type, which g i v e s u s a u n i f i e d way t o d e a l s i m u l t a n e o u s l y s u b s p a c e s of
several
with
grn(U).
I n [ 1 2 ] R e s t r e p o s t u d i e d t h e c l o s u r e o f t h e a l g e b r a of
poly-
n o m i a l s of f i n i t e type i n a Banach s p a c e o f a c e r t a i n k i n d , f o r t h e topology o f t h e uniform convergence of t h e f u n c t i o n and i t s d e r i v a t i v e on bounded s u b s e t s . I n [ l ] Aron and P r o l l a e x t e n d e d t h i s r e s u l t
to
a more g e n e r a l c l a s s of Banach s p a c e s , c o n s i d e r i n g t h e case m 2 2 and polynomial a l g e b r a s o f v e c t o r f u n c t i o n s weakly u n i f o r m l y
continuous
on bounded s u b s e t s . I n s e c t i o n 3 w e s t u d y i d e a l s of f u n c t i o n s weakly u n i f o r m l y c o n t i n u o u s on bounded s e t s , w i t h r e s p e c t t o t h e t o p o l o g y of t h e u n i f o r m convergence o f order m on bounded sets. I n s e c t i o n 4 , w e c o n s i d e r t h e topology
in
[lo].
T
C
introducedbyProlla
WHITNEYS SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
161
F i n a l l y , i n s e c t i o n 5, we use t h e r e s u l t s of s e c t i o n 4
t o es-
t a b l i s h some f a c t s a b o u t modules. The r e s u l t s o f t h i s p a p e r a r e t a k e n from t h e a u t h o r ' s D o c t o r a l D i s s e r t a t i o n a t t h e U n i v e r s i d a d e F e d e r a l d o R i o de J a n e i r o ,
written
under t h e guidance of P r o f e s s o r J . B . P r o l l a .
1. PmLIMINARIES
I n t h e sequel
stands
N
for
{0,1,2,...1,
m s t a n d s f o r a n e l e m e n t of
e l e m e n t s of
N.
Let
E
cal d u a l s E ' E' 8 F 9 8
a n d F'
For
E E +
X
IN
U
natural
{ml
and
respectively,
9(x)v
E
U
for
v
applications
F.
E
a real H a u s d o r f f l o c a l l y c o n v e x s p a c e , a function
unique) such t h a t , f o r
x E U,
Df(x)y = l i m
X
E
uniformly with respect to
Df : U
+
f :U + X
6(E;X) ( n e c e s s a r i l y
IR,
f ( x + XY)
A+O
A
-
f(x)
y o n e a c h bounded s u b s e t of
I n t h e same way, w e d e f i n e c - d i 6 6 e h e n t i a b i l i t y by
i, j,k
a non-empty open s u b s e t ,
s p a n n e d by t h e
d:(E;F)
F, p E E ' ,
E
C
i s c a l l e d b-diddehentiabLe i f there i s
b
integers
a n d F b e r e a l normed l i n e a r s p a c e s w i t h t o p o l o g i -
# 0
t h e l i n e a r s u b s p a c e of
v: x
set of
the
c a n d bounded by compact. W e o b s e r v e t h a t i f
space, b - d i f f e r e n t i a b i l i t y i s Frechet
E
E .
by
replacing
is
a
normed
d i f f e r e n t i a b i l i t y and c - d i f -
f e r e n t i a b i l i t y i s Hadamard d i f f e r e n t i a b i l i t y (Nashed [ 9 1 1 . Let
gy
T~
b d: ( E ; X )
denote the space
S ( E ; X ) endowed w i t h t h e t o p o l o -
of u n i f o r m c o n v e r g e n c e on bounded s u b s e t s o f
denote the space
f(E;X)
endowed w i t h t h e t o p o l o g y
E 7
C
and of
LC(E;X)
uniform
may d e f i n e c o n v e r g e n c e o n compact s u b s e t s of E . By i n d u c t i o n w e b k b b k-1 d:'(OEIF) = F a n d , f o r k 2 1, d: ( EIF) = d: (E;d: ( E I F ) ) . I n t h e same
162
GUERREIRO
way, replacing b by c, we have
LC(kEIF). Furthermore, let C(U;X)
denote the vector space of all continuous functions from U endowed with the compact-open topology The space
.
0
7
Gbm(U;F) and its topology
T~~
will
be
to
X,
defined
inductively as follows: For if
m = O , gbo(U;F)
=
C(U;F),
T~~
0
and we denote D f = f ,
= '7
f E C(U;F). gbl U;F) as the vector space of all € b
For m = 1, define
which are b-differentiable and such that
rbl
pology
Df
E
E
C(U;F)
C(U;d: (E;F)). The to-
is defined as the topology €or which the isomorphism
f E gbl(U;F)
+
(f,Df)
C(U;F)
x
C(U;Lb (E;F))
is a homeomorphism. For uniformity of notation, D1f = Df. Suppose we had already defined Eb(k-l) (U;F), 'Ib (k-1) Dk-l , &b(k-l) (U;F)
-+
C(U;lb(k-lEIF)), for some
and
2.
k
8b(k-1) (U;F) b k such that Dk-lf is b-differentiable and D(Dk-lf) E C(U;I: ( EIF)) Define Dk: gbk(U;F) -t C(U;eb(kEIF)) by Dkf = D(Dk-'f) and the toDefine
pology
-rbk
gbk(U;F) as the vector space of all
f
E
.
as being the only one for which the isomorphism
is a homeomorphism. Finally, define
ab"(U;F) =
n
kslN
the topology for which the isomorphism
is a
homeomorphism.
sbk(U;F) and consider as
b-
7
WHITNEY'SSPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
F = IR, w e w i l l w r i t e
For t h e c a s e The s p a c e
BCrn(U;F) and i t s t o p o l o g y
163
Ebm(U;F) = 8bm ( U ) .
i s defined
T~~
t i v e l y i n t h e same way, by j u s t r e p l a c i n g b by
induc-
c i n t h e above defi-
nition. k k There i s a n a t u r a l i d e n t i f i c a t i o n between L ( EIF) and L ( E;F) , t h e v e c t o r s p a c e o f c o n t i n u o u s k - l i n e a r maps from Ek t o F. b k
t h e r e i s a homeomorphism between b k d: ( E;F) ( r e s p e c t i v e l y
and
w i t h t h e topology
d: ( EIF)
d : C ( k E I F ) ) ,t h e space
(respectively
T'
(respectively
T
C
I n fact,
d:C(kEIF))
d:(kE;F)
endowed
1.
On t h e o t h e r hand, t h e n a t u r a l isomorphism between
Xs(kE;F)
t h e vector s p a c e of c o n t i n u o u s symmetric k - l i n e a r maps f r o m k
F , and
P ( E ; F ) , t h e s p a c e of c o n t i n u o u s k-homogeneous
from E
i n t o F,
Ek
,
to
polynomials
i s , a c t u a l l y , a homeomorphism, i f w e endow both spaces
w i t h t h e topology
T~
or both with t h e topology
T
C
.
cm
Moreover, g i v e n f b e l o n g i n g t o Cbm(U;F)or 8 (U;F), x E U, k z m , k k w e may a s s o c i a t e D f ( x ) w i t h a n e l e m e n t d k f ( x ) o f gs( E;F) which k may be i d e n t i f i e d w i t h a p o l y n o m i a l a k f ( x ) of P ( E ; F ) . bm I n t h a t case, t h e T t o p o l o g y may be d e f i n e d i n gbm(U;F) by t h e f a m i l y o f seminorms o f t h e form
K
C
U
a compact s u b s e t , k 5 m. cm
The t o p o l o g y
T
may be d e f i n e d i n
o f seminorms :
K
C
U, L C E
compact s u b s e t s ,
k
F o r d e t a i l s , see Nachbin [ 8
5 m.
1
.
LCm(U;F) by t h e
family
164
GUERREIRO
2 . IDEALS AND DIFFERENTIABILITY TYPES The c o n c e p t of holomorphy t y p e f o r complex f u n c t i o n s i s already
w e l l known (Nachbin [ 7 1 1 . The same d e f i n i t i o n may be a p p l i e d t o real s p a c e s (Aron and P r o l l a 11
DEFINITION 2.1:
P + II PII,
Pek (E;F) k
, which
E INl
t h e norm on e a c h b e i n g
denoted
s a t i s f i e s the following conditions:
i s t h e normed s p a c e o f a l l c o n s t a n t functions fran
Peo(E;F)
i)
F is asequence
A di6dekentiabiLity type dhom E ,to
of Bnnach s p a c e s by
1 1.
to F, i d e n t i f i e d w i t h F ; 8k ii) each P ( E ; F ) i s a v e c t o r s u b s p a c e o f E
iii) t h e r e i s a r e a l number
x E E
DEFINITION 2.2:
Let
0 b e a d i f f e r e n t i a b i l i t y t y p e from E
E
Pek(E;F) imply
j, k E IN
P
,jc
k,
i J P ( x ) E P e J ( E ; F ) and
pern(U;F) a s t h e v e c t o r s u b s p a c e of
t o F.We
gbm(U;F)
of
such t h a t , f o r x E U, k 5 m , w e have 2 f ( x ) EPek(E;F) -k Bk x E U + d f ( x ) E P (E;F) is c o n t i n u o u s .
f
and t h e mapping
W e endow
u 1. 1 s u c h t h a t
and
d e f i n e t h e space a l l functions
k P (E;F);
sem(U;F) w i t h t h e topology
em
d e f i n e d by t h e fam-
‘I
ily of seminorms; -i
p K I k ( f ) = sup IIld f ( x ) l l e ; x E K , 0
where
K
C
U
i s a compact s u b s e t and
I n t h e case
F = IR w e w i l l w r i t e
W e remark t h a t t h e s p a c e definition.
k E IN
5
i
5 kl,
I
k
5 m.
Eern(U;F)
= tZ e m ( U ) .
sbrn(U;F) i s a p a r t i c u l a r case of t h i s
WHITNEYS SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
DEFINITION 2 . 3 (Aron a n d P r o l l a [ 1 1 ) : from E
A d i f f e r e n t i a b i l i t y type
-3
F i s c a l l e d compact i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i -
to
t i o n s f o r each i)
166
k E IN:
k
Pf (E;F)
,
t h e v e c t o r s p a c e of c o n t i n u o u s k-homogeneous p l y -
n o m i a l s of f i n i t e t y p e , i s d e n s e l y c o n t a i n e d i n
v E F
ii) f o r e a c h
q
+
i s continuous
qk 8 v
from
* I l l t o ( Pkf ( E ; F ) , 11 * I l e ) ;
(E',lI
iii) i f
t h e map
Pek(E;F);
P E E ' 8 E,
then
For each
k E IN,
Q O P E Pek(E;F) f o r a l l
6k (E;F)
Q E P
and
EXAMPLES 2.4: k P f (E;F) i n
let
PCk(E;F)
k
be
the closure
of
6 = c i s a compact dif-
P (E;F) f o r t h e u s u a l norm. Then
f e r e n t i a b i l i t y t y p e c a l l e d cukhenZ compact t y p e . I f we c o n s i d e r , f o r each
k E IN, P N k ( E ; F ) , t h e Banach to
o f a l l n u c l e a r c o n t i n u o u s p o l y n o m i a l s from E
I/*l l N ,
n u c l e a r norm
F , endowed w i t h t h e
E h a s t h e approximation p r o p e r t y ,
then
i s a compact d i f f e r e n t i a b i l i t y t y p e called nuceeah type (see[ 2 1 ) .
9 = N
PROPOSITION 2.5:
Xy
and i f
space
.type ghom 16
P
Bk
E
Let
F b e a Banach npace and
F. k
to
(E;F) = P ( E ; F ) , k E I N , k
6 a di66eaentiabili-
5 m, t h e n
gbm(U;F) = Egm(U;F)
topoLogicalLy.
PROOF:
map
As w e h a v e ( P e k ( E ; F ) , 11. 11 ) a Banach s p a c e a n d t h e i n c l u s i o n 8
k
Pek(E;F) C P (E;F)
e q u i v a l e n t norms.
COROLLARY 2 . 6 :
is continuous, then
11
I1
and
11
- It6
are
0
Let E be a 6 i n i t e dimension nohmed bpace and
compac2 d i 6 6 e h e n t i a b i l i t g t y p e daom E t o
F.
9
a
GUERREIRO
166
k Pf(E;F) =
PROOF:
DEFINITION 2 . 7 : A
C
P 9k (E;F)
k
= P (E;F),
k E IN.
0
8 b e a d i f f e r e n t i a b i l i t y t y p e from E t o F and
Let
a e m ( U ; F ) a non-empty s u b s e t .
W e define:
i= where
n {A
+
m; k
I ( a , k ) ; a E U, k
I ( a , k ) = { f E Eem(U;F); $ f ( a )
=
E
0, 0 5 i
IN}
5
and:
k)
PROPOSITION 2 . 8 : A C
1 6 0 i d a di6dexentiabiLity t y p e 6hom em Eem(U;F) a n o n - e m p t y d u b b e t , t h e n in T - c l o d e d .
PROOF:
Fix
If every
g
f
a E U, k
E A,
T
t o F
and
E,
for
and c o n s i d e r
9 B(a,k) there is
E >
0
such t h a t
p(f
-
g) 2
where
Consider
em
5 m
E
V = {h E gem(U;F); p ( f
-neighborhood o f I f there exists
-
h) <
E
/2},
which
is
a
f.
h E V n B ( a , k ) , w e have
p(h
-
g) < ~ / 2
for
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
some
g E A . Then:
which i s a c o n t r a d i c t i o n . So
is closed.
B(a,k)
DEFINITION 2.9:
V
r)
B(a,k) = 4
which
C
Let
that
be a d i f f e r e n t i a b i l i t y t y p e from
8
to
5
F and
G C E' 0 E
a e m ( U ; F ) non-empty s u b s e t s . W e say that
and
proves
0
s a t i s f y i n g c o n d i t i o n ( i i i )o f D e f i n i t i o n 2 . 3 , A
107
V
C U
A o (glV) C
(A,G)
c o n d i t i o n (L) i f , given
bUti4dieb
a non-empty open s u b s e t such t h a t
have
(q),
,
P r o l l a [lo 1
,
. Let
E b e a s p a c e w i t h a Schauder basis {eo,el,.
and Pn t h e p r o j e c t i o n of
E
..,en' .
Given
x
E
on t h e v e c t o r
E l x = nz~ N Anen
,
let
compact d i f f e r e n t i a b i l i t y t y p e from E an a l g e b r a a n d c o n s i d e r quence
G
em t h e c l o s u r e b e i n g c o n s i d e r e d i n ( g (V;F) ,T'~).
EXAMPLE 2 . 1 0 :
(eo,el,.
we
g ( V ) C U,
S i m i l a r c o n d i t i o n s have been u s e d by L e s m e s [ 3 ] Llavona [ 4 1
g E
{gni ; i
.
E
IN
}.
subspace
gn(x) = An
to W
.
. .,en,. .. I
spanned
Let
such t h a t
by
a
be
0
gem(E)
is
I C g e m ( E ) , the i d e a l g e n e r a t e d by a s u b s e If
G = {Pn ; n E IN }
then (1,G)
satisfies
c o n d i t i o n (L)
T h i s example may be e x t e n d e d t o a s p a c e w i t h a g e n e r a l i z e d ba-
sis.
W e remark t h a t f o r t h e d i f f e r e n t i a b i l i t y t y p e s i n t r o d u c e d 2.4,
i s an a l g e b r a . More g e n e r a l l y , i f
Bem(U)
b i l i t y t y p e f r o m E t o IR
we have from
PQ
pei(5)
such that g i v e n
P e ( k + J ) ( E ) a n d t h e mapping
E x
Pel(,)
to
Pe(i+J) ( E ) , then
P
8 E
is a differentia
Pei(E)
(P,Q) * PQ gem(U)
in
and Q
E
-
pel(,)
is continuous
i s an a l g e b r a .
168
GUERREIRO
W e s a y t h a t E h a s p k o p e r r t y (B) i f t h e r e i s a
DEFINITION 2 . 1 1 : quence
{ P n ; n E IN } i)
Pnx
xI
+
C
x
-,P ,
ii) P o p n
E' 8 E
se-
such t h a t
E E
P E E'.
T h i s d e f i n i t i o n was used by R e s t r e p o [ 1 2 ] w i t h t h e condition t h a t the
Pn a r e p r o j e c t i o n s .
THEOREM 2 . 1 2 :
Let
8
IR 6 u c h t h a t
gem(U)
additional
be a compact d i d d e h e n t i a b i l i t y t y p e d h o m E t o
in a n a l g e b h a and l e t
S U p p O b e t h a t thehe
i b
1 c Sem(U) be a n idea+!.
G = {Pn ; n E N }
a bequence
E' 8 E
C
AUCh t h a t : i)
ha6 p k a p e h t y ( B ) with h e n p e c t t o
E
i i ) (1,G)
Then
G ;
6 a t i n & L e o c o n d i t i o n (L).
t h e -rem- cl?abuhe
.id
06
I in
fiem(U).
F o r t h e p r o o f w e n e e d s e v e r a l lemmas.
LEMMA 2.13:
Let
that
i n a n a l g t b r r a and ( E ' I U ) c g e r n ( U ) .
fiem(U)
let
El
C
0
E
be a d i 6 6 e h e n t i a b i L i t y t y p e dhom E t a
be a d i n i t e d i m e n n i o n a l v e c t o h o u b d p a c e , U1
a n o n - e m p t y open b u b n e t and c o n b i d e k
16
ideaC 0 6
R
+
'
g,U1 E
C %I
ElnU
bm
(U,).
ideaC t h e n t h e ~ ~ ~ - c C a h u r0 r6 e R ( 1 ) i n afl " gbm(U1). Maheauerr, id f E $em(U)l f E I , t h e n Rf belongs I
E
Lem(U) i n an
t o t h e ~ ~ ~ - c L o b u0 h6 e R ( 1 ) i n
PROOF:
R : g E gem(")
nuch
IR
Let
A = R(fiem(U)),
Lbm(U1).
which i s a s u b a l g e b r a of
Sbm(ul) b e c a u s e
is an a l g e b r a homomorphism. Now A s a t i s f i e s t h e h y p o t h e s e s of N a c h b i n ' s theorem
cause
1e A
and (El lU)
I t i s clear t h a t
C
gem(U)
R(I) i s
. Therefore
A is
[ 6 1
dense i n
a v e c t o r s u b s p a c e of
fibm(Ul).
be-
&?m(Ul). On t h e
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
o t h e r hand, if implies R(I)
*
R f E R ( 1 ) and
w e have
Rg E A
169
-
A c ~ ( 1 BY ) .c o n t i n u i t y o f m u l t i p l i c a t i o n , R(I)
and w e conclude t h a t
-
R(I) *
which
R(fg) E R(I),
.A c R(I)*A
R(I) , which completes t h e proof t h a t
C
R(1) is an ideal.
L e t now
v
f
E
there is
Definition 2.1,
orem, R f
E R(1).
LEElMA 2.14:
t o {p,; -to
F
,
ale
PROOF:
> 0, k E
n 2 no
let
'8
{P,;n
N, Ki
no E N
C
Pei(E;F)
E
huch t h a t
and d o h aLL
1 , Let
hehpect
compact hubA ef A , 1 5 i 5 k.
Q E Ki,
1 5 i 5 k.
.
L e t E be a hpace h a t i h d y i n g p h 0 p e h . t y (B) w i t h E
by
a n d , by t h e c l a s s i c a l W h i t n e y ' s t h e -
be a compact di6dehentiabiLiZy t y p e d h o m
See Aron and P r o l l a [ 11
LEMMA 2.15: t o
u 1. 1 i s g i v e n
IT
},
Thehe i h
dot
> 0. I f
L e t E be a hpUCe h a t i h d y i n g P h O p t h t Y (B) wLth
n E E
Rf E (R(1))"
E
such t h a t
g E I
-
So, w e h a v e -
k 5 m,
I , a E U1,
'8
hehpect
be u compact d i $ d e h e n t i a b i t i t y t y p e dkom
E
GUERREIRO
170
PROOF: L e t M L 1 be such t h a t llPnII I M , be such t h a t x
E
K , y E U, IIx
- yll< 6
n E IN, and l e t 0 < 6 < dist(K,E\U) -i k imply I1 d f ( x ) dif (y)1 I < E / ~ M,
-
O ( i 5 k .
By (B) and Lemma 2.14, t h e r e i s
no E N
s u c h t h a t , f o r n,no:
-
L e t r = 6/2M and, f o r e a c h x E K , B ( x , r ) = { t E U ; I l t
By compactness, t h e r e a r e
xl,..
.,x S
xII
such t h a t
E K
C o n s i d e r t h e non-empty open subset
V =
U
{ B ( x i , r ) ; 1 5 its].
a E V, IIa-x i 11 < r f o r some i. Then, i f n 1. no , IIxi - Pnall 5 6 IIa xi II + 7 < 6 , and w e have < IIpna P x. II + llPnxi - xi II < llp,ll n i If
Pn(V)
-
-
c u.
Finally observe t h a t f o r
x E K, 1 5 i 5 k ,
n 2 n0
:
171
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
LEMMA 2.16:
Let
P E E' @ E l
El
be a compact d i d d e f i e n t i a b i t i t y t y p e d'rom E t o F,
8
= P(E).
3avrach npace d o t each
PROOF:
c
=
IIQ
oPll,
i.) a
k E IN.
.
See Aron and P r o l l a [ 1 ]
v
I t i s clear t h a t
PROOF OF THEOREM 2.12:
I
nohm I I Q I l p
endowed with
Pk(E1;F)
T h e Apace
I C I . By P r o p o s i t i o n 2 . 8 ,
?. f E
Let
C
n C
no
U1
.
, 11
IIp
no E IN and C
U
P1(E1)
V
C
k 5 m,
I
a non-empty
U
P = Pn , E~ = P ( E ) ,
,
t h e n norm
15 i 5 k
) i s a Banach space b y Lemma 2.16,
e q u i v a l e n t t o t h e u s u a l norm Hence t h e t o p o l o g y
> 0.
E
open
By
subset
and
and c o n s i d e r
I f we define i n t h e n ( Pi ( E l )
a compact subset, k E IN
U
K C V, Pn(V)
Fix
K1 = P ( K )
K
t h e r e are
Lemma 2.15, such t h a t
:,
11
T~~
I1
in
-
u1
llQIIp and
=
u
n E~
,
I I Q o Pllgr
=
IIp
11
is
i
P (El).
may b e d e f i n e d i n
gbm(Ul)
by
the
f a m i l y of seminorms:
L
C
U 1 a compact s u b s e t , j E IN
,
j 5 m.
By u s i n g n o t a t i o n a n d r e s u l t s f r o m Lemma 2 . 1 3 , t h e r e i s such t h a t
Then :
g E I
GUERREIRO
172
-
Il;lif(x)
for all
x
(P/V)
,
h E I
(hlV)) < ~ / 3 .W e conclude t h a t
p K,k f
COROLLARY 2 . 1 7 : T
-
COROLLARY 2 . 1 8 :
h) <
I
I
=
?.
Sbl(U)
C
1
C
Theahem 2 . 1 2 , id
06
b e an i d e a t and duppohe t h e h e
nu&
E' 0 E
s a t i d d i e & condition
io t h e ~ ~ l - c L o s u h 06 e I in If
E
and
I
i h
i d
a
that:
(L). 8b1(U).
compact d i f f e r e n t i a b i l i t y t y p e from E ei 1 = P (E) = P (El. w e h a v e E ' = P:(E) bl By P r o p o s i t i o n 2 . 5 , Eel(") = S (U) t o p o l o g i c a l l y . PROOF:
that
had p h o p e h t y (B) w i t h h e b p e c t t o G .
ii) (1,G)
?
1 =
Let
-
such
0
G = {Pn ; n E IN
i) E
Then
to
I n .the dame c o n d i t i o n &
c k b n e d , .then
ncguence
a n d as (1,G) satis-
find
is possible
it
t h e p r o o f i s complete.
em
k.
~ ~ , ~ ( ( f l V g) o ( P l V ) ) < 26/3
(L)
-
5
i
-
So w e h a v e f i e s condition
5
0
K,
E
5
i i ( g O P ) (X)Il,
€I i s a n y
EXAMPLE 2.19:
If
I
C
gern(U) i s a n i d e a l and
s i o n a l , i t i s n o t always t r u e t h a t
f is
E
closed.
to
i s i n f i n i t e dimen Hence
R
-
Whitney's
173
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
-
theorem i s f a l s e i n t h e f o r m u l a t i o n
I =
?. W e
m = 1, remarking once more t h a t
example f o r
w i l l give gel(,)
l o g i c a l l y f o r any compact d i f f e r e n t i a b i l i t y t y p e
a counter-
= gbl(u)
topo-
0 .
L e t H be a real s e p a r a b l e H i l b e r t space o f i n f i n i t e dimension
and l e t
l e i ; i E IN}
Denote by
S
qi(x) =
(
x,ei
2
If
= H.
t h e i d e a l g e n e r a t e d by {gi ; i E IN 1 where
I C sbl(H)
, x
)
Pn : H
+
E H.
Explicitly:
d e n o t e s t h e p r o j e c t i o n of
H
{ei ; 0 5 i 5 n }
s p a c e spanned by
r e s p e c t t o G = {P ; n E IN} n d i t i o n (L)
I.
Consider t h e c l o s e d i d e a l
For a f i x e d
I.
I
+
a
I ( a , l ) = {f
If
g E I , then
as
gi E I , i
=
{f
a
+
f(a) = X a
0,
,
and
I
+
,
u E S
given
I(a,l) =
z f ( a ) = v. There i s
Then
t = gi h
f E I
+
h E Ebl(H), E
s a t i s f i e s con-
1
and
and, on t h e o t h e r
ag(0) E S
IN, d g i ( 0 ) = ei
w e claim t h a t
# 0 . Consider
(1,G)
(B) w i t h
a b l ( H ) ; f (0) = 0
E
t h a t Zg(0) = u . Then, i f a = O , I t I ( a , l ) = If If
has property
g b 1 ( ~ ) ;3 g E I , q ( a ) = f ( a ) , Zg(a) = i i f ( a ) l .
g(0) = 0
E
H
on t h e v e c t o r sub-
H,
E
E
then
H
and (Example 2 . 1 0 )
.
let us characterize
H .
t h e v e c t o r subspace a l g e b r a i c a l l y spanned by
H
C
{ei ; i E IN}. Then Consider
be an orthonormal b a s i s f o r
I, t ( a ) = A
there i s
bl
E G
g
hand, E
I such
( H ) ; f(O)=O, & ( O ) E S ] .
s b l ( ~ ) I.n
f a c t , let f E E~'(H),
g i ( a ) # 0 , because Xe. & ( a ) = - (1v - ~ ) . gi(4 gi(a)
i E IN s u c h t h a t
h ( a ) =and
gi ( a )
and
i t ( a ) = v,
which
proves
that
I(a,l).
W e conclude t h a t
Tbl-closed.
In fact:
f
= I
+
I ( 0 , l ) and w e c l a i m t h a t i t i s
not
174
GUERREIRO
I
the function I
+
+
,
I (0,l) $ .I
because f o r
f ( x ) = ( x , y ) belongs t o
9 S,
v E H, v
b u t does n o t belong
I.
to
I(0,l).
g E Io, i g ( 0 ) = v, there i s a
On t h e o t h e r h a n d , i f {si : i
IN
E
fi(x) = g(x) and
fi
1
C
-
S, (xIv
-
such
,x
si)
E H.
+
v
as
i
+
fi
as
i + m .
Then
Consider
m.
belongs to
Each
'Ib1
g, f o r t h e t o p o l o g y
-+
si
that
sequence
I
I
+
+
I(0,l)
I(0,l) = I o l
which e s t a b l i s h e s t h e c l a i m .
?.
I #
Furthermore d o e s not b e l o n g t o
plies
= I
f(x) = (x,x)
-
v
I = I . Moreover:
-
I
C
1,
belongs t o
I.
By C o r o l l a r y 2.18,
-I
In fact:
v
I =
7
C I.
=
-.
(I)
but
im-
0'
Then :
3 . IDEALS O F FUNCTIONS WEAKLY UNIFORMLY m-CONTINUOUSLY DIFFEREImIABLE
ON BOUNDED SETS
DEFINITION 3 . 1 : .tinUOUb E
>
0,
A function
f :E
+
i s c a l l e d Weahey unidohmey can-
F
a n bounded n e t n ( w u c b b ) i f g i v e n B C E a bounded subset
t h e r e are
x , y E B, ( p i ( x l
DEFINITION 3.2:
-
6 > 0
1
vi(yl
Let
B
< 6
and
,
p1
, ...
I
(pk E
1 5 i 5 k , imply
E'
Ilf(x)
-
be a d i f f e r e n t i a b i l i t y t y p e from
We d e f i n e :
i s wucbs,
k E IN
, k 5 ml.
such
and that
f ( y ) i I < E.
E
to
F.
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
W e endow
gem(E;F) w i t h t h e
d e f i n e d by t h e fam-
+:-topology,
W
175
i l y o f seminorms:
B
C
a bounded subset,
E
W e remark t h a t
k E IN
,
IF(E;F) = I
k
5 m.
bm
(E;F)
f o r a n y compact
type
of
0 , whenever E i s a f i n i t e d i m e n s i o n a l s p a c e . No-
differentiability
t i c e a l s o t h a t f o r 8 a compact d i f f e r e n t i a b i l i t y t y p e from E t o F , Om Pf ( E ; F ) C Iw ( E ; F ) (see Aron and P r o l l a [ 1 ] 1 .
DEFINITION 3 . 3 :
Let
b e a d i f f e r e n t i a b i l i t y t y p e from E
O
A C 8 p ( E ; F ) a non-empty
subset. W e define:
i n a s i m i l a r way a n d , by i n t r o d u c t i n g t h e na-
We may d e f i n e
t u r a l m o d i f i c a t i o n s i n Example 2 . 1 9 , +We m - c l o s e d subset of f o r any
A C
t o F and
is n o t always a
A -4
On the o t h e r h a n d , A
&:(E;F).
em Iw (E;F) a
w e see t h a t
non-empty
is
+:-closed
s u b s e t . The p r o o f o f t h i s f a c t
is
similar t o 2.8.
PROPOSITION 3 . 4 :
16
0
id
a di6dekentiabiLity t y p e 6kom
o a t i d 6 y i n g ( i i i ) 0 6 U e d i n i t i o n 2 . 3 , .then att
to
F
S p ( E ; F ) O P C S:(E;F),
doh
P E E ' @ E.
PROOF: then
E
Let
f E & p ( E ; F ) and
i k ( f o P ) ( x ) = ikf(Px) oP.
P E E ' 8 E.
If
k E IN, k
5 m, x
E
E,
176
GUERREIRO
Let
b e a bounded s u b s e t and
B C E
bounded subset, t h e r e are
- vi(Py)
Ivi(Px)
x,y E B,
IIGkf
(Px)
and
6 > 0
I
< 6 , 1:
-
i k f (Py) II
i
ql,...,ps
5 <
> 0. A s
E
Sr
P(B) E
is a
C E
such t h a t
E'
imply
€/I1 P I1k .
Then :
which p r o v e s that
x
E E
Let
DEFINITION 3 . 5 :
+
€I be
hk(f
oP) (x) E Pek(E;F)
i s wucbs.
a d i f f e r e n t i a b i l i t y t y p e from
s a t i s f y i n g (iii)of D e f i n i t i o n 2 . 3 , and l e t G
C
0
E
to
F
E ' 8 E and A C &$(E;F)
be non-empty subsets.
W e s a y t h a t ( A , G ) 6 a t i h d i e n c o n d i t i o n (L) i f g i v e n
have
A og
POlogY
C
em
A,
t h e closure being considered with respect t o t h e to-
'Iw *
{Pn ; n E IN }
C
E' 8 E
such t h a t
Let
IR nuch t h a t
& r ( E ) i6 an a l g e b h a and L e t
Suppobe thehe
v
o Pn
+
i b
a nequence
I
C
Then
ib
for all 9 EE'.
G = { P n ; n E IN } C E ' 8 E
ha4 p h o p e h t y (B*) N i t h h t h p e c t t o G ;
?
,
hatib6ie6
the
c o n d i t i o n (L).
Tp-tLobuhe
06
I i n
For t h e proof w e need t h e f o l l o w i n g lemmas:
.lE(:&
E
t o
8 P ( E ) b e an i d e a l .
that:
ii) ( 1 , G )
9
be a compact d i d b e h e n t i a b i l i t y Xype dhom
0
THEOREM 3 . 7 :
i) E
se-
W e s a y t h a t E had phopehty ( B * ) i f t h e r e i s a
DEFINITION 3 . 6 : quence
we
g E G
nuch
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
Let
LEMMA 3.8:
be a d i d i e h e n t i a b i e i t y .type daom
8
& P ( E ) in an aegebaa and
.that
R : g E Iwe m ( E l
16 i d e a l ad
I
+
C
C
m 1;
E
Thehe
id
Muheouek,
R ( 1 ) i n an
f
E
E P ( E ) , and
in
f E
I,
then
Rf
Sbm(El).
8 a compact d i d d e h e n t i a b i l i t y t y p e d h o m
id
IN, k 5 m , B C E
u bounded n u b n e t ,
E
henpect
E
t o
F,
> 0.
no E hl d u c h t h a t
S e e Aron a n d P r o l l a [ 1]
PROOF OF THEOREM 3 . 7 : Conversely, l e t
.
I t i s clear t h a t f E
:,
B
E
C
-
Y
I C I.
a bounded subset,
k
5 m,
and
be given.
By Lemma 3 . 9 ,
there is
no
E N
such t h a t
pBIk(f - f oPn) < ~ / 3 , n Fix
n
1. n 0
and l e t
P = P
a n d r e s u l t s from Lemma 3 . 8 , Rf in
connideh
l e t E b e a b p a c e 6 a t i b B y i f l g phopehty (B*) w i t h
f E s ~ ( E ; F ) ,k E
> 0
and
Analogous t o 2 . 1 3 .
t o {Pn ; n
E
nuch
EP(E).
g F ( E ) i n an i d e a e Xhen t h e rbm-cLonuhe 06
Ebm(E1).
LEMMA 3 . 9 :
PROOF:
rn
t o
glEl E gbm(El).
6eLongn t o t h e ~ ~ ~ - c k ? o b u06h e R ( 1 )
PROOF:
E
b e a d i n i t e dimenbionae llubbpace
El C E
Let
E'
I??
n
.
If
n0
.
= P(E)
belongs to t h e
Sbrn(El). F u r t h e r m o r e , P ( B ) C El
,
by u s i n g n o t a t i o n
r b m - c l o s u r e of
R(1)
i s a bounded subset, then a rela-
t i v e l y compact s u b s e t , a n d t h e t o p o l o g y by t h e f a m i l y of seminorms:
El
2
T~~
may be defined i n gh(El)
178
GUERREIRO
L C El
a compact subset, j
So, t h e r e i s
g
such t h a t :
E I
I i i i ( R f ) (Px) O P x E B ,
El, j 5 m.
E
-
ii(Rg) (Px) o P l l , < E / 3 ,
O ( i ( k ,
and u s i n g t h e f a c t t h a t ( 1 , G )
s a t i s f i e s c o n d i t i o n ( L ) , t h e r e is h E I
such t h a t
'B,k
(4 O P
-
h) < ~ / 3 .
Then :
x E B, 0
4.
-
i < k , which c o n c l u d e s the p r o o f .
IDEALS OF
0
Ecm(U)
DEFINITION 4.1:
For
A C ECm(U;F) a n o n - e m p t y
= n {A+I(a,k,L,E); a
E
U, k 5 m , L
C
subset
E compact,
E
we
define
> 0)
where I ( a , k , L , ~ l = { f E g C m ( u ; F ) ; I I i i f ( a ) v I I < E , v E L, 0
5 i 5
kl.
WHITNEYS SPECTRALSYNTHESIS THEOREM IN INFINITE DIMENSIONS
The d e f i n i t i o n o f
e x t e n d s n a t u r a l l y and obvious modifica
-
may b e f a i l t o b e TCm-closed. Bycontrast,
t i o n s i n 2.19 show t h a t
i s always
179
Tcm-closed.
The d e f i n i t i o n of c o n d i t i o n (L) f o r a p a i r ( A , G ) ,
G
C
E' 8 E
a non-empty s u b s e t , i s n a t u r a l l y e x t e n d e d t o o .
THEOREM 4.2:
be a n k k d and buppobe Ahetre 0 G
I C Ecm(U)
Let
C
E' 8 E
buch t h a t
i)
iE, t h e i d e n t i t y
E
06
,
betungb t o t h e ctobuhe
06
i n
G
F(E;E) ; ii) (1,G) b a t i b 6 i e b c o n d i t i o n ( L ) Then
LEMMA 4.3: VeCtOh
i b
Let
t h e Tcm-c.labuhe
I
U n El
C
16 we c a n b i d e h 06
06
then
Bcm(U1).
i,
K
C
giUl E gCm(U1)aU'l .the Tcm-dClbWze
06
acm(U),
f E
id
R(1)
i n
f E
1,
gCm(U1).
gbm(U1) = BCm(U1)
W e j u s t remark t h a t
I t i s clear t h a t
PROOF OF THEOREM 4.2: f E
dimevlshnd
to-
is a f i n i t e d i m e n s i o n vector space.
p o l o g i c a l l y b e c a u s e El
Let
+
Moheoveh,
Rf b e l o n g b t o t h e Tcm-C.tObWLe
PROOF: Analogous t o 2.13.
a dinite
E
C
a non-empty open bubbet.
Scm(U)
R :g
R ( 1 ) i b an i d e a l
ECm(U).
i n
be an i d e a l , El
C Ficm(U)
a u b b p a c e , U1
06 I
.
U
and
L
C E
i
C
i.
compact s u b s e t s , k
By Lemma 3 . 1 , P r o l l a a n d G u e r r e i r o [ l l ] , t h e r e are
5 m,
u E G
E
> 0.
and V C U
a non-empty open s u b s e t s u c h t h a t
Consider
El = u ( E ) , U1= E l
11 U,
K 1 = u(K) a n d
L1 = u ( L ) .
By
GUEAREIRO
180
u s i n g n o t a t i o n and r e s u l t s f r o m Lemma 4 . 3 , t h e r e i s
g
On t h e o t h e r hand (1,G) s a t i s f i e s c o n d i t i o n ( L ) h
E
I
acm(U)
-
THEOREM 5.1: C
aLL
so
there
is
L, 0 5 i 5 k .
x
T h i s shows t h a t
W
such t h a t
such t h a t
(x,v) E K
5.
E I
f E
7.
0
SUBMODULES OF
tCm(U;F)
Let F be a bpace with t h e apphoximation phopehtg
BCm(U;F) an
8m(u)-submodule s a t i n d y i n g :
(v
o W) 8 v c
tp E F', v E F .
Suppose thehe is i)
iE
G C E' Q E
duch t h a t :
6eLongn t o the ~ L o d u h e06 G in
LC(E;E);
if
and doh
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
Then
06
i b t h e Tcm-ctobuhe
W in
The p r o o f of 5 . 1 u s e s t h e f o l l o w i n g W
C
GCm(U;F) i s a n
T h e v e c t o h bubhpace
LEMMA 5.2:
. Moheoueh,
ECm(U)
&Cm(U)-submodule a n d
(9 o
W,G)
SCm(U;F). two
lemmas,
both ,
In
9 E F'.
9 o w = {p
hatib 6ieb
181
o g; g E W }
ad
an id&
i b
condition (L), id ( W , G )
hatib-
iieb condition (L).
PROOF:
If
h E gcm(U) a n d
= h(p og) E 9
g E W, then
O W . Therefore
9
OW
a n d , so
hg E W
i s an i d e a l .
Suppose now t h a t (W,G) s a t i s f i e s (L) a n d l e t be a non-empty open s u b s e t s u c h t h a t and
L C E
compact s u b s e t s , f
E
9 o (gh) =
g
and
E G
V C U
g ( V ) C U . I f we c o n s i d e r K
W , k 5 m,
E
> 0,
is
there
h
C
V
E
W
such t h a t
Then :
This proves t h a t
Suppobe t h a t
LEMMA 5.3:
G
doh borne
16
C
f E
(9 o
W) o ( g !V) C (9 o W l V )
.
0
iE beLong6 t o t h e c t o b u k e
G
06
in EC(E;E),
E' 8 E , and t h a t ( W I G )batid6ieb condition ( L ) .
GI
then 9 o f
beLong6 to t h e r c m - c l o b u h e
06
9 o
W
in
FhCrn(U).
PROOF:
Consider
f E
5,
a
E
U, k 5 m ,
E >
0
and
L
C
E
a
compact
182
GUERREIRO
s u b s e t . There i s
y
E
L, 0 5 i
5
g E W
such t h a t
9 o f E
k , which p r o v e s t h a t
, S i n c e Lemma 5.2
(9 o W)'
e n a b l e s u s t o a p p l y Theorem 4 . 2 , w e c o n c l u d e t h a t t h e TCm-closure of
q oW
PROOF OF THEOREM 5.1:
sets, k 5 m,
Then
E
>O
in
acm(U)
f
Let
E
i;,
and d e f i n e f o r
A = U {Ai;O
5
i
5 k)
a p p r o x i m a t i o n p r o p e r t y , t h e r e are that:
E
W
5
K C U, L i
5 k
be compact sub-
C E
t h e set
i s a compact s u b s e t of n
to
E
N,
'jEF',
"j
F.
E F
By t h e such
belongs t o
9.o f 3
9 .OW, 1
so
t h e r e are
w.
Consider
such t h a t
'K,L,k where
belongs
0
let 0
o f
n
By Lemma 5 . 3 , e a c h gj
.
q
E~
Let
= ~ / 3(1 +
(9.o f 7
-
9 . 09.) < E 3 3 1
n I: II vj 1 1 ) . j=1
n By h y p o t h e s i s , h = .E ( 9 . 09.)8 V ]=I 1 3 j'
t E W such t h a t
h
E
WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
183
-
P ~ , ~ , k ( ht) < E/3.
Then:
E/3
n
+
I:
(x,v)
E
This proves t h a t
p
+
EIIIV.lI
3
j=1 K
x
L,
KiLik
(f
4 3 < E,
0 5 i
- t) <
5 k.
E,
and so
f
E
g, as desired.0
I n o r d e r t o drop t h e approximation p r o p e r t y of t h e space F, we
w i l l i n t r o d u c e a new t o p o l o g y .
We w i l l d e n o t e by
DEFINITION 5 . 4 :
-rCm-* t h e t o p o l o g y
defined
in
Ccm(U;F) by t h e f a m i l y o f seminorms:
where
R
C
U, L
a r e compact subsets, and
E
C
Notice t h a t f o r any subset
9 E F', k
A C BCm(U;F) i t
above d e f i n i t i o n t h a t
f E BCm(U;F) b e l o n g s t o t h e
A i f , and o n l y i f ,
f
f o r each
PO
Let
k E IN.
follows f r o m
the
T ~ ~ c-l o*s u r e of
b e l o n g s t o t h e TCm-closure of V o A i n Ca(U),
p E F'.
DEFINITION 5 . 5 :
5 m,
A C CCm(U;F) be a non-empty
subset.
184
GUERREIRO
We define: A*
= n {A+I(a,k,L,lp,~); a E U, k
DEFINITION 5.6:
is
Let
C
E compact,
lp
E F',
E
>O},
is a non-empty subset, an argument similar to
If A c E'~(U;F) 2.8 shows that A *
5 m, L
A
T ~ ~ - closed. *
C
ICm(U;F) and
G C E'
8
E
be non-empty sub-
sets. We say that (A,G) b a t i d d i e 6 c o n d i t i o n (L*) if given g V
C U
E
G
and
a non-empty open subset such that g(V) C U, we have Ao(glV) C (A/V) I
the closure considered in
THEOREM 5.7: theae i d
G C
LC(E;E) a n d
Lcb
E'
@
W E
ECm(U;F) be a n
buch that
iE
Em(U)-6ubmodu~e.
b e l o n g 6 t o t h e cLoduhe
Suppode
06
G
in
( W I G ) 6ati6die6 condition (L*).
T h e n W* i 6 t h e
PROOF:
C
(Ecm(V;F), T'~-*).
T ~ - *cLa6uae
oh
W.
Apply Lemma 5.2 and Theorem 4.2.
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[ 11
R. ARON and J. B. PROLLA, Polynomial approximation of differentiable functions on Banach spaces (to appear).
[ 21
S.
DINEEN, Holomorphy types on (1971)I 241 - 288.
a
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WHITNEY'S SPECTRAL SYNTHESIS THEOREM IN INFINITE DIMENSIONS
31
185
J . LESMES, On t h e a p p r o x i m a t i o n of c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s i n H i l b e r t s p a c e s , Rev. Colombiana d e Matem.8
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217
- 223.
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B. MALGRANGE, I d e a L d v d diddehentiabte d u n c t i o n b , T a t a I n s t i t u t e o f Fundamental R e s e a r c h , Bombay, 1 9 6 6 .
1 61
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-
R.
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06
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T o p v P o g y o n dpac.eb
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M.
Z.
NASHED, D i f f e r e n t i a b i l i t y a n d r e l a t e d p r o p e r t i e s o f nonl i n e a r o p e r a t o r s : some a s p e c t s of t h e r o l e entials
i n nonlinear functional analysis, i n
Academic P r e s s , J. B.
( 1 9 7 1 ) , pp. 1 0 3
differ-
Nofinean
L. B. R a l l ) ,
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[lo]
of
- 309.
PROLLA, On p o l y n o m i a l a l g e b r a s o f c o n t i n u o u s l y
differ-
e n t i a b l e f u n c t i o n s , Rendiconti dell'Accademia Nazionale d e i L i n c e i , Serie 8, v o l .
[111
J . B.
PROLLA a n d C.
57 ( 1 9 7 4 1 ,
481-486.
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 spaces w i t h 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 for Mathematik 1 4 ( 1 9 7 6 ) , 251
- 258.
RESTREPO, An i n f i n i t e d i m e n s i o n a l v e r s i o n of a t h e o r e m B e r n s t e i n , P r o c . Amer Math. SOC. 23(1969) , 193 - 198.
1121
G.
[131
H . WHITNEY,
.
Math.
of
On i d e a l s o f d i f f e r e n t i a b l e f u n c t i o n s , Amer. J . of 70 ( 1 9 4 8 ) , 635
- 658.
This Page Intentionally Left Blank
Appro&mation Theory and Functional Analysis J.B. ProlZa ( e d . ) @North-XoZZand PubZishing Carrpany, 1979
RECENT PROGRESS IN BIRKHOFF INTERPOLATION G. G. LORENTZ t Department of Mathematics The University of Texas Austin, Texas, U.S.A.
s. D. RIEMENSCHNEIDER* Department of Mathematics University of Alberta Edmonton, Alberta, Canada
51. INTRODUCTION The first paper [ 3 1 on Birkhoff interpolation is due to G. D. Birkhoff himself, which he presented to the American Mathematicalsociety when he was only 19 years old. Its style is old-fashioned; the main interest is in identities, remainder formulas,
and mean value
theorems. Birkhoff w a s interested in the sign of the kernels which appear in these formulas, and proved the important and deep
theorem
about their number of zeros. In 1955 -58, TGran and his pupils studied the
"0 - 2
interpolation", which prescribes the values of Pn and knots. They studied a very special selection of knots rivatives of Lagrange polynomials
-
PA
lacunary at
the
zeros of de-
- and obtained many beautiful results
t Supported in part by Grant MCS 77-0946 of the National
Science
Foundation.
*
Research supported by Canadian National Research
A
- 7687.
187
Council, Grant
188
LORENTZ and RlEMENSCHNElDf R
(see [ 2
I,
[ 4 5 1 and [ 4 1 1 ) .
I n 1 9 6 6 , I . J . Schoenberg 1391 a s k e d when
the
interpolation
problem w i t h a g i v e n s t r u c t u r e i s s o l v a b l e f o r e p o s s i b l e
sets of
k n o t s . T h i s i s t h e problem of r e g u l a r i t y or p o i s e d n e s s of t h e i n t e r p o l a t i o n m a t r i x , which h a s proved t o b e e x c e e d i n g l y Atkinson and A. Sharma [ l ] and D . Ferguson [ 7 ]
difficult.
gave t h e b a s i c t h e -
orems o f r e g u l a r i t y , K a r l i n a n d Karon I131 c o n t r i b u t e d a b o u t c o a l e s c e n c e , and L o r e n t z (1181
,
[ 191
X.
, [ 221 ) ,
the
theorem
theorems of s i n g u -
l a r i t y . Among t h e a p p l i c a t i o n s of B i r k h o f f i n t e r p o l a t i o n , w e mention t h e u n i q u e n e s s problem f o r monotone a p p r o x i m a t i o n [29 1 , R. A. L o r e n t z [ 30 1 1 , and t h e B i r k h o f f
(Lorentz -Zeller
quadrature formulas (Lormtz
and Riemenschneider [ 2 4 1 ) . I n r e c e n t y e a r s s e v e r a l p a p e r s have d e a l t w i t h t h e Birkhof f i n t e r p o l a t i o n problem f o r s p l i n e f u n c t i o n s ( K a r l i n Karon 1131 and o t h e r s ) . The p r e s e n t r e p o r t attempts t o g i v e a n e x p o s i t i o n of this thmry f o r polynomial i n t e r p o l a t i o n . F o r t h e s a k e of b r e v i t y , weomit s p l i n e i n t e r p o l a t i o n , and " l a c u n a r y i n t e r p o l a t i o n " w i t h s p e c i a l k n o t s . T h i s paper
i s b a s e d on t h e 1975 r e p o r t [ 2 0 1 of o n e of u s t o t h e c e n t e r o f
Numerical A n a l y s i s , U n i v e r s i t y o f Texas i n A u s t i n . A l a s t remark : t h e name "Birkhof f" i n t e r p o l a t i o n problem (rather
t h a n " H e r m i t e - B i r k h o f f " ) seems t o be c o m p l e t e l y j u s t i f i e d
from
all
p o s s i b l e m a t h e m a t i c a l p o i n t s of v i e w ; b o t h h i s t o r i c a l as w e l l as t b s e of s u b s t a n c e .
92. BASIC DEFINITIONS AND THEOREMS 2.1.
DEFINITIONS
Let
S = {go,gl,...,gN)
be a system of
n
times
c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s on a s e t A which i s e i t h e r a n i n t e r v a l [a,b] or t h e c i r c l e
T . A l i n e a r combination
w i l l be c a l l e d a podynomiat i n t h e s y s t e m S . A matrix
P =
N
Z
j=O
'3
RECENT PROGRESS IN ElRKHOFF INTERPOLATIOW
189
(2.1.1)
i s a n i n t e h p o t a t i o n mathix d o h
S i f i t s elements
o n e and i f t h e number of o n e s i n
i s equal t o
E
N
I n g e n e r a l w e do n o t a l l o w empty r o w s r t h a t i s a n k=O,...,n
.A
bet
04 h 0 t b
X = {xlI...,x
p o i n t s of t h e s e t A . The e l e m e n t s fined for
+
= N
1.
i f o r w h i c h eik=O,
m
and t h e d a t a
distinct
cik
(de-
eik = 1) d e t e r m i n e a Bihhhadd intehpaeatian problem which
c o n s i s t s i n f i n d i n g a polynomial
satisfying
P
P ( ~ (xi) ) = cik
(2.1.2)
The s y s t e m ( 2 . 1 . 2 )
N + 1
+ I, 1 E I
m 1 c o n s i s t s of
E, X I S
or
eik are z e r o
(eik = 1).
c o n s i s t s of
N
+
1 l i n e a r equations
with
a
The p a i r E l X i s c a l l e d hegulah i f e q u a t i o n s j' ( 2 . 1 . 2 ) h a v e a ( u n i q u e ) s o l u t i o n f o r e a c h g i v e n s e t o f cik ; o t h e r w i s e unknowns
the pair
El X
i s ALngueah. A p a i r
E l X i s r e g u l a r i f and
only
if
t h e d e t e r m i n a n t of t h e s y s t e m
(2.1.3)
gives a row of t h e
i s d i f f e r e n t f r o m z e r o . Formula ( 2 . 1 . 3 ) nant c o r r e s p o n d i n g t o t h e e n t r y
eik = 1 i n E ;
t h e o r d e r o f t h e rows
i n (2.1.3) i s taken a s ' t h e lexicographical ordering of By
A(E;X) w e denote t h e (N
+
1)
x
(N
+
determi-
( i l k ) , eik=l.
1) m a t r i x g i v e n by ( 2 . 1 . 3 ) .
The basic n o t i o n of t h i s report i s t h a t of a p o i b e d
or hegdah
m a t r i x E . An i n t e r p o l a t i o n m a t r i x E h a s t h i s p r o p e r t y i f t h e E l X i s r e g u l a r f o r each set of k n o t s
X
i n a given class.
c o n s i d e r several types o f r e g u l a r i t y ; ohdctr h e g u t a k i t y , i f a n d t h e k n o t s must s a t i s f y
a
5 x1
( c o m p l e x h & g u l a h i t y ) when t h e k n o t s
<
... <
are
xm 5 b; arbitrary
One
pair can
A = Ia,bl
t e a l mc?gutaJLitq distinct
real
L O R E N 1 2 and RIEMENSCHNEIOER
190
(complex) numbers; a n d , t k i g o n o m e - t l r i c negueafii-ty which i s o r d e r regul a r i t y on t h e c i r c l e ,
-
... <
5 x1
71
means t h a t t h e d.eterminant
D(E,X) # 0
k n o t s l w h i l e s i n g u l a r i t y means t h a t
x
m
c 71.
for
The r e g u l a r i t y o f
all
( a d m i s s i b l e ) sets of
vanishes f o r
D(E,X)
E
s h a l l d i s t i n g u i s h between d . t h o n g n i n g u l a k i t y 16 1
some
X.We
when D ( E , X ) t a k e s
v a l u e s o f d i f f e r e n t s i g n , and weak n i n g u L a h i R y , when
D(E,X) v a n i s h e s
w i t h o u t a change of s i g n . A m a t r i x i s s i n g u l a r i f and o n l y i f some non-trivial polynomial
P
is annihilated b y
El X
f o r some a d m i s s i b l e X ; t h i s means t h a t P
s a t i s f i e s t h e homogeneous e q u a t i o n s
A(E,X).
.the d e d e c L
06
eik = 1. For
for
r ( E ) , t h e lowest p o s s i b l e r a n k o f t h e
a singular matrix we consider matrix
P ( k ) ( ~ i )= 0
Then
E , i s t h e l a r g e s t p o s s i b l e dimension o f t h e
subspace
o f p o l y n o m i a l s P I a n n i h i l a t e d by E l X f o r some X .
EXAMPLES:
A Lagkange i n t e k p o L a t i a n rnattix h a s
i n t h e column
0.
m
= N
A T a y L a k L n t e h p o L a t L o n rnathix i s a n
+
1 and
1
x
(N
ones
+ 1) ma-
t r i x c o n s i s t i n g of a s i n g l e row o f o n e s . A Hehmite m a t h i x h a s b l o c k s eio
--
... = ei l k i
= 1 of o n e s i n e a c h r o w w h i l e t h e r e m a i n i n g e n t r i e s
a r e a l l z e r o s . An A b e l m a t k i x p r e s c r i b e s e a c h d e r i v a t i v e
P ( ~ )
at
e x a c t l y o n e p o i n t and t h u s h a s o n l y a s i n g l e one i n e a c h column.
2.2. THE ALGEBRAIC CASE
(2.2.1)
Here
N
g o ( x ) = N!
I
...
I
A = [a,b]
gN-l ( x ) =
,S X T
c o n s i s t s of t h e functions
I
g N ( x ) = 1,
and P are t h 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 k > N , w e may assume t h a t
. Since
n 5 N , and by a d d i n g columns of
P(k)
Z O
zeros
for to
RECENTPROGRESS IN BIRKHOFF INTERPOLATION
E , w e may s e t
that
N = n
191
such matrices a r e c a l l e d nohmae. W e s h a l l a s s m
n = N;
i n what f o l l o w s .
Now f o r m u l a (2.1.3) becomes n- k- 1
-k
-.(X-ik ) ! '
(2.2.2)
i f w e agree t o r e p l a c e l/pl
PROPOSITION 2 . 1 :
p < 0.
by z e r o i f
The detehminant
eik
i d a homogeneous p o L y n o m i d
D
... + n
- X eik=l
k = p luhich
(2.2.3)
(2.2.4)
ax
= {axl
, ...
,axmll
x
t a = (xl+u
, ... , x m + a l .
I n p a r t i c u l a r , it f o l l o w s t h a t t h e r e g u l a r i t y ( s i n g u l a r i t y ) o f E d o e s n o t depend on t h e c h o i c e o f t h e i n t e r v a l
[a,b].
F o r a normal m a t r i x E , l e t m(k) be t h e number o f o n e s i n k column k a n d l e t M(k) = X m ( r ) b e t h e number of o n e s i n columns r=O 0, k F o r example, M(n) = n + 1 , w h i l e M ( 0 ) > 0 means t h a t there
..., .
a r e o n e s i n column
dunctionb 0 4
(2.2.5)
E
0.
. The
The f u n c t i o n s
m ( k ) , M(k) a r e c a l l e d t h e po'&a
condition
M(k) 1. k
+
1,
k = 0
, ...
,n
LOREN12 and RIEMENSCHNEIDER
192
i s c a l l e d t h e PoLya c o n d i - t i o n , and t h e c o r r e s p o n d i n g m a t r i x i s c a l l e d
a Polya m a t h i x . S i m i l a r l y , a BinkClod6 m a t k i x is a matrix E whose P 6 l y a f u n c t i o n s a t i s f i e s t h e Rihhhodd c o n d i t i o n
M(k) 1. k
(2.2.6)
+
-
k = O,l,...,n
2,
1.
These c o n d i t i o n s p l a y t h e f o l l o w i n g r o l e . I t i s d i f f i c u l t t o when
E i s s i n g u l a r , t h a t i s when
e a s y t o see when
THEOREM 2.2:
minant
D(E,X)
D(E,X)
i s zero f o r a l l
(G. D. B i r k h o f f ,
I).
= 0
f o r some X
,
decide
b u t it i s
X:
Ferguson, and B. Nemeth)
id
D(E,X) i6 n o t i d e n t i c a l l y zeho i6 and o n l y
E
The d c t e k -
sa-tindies
t h e PoLya c o n d i t i o n . If
+
k
M(k)
1 f o r some k, t h e n t h e r e i s a n o n - t r i v i a l p l y -
P o f d e g r e e k which i s a n n i h i l a t e d by
nomial
.
E
This
proves t h e
n e c e s s i t y o f t h e c o n d i t i o n . The s u f f i c i e n c y was proved i n c o r r e c t l y by G.
D.
Birkhoff [ 3 1
and l a t e r c o r r e c t l y proved i n d e p e n d e n t l y by m t h
I331 a n d D. Ferguson [ 7 1
.
F o r normal matrices, c o n d i t i o n (2.2.5) i s e q u i v a l e n t t o t h e assumption t h a t any
(2.2.7)
n
C
k=nl
s l a s t columns c o n t a i n a t most m(k)
5 n - n1 +
1,
0
s ones:
5 n1 5 n -
I t i s (2.2.7) t h a t w e c a l l t h e P o l y a c o n d i t i o n
for arbitrary
normal) matrices. C o n d i t i o n (2.2.7) h o l d s i f and o n l y
if
(not can be
E
made i n t o a normal P 6 l y a m a t r i x by t h e a d d i t i o n of one or more
sup-
p l e m e n t a r y rows. A normal m a t r i x
E
is decomposable, [ 1]
,
c a n be s p l i t v e r t i c a l l y i n t o t w o normal matrices
E = El @ E2
,
i f it
El ,EZ. A m a t r i x is
indecomposable i f and o n l y i f e i t h e r i t s a t i s f i e s the Birkhoff condition,
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
193
o r i t c o n s i s t s o f a s i n g l e column. F o r e a c h P 6 l y a m a t r i x E, t h e r e i s t h e maximal c a n o n i c a l d e c o m p o s i t i o n
E = E
(2.2.8)
1 fB
... @
EU
,
i s e i t h e r a one column o r a B i r k h o f f m a t r i x . j E = El @ E2, t h e n t h e m a t r i x A ( E , X ) which appears i n (2.2.2)
where e a c h m a t r i x E
If
c a n be w r i t t e n , a f t e r a p r o p e r r e a r r a n g e m e n t of r o w s , as r + l
A(E,X)
(2.2.9)
{)>-(
r + l n - r
=
A s a corollary, we obtain
(2.2 * 10)
D(E,X) =
f
D(E1,X)D(E2,X).
Hence ,
THEOREM 2.3:
.LA h e g u d a h
( A t k i n s o n and Sharma [ 1 1
.L6 and o n l y i6 b a t h
2.3. REGULAR MATRICES
a6
i t b
A decompobable rncukix E =El @ E2
components ahe h e g u l a h .
By Theorem 2.2, t h e P 6 l y a c o n d i t i o n
(2.2.5)
i s n e c e s s a r y f o r r e g u l a r i t y . To o b t a i n a w o r k a b l e s u f f i c i e n t
t i o n , w e need t h e f o l l o w i n g n o t i o n . By a s e q u e n c e i n a r o w
condi-
i of t h e
m a t r i x E , w e mean a c o n t i n u o u s b l o c k of o n e s e ik =
(2.3.1)
,.. -- eill
= 1
which i s maximal. T h e r e f o r e , f o r a s e q u e n c e e i t h e r e i ,k-1
= 0,
and e i t h e r
d
=
n
or
ei,l+l
= 0
.
k = 0
A sequence
or is
else odd
LORENTZ and RIEMENSCHNEIDER
194
(or euen) i f i t h a s a n odd ( e v e n ) number o f o n e s . A sequence (2.3.1) i s b u p p a a t e d i f t h e r e e x i s t two o n e s
t o the
NW and
SW of
,
positions (il,kl)
in
E
eik = 1, i n o t h e r words, i f t h e r e a r e o n e s i n
(i2,k2) i n
Already G . D . B i r k h o f f [ 3
E with
il < i , k
k; i 2> i tk 2
1
1 n o t i c e d t h a t odd s u p p o r t e d s e q u e n c e s a r e
n o t good f o r r e g u l a r i t y : h e c a l l e d a m a t r i x E COnbehVativ&,
if
it
h a s no odd s u p p o r t e d s e q u e n c e s .
THEOREM 2.4: (Atkinson and Sharma [ 1
1
A nohmae PoLya m a t k i x E i b
)
a5deh &egul.uk id it kab no a d d d u p p o t t e d h e q u e n c e b . The s i m p l e s t proof of t h i s theorem depends o n a n e x t e n d e d form of R o l l e ' s theorem; a n o t h e r proof
i s b a s e d on t h e
B i r k h o f f 's k e r n e l , (see S e c t i o n 7 . 3 ) If
.
i s c o n t i n u o u s l y d i f f e r e n t i a b l e on
f
any two c o n s e c u t i v e zeros o f z e r o s of t h e d e r i v a t i v e b e r o f z e r o s of
f'
a, B o f f'
properties
f
. Hence,
la,b],
then
of
between
i s a n odd (or i n f i n i t e ) number i f between
c1
and B some num-
o f e v e n m u L t i p R i c i t y are known t o e x i s t , t h e n f '
must have a n a d d i t i o n a l z e r o ( w i t h a new v a l u e o r
as an
additional
m u l t i p l i c i t y of a known z e r o ) . Using t h i s and i n d u c t i o n , one p r o v e s
THEOREM 2 . 5 :
16 f
Z i o n a n n i k i e a t e d by
i b
an n-timed COntinUOUb~y did6ekentiable
bunc-
a P z l y a i n t e h p o e a t i o n mathix E c o n t a i n i n g no add
b u p p a h t e d beqUenCed a n d a
b e t 06 kn0tb
X = {xl,...,xm},
then
(2.3.2)
Applying t h i s theorem t o a polynomial
we obtain
Pn
=
0
Pn a n n i h i l a t e d by
E,X,
and t h u s p r o v e Theorem 2.4.
S i n c e 2-row matrices have no odd s u p p o r t e d s e q u e n c e s , w e o b t a i n THEOREM 2.6
( P 6 l y a [ 361 , W h i t t a k e r 1 4 6 1 ) :
A
2
x
(n
+ 1) i & e h p o e U t i o n
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
195
m a t h ix i d h e g u t a h i 6 and onl?y id it h a t i n 6 i e n t h e PZl?ya c o n d i t i o n . The problem of o r d e r - r e g u l a r i t y i s much more c o m p l i c a t e d i f the number o f rows i s more t h a n t w o (even i f i t i s o n l y t h r e e , see F o r cornpLex-hc?guRakity, when t h e k n o t s
xi may be a r b i t r a r y
numbers, t h e s i t u a t i o n i s d i f f e r e n t . D . Ferguson [ 7
1
56).
complex
characterized
c o m p l e x - r e g u l a r B i r k h o f f m a t r i c e s ( f o r a n o t h e r p r o o f see [ 1 3 1 ) . Comb i n i n g t h i s w i t h t h e d e c o m p o s i t i o n f o r m u l a (2.2.8)
THEOREM 2 . 7 :
A PzLya mathix E
canonical? d e co m poni t i on (2.2.8) 06
i h
one o b t a i n s
complex 4eguLah id and o n l y
,lLd
ConhihtA anLy 0 6 Hehmite mathiced and
m a t h i c e d w i t h a t mont t w o non- z et a h o w d .
U n f o r t u n a t e l y , t h e known p r o o f s a r e n o t s i m p l e . If
E
i s n o t r e g u l a r , i t s " d i s t a n c e from r e g u l a r i t y "
measured by i t s d e f e c t , g i v e n by f o r m u l a (2.1.4). from [ 5
Using a n
can
be
argument
1 w e can prove
THEOREM 2 . 8 :
F o h a no4maL P o l y a m a t k i x i u i t h e x a c t l y p o d d duppohted
Af?quencLn,
(2.3.3)
T h i s i n e q u a l i t y c a n n o t be improved.
2.4.
EXAMPLES, SYMMETRY, AND TRIGONOMETRIC INTERPOLATION
Theorem 2.4,
w e see t h a t a l l H e r m i t i a n ( h e n c e a l l Lagrange
Applying and
a11
T a y l o r ) matrices a r e r e g u l a r : t h e y do n o t have odd s u p p o r t e d swences. Abel m a t r i c e s a r e r e g u l a r by Theorem 2 . 3 s i n c e t h e y decompose
into
one column m a t r i c e s , e a c h w i t h a s i n g l e e n t r y e q u a l t o o n e , a n d t h e s e
are r e g u l a r . Computing t h e d e t e r m i n a n t
D(E,X),
o n e sees t h a t i n
106
MAENTZ and RIEMENSCHNEIDER
1 0 0
(2.4.1) E l = (
0
1
1 0)
1
0
0
1
0
,E2=( 0 1 0 1 0)
1 0 0
,E3=
1
0
0
0
0
,
1 0 0 1 0)
(0
1 1 0 0 0 0
1 0 0 0 0
t h e m a t r i x El is s t r o n g l y s i n g u l a r , t h e m a t r i x E 2 i s weakly s i n g u l a r , while t h e m a t r i x E j i s r e g u l a r i n s p i t e of t h e f a c t t h a t it has
( t w o ) odd s u p p o r t e d sequences. Thus, Theorem 2.4 o f A t k i n s o n - S h a r m a cannot b e i n v e r t e d . N e v e r t h e l e s s , t h i s i n v e r s i o n
is
"usually" t r u e .
M a t r i c e s which have e x a c t l y one odd s u p p o r t e d s e q u e n c e i n one o f t h e rows ( w i t h o t h e r s e q u e n c e s o f t h i s r o w b e i n g even or
not supported)
a r e n e c e s s a r i l y s i n g u l a r (see Theorem 5 . 1 ) . There are a l s o o t h e r res u l t s in t h i s d i r e c t i o n .
In h i s " l a c u n a r y i n t e r p o l a t i o n " , P . T i r a n h a s s t u d i e d symmetric matrices. A
bymmc3t'uk
mazhix E s h o u l d have a n odd number, 2 m + 1 ,
rows ( t o a s s u r e g e n e r a l i t y , w e a l l o w h e r e
a central
o n l y of z e r o s ) . T h e m a t r i x E i s symmetric i f and a l l k
.
A symmetric m a t r i x E i s a y m m e t h i c a e t y
i s r e g u l a r f o r e a c h symmetric s e t of k n o t s xo --0 ,
with
row c o n s i s t i n g
e-i,k = e i l k ,
hLegU&Ih
i f the
i=l,...,m
p a i r E,X
X = ~ ~ - ~ , . . . , x ~ , . . . , x ~ ~
x - ~ = - x i . However, t h i s n o t i o n c a n b e r e d u c e d t o
l a r i t y ( 2 2 1 : E i s s y m m e t r i c a l l y r e g u l a r i f and o n l y i f b o t h E2 a r e r e g u l a r , where El
(or
E2) c o n s i s t s of row 0 o f
E
a l l e l e m e n t s i n odd ( o r even) p o s i t i o n s have been r e p l a c e d w h i l e t h e o t h e r e l e m e n t s a r e l e f t unchanged,&
If t h e matrix
E
of
regu-
El and i n which by
ones
of r w s l , . . . , m of E.
h a s some measure o f symmetry, one
can
find
some s i m p l e n e c e s s a r y c o n d i t i o n s f o r r e g u l a r i t y which complement t h e P6lya c o n d i t i o n . For example, a g a i n l e t t h e rows o f
-m,...,-l,O,...,m, which row
j =I,...,m,
jk = 1. A l s o , l e t pe , p o E i n even o r odd p o s i t i o n s .
e-j,k
0 of
and l e t q = e
E
b e numbered
b e t h e number of k ' s
for
be t h e number o f z e r o s i n
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
PROPOSITION
2.9
197
[ 231 : U n d e t t h e a b o v e a d b u m p t i v n b ,
the
i n e q u a l i t y i n a n e c e n n a t y c v n d i t i o n d o t t h e t e g u l a t i t y ad
6vllvwLng E
(2.4.2)
Only a little is known about Birkhoff trigonometric interpolation. Here the system of functions
(2.4.3)
S =
( 1 , cos x, sin x , .
The polynomials in
S
TN(x) =
S
is
. . , cos Nx, sin Nx 1 ,
x E
[-TI,
1 7 .a +
N .. 8
(a, cos kx + bk sin kx).
k=l m
x
(n + 1) matrix
minant
of
TN
n with i=l,k=O n = 2N since
E = (eik)
2 N + 1 ones. There is no a phiohL reason to assume that
Tik)
.
are the trigonometric polynomials
An interpolation matrix is a
derivatives
n)
of all orders are non-trivial. The deter-
D ( E , X ) for E is translation invariant, D(E,X
+
a)
= D(E,X)
for all a. The P d l y a condition ( 2 . 2 . 5 ) i s replaced here by the condition
M(0) > 0
(2.4.4)
.
Also, an Atkinson-Sharma theorem holds for trigonometric terpolation. However, now one should consider cylindrical
in-
matrices
(with row m of E proceeding row 1). In this case, when defining the support for a sequence of row i, one can take supporting ones
from
the same row.
THEOREM 2 . 1 0 :
A mattix with.
m
F
2
how6
b a t i h d y i n g condifion ( 2 . 4 . 4 )
i n t t i g o n v m e t h i c a l l y h e g u l a h L6 it has nu o d d nequenced e x c e p t t h u n e
LORENTZ and RIEMENSCHNEIDER
198
b e g i n n i n g i n column 0 . As f u r t h e r examples o f r e s u l t s t h a t h o l d f o r t r i g o n o m e t r i c i n t e r p o l a t i o n , w e mention:
PROPOSITION 2 . 1 1 :
(i) ex
il,...,i
phime. L e t
P R1
be d i d d e h e n t hawn (R2)
04
E and 6 u p -
b e t h e n e t ad e v e n
(odd)
w h i c h t h a t m e onEd in p o n X a m (i k), j =l,...,p.
jf T h e n t h e doLeowing i n e q u a e i t i e b u h e MeCQAbahLj doh t h e hegu-
Lahity
06
E
( J o h n s o n [ 1 2 1 1 . A one hou m a t h i x
id it had
N
+ 1 one4 i n
E
i d
h e g u t a h id and only
e v e n pObitiOnA and N OneA i n odd
pObitiOMd.
53. COALESCENCE OF MATRICES 3.1. LEVELING FUNCTIONS AND COALESCENCE
The i m p o r t a n t
concept
of
c o a l e s c e n c e f o r t w o a d j a c e n t rows of a m a t r i x was i n t r o d u c e d by Karlin
al-
a n d Karon [ 1 3 1 . They a l s o g a v e t h e T a y l o r f o r m u l a , Theorem 3.3, though i t was L o r e n t z and Zeller [ 2 9 1 C
# 0
who f i r m l y e s t a b l i s h e d
i n t h a t f o r m u l a . R e c e n t l y , L o r e n t z 1 2 2 1 p u t t h e method
that on
a
b r o a d e r b a s i s which a l l o w s m u l t i p l e c o a l e s c e n c e . V a r i o u s a p p l i c a t i o n s
of t h i s method can b e f o u n d i n 1131, 1191 a n d [ 2 2 1 (see a l s o 5 3 . 3 ,
55
and 5 6 . 2 ) .
Let
m x ( n + l ) matr x , n o t n e c e s s a r i l y n o r m a l , s a t i s -
E be an
f y i n g t h e P6lya c o n d i t i o n (2.2.7)
see ( 3 . 1 . 1 ) b e l o w ) . W e
E a s a v e r t i c a l g r i d of boxes.
eik = 1, t h e n a b a l l o c c u p i e s t h e
i
-th
box i n t h e
k
-th
If
column. W e p l a c e a t r a y of
interpret
n + l boxes u n d e r
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
199
t h e columns o f t h e g r i d . Then t h e b a l l s a r e p e r m i t t e d t o
fall
from
t h e g r i d i n t o t h e boxes o f t h e t r a y i n s u c h a way t h a t if t h e boximm e d i a t e l y below i s o c c u p i e d , t h e n t h e b a l l r o l l s t o t h e f i r s t a v a i l a b l e box on t h e r i g h t . The c o n d i t i o n ( 2 . 2 . 7 )
a s s u r e s u s t h a t no b a l l
w i l l r o l l o u t o f t h e t r a y . The d i s t r i b u t i o n o f b a l l s i n t h e t r a y cons t i t u t e s t h e one row m a t r i x o b t a i n e d by c o a l e s c e n c e o f t h e m rows o f E.
I t is t o be expected t h a t t h e f i n a l arrangement o f t h e
balls
in
t h e t r a y i s i n d e p e n d e n t o f manner i n which t h e b a l l s w e r e a l l o w e d t o f a l l . Here i s a n example o f c o e l e s c e n c e of a two row m a t r i x :
. .
1st row
1
0
1 1
1
0
0
1
0
0
0 .
2nd row
0
1
1
0
0
0
1
1
0
o . . .
coalesced row
1
1
1 1
1
1
0
1
1
1
o . . .
p r e - c o a l e s c e d 1st row
1
0
0
1 1
0
0
0
1
0
0
1
. ..
F i g . 1.
Let
m(k) = m k ,
M(k) =
%
denote t h e P6lya f u n c t i o n s
of
some
i n t e r p o l a t i o n matrix s a t i s f y i n g t h e P6lya condition
n
(3.1.1
W e s h a 1 u s e c a p i t a l l e t t e r s t o d e n o t e t h e sum o f a f u n c t i o n , e.g. k G ( k ) = c g ( r ) . The e e v e L 6unctionn m o , Mo o f m and M are the largr =O e s t f u n c t i o n s g , G w i t h i n t e g r a l v a l u e s which s a t i s f y
(3.1.2)
0 5 g(k)
5
1, G ( k ) =
k
E
r=O
g ( r ) 5 M(k), k = O f . . . , n .
This is equivalent t o t h e following: i f (3.1.3)
p k = (..
. ( ( m , - 1 ) ++ m l - 1 ) ++ . . . + s - ~ - l ) + +
,
k =Of..
., n r
200
LOREN12 snd RIEMENSCHNEIDER
then
(3.1.4)
The c o e a d i c i e n t
06
co4?4?iAion a(M) = a ( E ) o f M,orof t h e m a t r i x
E l measures t h e d i s t a n c e of
Mo and i s
M t o t h e l e v e l function
de-
f i n e d by
(3.1.5)
( I n t h e above i n t e r p r e t a t i o n of c o a l e s c e n c e , t h i s i s t h e d i s t a n c e t h a t t h e b a l l s m u s t roll.) A m a t r i x
huh C o ~ k ? i A i O n hi f
E
a(E) > 0 .
The
b a s i c p r o p e r t i e s of l e v e l f u n c t i o n s a r e g i v e n i n t h e f o l l o w i n g theo-
rem.
THEOREM 3.1: ( M ~+ M ~ ) ' =
(i)
(MY +
(MY +
M ~ ) ' =
M;)'
( i i ) ( ( M ~+ M ~ ) ' + M ~ ) = ( M + ~ ( M ~+ M ~ ) ' ) ' 0
(iii) 16 t h e d u n c t i o n MY
(iv)
+
M2
.
a(Ml t M 2 )
= a(M1)
+
a(MY
a(M1)
+
a(M2)
=
The
1
x
M1 + M 2 4 a t i n d i e b (3.1.1)
+
= ( M +~ M +~ M ~ O)
,
then
bo
doeb
M2)
+
0
a(M1
+
Mz)
.
(n + 1) m a t r i x E o w i t h P6lya f u n c t i o n s m o l Mo i s c a l l e d
t h e coa.teAcence o d t h e mathiw
E t o one h o w .
More
generally,
if
U E2 i s a decomposition of E i n t o two d i s j o i n t s e t s o f r o w s , 1 t h e n t h e c o a l e b c e n c e i n E 0 6 t h e hvw4 El do o n e &ow i s thematrix
E = E
EY
U
E2
.
From Theorem 3.1 w e have
20 1
(3.1.6)
and
(3.1.7)
Noreover, the P6lya condition (3.1.1) is passed on to El
U
E:
u E~
from
E2.
In particular, we can consider the coalescence of two rows the matrix E , say
Ei-
in
- the j -th row. The j replaces these two rows by a single row, and the i-th row and
coalescence (E U E . I i 3 has collision coefficient
E
= a(Ei U E.). For a horizontal subma3 ij trix El of E, we consider the coalescence (E U E;I0 of El with 1 one row Ei produced by coalescing a disjoint horizontal submatrix
E2
c1
to one row. The coefficient of collision
crease with the number of ones in For
E2 = E \ El
E;
a(E1
i.e. as
E2
U
E i ) can only in-
increases in size.
,
is called the coe44icien.t
06
maximal c o l t i d i o n
06
El in the
matrix
E. We have deliberately not mentioned any ordering of the rows in E. Indeed, the disjoint sets of rows, El and E 2 , could intertwine.This is good for considerations involving real (or complex) regularity,but we shall need to consider the order of the rows for applications to order regularity.
LOREN12 and RIEMENSCHNEIDER
202
3.2. SHIFTS AND DIFFERENTIATION OF DETERMINANTS:
matrices El
and E 2 o f a m a t r i x E l
-+
t h e P d l y a f u n c t i o n (M1
MZ)O
-
U
.
Mi
t h e 1 x ( n + 1) m a t r i x
El
and
Ei
E i ) = 0 , and w e o b t a i n
U
i1
the
t h e pte-cvalebced matrix with respect to
I f w e denote t h e p o s i t i o n s of t h e ones i n .ti <
El
having
E 2 1 0 from t h e s e t w o by s i m p l e a d d i t i o n ( s e e F i g . 1 f o r an
example). W e c a l l
and
can b e ob-
E2)'
U
Then t h e t w o r o w s
"(El
w i l l be without c o l l i s i o n s , i . e .
row ( E l
t h e row (El
i1 b e
t a i n e d i n t h e f o l l o w i n g way. L e t
For horizontalsub-
...
<
respectively, then
II'
P i s formed from EY
0
El
and El by 11<...
Lj 5 R;
E~ u E;
lP and
j = l,...,p
I
by s h i f t i n g t h e o n e s from p o s i t i o n s R
The c o e f f i c i e n t o f c o l l i s i o n for
E2.
-
j
to k '
j'
is
(3.2.1)
S i n c e t h e o r d e r of t h e rows i s i m p o r t a n t f o r s t u d y i n g t h e order r e g u l a r i t y of m a t r i c e s , w e now c o n s i d e r t h e placement of thecoalesced rows i n E and i t s e f f e c t on t h e d e t e r m i n a n t . L e t and E
j
b e t w o r o w s of
a l e s c e d row
Ei
E
I
Ei
and l e t
with respect t o
E
(a;,
. . ., a P' )
The new ( m - 1 )
j '
o b t a i n e d from E by o m i t t i n g row
=
Ei
=
be t h e p r e x
(n+l)
- co-
matrix
and replacing row E . by (E. U E . 1 ' 3 1 3 i s t h e m a t r i x of cvaeebcence ad &ow Ei Zv )Low E i n E. I f X is Ei
I
j
t h e set o f k n o t s X w i t h
xi
omitted, we can o b t a i n t h e
a s f o l l o w s . The rows o f t h e m a t r i x
D(E,%)
which c o r r e s p o n d t o
Ei
determinant
A ( E , X ) a p p e a r i n g in(2.1.3)
a r e g i v e n by d e r i v a t i v e s of o r d e r s El,..
w e r e p l a c e t h e s e d e r i v a t i v e s by d e r i v a t i v e s o f o r d e r s . t i , . . . l I I ' replace
xi by x j
. The
The i n t e t c h a n g e numbex bring t h e r o w s of
.,R P;
P
and
new m a t r i x w i l l have d e t e r m i n a n t ( - l ) ' D ( i l g ) . 0
i s t h e number o f i n t e r c h a n g e s r e q u i r e d
A ( E , X ) i n t o t h e l e x i c o g r a p h i c order o f
to
fromthe
order i n h e r i t e d from E . S i m i l a r l y , if
Ei
= (.t~,,..,L*)
P
= E"i i s t h e p r e - c o a l e s c e n c e
of
RECENT PROGRESS IN BIAKHOFF INTERPOLATION
El w i t h r e s p e c t t o
t h e n w e o b t a i n t h e rnathix
E\Ei,
coakebcence by o m i t t i n g row
,
Ei
*
s e l e c t i n g some
E*
06
maximae
j f i , and r e p l a c i n g
by E j U Ei ( s o m e t i m e s r e f e r r e d t o as w d ~ c e n c c o ~ a oEiw Aa aou j i n & k L t y ) . F o r p r a c t i c a l p u r p o s e s c o n c e r n i n g r e g u l a r i t y , t h i s ma-
row E. 7
203
E
a.t
t r i x i s e s s e n t i a l l y independent of
c o a l e s c e n c e , t h e columns Furthermore, i f submatrix of
. , RP*
R:,.
j . I n d e e d , by t h e n a t u r e o f
in
c o n s i s t only of zeros.
E \ Ei
satisfies t h e P6lya c o n d i t i o n th en each
E
E \ Ei
o f columns
the
vertical
( a 0* =-l), is a PBlya
k , R G < k c Rl+,,
m a t r i x . T h e r e f o r e , E* h a s a d e c o m p o s i t i o n i n t o P d l y a matrices h a v i n g s i n g l e column components i n p o s i t i o n s
j
.
M o t i v a t e d by t h e need t o b r i n g a row E:
-
w e d e f i n e a 4h.id.t
Ell
A :k
1 of a s u b m a t r i x El
ei,k+l = 1. A s h i f t is d e f i n e d on
R
, El
R'
R'
ci
9'
of
t o a p r e - c o a l e s c e d form
eik = 1, of some row
an example, a g a i n l e t
As
+
k
+
lows: a s h i f t moves a o n e , position
The fornula (2.2.10)
D(E*,X*) i s i n d e p e n d e n t ( u p t o s i g n )
shows t h a t t h e d e t e r m i n a n t t h e c h o i c e of
. , & *P .
R;,..
i in
o f E as f o l El
into
the
eik = 1 only i f ei,k+l=O.
represent t h e p o s i t i o n s of onesin
i s t h e l a r g e s t R' w i t h R' # L , and qo q q q then R i s t h e f i r s t o n e of t h e s e q u e n c e i n EY ending i n R if 90' CIl t h e r e i s a s h i f t i n some row of El w i t h k s u c h t h a t llql 5 k 5 R
EY
respectively; i f
qo
which e i t h e r i n c r e a s e s
R
or decreases
q new m a t r i x (AE,)
AE1,
of
AE1
by o n e u n i t w i t h o u t c h a n g i n g theremaining
R
qo by o n e u n i t w i t h
a(E1)
reduces t h e
replaces
, EY
El
CIO
coefficient
unchanged. The (3.2.1)
when
Such a s h i f t i s c a l l e d a a e d u c i n g A&@
El.
n
A muetipee b h i d t
=
simple s h i f t s . I t transforms
PROPOSITION 3 . 2 :
Ei
.
collision
Rql,...,R
=
Let
Ei
(.Li,...,L') be t h e P
=
...
A1 El
Aa
of o r d e r
i n t o a matrix
(el, ...,eP )
phe-coaleocence
and E
06
Ei
j
B i s a product of B I
h E1 .
b e Rtuo a o w d
06
w i t h Renpect t o
E, and
Ej
.
204
LORENTZ and RIEMENSCHNEIDER
a system S of differentiable functions and the matrix A(E,X) associated with
E, X , S , we want to find the partial derivatives
the determinant
D(E,X) of (2.1.3).
T o differentiate
D(E,X)
of
with
respect to one of the parameters xi, we have to differentiate
each
row of (2.1.3) which contains xi. This leads t o
(3.2.2)
(and a similar formula for mixed derivatives), where the sum is taken over all representations of the multiple shifts of order i - th
f3
in
the
row of E. We would like to examine the behaviour of
D(E,X) as a function
of xi as x approaches another knot x This behaviour can be dei j' termined through the relationships between shifts, collisions, and coalescence.
THEOREM 3.3: (3.2.3)
FoZ
xi
+
x
j'
D(E,X)
has t h e TagLon. e x p a n h i o n
206
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
whehe C i n d e d i n e d i n P h o p o 6 i t i a n 3 . 2 ( b ) ,
ci = CY.
=
ij
a(Ei
U
E.), and 3
u in t h e i n t e h c h a n g e numbea. F o r p o l y n o m i a l i n t e r p o l a t i o n , when
S i s t h e system ( 2 . 2 . 1 ) , w e
have
THEOREM 3.4:
( i ) 7 6 El
i6 a 04
i b
a hohizontaL bubmathix
p o l y n o m i a l i n t h e u a h i a b l e b xi
j 0 i n . t d e g h e e n o t gheateh t h a n
(ii) I n a bingLe v a h i a b l e D(E,X)
(3.2.4)
E , t h e n D(E,X)
cohtrehponding t o
y(EL).
c o h h e b p o n d i n g t o a h o w Ei i n E ,
ha6 t h e h i g h e b t team
D(E,X) =
with
xi
06
y = y(Ei)
Xl
y !( -
c* D(E*,x*)
+
...
a n d E* d e d i n e d b y t h e maximal c o d u c e n c e .
3.3. APPLICATIONS OF COALESCENCE g u l a r i t y , w e t a k e rows Ei
i)'*
F o r o r d e r r e g u l a r i t y or o r d e r s i n -
t o be a d j a c e n t i n (3.2.3) w h e r e a s j f o r r e a l o r complex r e g u l a r i t y t h i s i s n o t n e c e s s a r y . T h i s r e m a r k a p and E
p l i e s throughout t h i s s e c t i o n .
3.3.1.
Suppose t h a t
E
i s a normal P 6 l y a m a t r i x . W e c a n g i v e a very
s i m p l e p r o o f [ 1 9 ] of Theorem 2.2. If i n (3.2.3), t h e n t h e same i s t r u e o f
D(E,%) i s n o t i d e n t i c a l l y z e r o D(E,X). By r e p e a t e d c o a l e s c e n c e s
of t w o r o w s , w e f i n a l l y r e d u c e E t o t h e one r o w form
~1,1,...,1~,
for which t h e d e t e r m i n a n t i s t h e Vandermonde d e t e r m i n a n t of t h e s y s -
tem
S = {go,...,gn}.
THEOREM 3.5:
16
Therefore,
E i b a notrmal P a l y a ma&ix
tetrminant 06 t h e bybtem S not identicaCCy
ZehO.
i d
a n d t h e Uandenmonde d e -
n o t i d e n t i c a l l y zeho, t h e n
D(E,X)
ib
LORENTZ and RIEMENSCHNEIDER
206
I f t h e determinant
3.3.2.
changes s i g n , t h e n so does
14 o n e
THEOREM 3 . 6 :
dingutah, then
06
or
D(I?,X)
D ( E , X ) . Hence [ 1 3 1
,
W e can e x p l o i t t h e i n t e r c h a n g e number
(3.2.3)
(and even i n
[ 22 ] )
,
nthongly
o h E* i d
E.
3.3.3.
(3.2.4)
[ 191
E
t h e coatebced mathiceb
t h e ohiginat mathix
ho i d
in (3.2.3) or (3.2.4)
D(E*,X*)
o occuringinformulas
by comparing t h e m a f t e r coalescence
of s e v e r a l rows i n d i f f e r e n t ways.
THEOREM 3.7:
q 2 3
Let
hVW6
F1,...,Fq
t i m e d , i n t w o d i d d e h e n t wayd, t o pfioduce t h e O1I
and
*.
.loq$
a;,
. ..
,5'
q- 1
-
h i n g t c ? hou.
1 76
a h e t h e cohhedponding i n t e h c h a n g e m b m ,
u1
t
... +
u
9
q- 1
t h e n E i d A t h o n g e g heaL b i n g u t a h
c l u s i o n hotdd
how
AaMC
g
id
(3.3.1)
F1,
E be c o a l e h c e d
06
...,F9
doh
"i t ,304
... t
5'
q- 1
(mod 2 )
any d g d t e m S . T h e name
bthLong o h d e h d i n g u l a h i t y . L h l i n a d d i t i o n ,
con-
thehowd
a t e a d j a c e n t and a t l c o a l e n c e n c e d a t e i n o n e d i r r e d o n ( L e e . i + l ,o h h a w
i t o how
i
+ 1
i].
t o how
The l a s t s t a t e m e n t i s r e q u i r e d s i n c e t h e c o a l e s c e n c e of r o w t o row
i t 1 c o n t r i b u t e s t h e s i g n from (xi
-
x i+l) " .
This
i
contribu-
t i o n i s t h e same on b o t h s i d e s i f a l l t h e c o a l e s c e n c e s have t h i s sam d i r e c t i o n . I n g e n e r a l , a less s i m p l e s t a t e m e n t , t a k i n g i n t o t h e c o l l i s i o n s f o r coalescences t i o n s of t h e
q
-1
i to
i + 1 , i s t r u e when t h e d i r e c -
c o a l e s c e n c e s a r e f r e e (see 5 6 . 2 ) .
W e g i v e a more e x p l i c i t f o r m u l a t i o n o f Theorem 3 . 7
r o w s of
E;
F1 =
account
(!2i,... ,Xi),
,...,R")q
F 2 = (!I:
for
three
and F 3 . By ( F ) s l w e
mean t h e p o s i t i o n s of t h e o n e s of row F p r e - c o a l e s c e d w i t h
respect
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
-
t o row Fs, and by (F)st
t h e p r e c o a l e s c e n c e w i t h r e s p e c t to (Fs
mean t h a t t h e i r e l e m e n t s s h o u l d
be w r i t t e n o u t i n t h e
o r d e r . By c o n s i d e r i n g t h e i n t e r c h a n g e numbers f o r ((F1
U
F2)'
U
ta.iVIA
thhee
and
F3)'
PROPOSITION 3.8: haw6
U
Ft)'.
t h e convention t h a t t w o sequences f o llo win g
Further, w e adopt other
207
(F1 U ( F 2
The m a t h i x
E
i h
U
each given
t h e coalescences
we obtain [22]
F3)')',
h t h a n g e y h e a l b i n g u l l a h id it c a n -
6 a h w h i c h ,the t w o b e y U e n C e A
(
(a;,
...,R')P 2 '
RY,.
.,
,&")
9 3
and
b e l o n g .to d i d d e h e n t pehmutatian C e a b b e b . Thus, a m a t r i x c a n h a v e t h r e e r o w s t h a t are so bad t h a t it s i n g u l a r f o r any a r r a n g e m e n t of o n e s i n t h e o t h e r rows, and for systems
3.3.4.
all
S.
By p r o p e r l y s e l e c t i n g k n o t s
PROPOSITION 3.9:
(3.3.2)
i b a d d , wheat 06
is
c o e L i b i o n 60a
Theorem 3 . 3 and 3.4 g i v e us
16
yi-
yi
xi,
= y(Ei)
haw i in
p o l y n o m i a e intehpoeatio n .
Z a
j+i
and
ij
aij
= a ( E i U E.) ahe t h e cveddicien-tA
I
E , t h e n E LA A t h V n g l y h e a l b i n g u t a h d a h
LOREN12 end RIEMENSCHNEIOER
208
3.3.5. D(E,X)
W e have restricted o u r s e l v e s i n (3.2.3) t o t h e e x p a n s i o n
i n one v a r i a b l e f o r s a k e o f s i m p l i c i t y . I f s e v e r a l k n o t s
approach
x
of xi
w e o b t a i n m u l t i p l e c o a l e s c e n c e . The e x p a n s i o n w i l l then
j r
c o n t a i n , as i t s main t e r m , a form o f o r d e r
a i n s e v e r a l variables.If
t h i s form changes s i g n , t h e m a t r i x E must be s t r o n g l y s i n g u l a r . T h i s r e q u i r e m e n t i s p a r t i c u l a r l y meaningful f o r r e a l s i n g u l a r i t y when t h e v a l u e s o f t h e v a r i a b l e s of t h e form are u n r e s t r i c t e d .
Let
3.4. EXAMPLES:
E
t r i c e s E and
k
b e o b t a i n e d from E by c o a l e s c e n c e .
The
c a n be r e g u l a r , weakly s i n g u l a r or s t r o n g l y s i n g u l a r
i n l o g i c a l l y n i n e p o s s i b l e c o m b i n a t i o n s . Theorem 3 . 6 r u l e s
E
combinations:
ma-
strongly singular with E being regular
out
or
two
weakly
s i n g u l a r . All t h e o t h e r c o m b i n a t i o n s c a n o c c u r . I n d e e d , by c o a l e s c i n g the matrices
E
El
and E3 of (2.4.1) t o t w o row form, w e see that
,E2
may be r e g u l a r when
examples of
G = ( O
1
(
0
E5=
The matrices
E
The
following
Kirnchi and N . Richter-Dyn [16] a r e less t r i v i a l .
E.
0
(3.4.1)
is any o f t h e t h r e e t y p e s .
E
1
1
0
0
0
0
1
0
1
0 ) , E 4 =
1
0
0
0
0
1
1
0
0
(' 0
1
1
0
0
0
1
0
0
0
l0 o0 )
1
0
0
0
0
0
') (' ') 0
0
1
0
0
0
0
,E6=
0
1
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
1
1
0
0
0
0
a n d E 5 are weakly s i n g u l a r ; t h e m a t r i x E4 i s r e g u -
lar; and t h e matrix E s i s s t r o n g l y s i n g u l a r .
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
209
5 4 . INDEPENDENT KNOTS The c o n n e c t i o n between t h e c o n c e p t o f an odd s u p p o r t e d sequence a n d t h e e x t e n d e d form of R o l l e ' s Theorem w a s e x p l o i t e d i n
52.3
to
o b t a i n a s i m p l e p r o o f o f t h e Atkinson-Sharma theorem. T h i s s i m p l e c o n n e c t i o n s u g g e s t s t h a t a more d e t a i l e d s t u d y of t h e i n f o r m a t i o n gained from R o l l e ' s theorem i s w a r r a n t e d . The method o f i n d e p e n d e n t
w a s f o r m u l a t e d by L o r e n t z and Zeller [ 2 8 1 and d e v e l o p e d
knots
further
L o r e n t z ((181 , [ 2 0 1 1 , i n order t o s t u d y s i n g u l a r i n t e r p o l a t i o n
by ma-
trices. m
E be a n
Let
x
n +1
d i f f e r e n t i a b l e f u n c t i o n on X = (x,,..
f
[a,b]
i s a n n i h i l a t e d by
which
E
and
and i t s d e r i v a t i v e s s p e c i f i e d by (4.11, w e
can
.,xm) C [a,b]
From t h e zeros o f
i n t e r p o l a t i o n m a t r i x , and f be an n-tires
,
that is, let
f
satisfy
d e r i v e f u r t h e r z e r o s by means of R o l l e ' s theorem. A s e l e c t i o n
a
of
complete s e t o f s u c h z e r o s i s c a l l e d a "Rolle set" of zeros. A RaLLe b e t
Rk
lection
,
k=0,1,
ties specified)
bat
a s f o l l o w s : The s e t (4.1).
If
doh a
R
ROl...,Rk
dunction
...,n ,
each
Ro
06
f
a n n i h i l a t e d by
E
,X
isacol-
o f RoLLe d e t b 0 6
zehob
t h e dehivativeb
f ( k ) s e l e c t e d inductively
c o n s i s t s of t h e zeros of
(with m u l t i p l i c i -
f as s p e c i f i e d
in
have a l r e a d y been s e l e c t e d , t h e n w e s e l e c t Rk+l
according t o t h e following r u l e s :
19
A zero of f ( k ) i n
a l s o a zero o f 2Q
All zeros of
R
k
o f m u l t i p l i c i t y g r e a t e r t h a n one
f (k+l) w i t h i t s m u l t i p l i c i t y reduced by one. f (k+l)
( i n c l u d i n g m u l t i p l i c i t i e s ) as specified
by (4.1) are i n c l u d e d i n 39
is
For any a d j a c e n t z e r o s
Rk+l a
,B
select, i f possible, a zero of
. of
f ( k ) belonging t o Rk, we f ( k + l ) between them subject
LORENTZ and RIEMENSCHNEIDER
210
to the restrictions: ( a ) If the new zero is one of the
xi' then it is not listedin
(4.1), or
(b) there is an additional multiplicity of xi as a zero of f(k+l) which is not acknowledged by (4.1). (c) In the event of (b) if t is the multiolicity of xi as a f (k+l) given by (4.1)
zero of
then the multiFlicity of xi
in Rk+l is defined as follows. We add to (4.1) the equation f(k+t) (xi) = 0, and determine the multiplicity of xi
as a zero of
f (k+l) from these equations. This may connect
two sequences in E and prescribe than
a
multiplicity
larger
+ 1.
t
If a zero does not exist subject to the restrictions in3Qr then we say that a
eodd
occurs at step
k + 1. A Rolle
set
constructed
without losses in any of its steps is called rnaxirnae. The function f may have many Rolle sets, some of them may be maximal, while
others
are not maximal. Some properties of Rolle sets are immediate consequences ofthe selection procedure. First of all, the only multiple zeros of f ( k + l ) in
Rk+l
are among the points xi in X . Secondly, the extended fonn
of Rolle's theorem shows that a loss will not occur if the rows of E corresponding to xi between adjacent zeros of Rk contain nooddsup ported sequences. (This was the connection used in 52.3.) We have
LEMMA 4.1:
RoUe
detd
7 6 t h e mathix
06
a bunction
E had no o d d d u p p o t t e d dequenced, ,then&
f , annihibated by
E
, X , ahe
maxirnab.
The number of Rolle zeros in a maximal Rolle set can be determined by induction: LEMMA 4.2:
16 f
i d a n n i h i b a t e d by
E,X, t h e n d o t
each k, k = 0 , 1,...,n,
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
t h e numbeh
RoCLe zeho.4 a d
06
d u n c t i a n f i n a t Lean2
M(k)
f(k)
-
21 1
i n a maximal R o l L e n e t
the
doh
k.
S = {gol...lgn) be a system of n-times continuously dif-
Let
ferentiable functions which are linearly independent on subinterval of
.
[a,bI A set of knots
each
open
X C [a,b] is called independeat
with respect to the system S if for each interpolation matrix every polynomial P in
S
annihilated by
E,X
El
has a maximalRolle set
of zeros. Using a weak form of Markov's inequality, which is
valid
for (k+l) P
each system S I it is possible to show that Rolle zeros for can be selected away from the zeros of
P ( k ) . More precisely,
(see
[ 3 7 1 for algebraic polynomials)
For simplicity and without loss of generality,we take [a,b]=[-1J]. From Lemma 4 . 3 , one derives
each
Foh
with
having t h e d o l t o w i n g phopeJLtq. l e t
[-p,p]
p <
y1
U {f
ysl, and
E
p
I
be a
X
{*ysjs=l Aubbet
b e an i n t e h p o l a t i a n m a t h i x w h i c h had
ouppohted nequencen i n t h e how6 cohhenponding t o h n o t d Then each polynomial
W
0 < p < 1 , t h e h e i b a nequence
THEOREM 4 . 4 :
P i n S , a n n i h i l a t e d by
E
I
06
no a d d
xi, -pLxi(p.
X, han a
maximal
Rolle net. For the proof] the points y, are chosen inductively very close to 1 so that the selection of Rolle zeros in step 3 9 is always sible. It is essential for the proof of Theorem 5.1 the main idea
-
-
pos-
indeed, it is
that the "harmless" knots xi can be made variablein
LORENTZ and RIEMENSCHNEIOER
212
an interval
(- p
,p ) ,
arbitrarily close to ( - 1, 1). Clearly, any knot
I+ ysl
set X contained in
is independent with respect to the sys-
tem S. Theorem 4 . 4 gives another simple proofof Theorem 2.2 (Windhauer 1 4 7 1 or [ 2 0 ] ) . Assuming that E is a normal Pdlya matrix,
X
C
{k ys}
and show that the pair
nomial Pn annihilated by
we
take
E, X is regular. Indeed, a poly-
E, X is identically zero by a standardap
plication of Lemma 4 . 2 . As has been pointed out in [191, Theorems 2 . 2 and 2 . 4 extendto equations of the form
DODl
where D
j
... Dk f(xi) = cik
(eik
=
11,
are certain differential operators of order 1, and Sisthe
Chebyshev system connected with these operators (for a definition of S, see [ 1 5 , p. 9, p. 378- 3791).
55. CLASSES OF SINGULAR MATRICES
The Atkinson-Sharma theorem provides only a sufficient
condi-
tion for the regularity of matrices; the condition is not necessary. However, a good guiding idea is that.this condition
is
"normally"
necessary, or at least necessary under some simple additional conditions. All theorems of this section refer to intehpoLation b y
aege-
b h a i c p o l y n o m i a b and ohdeh b i n g U i h h i t y .
THEOREM 5.1:
Lat i d it
An
m
COntaivlb
otherr dequenced
06
x
(n + 1 ) n0hma.t Bihkhodd m a t t i x in b t h o n g . t y bingu-
a h o w w i t h p h e c i n e e y one o d d duppo1Lted dequence t h i n h o w being e v e n
OR.
n o t nuppoktcd).
The simplicFty of the statement of Theorem
5.1
belies
the
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
21 3
d i f f i c u l t i e s of i t s p r o o f . T h i s theorem w a s f i r s t proved i n t h e spec i a l case when t h e i n t e r i o r row c o n s i s t s o f a s i n g l e s u p p o r t e d o n e b y L o r e n t z and Zeller [28 1
,
and i n f u l l g e n e r a l i t y by L o r e n t z
in
[ 181
and i n [20]. The theorem also a p p e a r s i n K a r l i n and Xaron [ 1 3 ] as
a
consequence of t h e method of c o a l e s c e n c e , b u t t h e p r o o f o f f e r e d t h e r e c o n t a i n s a n e s s e n t i a l gap.
A p r o o f may b e b a s e d [ 2 0 ] on Theorem 4.4 o f t h e method o f dependent k n o t s . The r o w c o n t a i n i n g a s i n g l e odd s u p p o r t e d is a s s i g n e d a
variable k n o t
t a k e n among t h e zeros of
+ys
p ( k ) (x) =
(-
-p
sequence
< y < p , w h i l e t h e o t h e r knots are
o f Theorem 4 . 4 . I t i s n e c e s s a r y t o s t u d y
- P ( x , y ) , where ax
depending on t h e p a r a m e t e r the root
y,
in-
P i s some p o l y n o m i a l i n
y , and t o show t h a t f o r some
x = y . T h i s i s done by examining a l l zeros of
y
the
x,
it
has
P ( ~ ()x )
in
1,l).
i n (2.4.1) shows t h a t Theorem 5 . 1 f a i l s t o h o l d E3 f o r matrices w i t h t w o odd s u p p o r t e d s e q u e n c e s i n a row. F o r matrices The m a t r i x
w i t h a n odd number o f odd s u p p o r t e d s e q u e n c e s i n one row some special c a s e s a r e known (see S e c t i o n 6 . 2 ) . However, w e have t h e example
of
the matrix
E7
=
(
1
1
0
0
0
0
0
0
1
0
1
0
1
0
1
1
0
0
0
0
0
)
which i s weakly s i n g u l a r . T h i s example a p p e a r s i n [ 5 1 and h a s maxi-
m a l d e f e c t , d = 2 (see ( 2 . 3 . 3 ) ) .
I t would b e i n t e r e s t i n g t o know whether
a t h r e e r o w B i r k h o f f m a t r i x w i t h a n odd number o f odd s u p p o r t e d
se-
quences c a n b e r e g u l a r .
A second c l a s s of s i n g u l a r matrices i s found by r e s t r i c t i n g all b u t o n e r o w . A r o w of t h e i n t e r p o l a t i o n m a t r i x E i s d i m p e e i f i t c o n t a i n s a s i n g l e e n t r y o n e and r e m a i n i n g e n t r i e s z e r o . The m a t r i x E is
214
LOREN12 and RIEMENSCHNEIDER
almadt dimple i f all i t s rows b u t a t most one a r e s i m p l e . I n t h e c a n o n i c a l decomposition ( 2 . 2 . 8 )
of an a l m o s t s i m p l e ma-
t r i x , e a c h component m a t r i x i s a l m o s t s i m p l e . More i n t e r e s t i n g l y , t h e almost s i m p l e matrix E i s s t r o n g l y s i n g u l a r i f (and o n l y i f ) one of i t s component m a t r i c e s i s s t r o n g l y s i n g u l a r . T h i s o b s e r v a t i o n i s used
i n t w o ways. F i r s t of a l l i t i s enough t o s t u d y t h e s t r o n g singularity of almost s i m p l e B i r k h o f f matrices. Secondly, one c a n assume t h e exi s t e n c e of a "smallest" r e g u l a r o r weakly singular ahmst simple B i r k b f f m a t r i x Eo c o n t a i n i n g odd s u p p o r t e d s e q u e n c e s . Theorem 5 . 1 andmethods of c o a l e s c e n c e are used t o o b t a i n a c o n t r a d i c t i o n f o r t h e m a t r i x Eo.
I n t h i s way one o b t a i n s ( L o r e n t z [191):
THEOREM 5 . 2 :
An aLmabt b i m p e e B i t h h a d d mathix
i b
t e g u L a t id it
had
no odd buppohted bequenceb and i d b t h o n g l g bingU.tah o t h t h w i d e .
An i n t e r e s t i n g s p e c i a l c a s e of t h i s was e s t a b l i s h e d b y K . Z e l l e r i n 1 9 7 0 . L e t E be a normal i n t e r p o l a t i o n m a t r i x w i t h a t l e a s t t h r e e
rows and which h a s non-zero e n t r i e s o n l y i n a n i n t e r i o r row io a n d i n column z e r o o f t h e o t h e r rows. L e t p o s i t i o n s of a l l zeros i n r o w
5
ko <
...
n be the P i o . AS a c o r o l l a r y t o Theorem 5 . 3 , w e 0
k
have
THEOREM 5.3:
A mathix
o n l y i d a l l di66ehenceb
06
t h e .type j u d t d e d c h i b e d kj
-
kj-ll
j = l,...,p
i d
hegulah id and
ate odd.
A s l o n g as t h e q u e s t i o n of r e g u l a r i t y of an i n t e r p o l a t i o n
ma-
t r i x h a s n o t been solved, i t i s n a t u r a l t o a s k whether it can be solved
"most" of t h e t i m e . That i s , what i s t h e p r o b a b i l i t y t h a t a n o r m a l interpolation matrix is r e g u l a r ? A f i r s t step i n t h i s direction i s t o c o u n t t h e number of P 6 l y a matrices, or B i r k h o f f matrices, among
m
x
(n
+ 1) normal
i n t e r p o l a t i o n matrices. L e t
B ( m , n ) denote t h e number of a l l
m
x
A(m,n)
,
P (m,n)
all and
( n + 1) normal i n t e r p o l a t i o n , P 6 l y a
RECENT PROGRESS IN EIRKHOFF INTERPOLATION
and B i r k h o f f m a t r i c e s r e s p e c t i v e l y . Here
216
we a l l o w some rows
of t h e
e shall outline m a t r i x t o c o n s i s t o n l y o f z e r o s , a n d assume 1 z m l n . W t h e r e s u l t s of L o r e n t z a n d R i e m e n s c h n e i d e r [ 25 1
THEOREM 5 . 4 :
-
huh t h e e q u a t i v n h
(5.2) A(m,n) =
n + l
I n p a r t i c u l a r , one has t h e s t r o n g asymptotic r e l a t i o n s
(5.3)
The f i r s t r e l a t i o n
( 5 . 3 ) shows t h a t ( f o r l a r g e
n)
a l m o s t a l l normal
matrices do n o t s a t i s f y t h e P 6 l y a c o n d i t i o n ; t h e s e c o n d r e l a t i o n shows t h a t t h e r e i s a l a r g e p r o p o r t i o n of B i r k h o f f matrices among t h e P6lya m a t r i c e s . Theorem 2 . 2 i m p l i e s now t h a t a l m o s t a l l normal m a t r i c e s are s i n g u l a r . However, w h e t h e r a g i v e n m a t r i x s a t i s f i e s B i r k h o f f c o n d i t i o n i s e a s y t o d e t e r m i n e . Thus, s h o u l d b e : What i s t h e p r o b a b i l i t y t h a t a normal
the
P6lya
t h e proper
m
x
(n
+
or
question 1)
Pdlya
( o r Birkhoff) matrix is regular ?
We assume t h a t o u r
x
n
normal matrices s a t i s f y t h e c o n d i t i d n
(1 + 6 ) n / l o g n
(5.4)
where
m
6 > 0
5 m 5 n,
is a constant.
THEOREM 5 . 5 [ 2 5
I
:
F o h each
E
> 0, t h e h e i n an
no = n o ( E ) w i t h t h e
LOREN12 and RIEMENSCHNEIDER
216
B(m,n) B i h h h o d d m a t h i c e s
6 o l l o w i n g phopekty. Among a l e (5.4) w i t h
but
n 2 no
EB(m,n)
ahe h e g u l a h . What i d m o t e , a&?
eB(m,n) ad them have nuppotrted s i n g e e t o n s .
THEOREM 5 . 6 [ 2 5 ] : 604
, a t mod2
nutindying
n 2 no
mosZ
F o h each
, among a l l
E
> 0 , thehe i b an
no = n o ( € )
P(m,n) Poklya m a t h i c e d n u t i n d y i n g
no . t h a t
(5.4).
at
EP(m,n) a t e / r e g u t a t .
5 6 . THREE ROW MATRICES
6.1.
I t i s n o t clear i n what r e s p e c t t h e
ALMOST HERMITIAN MATRICES
t h e o r y o f r e g u l a r i t y becomes s i m p l e r f o r t h r e e row matrices. T h e t h e orems on c o a l e s c e n c e a r e n o t s t r o n g enough t o r e d u c e t h e g e n e r a l c a s e
t o t h i s one. F u r t h e r m o r e , w e s h a l l see t h a t e v e n v e r y s i m p l e t h r e e row
matrices p r e s e n t c o n s i d e r a b l e d i f f i c u l t i e s . The r e s u l t s of
56 refer
t o order regularity.
W e s h a l l study
3
x
(n
+
1)
normal B i r k h o f f matrices w i t h t h e
f o l l o w i n g placement o f ones
elk
= 1,
0
5 k
< p ; e 3k = 1 , 0
5 k < q;
(6.1.1)
p tq +1 =n
Then
also assume t h a t
x
=
I- 1, X I
and
k2 < n; w i t h o u t loss o f g e n e r a l i t y , w e
kl < k 2
-
1, p
shall
5 q. F o r t h e k n o t s e t , w e shall take
11.
One of t h e smallest matrices of t y p e (6.1.1), E3 of (2.4.1Xhas s e r v e d t o show t h a t r e g u l a r matrices c a n have odd s u p p o r t e d sequenes. G e n e r a l i z i n g t h i s example, s e v e r a l a u t h o r s ( D e V o r e , CZeir a n d [6
1
,
L o r e n t z and Z e l l e r i n [ 1 9 1 , and L o r e n t z , S t a n g l e r , and
Sharma Zeller
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
[ 2 6 ) s t u d i e d matrices of t h e form ( 6 . 1 . 1 ) .
21 7
I t was hoped t h a t i n t h i s
way t h e problem o f r e g u l a r i t y c o u l d b e c o m p l e t e l y s o l v e d f o r a t l e a s t one n o n t r i v i a l case. The i n c o m p l e t e s u c c e s s of t h i s a t t e m p t l e a d s o n e
t o believe t h a t i t i s h a r d l y p o s s i b l e t o e x p r e s s t h e p r o p e r t y of regu-
eik of a m a t r i x E .
l a r i t y i n terms of s i m p l e p r o p e r t i e s o f e l e m e n t s The method o f t h e p a p e r [ 6
classical J a c o b i polynomials
1 w a s t o a p p l y known f a c t s a b o u t the B, ( x )
'L P
. In
[ 1 91
,
the
alternation
p r o p e r t i e s o f zeros of d e r i v a t i v e s of t h e polynomial (1 + x I p ( 1
-
x)'
were used. The f i r s t method g i v e s more d e t a i l e d i n f o r m a t i o n w h i l e the second method i s a p p l i c a b l e t o w i d e r classes o f matrices.
THEOREM 6 . 1 [ 2 6 1 :
I n o h d e h .that t h e mathix (6.1.1) b e h e g u t a h ,
necebbahy t h a t
kl -+ k 2 = p
(6.1.2)
+
q
+
1,
k2
'q
O h
(6.1.3)
I n t h e cane ( 6 . 1 . 2 ) .
(6.1.3),
E
i n hegudah
.the mathix c a n b e
eithea
and o n d y id
p = q . I n t h e cane
bingudah,
h q U d U h oh
b u t t h e hegu-
k?a/rity o b t h e mathix (6.1.1) i m p L i e b t h e h e g u t a h i t y 0 6 a thix
w i t h pahametehn
ki, ki
( b ) k i = kl + 1,
ka = k 2
( N o t e that i n e q u a l i t i e s
p a p e r 1 2 6 1 , namely w i t h
i n n - t e a d 06
+
kl,
k2
bimitah
ma-
i6
1 1 q.
( a ) have been s t a t e d i n c o r r e c t l y i n the
ki 5 kl < k 2 5 k;.
T h i s error o c c u r e d i n the
l a s t l i n e s o f t h e p r o o f i n [ 26, p . 4 3 5 1 . The i n e q u a l i t y ( 5 . 5 ) s h o u l d
21 8
LORENTZ and RIEMENSCHNEIDER
b e r e p l a c e d by t h e r e v e r s e one: "(5.5) y '
i+l( A )
2
y ; ( h ) f o r some i , " )
The proof o f t h i s theorem i s by t h e " c h a s e method". As a d i d a t e f o r t h e n o n t r i v i a l polynomial a n n i h i l a t e d by
we l e t
X change c o n t i n u o u s l y from
t o t h o s e zeros of
P
(kl)
and
P
(k2)
-0
to
+m
can-
E, X , w e t a k e
and s t u d y whathappens
whose e x i s t e n c e is g u a r a n t e e d by
R o l l e ' s theorem. The m a t r i x i s s i n g u l a r e x a c t l y whenone o f t h e s e zeros (kl) (k2) o v e r t a k e s t h e o t h e r a t some x o , f o r t h e n P (xo,h) =P (xo,h) -0. The second p a r t of Theorem 6 . 1 means t h a t i n t h e t r i a n g l e given p 5 x, y
by
5 q, x + 2 5 y , t h e r e e x i s t s a monotone i n c r e a s i n g curve
y = X ( x ) , w i t h slope a t most o n e , so t h a t on t h e c u r v e , and r e g u l a r below i t . For c o v e r e d i n [ 6 1,
and
p = 1, t h i s c u r v e w a s
dis-
and was shown t o be t h e upper b r a n c h of t h e e l l i p s e
(q + 2 ( x
(6.1.4)
i s s i n g u l a r above
E
+
y
-
112
-
4(q
+
1 ) x y = 0;
moreover, E is weakly s i n g u l a r on t h i s c u r v e and s t r o n g l y
singular
above. F o r some v a l u e s of t h e p a r a m e t e r s , t h e statements
Theorem
of
6.1 w e r e proved a l s o i n [ 6 1 and [19]; i n a d d i t i o n , it w a s
possible
t o d i s t i n g u i s h between s t r o n g and weak s i n g u l a r i t y . One g e n e r a l c a s e of weak s i n g u l a r i t y h a s been found t o d a t e , namely when
kl
+
k2 = p + q
+
1. For more d e t a i l s , c o n s u l t t h e p a p e r of
Meir and Sharma I 6 1
6.2.
q = p
.
C R I T E R I A BASED ON COALESCENCE
P 6 l y a m a t r i x . For t h e k n o t s e t
Let
E be a
3 x ( n + 1)
X = {O,x,l}, t h e determinant
is a polynomial i n x. C l e a r l y , E w i l l be s t r o n g l y s i n g u l a r s i g n of
D(E,X)
+
1,
DeVore,
normal
D(E,X) if
the
i s d i f f e r e n t i n ( 0 , ~ )and ( q , l ) ( c , 1-0sufficiently smll).
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
219
This simple obs e r v a t i o n i s t h e e s s e n c e of s e v e r a l c r i t e r i a s t r o n g o r d e r s i n g u l a r i t y of
E,
although t h e statements
for
the
themselves
appear t o t a l l y unrelated. T h e r e are s e v e r a l e q u i v a l e n t forms i n which t h i s c o m p a r i s o n of s i g n s c a n b e c a r r i e d o u t . One o f them i s g i v e n by t h e s p e c i a l c a s e o f P r o p o s i t i o n 3.8 when t h e m a t r i x E c o n s i s t s j u s t o f t h e t h r e e ordered
rows
F1,
is not
F 2 , F3 ( o f c o u r s e , t h e i n t e r e s t o f P r o p o s i t i o n 3.8
l i m i t e d t o t h i s case). A n o t h e r form i s o n e g i v e n by K a r l i n a n d Karon "13,
Theorem 2 . 3 1 ) :
Let
ones i n row 2 , and l e t
(11,L2,...,t 1
(1, ,
.. .
t i o n s i n t h e pre-coalescence
9
)
=
F2
b e t h e p o s i t i o n s of the
. ..
-
3 ,L 1) and (el, ,L 3 be t h e i r p o s ' i 9 9 of r o w 2 w i t h r e s p e c t t o r o w 1 a n d r o w
3 respectively.
PROPOSITION 6 . 2
( K a r l i n and
rna-lkix, .then E
i b
(6.2.1)
b.thong&y
j =1
' I n {ac.t, t h i n bum n e e d o n l y
b e t a k e n owe4 j
doh. w h i c h t h e
lj
ahe
The method o f K a r l i n a n d Karon w a s t o a n a l y z e t h e s i g n s o f t h e d e t e r m i n a n t s i n v o l v e d by u s i n g a r g u m e n t s from t h e
theory
p o s i t i v i t y due t o S . K a r l i n . F o r t h e l a s t s t a t e m e n t o f t h e one v e r i f i e s t h a t a d j a c e n t ones, contribute
0 mod
2
=
lj+l
o€
total
theorem,
L . + 1, or u n s u p p o r t e d o n e s , 3
i n (6.2.1).
Both P r o p o s i t i o n 6 . 2 a n d P r o p o s i t i o n 3.8 are c o n s e q u e n c e s
of
c o a l e s c e n c e a n d t h e u s e o f t h e T a y l o r ' s f o r m u l a ( 3 . 2 . 3 ) . T h i s c a n be. e x p l a i n e d b e s t o f a l l i f w e d e f i n e "directed c o a l e s c e n c e " as f o l l o w s . If
F1, F2
alescence
a r e t w o a d j a c e n t rows of F1
=*
Fa
E
, we
define the directed
a s t h e m a t r i x d e r i v e d from E by r e p l a c i n g
corow
LORENTZ a d RIEMENSCHNEIDER
220
f,,
F1 by i t s p r e - c o a l e s c e n c e ,
number, a l r 2 ,
with respect to
of t h e c o a l e s c e n c e
F1 * F2
F2. The
is t h e nunber of
changes needed t o b r i n g t h e sequence of i n t e g e r s (Here a r o w
order.
interchange
F1,
inter-
i n t o natural 2 F i s r e p r e s e n t e d by t h e p o s i t i o n s of t h e o n e s a s
i n 5 3 . ) I n a similar way,
F,
a l e s c e n c e w i t h respect t o F1
C
F2
r e p l a c e s row
F
F2 by i t s p r e - c o -
and h a s t h e i n t e r c h a n g e number
O2,l‘
Then (by ( 3 . 2 . 3 ) )
u2,1 E a l I 2 + a l r 2 ( m o d 2 1 ,
(6.2.2)
where
a
112
= a(F1
U
F2) i s the c o e f f i c i e n t o f c o l l i s i o n . For c a l c u -
l a t i n g t h e i n t e r c h a n g e numbers of f u r t h e r d i r e c t e d c o a l e s c e n c e s , t h e p o s i t i o n s of t h e o n e s i n
F1 e F2
and
F2
9
F1
a r e assumed t o b c i n
t h e i r n a t u r a l o r d e r ; t h e n , f o r example, u d i r e c t e d coalescences
.
tively
Let
F3 * (F1 e
=lY
3, (2,1) 3,(112) F2) and F3 * (F1 * F2)
for t h e
respac-
6 be t h e sum of t h e exponents of powers of ( - 1) giving t h e
s i g n s mentioned a t the b e g i n n i n g of t h i s s e c t i o n . We can g i v e several equivalent expressions f o r
6
(mod 2 ) by means o f d i r e c t e d
coales-
c e n c e s and Theorem 3.3. For example, t o o b t a i n P r o p o s i t i o n 3 . 8 ,
estimate t h e s i g n of (F1
=*
D(E,X)
F2) * F3, and near
near
0 by means
1 by means of
F1
=*
of
the
we
coalescences
(F2 * F3) and o b t a i n i n
t h i s way
(6.2.3)
-
To o b t a i n P r o p o s i t i o n 6 . 2 , we c o n s i d e r t h e c o a l e s c e n c e s and
F1
(Fa
=*
F3); t h i s gives
(F, * F2) * F3
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
( t o see t h e e q u i v a l e n c e of ( 6 . 2 . 3 ) (6.2.2)
and (6.2.4)
a n d a n e x t e n d e d f o r m of ( 3 . 1 . 7 ) ) .
221
directly,
Equations
one (6.2.1)
6 : 1 (mod 2 ) c a n be shown t o b e e q u i v a l e n t by t h e c a r e f u l
t i o n of t h e c o l l i s i o n and i n t e r c h a n g e numbers o f
(6.2.4)
uses and
computa-
by means of
t h e q u a n t i t i e s i n (6.2.1). S i m i l a r i d e a s g i v e a s p e c i a l c a s e of P r o p o s i t i o n s 3 . 8 and E be a
found by Sharma and T z i m b a l a r i o [ 4 2 1 . L e t Birkhoff matrix with ones i n p o s i t i o n s and l e t
kl <
... <
PROPOSITION 6 . 3 :
(6.2.5)
k
Id P
Z
j =1
P+r
F =
3
x
6.2
( n + 1) normal
(.li,...,k?'), F 3 = ( L Y , . . . , , L ; ' " ) ,
1 P I: b e t h e p o s i t i o n s of t h e z e r o s i n r o w 2.
kl > m a x ( L '
P
(kr+j
- k 3. )
-
p , L;;'
+
-
r ) and .id
p r z 1 (mod 2 1 ,
Here w e u s e the c o a l e s c e n c e s (F1 * F2) .= F 3
a n d F1 * (F2 .= F 3 )
to o b t a i n
Under t h e a s s u m p t i o n s o f t h e t h e o r e m , i t i s e a s y t o v e r i f y t h a t (6.2.5) and
6 I 1 (mod 2 ) are e q u i v a l e n t by ( 6 . 2 . 6 ) . Sharma and T z i m b a l a r i o
p r o v e d P r o p o s i t i o n 6 . 3 by u s i n g p r o p e r t i e s of special d e t e r m i n a n t s . An i n t e r e s t i n g s p e c i a l case of P r o p o s i t i o n 6 . 3 is whenthe f i r s t and l a s t rows o f
E are H e r m i t i a n . I f , i n a d d i t i o n , t h e r e i s o n l y o n e
element i n t h e Hermitian f i r s t row, t h e last r o w is Hermitian,
and
there i s a n odd number of odd s u p p o r t e d s e q u e n c e s i n E l t h e n Propos i t i o n 6 . 3 i m p l i e s t h a t E i s s t r o n g l y s i n g u l a r (see Passow [ 3 5 1 1 .
LORENTZ and RIEMENSCHNEIDEA
222
57. BIRKHOFF'S KERNEL
7.1.
DEFINITION AND PROPERTIES
Birkhoff's kernel, K ( t ) = K E ( X , t )
a s p l i n e f u n c t i o n (piecewise polynomial) i n t i m a t e l y connected
is
with
B i r k h o f f i n t e r p o l a t i o n by a l g e b r a i c p o l y n o m i a l s . I t a p p e a r s i n the integral identities (7.1.4)
and (7.1.6); t h e second o f t h e t h e s e g i v e s
a r e m a i n d e r f o r t h e i n t e r p o l a t i o n formula. The d e e p e s t
theorem
of
B i r k h o f f i n h i s famous p a p e r ( 3 ] w a s a n estimate o f t h e number of zer o s o f h i s k e r n e l . I t i s g i v e n h e r e as a theorem a b o u t t h e number of changes o f s i g n o f s p l i n e s w i t h d i s c o n t i n u i t i e s l i n k e d t o t h e m a t r i x E.
As u s u a l , l e t u
ur
r 2 0
for
mean
ur
5 0 , e x c e p t t h a t it is n o t d e f i n e d i f both
the kernel
KE(X,t)
from t h e d e t e r m i n a n t
if
u
5
0 , and
0
u = O , r -0. W e
D(E,X)
of ( 2 . 2 . 2 )
if
obtain by
re-
::
p l a c i n g the e l e m e n t s of t h e f i r s t colunn i n (2.2.2) by .(xi-t)Fk-l/(n-k-l)
If
Dik(X)
a r e t h e a l g e b r a i c components o f t h e f i r s t column e l e m e n t s
of t h e determinant (2.2.21,
defined f o r
e i k , = 1, t h e n
(7.1.2)
I f t h e k n o t s a r e o r d e r e d , x1 <
i s a polynomial i n
...
xm'
then the determinant
t i n e a c h of i n t e r v a l s (-m,xl) I
hence a s p l i n e . One sees t h a t
(%,X2),
K ( t ) i s z e r o o u t s i d e of [ X
..- ,(x~,+oJ),
~ , X ~ ] {The .
same a p p l i e s t o t h e d e r i v a t i v e s of the kernel. Thus, K ( 1 ) (t)I j = O l e
is zero outside of t h e l a r g e s t ) of t h e
..dI-1,
, where A j ( o r B 7. ) i s t h e smallest (or 7 w i t h eik = 1 f o r s o m e k 2 n - j - 1 1
[ A . ,B . ]
1 xi
K(t)
Integrating (7.1.2), we obtain
.
223
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
X
(7.1.3)
K(t)dt = D(E,X).
x1 L e t An
denote t h e c l a s s of a l l ( n - 1 ) - t i m e s c o n t i n u o u s l y d i f -
f e r e n t i a b l e functions
f on
[ a , b 1 f o r which
f (n-l)
is
absolutely
continuous.
THEOREM 7 . 1
(Birkhoff‘s Identity [ 3 ]
~ e h 0 . 4i n t h e L a n t column. F o t e a c h [ a,bl
) :
1e t
E
be a nohmd mathix w k h
f E An and e a c h o e t
06
k n a f b X in
,
(7.1.4)
X
eik=l
D i k ( X ) f ( k ) (xi) - jbf‘”’ ( t ) K ( t d t . a
The s i m p l e s t s p e c i a l case of ( 7 . 1 . 4 ) is T a y l o r s formula
with
i n t e g r a l remainder. From t h i s theorem, w e can o b t a i n mean f E Cn[a,bl. I f
value
formulas.
Let
K(t) does n o t change s i g n , t h e n by u s i n g (7.1.4) and
(7.1.31, w e can o b t a i n
n)(E)D(E,X),
K i s of a r i t r a r y s i g n , b u t
The same i s t r u e i f
of degree n o t exceeding n. e
ik
x l < < < x m ( x l < x < Xm’.
f
is a
polyno~ial
I n both cases, t h e r e l a t i o n s f ( k ) ( x i ) = a ,
= 1 and D(E,X) # O imply f ( ” ) ( C ) = O f o r some 5, x1 < 5 < xm.
Suppose now t h a t
E
i s a normal P6lya m a t r i x w i t h o u t any
s t r i c t i o n s and X i s a s e t of k n o t s f o r which D ( E , X ) # 0 . If t h e r e e x i s t s a polynomial Pn of degree a t m o s t p ( k ) (xi) = f ( k ) ( x i ) ,
n
eik = 1.
n for which
re-
f € AMlr
LORENTZ and RIEMENSCHNEIOER
224
W e would l i k e t o g e t a formula f o r t h e d i f f e r e n c e
E
extend E t o
xi,
=
K ~ ( 2 , t ) is
THEOREM 7 . 2 ( B i r k h o f f ) :
x be d i f f e r e n t fran
t h e Peano h e x n e e of t h e i n t e r p o l a t i o n .
One has 404 X C [ a , b ]
( A s i m i l a r formula h o l d s f o r t h e d i f f e r e n c e f (k)( x )
- P F ) (x) ,
i f we i n s e r t t h e one i n t h e new 0-th row i n p o s i t i o n k, i . e . e
7.2. 13
NUMBER OF ZEROS O F SPLINES
I c o u n t s t h e number o f
hangeb
06
b i g n of a k e r n e l
c o n c e r n s t h e numbeh
e r a l i z a t i o n by L o r e n t z 1 2 1 A f u n c t i o n S on (-
,+=)
O3
KE(X,t).
This
1 1.
gen-
.. <
I
i , and is z e r o o u t s i d e o f
has e x a c t l y degree
S vanishes.
THEOREM 7 . 3 [21]:
be a bpt.ine
E = (eik) be an
i =1,...,m,
at i b
xi.
16
and t h a t
x
06
Oneb
theih muttiplicitieb)
06
eik =1 Lvheneueh
04
add
i n E , and 06
S
Z
for
deghee n - 1 Lclith knob xl<
( n +1) mathix ad Z e h o b a d oned do
p i d t h e numbet
t h e numbeh
(7.2.1)
m
n
at
( x , , x m ) . L e t [ a , b ] be the smallest
i n t e r v a l o u t s i d e of which
Let
zehob of s p l i n e s .
so t h a t S i s a polynomial of de-
x
x1 < m g r e e 5 n on each i n t e r v a l (xilxi+l)
S
06
A
i s a s p l i n e of f i n i t e s u p p o r t of degree n
i f t h e r e are p o i n t s
Let
=l.l
0 Ik
The d e e p e s t theorem of B i r k h o f f i n
estimate i s a l s o v a l i d f o r o t h e r s p l i n e s (D. Ferguson [ 8
l e a s t one
We
Pn(x).
2 is the s e t of k n o t s o b t a i n e d by a d d i n g x t o X.The
then
k(t)
kernel
-
by adding a 0-th row w i t h o n l y a s i n g l e o n e , eO0=1,
and by a d d i n g an ( n + 1 ) - s t column o f z e r o s . L e t the
f (x)
sequences a 6
t h e numbeh ad
i n (a,b), then
Z z N - n + p .
<Xm.
ei,.,=O,
S ‘ J ) , j = n - k -1, hub a jump
bUppOh2kd
i b
...
ZehOb
El N + 1
(counting
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
226
A n e s s e n t i a l p a r t of this t h e o r e m i s t h e d e f i n i t i o n of t h e mul-
a of a zero
tiplicity o n which
z
then
z,
A zero
c1
a i n [21]
,
= min(B,y)
where
1
z.
a r e t h e m u l t i p l i c i t i e s of
y
(In [21]
min, b u t this w a s c o r r e c t e d i n [20 1 . I I f
z , then
1
is a f o l l o w s . I f S i s c o n t i n u o u s B and
S to t h e r i g h t and t o t h e l e f t of
for
may be a maximal i n t e r v a l (x., x , )
z
S v a n i s h e s , o r a p o i n t , which may o r may n o t b e one o f t h e
k n o t s . The d e f i n i t i o n o f on
.
ci = 1
if
S changes s i g n on
,
z
a
,
max
S is d i s c o n t i n u o u s on
o t h e r w i s e . I n [ll],
= 0
a . A d i f f e r e n t ver-
Jetter gives another apparently l a r g e r count f o r s i o n of Theorem 7 . 3 i s g i v e n i n Schumaker [ 4 0
a . But also h i s
d e f i n i t i o n of
p i n (7.2.1)
stands erroneously
1 , with
still
another
i s d i f f e r e n t , so that his
f o r m u l a t i o n d o e s not c o n t a i n Theorem 7.4. Of c o u r s e , f o r p o i n t z e r o s
z
of
S that are not also knots, a
i s t h e o r d i n a r y m u l t i p l i c i t y ; and e a c h c h a n g e o f s i g n o f
S
is a zero.
The p r o o f o f Theorem 7.3 u s e s a t y p e o f R o l l e ' s t h e o r e m f o r s p l i n e s .
7 . 3 . APPLICATIONS OF THE BIRKHOFF KERNEL
Let
E
again be a
normal
B i r k h o f f m a t r i x . From Theorem 7.3 w e d e r i v e
THEOREM 7 . 4
(Birkhoff) :
t h e kehnet
KE(X,t)
AeqUenCeb Let
06
T h e numbeh
06
changeb 0 6 b i g n L M
d o e n n o t exceed t h e numbet
p
05
add
[a,b]
buppohted
E.
p =O
and
X be given. I f
from ( 7 . 1 . 3 ) w e see t h a t
K(X,t)
i s n o t i d e n t i c a l l y zero,
D ( E , X ) # 0 . Thus, t h e p a i r E , X i s r e g u l a r .
S i n c e a s i m p l e a d d i t i o n a l a r g u m e n t shows t h a t t h e case
K(t)
i m p o s s i b l e , this g i v e s a new p r o o f of t h e Atkinson-Sharma Theorem 2 . 4 .
06
0
is
Theorem,
Thus , t h i s t h e o r e m i s e s s e n t i a l l y c o n t a i n e d in Birkhoff's
r e s u l t s of 1 9 0 6 .
An i n t e r p o l a t i o n m a t r i x E i s c a l l e d A t h a n g t y Xegutah each
X, t h e k e r n e l
K(X,t)
i s of c o n s t a n t s i g n a n d d o e s n o t
i d e n t i c a l l y . W e have t h e i m p l i c a t i o n s :
if
for
vanish
LOAENTZ and AIEMENSCHNEIOER
226
E conservative
* E strongly regular * E regular,
and they cannot be i n v e r t e d . For t h e l a s t two s t a t e m e n t s , a n example is g i v e n by t h e m a t r i x E3 of
(2.4.1)
which i s r e g u l a r but not strong-
I
l y r e g u l a r . Another example ( J a f f e 1 1 0 1 ) i s t h e s t r o n g l y r e g u l a r matrix
(. 1
E =
(7.3.1)
1
")
1
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
which i s n o t c o n s e r v a t ve ( c o n t a i n s odd s u p p o r t e d sequences,. See [lo] f o r o t h e r examples. I t s h o u l d be n o t e d t h a t t h e s t r o n g r e g u l a r i t y of
E i s equiva-
l e n t t o t h e v a l i d i t y of t h e e x t e n d e d R o l l e ' s theorem (Theorem for E l
for a l l
X, all
f E C ( n ) [ a,b]
,
and some
2.5)
a < 5 < b.
$8. APPLICATIONS O F BIRKHOFF INTERPOLATION 8.1. RESTRICTED RANGE APPROXIMATION a function
I n t h e uniform approximationof
f E C [ a , b ] by a l g e b r a i c p o l y n o m i a l s , w e may want t o
re-
s t r i c t t h e approximating polynomials i n some way. For example, wemay assume t h e approximating polynomials a l l y , w e may r e s t r l c t t h e
where
Pn are i n c r e a s i n g . More gener-
Pn t o s a t i s f y [ 2 7 1
...
= k 1 a r e g i v e n s i g n s , and 1 5 kl < k < n are j P g i v e n i n t e g e r s . I n analogy w i t h t h e case k = k = 1, t h i s is s t i l l P 1 c a l l e d t h e problem of monotone approximation. E
Even more g e n e r a l l y , one can r e s t r i c t the ranges of the derivatives
RECENT PROGRESS I N B I R K H O F F I N T E R P O L A T I O N
For t h e bounding f u n c t i o n s that either
u. i 1
and t h a t e i t h e r
+
Lj
I
or that
ml
-
227
u j l i t i s n e c e s s a r y [381
to
u
a < x < b,
j
is differentiable for
assume
m, o r t h a t L. is d i f f e r e n t i a b l e . Problems 3 7 c o n c e r n i n g t h e uniqueness of a b e s t r e s t r i c t e d a p p r o x i m a t i n g polyno-
II. E
mial a r e i n t i m a t e l y r e l a t e d t o Birkhoff i n t e r p o l a t i o n .
THEOREM 8.1:
0 6 .the
Ceabb
Foa each (8.1.1)
06
f
E
C [ a , b ] t h e h e e X i b t b a u n i q u e polynomiab
be.n.t u n i 6 o k m appkoximatian .to
T h i s was proved by L o r e n t z and Zeller [ 2 7 1 f o r R.
A . L o r e n t z 1301 f o r
p > 1.
For t h e
proof,
one
f . p =1, first
and
by
finds
a
Kolmogorov-type theorem, c h a r a c t e r i z i n g polynomials of b e s t a p p r o x i mation of t h e f a m i l y ( 8 . 1 . 1 ) . These polynomials p o s s e s s c e r t a i n
ex-
t r e m a l p r o p e r t i e s ; among them t h e r e e x i s t minimal palynamiaLhof b e s t a p p r o x i m a t i o n , w i t h t h e smallest number of e x t r e m a l p r o p e r t i e s .
For
t h e proof of Theorem 8 . 1 i t i s s u f f i c i e n t t o show t h a t t h e difference, Q , between any b e s t approximation polynomial and minimal
polynomial
i s z e r o . T h i s i s a c h i e v e d by means o f t h e Atkinson-Sharma Theorem i f
p = I, b u t r e q u i r e s a m o r e s o p h i s t i c a t e d p r o o f by i n d u c t i o n f o r p > 1 [301. A s i m i l a r proof can be g i v e n [ 3 8 1 t o e s t a b l i s h t h e
i n case o f t h e m o r e g e n e r a l problem ( 8 . 1 . 2 ) . r e s t r i c t i o n s a r e required for
.tjl u
j
if
uniqueness
I n [3Ol,some a d d i t i o n a l
kj+l
=
k. 1
+
1. There i s an
i n d i c a t i o n i n [17] t h a t t h e s e r e s t r i c t i o n s can be a v o i d e d .
8.2.
SIMULTANEOUS APPROXIMATION
a function
The s i m u l t a n e o u s a p p r o x i m a t i o n
f E C k + l [ a I b ] and of i t s d e r i v a t i v e s of o r d e r
kj
of
by a
LORENTZ and RIEMENSCHNEIDER
228
polynomial Pn and i t s r e s p e c t i v e d e r i v a t i v e s means a p p r o x i m a t i o n i n t h e metric
(8.2.1)
W e assume
1 5 kl <
...
k
$ k. T h i s problem i s a g a i n r e l a t e d
P
B i r k h o f f i n t e r p o l a t i o n . The s e t
to
B ( f ) of t h e polynomials of b e s t ap-
p r o x i m a t i o n does n o t n e c e s s a r i l y c o n s i s t o f one p o i n t . Using the m i n i m a l polynomials of
(see Chalmers [ 4 1
(8.1.1),
, R.
one can f i n d 1311 t h e dimension of
A . L o r e n t z [ 3 1 ] , and a l s o [ 1 7 1 ) .
8.3. CHEBYSHEV SYSTEMS ON [- 1, + 1 1 P i h a is the SySten S ={Xkl ,Xk2 a Chebyshev system on
B(f)
,.. .
,X
‘11
[-1, + 1 ] ? Using Theorem 5 . 3 , w e o b t a i n a l m o s t
immediately [181 , (see a l s o Passow [ 3 4 1 ) :
THEOREM 8 . 2 :
Fall ifitegeJLA 0 5 kl <
C h e b y ~ l zb~y s~t e m an 6e4enCeA
kj+l
- k
j
... <
k
[-1,+ 11 .id and o n l y .id
P’
.the
Ay6bem
S
LA
a
kl = 0, and i6 aLe d i d -
a4e o d d .
8.4. OTHER TYPES OF INTERPOLATION
Among o t h e r t y p e s o f i n t e r p o l a t i o n
which are of c o n s i d e r a b l e i n t e r e s t and s h o u l d be i n v e s t i g a t e d , w e mt i o n B i r k h o f f i n t e r p o l a t i o n by p o l y n o m i a l s o f s e v e r a l v a r i a b l e s , and B i r k h o f f i n t e r p o l a t i o n by i n t e g r a l f u n c t i o n s . For t h e l a s t there is the d i s s e r t a t i o n
8.5.
[9 ]
.
BIRKHOFF QUADRATURE MATRICES
Corresponding t o e v e r y
problem,
Birkhoff
i n t e r p o l a t i o n problem, t h e r e i s a q u a d r a t u r e problem f o r m u l a t e d
follows :
as
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
Given t h e
m
X
229
( n + 1 ) i n t e r p o l a t i o n m a t r i x E w i t h N + 1 ones,
a s e t of k n o t s
X = {x, <
.. .
< xmJ
C
[
0,ll
,
and a measure
dg on [0,1], when does t h e r e e x i s t a q u a d r a t u r e formula of
(8.5.1)
t h e form f (x)dg =
eik=l
cik f ( k ) ( x i )
which i s e x a c t f o r polynomials of d e g r e e n ?
The l i t e r a t u r e on t h i s s u b j e c t i s meagre; b e s i d e s t h e
inter-
e s t i n g p a p e r of M i c c h e l l i and R i v l i n [321, which t r e a t s s o m e s p e c i a l c a s e s , w e have an a t t e m p t o f a g e n e r a l Riemenschneider [ 2 4 b a s e d on
theory
in
Lorentz
and
1 and S t i e g l i t z [421 , [431. What f o l l o w s is mainly
[24 I .
A pair
E
X i s c a l l e d q-hegutah w i t h h e d p e c t t o
I
problem (8.5.1) i s s o l v a b l e . L e t d g , and l e t
be t h e moments of t r i x o b t a i n e d from o f t h e moments
A(E,X)
(pnr...,p0).
p j =jixJ/J!
dg
if
the
d g ( x ) , j = Oflf...l n ,
B(E,X,dg) be t h e ( N + 2 )
x
( n + 1 ) ma-
of (2.2.2) by a d d i n g a l a s t r o w c o n s i s t i n g The f o l l o w i n g r e s u l t compares, i n
par-
t i c u l a r , r e g u l a r i t y and q - r e g u l a r i t y :
PROPOSITION 8.3:
A pai& E l X
i n q - t e g u l a h w i t h h e d p e c t t o dg id
and o n l y id r a n k A ( E I X ) = rank B ( E , X , d g ) ;
(8.5.2)
i n p a h t i c d a h , id E l X i b h e g u l a h , * h e n A
N =n
and E l X 0 q-hegdcm.
matrix E is q-hegukkh w i t h h e d p e c t t o dg i f t h e p a i r
i s q - r e g u l a r f o r any s e t o f k n o t s restriction
X with
0 =xl <
x1 = 0 , xm = 1 i s n e c e s s a r y t o o b t a i n
.. . < a
El X
xm = 1. The new c o n c e p t ;
LORENTZ and RIEMENSCHNEIDER
2T)
without t h i s r e s t r i c t i o n , q-regularity
( a t l e a s t when
N =n)
would
b e i d e n t i c a l w i t h r e g u l a r i t y . The r e g u l a r m a t r i c e s a r e s t r i c t l y cont a i n e d i n t h e c l a s s o f q - r e g u l a r matrices. I n d e e d , t h e f o l l o w i n g sing u l a r matrices (Theorem 5 . 1 ) a r e q - r e g u l a r w i t h r e s p e c t
to
dx
on
[0,11
(i
0
(8.5.3)
E8=
1
()I
0
0
The emphasis on t h e measure
1
0
1
0
0
E 9 = (1
0
1
0
0
1
0
1
0
0
1,)
.
0
dg i n t h e d e f i n i t i o n of a q-regu-
l a r normal m a t r i x i s e s s e n t i a l ( [ 241
,
[ 4 3 ] ) . The m a t r i x E i s q-regu-
1 l a r w i t h r e s p e c t t o a l l non-zero measures dg = w(x)dx 2 0, w(x) E L [ 0,1] i f and o n l y i f A pair
less t h a n
n
E
i s a regular interpolation matrix.
E, X can be q - r e g u l a r even i f t h e m a t r i x
+1
ones
- it
E,
THEOREM 8.4:
I6
in
n o t noLeLy
can be q - r e g u l a r w i t h o u t t h e m a t r i x
X
E s a t i s f y i n g t h e P6lya c o n d i t i o n
. We
contains
i s s u f f i c i e n t t o remember G a u s s i a n q u a d r a -
t u r e f o r m u l a s . Thus, a p a i r
q-regulari ty
E
-
Theorem
is not
2.2
valid
for
have , however ,
dg ( f 0 ) i n a n o n - n e g a t i v e meabufie o n C0,l 1 ,
nuppatlted o n t h e t w o p o i n t n e t
[0,11 which
t h e n any r n m x
dg, m u d t b a t i d d y t h e P 6 L y a c o n d i t i o n .
E , q-fieguLafi w i t h X e b p e C t t o
T h i s was f i r s t proved by S t i e g l i t z [ 4 2 ] f o r
dg = dx. The gen-
e r a l c a s e needs a d i f f e r e n t p r o o f , which i s b a s e d , among other things, on d i f f e r e n t i a t i o n of d e t e r m i n a n t s s i m i l a r t o
D(E,X)
.
A normal P 6 l y a m a t r i x may b e s o b a d l y s i n g u l a r t h a t i t i s
not
q - r e g u l a r f o r any r e a s o n a b l e measures. F o r s i m p l i c i t y , w e assume t h a t t h e measures a r e o f t h e form (8.5.4)
dg = w ( x ) d x ,
W(X)
> 0,
W(X)
E L
1[0,1].
RECENT PROGRESS IN BIRKHOFF INTERPOLATION
Let
THEOREM 8 . 5 [ 2 4 1 :
b e a nohmaL P o t y a m a t h i x w i t h exuactty p odd
E
s e q u e n c e s i n t h e h O W b cohhenponding t o
an
x
231
xi,
0 < xi
in
< 1. 16 t h e h e
nuch t h a t
(8.5.5)
rank A ( E , X )
then El X If
id
E
+
= n
1
-
[y],
n o t q - h e g u t a h w i t h h e n p e c t t o a n y meabuhe ( 8 . 5 . 4 ) .
c o n t a i n s e x a c t l y one odd s e q u e n c e i n i t s i n t e r i o r
which i s a l s o s u p p o r t e d , t h e n Theorem 8 . 5 and 5 . 1 imply n o t q - r e g u l a r f o r any measure ( 8 . 5 . 4 ) . t h a t t h e theorem does n o t h o l d i f
rows
that
is
E
The examples i n ( 8 . 5 . 3 )
show
c o n t a i n s o t h e r (non - s u p p o r t e d )
E
odd s e q u e n c e s i n i t s i n t e r i o r rows. A matrix
i s c a l l e d Gaunnian i f
E
i s q - r e g u l a r f o r some s e t
PROPOSITION 8 . 6 [ 8 ]
,
16
and y e t t h e p a i r
X o f k n o t s and some measure
E
E,X
dg.
b e Gaubdian w i t h k n o t b e t
X
doh
p a d d i t i o n a t o n e n ahe hequitred d o t h a t
E
Let
[24]:
dome meanuhe ( 8 . 5 . 4 ) .
N
would noZ h a v e o d d neyuencen i n
now^
cohhebponding
0 < xi
to
< 1,
then N z n - p .
(8.5.6)
1 ~ (
N
I n pahticuk?ah,
-
n1 ) .
A d e e p e r and h a r d e r problem i s t o d e t e r m i n e classes of G a u s s i a n
m a t r i c e s . The known c a s e s i n v o l v e H e r m i t i a n
(see [ 3 2 1 , [ 1 4 ] )
,
and quasi-Hermi"Lian mtrices
when P r o p o s i t i o n 8 . 6 i s e s s e n t i a l l y i n v e r t i b l e . The
f o l l o w i n g i n t e r e s t i n g theorem i s due t o M i c c h e l l i and R i v l i n
THEOREM 8 . 7 :
I
J
I
=
1, and
I,
Let k
+
d
J
-
be n u b n e t n 2s = n
+
06
1. 16
IO, 1
,..., n 1
with
I
w ( x ) i n an i n ( 8 . 5 . 4 ) ,
[321
11 =
. k,
then
LORENTZ and RIEMENSCHNEIDER
232
thehe exidtd a unique dekection 0 6 knotn
0 < x1 <
...
xs < 1
doh
w h i c h t h e da5muka
(8.5.7)
I
1 f ( x ) w ( x ) d x=
c)
id
a.f(i)(0)+ Z
E
jEJ
iE1
( u n i q u e k y ) b o k v a b k e .to b e e x a c t
mod2
604
’
b . f ( J ) ( l )+
crf(xr)
r =1
a . f l p o k y n o m i a k h o ~ deghee
a2
n.
REFERENCES
[ 11
K . ATKINSON a n d A.
SHARMA, A p a r t i a l c h a r a c t e r i z a t i o n of poised
H e r m i t e B i r k h o f f i n t e r p o l a t i o n p r o b l e m s . SIAM J . Numer. Anal. 6 ( 1 9 6 9 ) [ 21
J . BALAZS and P .
230
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11,
111,
IV.
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[ 31
,
G.
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[ 41
B.
CHALMERS, Uniqueness o f a p p r o x i m a t i o n of a f u n c t i o n and i t s d e r i v a t i v e s . J . Approximation Theory 7 ( 1 9 7 3 ) , 213 - 2 2 5 .
[ 5
1
B.
CHALMERS, D. J . JOHNSON, F. T . METCALF and G. D. TAYLOR, Remarks on t h e r a n k o f Hermite-Birkhoff
N u m e r . A n a l . l l ( 1 9 7 4 ) , 254 [ 6]
R.
interpolation. SIAM J.
DEVORE, A. M E I R and A. SHARMA, S t r o n g l y and weakly wn-poised H-B
1040 [ 71
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D. FERGUSON, The q u e s t i o n o f u n i q u e n e s s f o r G. D. B i r k h o f f i n t e r p o l a t i o n p r o b l e m s . J . Approximation Theory
1
- 28.
2(1969) ,
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[ 81
D. R. FERGUSON, S i g n c h a n g e s a n d m i n i m a l s u p p o r t p r o p e r t i e s o f Hermite-Birkhoff
s p l i n e s w i t h compact s u p p o r t . S T A M
Numer. Anal. 1 1 ( 1 9 7 4 ) , 769 [ 91
233
J.
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C. S . FLOWERS-SHULL, Mean i n t e r p o l a t i o n a n d i n t e r p o l a t i o n of e n t i r e f u n c t i o n s . D i s s e r t a t i o n , Texas A 8 M U n i v e r s i t y , December 1 9 77.
[lo]
L. JAFFE, R o l l e r e g u l a r B i r k h o f f matrices. I n Appoximation
ofiy,
The-
11. Academic P r e s s , N e w York 1 9 7 6 , 397 - 4 0 4 .
[ l l ] K . JETTER, Duale Hermite-Birkhoff-Probleme,
Theory 1 7 ( 1 9 7 6 ) , 119
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Approximation
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[12]
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[13]
S. K A R L I N a n d J . M. KARON, P o i s e d a n d non-poised Hermite-Birkhoff
[14]
S . KARLIN a n d A.
i n t e r p o l a t i o n s . I n d i a n a Univ. Math. J . 21(1972) , 1131-1170. PINKUS, G a u s s i a n q u a d r a t u r e f o r m u l a e withmul-
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Interscience Publishers,
N e w York 1 9 6 6 . [16]
E. K I M C H I a n d N . RICHTER-DYN,
An example o f
a non-poised inter-
p o l a t i o n problem w i t h a c o n s t a n t s i g n d e t e r m i n a n t . A p p r o x i m a t i o n Theory 1 1 ( 1 9 7 4 ) , 361 [17]
E. K I M C H I a n d N . RICHTER-DYN,
J.
- 362.
On t h e u n i c i t y
in
simultaneous
a p p r o x i m a t i o n by a l g e b r a i c p o l y n o m i a l s . J . A p p r o x i m a t i o n
T h e o r y 1 8 ( 1 9 7 6 ) , 388 [18]
G.
- 389.
G. LORENTZ, B i r k h o f f i n t e r p o l a t i o n and t h e p r o b l e m of f r e e matrices. J . A p p r o x i m a t i o n T h e o r y 6 ( 1 9 7 2 ) , 283 - 290.
LO RENT2 and R IEMENSCHNE I DE R
234
[191
G.
G.
LORENTZ, The B i r k h o f f i n t e r p o l a t i o n p r o b l e m : N e w m e t h o d s a n d r e s u l t s . I n Phoceedingd
I n t . condehence ObekwoLdach,
B i r h a u s e r V e r l a g , B a s e l , 1 9 7 4 (ISNM251, 4 8 1 - 5 0 1 . [20]
G . G. LORENTZ, Bihkhodd
IntehpoLation PhobLem. CNA R e p o r t - 1 0 3 ,
The U n i v e r s i t y of T e x a s a t A u s t i n , J u l y 1 9 7 5 . [21]
G.
G . LORENTZ, Zeros of s p l i n e s a n d B i r k h o f f ' s k e r n e l . M a t h . Z .
142 (19751, 1 7 3 [22]
- 180.
G . G. LORENTZ, Coalescence of matrices, r e g u l a r i t y a n d s i n g u -
l a r i t y o f B i r k h o f f interpolation problems. J.Approxima20(1977) , 178
tion Theory [23]
G.
- 190.
G . LORENTZ, Symmetry i n B i r k h o f f matrices.
I n t h e PhOCeedingb
ad a Condehence o n A p p h o x i m a t i o n , Durham, E n g l a n d , J u l y 1 9 7 7 . To appear b y A c a d e m i c P r e s s . [24]
G.
G.
LORENTZ a n d S . D.
matrices.
In
Birkhauser [25]
G.
G.
RIEMENSCHNEIDER, B i r k h o f f
dpaced
Lineah
Verlag,
LORENTZ a n d S. D.
Basel,
and
1978
quadrature
Apphoximation,
(ISNM 4 0 1 , 3 5 9 - 3 7 4 .
RIEMENSCHNEIDER, P r o b a b i l i t y o f s i n g u -
in
l a r i t y i n B i r k h o f f i n t e r p o l a t i o n . To appear
Acta
Xath. S c i . H u n g a r .
[26]
G. G. LORENTZ, S. S. STANGLER a n d K . L . ZELLER, R e g u l a r i t y
of
some s p e c i a l B i r k h o f f matrices. I n A p p h o x i m a t i o n Theohy, 11. Academic P r e s s , N e w York 1 9 7 6 , 4 2 3 - 4 3 6 .
[27]
G.
G. LORENTZ a n d K .
[28]
G. G. LORENTZ a n d K . L.
Numer. A n a l . 1291
T r a n s . Arwr. Math. Soc. 149 (1970) , 1-18.
ZELLER, B i r k h o f f i n t e r p o l a t i o n . SIAMJ.
8(1971), 43
G . G. LORENTZ a n d K . L .
- 48.
ZELLER, B i r k h o f f i n t e r p o l a t i o n problem:
C o a l e s c e n c e o f rows. A r c h . M a t h . (Basel) 26 (1975) [30]
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polynomials.
al-
L . ZELLER, M o n o t o n e a p p r o x i m a t i o n b y
g e b r a i c polynomials.
by
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,
I
-
189 192. monotone 401-418.
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[31]
236
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and 11
A. B. NEEiETH, Transformations of the Chebyshev systems, Mathematica (Cluj) 8(31) (19661, 315 - 333.
[ 3 4 ] E. PASSOW, Alternating parity of Tchebycheff systems. J. proximation Theory 9(1973), 295 -298.
Ap
-
[35 ] E. PASSOW, Hermite-Birkhoff interpolation: A class of mn-poised matrices. J. Math. Anal. Appl. 62(1978) , .140-147. [36 ]
G. P6LYAl Bemerkungen zur Interpolationund zur Naherungstheorie
der Balkenbiegung. 2. Angew. Math. Mech. ll(1931) 445-449. [ 371
M. RIESZ, Eine trigonometrische Interpolationsformel und einige Ungleichungen fur Polynome. Jahresber. Deutsch. Math. Verein. 23(1914), 354 - 368.
[38]
J. A. ROULIER and G. D. TAYLOR, Approximation by polynomials with restricted ranges of their derivatives. J. Approximation Theory 5(1972) , 216 - 227.
[39]
I. J. SCHOENBERG, On Hermite-Birkhoff interpolation, J. Math. Anal. Appl. 16(1966), 538 -543.
[ 4 0 ] L. L. SCHUMAKER, Toward a constructive theory of generalized spline functions. In S p L L n e F u n c t i o n d , K a t L o t u L e , 1975, Lecture Notes in Mathematics 501, Springer-Verlag,Berlin 1976, 265 - 329. [ 4 1 ] A. SHARMA, Some poised and non-poised problems of interpolation. SIAM Review 14 (1972), 129 - 151. [ 4 2 ] A. S H A M and J. TZIMBALARIO, Some strongly non-poised I1 -B problems. J. Math. Anal. Appl., 6 3 (19781, 5 2 1 - 5 2 4 .
236
[431
LORENTZ and RIEMENSCHNEIDER
M.
STIEGLITZ, Beste Q u a d r a t u r f o r m e l n
fur
Inzidenzmatrizen J . Ap-
ohne u n g e r a d e g e s t u t z e Sequenzen. To a p p e a r i n proximation Theory. STIEGLITZ, Beste Q u a d r a t u r f o r m e l n f i i r l n t e g r a l e
[441
M.
[45]
J . SURANYI and P. T U R h N ,
[46 1
J . M.
mit
G e w i c h t s f u n k t i o n . Monatsh. Math. 8 4 ( 1 9 7 7 ) , 247
einer
- 258.
Notes o n i n t e r p o l a t i o n . I . A c t a . Math. 79.
Acad. S c i . Hungar. 6 ( 1 9 5 5 ) , 67
-
WHITTAKER, I n t e k p o e a t o h y F u n c t i o n T h e o k y . Cambridge Univ e r s i t y P r e s s , London 1 9 3 5 .
[47]
H . WINDAUER,
On B i r k h o f f i n t e r p o l a t i o n :
Free Birkhoff n0des.J.
A p p r o x i m a t i o n Theory 1 1 ( 1 9 7 4 ) , 1 7 3 - 1 7 5 .
Approximation Theory and Functional Analysis J.B. Prolla (ed.) @North-HoZZand Publishing Company, 1979
APPROXIMATION POLYNOMIALE PONDBREE ET PRODUITS CANONIQUES
PAUL MALLIAVIN I n s t i t u t Henri Poincarg P a r i s , France
Soint sur
E.
E
une p a r t i e f e r m i e iie
On d i t que
R+, p ( x ) une f o n c t i o n
dgfinie
p ( x ) e s t un p o i d b d e B e h v i b t e i n s i n o t a n t par Co(E)
les f o n c t i o n s c o n t i n u e s sur E e t t e n d a n t v e r s z e r o d l ' i n f i n i
xnp(x) E c ~ ( E )
{ x n p ( x )1
On n o t e p a r
pour t o u t e n t i e r
e s t t o t a l e dans
H(q,E)
n >
o
alors
et
Co(E).
l a classe d e s f o n c t i o n s rgelles
h harmo
-
n i q u e s dans l e complgmentaire d e E e t v e r i f i a n t
On s a i t que [ 3 1 Ixn+2p ( X I
In> 0
H(q,E) # @
e s t non t o t a l e d a n s
avec
Co(E).
l e l i e n e n t r e les p o i d s de B e r n s t e i n s u r d u i t s c a n o n i q u e s d e l a forme
237
q =
-
log p
entraine
On se p r o p o s e
que
d'studier
E e t l a c r o i s s a n c e de pro-
MALLlAVlN
238
06
dp
e s t une mesure d e Radon s u r
E
.
[ 3 ] p o u r une premi&e
(Cf.
Btude d a n s c e t t e d i r e c t i o n . ) L e s deux problgmes p a r a i s s e n t d i f f e r e n t s , i l s
a s s e z l i 6 s . I1 e s t d ' a b o r d 6 v i d e n t que n i q u e dans dier si
C \ E
H(q,E) #
sont
toutefois
Wp(z) e s t une f o n c t i o n h m -
e t p a r s u i t e q u ' e l l e f o u r n i t un c a n d i d a t pour e t u -
8.
L e s r g s u l t a t s i n t g r e s s a n t s vont dans l ' a u t r e d i -
c e c t i o n . L e s Th6orGmes p r i n c i p a u x s o n t 6nonc6s c i d e s s o u s . On d i r a q u ' u n p o i n t fonction continue blgme d e D i r i c h l e t
x
0
de E e s t h e g u e i e h s i 6 t a n t donn6 une
h ( x ) quelconque d s f i n i e s u r
E
l a s o l u t i o n du pro-
H ( z ) pour c e t t e f o n c t i o n s a t i s f a i t 5
l i m H(z) = h ( x o ) . z=x0 L'ensemble d e s p o i n t s i r r g g u l i e r s d e E e s t de c a p a c i t 6 n u l l e .
On a
a l o r s 1'6nonc6:
W'(x)
2 q(x)
pouh t o u t
x
E
L e n i g n e d ' e g a L i t g a y a n t L i e u q u a d i p a ~ t t o u th u h
E,
E.
En a p p r o c h a n t p p a r une mesure s o m e d e masses d e D i r a c onobtient
1 . 2 PROPOSITION:
+
S o i n t E une p a h t i e d e R & Oupponom
Q U H(-log ~
p,E)
APPROXIMATION POLYNOMIALE PONDEREE ET PRODUITS CANONIQUES
n o i t n o n v i d e ( -log p(e') dinckete
El
C
E
e t a n t convexe) .
t e L 1 e que p ne
Abhb
iL e x i n t e une pam2e
pan u n p o i d o
boif
239
El.
nuh
On c o n s i d g r e l ' e n s e m b l e d e s f o n c t i o n s c r o i s s a n t e s c o n s t a n t e s s u r l e comple'mentaire d e m E H+(E) [ r e s p m E H-(E)]
pour t o u t
x
si
on d i r a q u ' u n e telle f o n c t i o n
E;
h(x)
2
localement
0
p o u r t o u t x E E [resp. m(x) '0
E El.
On n o t e d ' a u t r e p a r t p a r
pE
la f o n c t i o n c a r a c t e r i s t i q u e d e E l
Par
oii A d g s i g n e une c o n s t a n t e f i x g e . D ' a u t r e p a r t on a u r a
5
supposer
dans c e r t a i n s cas q u e
On a les 6noncGs s u i v a n t s d o n n a n t d e s c o n d i t i o n s n k e s s a i r e s e t des c o n d i t i o n s s u f f i s a n t e s dgpendant de f o n c t i o n s a r b i t r a i r e s d e
e t H-
.
1.4.
~ u p p o d o n nq u e p n a t i o i j a n o e 2 ( 1 . 3 ) e,t q u ' o n
m
E H+(E)
tee
puinne
H+
thouuetr
que
m
aloko o n peut t k o u v e k une p a t i e d i h c k z t e
El
C
El
t&e
que W ( E l l p , l )
h o i t non v i d e .
1.5.
1.5.1.
SUppOhOnb q u ' o n p o i n n e t h o u v e h
m E H-(E)
t e e que
240
aeOhh
MALLlAVlN
W(E,p,O) t o t V i d e .
REMARQUE:
On p e u t r e m p l a c e r 1 . 5 . 1 p a r l a c o n d i t i o n
l i m inf
1.6.
-1 log p (XI
dt
L e s 6noncgs 1 . 4 e t 1 . 5 ramsnent l e problsme
de f o n c t i o n s d ' g p r e u v e d e s classes
H+ e t
H-.
-
5 l a dgtermination
C e s f o n c t i o n s pourront
Gtre c h o i s i e s a d 6 q u a t e s a u c a s p a r t i c u l i e r c o n s i d 6 r 6 . D o n n o n s p l q w s exemples : D'abord un r g s u l t a t g i n g r a l .
stre
C e rgsultat peut
p r g c i s 6 moyennant d e s h y p o t h g s e s s u p p l &
mentaires.
1.6.2.
SUppObanb
1.6.3.
Suppanonn q u e
tee
Q
U
~e
Q
U
~
l i m a ( x ) = 0 , panonn
l i m sup a ( x )
aX(x)
1, aeohd a n peut t h o u v e h
c1 >
1
H+.
C e c i montre q u ' u n r e s u l t a t d e K a t z n e l s o n [ 2
1 ne p e u t pas
Etre
sensiblement am6lior6.
1.6.4.
Suppodonn q u e E que
h o i t
La h e u n i o n d ' i n t e h v a L e e s [a,, b k l , t e e n
APPROXIMATION POLYNOMIALE PONDEREE ET PRODUITS CANONIWES
Ce r g s u l t a t est
241
5 rapprocher de ( 4 . 1 ) .
2 . BALAYAGE AVEC LE NOYAU DE WEIERSTRASS
La t h g o r i e d e b a l a y a g e e n p o t e n t i e l l o g a r i t h m i q u e e s t dgvelopp6e s o i t p o u r d e s compacts, s o i t e n u t i l i s a n t l e s f o n c t i o n s d e Green On se p r o p o s e d ' u t i l i s e r i c i l a t h b
p a r exemple c e l l e du demi-plan.
r i e c l a s s i q u e p o u r un cas non compact l e noyau g t a n t Si
p
log
I1
-2t-11.
e s t une mesure, on c o n s i d g r e r a 1' n t g g r a l e (que l ' o n s u p p o s e r a
t o uj o u r s absolument c o n v e r g e n t e )
WU(Z)
=
I l o g 11 -
zt-1
J
D'autre p a r t si
)J
dp(t).
e s t 5 s u p p o r t compact on p e u t c o n s i d 6 r e r ggal-nt
U ~ ( Z )=
J l o g 1-1
1
dp(t).
On a a l o r s
WP(Z) = UV(0)
-
UU(2).
Nous a l l o n s commencer p a r 6 c r i r e la f o n c t i o n 2.1. que UbbeZ
2.1.1.
LEMME d e f a c t o r i s a t i o n : q(e')
bait
q(x)
SuiX
une d o n c t i o n c o n v e x e , 2
q come un p o t e n t i e l .
une doncZion pahi,tive,
ChOihbaKCe
einzaihe
g h a n d , e t t e e e e que
lo
1
q ( t ) t-3'2 d t <
a.toJLb o n p e u t t h o u v e h une d a n c t i a n
m
4
p o h i t i v e vzhidiant
Z&e
pouh
x
242
MALLlAVlN
j log I 1 -
2.1.2.
x t - l / $t)T d t = q(x),
L ' h y p o t h s s e de c o n v e x i t 6 f a i t e s u r
PREUVE:
2.1.3
2.1.1.
oii
dt
m(t)
x > 0.
q i m p l i q u e que
est c r o i s s a n t e .
implique que
Q(z) l a transformge de M e l l i n de q .
Soit
mellement l e s deux membres de 2 . 1 . 2 .
1
m
0
l o g 11
-
uI uZ-ldu =
-
Transformons
for-
p a r M e l l i n , remarquant que
cotg
-
712
Z
1 < Rez < 0
on o b t i e n t
2.1.3.
donne, n o t a n t p a r
h ( x ) l a fonction ayant
Notons p a r
%,
M l a transform6e de M e l l i n - S t i e l j g s de
-
< Rez <
1 2
pour
transform6e
de
dm
Mellin
h ( x ) se c a l c u l e p a r r g s i d u s e t on t r o u v e
h ( x ) = log11
+
x1l21
- log11
s u r c e t t e e x p r e s s i o n il e s t g v i d e n t que
-
x1 / 2
I
APPROXIMATION POLYNOMIALE PONDBREE ET PRODUITS CANONIQUES
h(x) > 0
2.1.4.
2.1.5.
06
L
(Indgalitd fondamentale)
%,a
2.1.6.
h E Lm
1,a
mesure
243
- -'2
< a < -1
- -l
< a 2 -1
2
2 -
fa
2
ddsigne l'espace des fonctions sommables par rapport 5 la -a l'espace des fonctions borndes par x .Ceci xa-1 dx; a
dtant, justifions les opsrations formelles effetudes ci-dessus. Posons :
dm
Alors
E
Mc1
espace des mesures sommables pour xa
d'oii d'aprss 2.1.5.
2.1.8.
est bien d6fini et
C o m e d'autre part
duit de composition k
- -
2
h*dm
:=
2.1.7.
< B < 0.
=
log11
log 11
-
- XI
XI
%,a
LlIa, - 1 < a < 0, le
E
*
s
1 Si a > - 7 ; - - 1< a < - .1 2 2
4
est bien d6fini et
On a enfin
K(z) =
- cotgnz
. 6(z)
si
Re2 = 6,
-
1 7
< B < 0,
et
(z) K ( z ) =- M 2
k(t) =
jot
les deux membres &ant
dx
et
presque partout
continus ceci vaut partout d'oc 2.1.2.
E
proL
1, B
MALLlAVlN
244
Posons
il rgsulte de 2.1.8. que
lim r(x)
2.1.9.
existe
X=m
Nous allons monter un lemme glbmentaire sur l’allure d’un potentiel d’une mesure portbe pour l’axe ri?el.
lim y=o
exibte e t
+ iy)
b0it
dinie
.
ALohd o n u
lim U” (xo + iy)
y=o
PREWE:
=
up(x0).
Up (x) est semi-continue infbrieurement donc
D‘o6 l’intggrale
- .f log11
-
xot-1I d p (t) est convergente. ReMlrquant
que les points rgguliers de E lim h(x y =o d’oG en utilisant 2 . 2 .
+
iy) = q(x)
APPROXIMATION POLYNOMIAL€ PONDIRE€ ET PRODUITS CANONIQUES
en tous les points rsguliers de E l tout dense sur E et
246
ceux-ci formant un ensemble par-
Wp(x) 6tant semi-continue supgrieurement, q(x)
continue, on obtieni
3.
3.1.
Nous nous proposons dans ce paragraphe de d6montrer 6nonc6s 1.2.
THEOREME:
u n e mebuhe dX
DEMONSTRATION:
Si
H(E,
- logp)
e b t non
v i d e , aeohn o n p e u t &ouve,t
a y a n t pouh buppotrt u n enbemble d i b c h e t
Soit
H ( E , -1ogp)
El
# 6. I1 existe d'aprgs
I1 r6sulte du fait que cette int6grale est >
-
m
que
t
C E,Z&
que
1.1.
dp=p(t) est
0
une fonction continue. Soit n(t)
=
et soit exp [
-
partie entigre de
1
log(1
-
zt-')dn(t)]
~(t)
= F(z).
F ( z ) est une fonction m6romorphe n'admettant que des pzles simples.
D'autre part, posons s(t) = II
3.1.3.
j log11 - zt-l
MALL1AVlN
246
=
a/x
Jo
Lx 1/2
+
2
+
J1/2
+
Jim.
La premisre int6grale est i n f 6 r i e u r e 5
L a seconde d
+
log x
.
0 (1)
.
La d e r n i s r e d
(1 (1)
R e s t e d 6 v a l u e r l a 3sme i n t 6 g r a l e s = s
1
+ - -1
I
1 Isl! 2 T
La p r e m i s r e i n t 6 g r a l e <
I
-
011
le
fera
en
posant
d'oii
l o g - r +0(1), l a s e c o n d e e k l l a t r o i s i 6 m e s o n t
O(1) , d'oG e n t e n a n t compte d e 3 . 1 . 1 .
S o i t r un e n t i e r > A
+
2, bl,
. . , ,br
,r
a l o r s on p e u t t r o u v e r une f r a c t i o n r a t i o n n e l l e pour p 6 l e s s i m p l e s e t t e l l e que
F
1
(2)
= F(z) H(z)
v6rifiera
p o i n t s d e E distincts; H ( z ) a y a n t les
bk
APPROXIMATION POLYNOMIALE
PONDEREE ET PRODUITS CANONIOUES
247
3.1.3.
On a
OG
E R6sidus de F1(x) < t. e Le m- r6sidu a v6rifiant
p(t)
=
D‘autre part on a
-
d’oc .f
06 yn
t2
<
n(t) = O(t1’2) m.
en vertu de la formule de Carleman
Par suite si l‘on pose
est le premier moment de dp diffgrent de zgro. Prenons z=x+i,
x E E ; cette egalit6 contredirait 3.1.3.
4.
FACTORISATION DU NOYAU
log11
-
c.q.f.d.
U/
Nous allons d6composer dans l’alggbre de composition sur (0,m) le noyau
log11
- u / dans le produit d’un
n u y a u p u d i t i d et d‘un op&
rateur diffgrentiel. C‘est un fait bien connu que l‘gvaluation
des
produits de Weierstrass est compliqu6 par le fait gue le noyau bgll-ul est
( 0
si u) positif sinon la &partition
localement et globalement
: globalement
li6 par Nevanlinna 2 la moyenne
des
z6ros
par la fonction
intervenant
I”
mR sur le cercle R. localement
dt par
les perturbations au voisinage des zeros 5 l’evaluation donnde par la moyenne
m R’
248
MALLlAVlN
Nous a l l o n s donner une f a m i l l e d e t e l l e s f a c t o r i s a t i o n s du noyau
logll- uI ddpendant d ' u n e f o n c t i o n a r b i t r a i r e .
4.1.
PROPOSITION:
nze, dzdinie
Suit
but [ O f
n ( t ) u n e d o n c t i o n 2 v a h i a t i o n klocdement b o t -
+ a [ ,
t e U e que
n ( t ) = O(t1'2)
,
t
+ a,
et
b0i-t
xo t e e que
Soit
s ( t ) une
d e pCub q u e
PREUVE:
doncXion b a t i b d a i b a n t aux
s(t)
+
+
m,
que
L'hypothgse 4 . 1 . 1 .
n(t)
bOit
miimeb
c o n d i t i o n b . SUppObOnb
n u t a u uoi.binage de z z o . S o i e n t
permet d ' g c r i r e
APPROXIMATION POLYNOMIALE PONDCREE ET PRODUITS CANONIQUES
xo-
E
v(x0) = l i m
+
'0
E=O
+ m
-1
dt
xn(t) x - t
IX0+E
s ( t ) dt = F ( x , x +
249
t
E)~(x+E)
X+ E
+ I+
JX+E
~ ( x =) l i m [ F ( x , x + E ) p ( x + E )- F ( x , x - ~ ) p ( x-
F(x,t)dp(t)
E)]
+
E =0
';1
+
lim
E =0
p(x) s a t i s f a i t en
F(x,t)dp(t) X+ E
x
0
l a condition 4.1.1.
ce q u i p e r m e t d ' 6 c r i r e
le
premier crochet
p(x) l i m [ F ( x , x +
E)
E =0
d'ofi
4.2.
-
F(x,x
-
E)]
4.1.
COROLLAIRE:
PREWE:
On a
Appliquons l a p r o p o s i t i o n 4 . 1 .
avec
s(t) =
t1l2,0 < t
A l o r s une i n t g g r a t i o n p a r r 6 s i d u s donne
y ( x ) = V.P.x
I, - U 1/2
x-u
du = 0, u
x > 0.
D'autre part
<+a.
260
MALLlAVlN
e n t e n a n t compte d e l a deuxigme e x p r e s s i o n donn6e d e P dans l'&ncG. L e f a i t que
P
y(x) = 0
montre que l a deuxigme e x p r e s s i o n props& pour
e s t 6 g a l e 1 l a premigre. L ' a p p l i c a t i o n de 4 . 1 .
Gtablit alors 4.2.
Nous a l l o n s donner une p r e m i s r e a p p l i c a t i o n du r 6 s u l t a t de factorisation 4.1.
4.3.1.
Soit
ait
s(t) E
une d o n c t i o n c t o i n b a n t e b a t i d d a i b a n t 1.9.1,
p o u t buppoht e t
t&e
que & ( t )
Q U ~
x € E .
Suppooonb
Q U ~
4.3.3. E Alotb
il e x i b t e
Une
donction
q ( z ) < 0 , hahmonique danb
l e comp.Lben-
t a i h e d e E , t e l l e que
REMARQUE:
S i E e s t compos6 d'intervalles de longueur logarit-hmique > a > 0,
APPROXIMATION POLYNOMIALE PONDERdE ET PRODUITS CANONIOUES
a l o r s on p e u t d a n s 4 . 3 . 3 , p r e n d r e
c1
l e p r o d u i t canonique c o n s t r u i t avec
( x ) = 1; d e p l u s 4 . 3 . 2 a l i e u si ds
e s t simplement p o s i t i f .
Posons
PREUVE:
Alors
dn a p o u r s u p p o r t E
,
et
= o
t # E .
Escrivons 2 . 1 , remarquant
oc
oii
251
p(x) = a ( 1 ) e t u t i l i s a n t 4 . 3 . 2 ,
0 > 0 , e t une 6 v a l u a t i o n a n a l o g u e p o u r
et
Prenons
€I2 sont
n1 =
e -1 3 n
t
E
3 [ x , XI~
d’oc
deux c o n s t a n t e s num6riques p o s i t i v e s d ’ o c
e t posons
262
MALLlAVlN
03
Le r i s u l t a t s u i v a n t c l a s s i q u e p o u r les f o n c t i o n s e n t i s r e s d'or1 s'etend 5
dre
\o(z): i l e x i s t e une s u i t e i n f i n i e de cercles
Rk
t e l s que
q(Rkeie)
Dans
tisre,
{lz
donc
I
-+
-
m
uniformgment e n
0 .
n C E , \ o ( z ) e s t harmonique n 6 g a t i v e s u r l a fron-
~ ( z <) 0
q u e l que s o i t
z
.
5. Nous a l l o n s donner dans ce p a r a g r a p h e des c o n d i t i o n s pour que la' s u i t e
{ x n p ( x )1
suffisantes
s o i t non t o t a l e dans l ' e s p a c e
Co(E) des
fonctions continues s u r E n u l l e s 5 l ' i n f i n i . E t a n t donne x E Ix , I x
C
E.
x
E E
s o i t Ix l e p l u s g r a n d i n t e r v a l l e t e l
que
Posons
a * ( x ) = i n f { 1,
5.1. PROPOSITION:
L e A n o t a t i o n d z t a n t c e e e e ~d e 4 . 3 , A U p p O A O n A
eeA h y p o t h h e d d e 4 . 3 , A o n t ~ a t i A 6 a i t e A p o u t pLuA
AUppOAOnA
5.1.1. ou b i e n que
5.1.2.
ou b i e n que
pl(x) = log p ( x ) ;
que de
APPROXIMATION POLYNOMIALE PONDER~E ET PRODUITS CANONIQUES
283
ou b i e n que
5.1.3.
PFtEuvE:
Nous a l l o n s c o n s t r u i r e un p r o d u i t cononique
t e l que l a f o n c t i o n conjuguge s o i t uniforme
modulo
p l g m e n t a i r e de E. Supposons p a r exemple que 5.1.3,
27~ dans leccun-
est
satisfait.
Posons :
03
1
A(t)
sera une f o n c t i o n p o s i t i v e localement c o n s t a n t e s u r E c ' e s t
d i r e c o n s t a n t e s u r chaque
I,. On a puisque
p1 e t s s a t i s f o n t
5
1.9.2,
cette q u a n t i t 6 t e n d a n t v e r s
-
m
on p e u t d g t e r m i n e r l a f o n c t i o n X tel-
l e que dn = e n t i e r ,
A(x)
+
1,
x
+
X
Soit t e l l e que
g ( z ) l a f o n c t i o n holomorphe dans l e complsmentaire de
log
I g(z)I
= v(z).
A l o r s l a formule 4 . 1 ,
donne
E
264
MALLlAVlN
N 19 ( X )
I
< p(x),
pour
N
e n t i e r f i x 6 assez grand, x
E
E
classi-
I n d i q u o n s r a p i d e m e n t comment m o d i f i e r un raisonnement que p o u r c o n c l u r e N 2 -
dt
t
Oii
D ' a u t r e p a r t l a formule d e Nevanlinna donne
log
Puisque
I g ( r ei
n(t)
+
B
-
)Id0 =
m
1
2
9
dt.
cette dernigre q u a n t i t g tend v e r s
-
S o i t q l e p r e m i e r e n t i e r t e l que
alors
log
I
gN(reie)
e t d'autre part
I
<
- ( q - 1)log r + l o g ( 1 + r -1y -1b )
+log
1
aq
I
m
.
.
-
APPROXIMATION POLYNOMIALE POND~AEE ET PRODUITS CANONIQUES
ce q u i c o n t r e d i r a i t l a formule de N e v a n l i n n a . thogonale
: tn(n ?
0); d ' a u t r e p a r t
e s t non t o t a l e d a n s
Co(E)
e s t non t o t a l e d a n s
Co(E).
2
5.1.2,
71
Idn
tn
d'oc
e t 5.1.2,
p(t)
s o n t s a t i s f a i t e s on &-
n ( x ) a v e c l a mgme f o r m u l e , les h y p o t h s s e s
a y a n t pour e f f e t
telle que
f ( t ) d t e s t a i n s i or-
f ; ( t )= t p ( t ) ce q u i entraine que t n p ( t )
oc
Lorsque l e s h y p o t h e s e s 5 . 1 . 1 , f i n i t encore
f(t) <
266
determiner l a fonction
que l ' o n p e u t
= e n t i e r e t que
X ( t ) s o i t born6e s u r ( 0 ,
On p e u t d ' a u t r e p a r t remarquer que l e s c o n d i t i o n s 5.1.3 remplacent l a c o n d i t i o n 2.3
et
5.1.1,
+
X(t)
-1.
5.1.2
q u i n ' a plus besoin a l o r s
et
d'stre
v6rifiie. Nous a l l o n s m a i n t e n a n t donner d i v e r s e s 6 v a l u a t i o n s d e l a fonction
s ( t ) q u i cornbinges a v e c 5 . 1 , d o n n e r o n t d e s c o n d i t i o n s
saires p o u r que
5.2.
xnp(x) s o i t totale dans
PROPOSITION:
Suit U ( x ) =
-+ j
En
dae dam 5.1.
PREUVE:
soit
P(X)
-
X
1/2
dp = 0
d'aii,
P < 0 , on c o n c l u t
x
E
[OiXI
E
-
Co(E)
e t appliquons ,si
n6ces
dt t '
4.2.
aeoab o n p u t p a e n -
Alors
266
MALLlAVlN
La d u i t e
xnp(x)
ebt
non t o t a l e dann
Co(E).
Nous allons donner s o u s des h y p o t h g s e s s u p p l g m e n t a i r e s d e m e i l l e u r e s 6 v a l u a t i o n s des
s(x)
possibles.
Posons
5 . 3 . PROPOSITION:
si
l i m ;(XI
panonb
= 0,
aLahn Q U ~ L que L ~ h o i t La c o n n t a n d e
X on p e u t dann 5 . 1
s ( x ) = e Av(x)
PREUVE:
Posons P(X)
= e
Av ( x ) .-1/2
alors
06
on e n d g d u i t que
phendhe
APPROXIMATION POLYNOMIALE PONDEREE ET PRODUITS CANONIOUES
0 < b <
X (-log
Remarquant que l e noyau P i n t r o d u i t e n 4 . 2
et
decroissant sur
257
a ) -1
est c r o i s s a n t S U L . [ ~ , + ~ ]
1 1 , on a
[ 0,
a
e-5
La premigre i n t s g r a l e e s t >
-
t i e r p o s i t i f ou n i g a t i f )
x n = 3n x ( n en-
d'autre part si
p(x)X
-3n+1
123"
+
P(t-1)5-3/2dS(1
-
o(l))
2
e t t o u t e s ces i n t g g r a l e s s t a n t p o s i t i v e s d ' o i i N d g s i g n a n t un e n t i e r fixb,
x -+
on a
w
c e t t e i n t 6 q r a l e se c a l c u l e e t e s t assez g r a n d e t
REMARQUE:
pu p r e n d r e
N > X
,
d'oE
Au l i e u d e f i x e r N = N(x)
. Ce
>
d'oc
N
P(f)dp(t) > 0
si x
e s t une f o n c t i o n p o i d s . c.q.f.d.
N
et d e f a i r e t e n d r e
c a l c u l ne p e u t
stre
x
+ m
on
aurait
men6 5 b i e n que myennant
des hypothGses s u p p l 6 m e n t a i r e s s u r E . On o b t i e n t alors d e s f o n c t i o n s
p o i d s d e l a forme
e x ( x ) v ( x ) 03
h(x)
+
. Quel q u e
s o i t E,ce prod&
MALLlAVlN
268
X (x) croissant plus vite que logx.
de calcul ne permet pas d'obtenir
-a(x)
On aurait pu d'autre part ddfinir en considgrant l'intervalle x 3x [ xB-', xB] au lieu [ y , 2 1 03 B ddsigne un nombre fix6 B > 1. Un cas indressant est celui oii
ceci est en particulier le cas 03 la s6rie
<
m
06 In ddsigne le ne intervalle de l'ensemble
E et
TdtI .
l o g In = l o d l
In On pourrait comparer cette condition avec la condition de Wiener d'effilement 5 l'infini. On a l'amglioration suivante de 5.2.
5.4. PROPOSITION:
5.4.1.
adffhb
Si
lim sup
;(XI
=
on p e u t thffUVeh une c a n a t a n t e s(x) = e xu (XI
PREWE :
alors
e
< 1
XI
X > 1 teLde
Q U ~
u h ~ i d i e 5.2.
APPROXIMATION POLYNOMIALE PONDEREE ET PRODUITS CANONIQUES
269
et d'autre part
Eva1uons
choisissant y vgrifiant
y < e
-02 log 3
OG
et
I1
+
I2
I1
l'intggrale sur intervalles
[ xT , 23x 1
sera positive; il en sera de m & n e p u r les
3nx , 7 3"+L I d'oc [F
5.5. REMARQUE:
est une fonction s
.
On dira que l'ensemble E est dense 5 l'infini
dans
F si posant
on a, B fix&, 5.5.1.
Posons
Alors si 5.5.1. vaut,
e A!J*(x) est une fonction poids pour E, qwlle
MALLlAVlN
260
que s o i t la c o n s t a n t e A
I
vgrifiant
X < 1. La d g m o n s t r a t i o n
s'ef-
f e c t u e c o m e en 5 . 4 .
lim
6.1.1.
PREUVE:
wr ( - x ) = + m . q (XI
Supposons que 6 . 1 . 1 .
ne s o i t pas s a t i s f a i t e l a l i m i t e i n f g -
r i e u r e du p r e m i e r membre de 6 . 1 . 1 .
sera 6 g a l e 5
c o n s t r u i r e une f o n c t i o n q1 t e l l e que vide. S o i t
CJ
q = O(ql)
Nous
allons
b <
m.
et
H(E,ql) soit non
la mesure harmonique du complgmentaire d e E d a n s
42
alors
Soit
rl
( x ) une f o n c t i o n c r o i s s a n t e t e n d a n t v e r s l ' i n f i n i
on p r e n d r a q1 t e l que
q i = rlq'
que
telle
.
Alors
6.1.2.
Soit
l i m inf
wr(-x)
= 0.
q i (XI
h ( z ) une f o n c t i o n harmonique a p p a r t e n a n t 5
H(E,ql),
et
p
la
mesure associge p a r . 1.1.
Remarquons que le maximum de Wf s u r
x < 0 , s o n minimum s u r
1
z
I
x > 0 , on o b t i e n t
= R
est a t t e i n t sur l'axe
qu'il
existe
une
suite
APPROXIMATION POlYNOMlALE PONDERBE ET PRODUITS CANONlClUES
R + k
261
telle que
D’autre part on a sur E
d’oG en remarquant que Wr et W’
sont hmniques dans { z ; [ z / <%117C\E.
On obtient
k &ant
arbitrairement grand Wr(z) 5 0
.
Appliquons la formule de Nevanlinna
0 <
lo
t
r ( u ) e = Moyenne de W’
sur
z
i
=
t < 0
contradiction. On peut remarquer que l’on aurait pu
renplacer
simplement
l’hypothsse H(E,q) non vide par celle qu’il existe une fonction F ( z ) , holomorphe dans le complgmentaire de E
,
telle que IF(x) I <Wq(x),x E E.
I1 suffisait de remplacer dans le raisonmement ci-dessus la minorante harmonique de
6.2. PROPOSITION:
DEMONSTRATION:
On cl
-
W‘(-X)
lOg[F(z)1 .
< 2
dt r(t)t
-
Wr(X).
W’(z)
par
262
MALLIAVIN
Wr(-x)
L'inGgalitG.
-
-
j0
X
log 1 1
dt
- vi
=-jo
X
> log
+a
X r ( t ) d t + j x+t r x+ t X
i
1 + vI
dt
( t ) t
donne
BIBLIOGRAPHIE
,
[ 11
G . COULOMB-COURTADE
[21
Y. KATZNELSON, C o m p t e s R e n d u s 246 ( 1 9 5 8 1 , p . 2 8 1 .
[ 3
1
I .
Th&e
P a r i s 1976.
P . MALLIAVIN e t S . MANDELBROJT, S u r l ' g q u i v a l e n c e de d e u x pro-
blsmes de l a thsorie c o n s t r u c t i v e des f o n c t i o n s , S c i . Ecole N o r m . S u p . ( 3 ) 7 5 ( 1 9 5 8 ) , p . 49 - 5 6 . [4
1
S.
Ann.
MANDELBROJT, Genehd theohemmn ad CLohWLe, Rice I n s t i t u t e P a m p h l e t .
Special i s s u e ( 1 9 5 1 ) .
[5 1
E L e m e n t A 0 6 Apphoximation T h e a n y , D. van Nostrand Co., I n c . 1 9 6 7 . R e p r i n t e d by R. Krieger Co., I n c . 1 9 7 6 .
L. NACHBIN,
Approximation Theory and Functional Analysis J . B. Prolla (ed. ) 0 North-Hol land Publishing Company, 1979
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
REINHOLD MEISE Mathematisches Institut der Universitat D-4000 Dusseldorf, Universitatsstr. 1 Bundesrepublik Deutschland
PREFACE
When the author started to deal with the subject
of
the
present article, he did not know too much of the various ways howone can do calculus in (real) topological vector spaces. He was irainly interested in a generalization of the theory of distributions to infinite dimensional locally convex (1.c) spaces, by means
of
duality
theory and with relations to infinite dimensional holomorphy. Therefore he began studying spaces of differentiable functions on
1. c.
spaces by analyzing the definitions of Cn-functions used by Aron [ 3 1, Bombal Gord6n and Gonzhlez Llavona [lo 1 andYamamuro [ 2 4 1 .Then he (re-) invented the notion of n times continuously y-differentiable func tions on an open subset R of a 1.c. space E ( 2 . 4 )
,
where y is
an
arbitrary covering of E by bounded subsets. As he realized later, this notion had been introduced with a slightly different definition
by
Keller [18] already. Further investigations showed that many results known to be true on open subsets of
IRN carry over at least to Frkhet
spaces or strong duals of Frgchet-Monte1 spaces if one takes as ythe system of all compact subsets of E . In order to be precise, we will now sketch the main results of the article. In the first section most of the definitions are stated aswell as some results which will be used in the sequel. In the second part 263
264
MEISE
we i n t r o d u c e t h e 1.c. space Cn(RIF) o f
Y
n t i m e s continuously
f e r e n t i a b l e f u n c t i o n s on a n open s u b s e t R v a l u e s i n t h e 1 . c . s p a c e F , where
of t h e 1 . c . s p a c e E w i t h
n i s a n a t u r a l number
i s a s y s t e m a s i n t r o d u c e d above. For open s u b s e t s
y
complete 1 . c . s p a c e E l w e show t h a t
y-dif-
Cmo(Q)
or
* and
R i n a quasi-
i s a Schwartz s p a c e i f
EAo i s a Schwartz s p a c e .
and o n l y i f
In the third section we give
a s u f f i c i e n t condition
.
for
the
a p p r o x i m a t i o n p r o p e r t y of Cn ( Q ) The p r o o f of t h e c o r r e s p o n d i n g t h e co orem 3.5 i s a g e n e r a l i z a t i o n o f t h e proof g i v e n by Bombal Gord6n and Gonzslez Llavona [lo 1 i n t h e c a s e o f Banach s p a c e s . ( A c t u a l l y a n a n a l y s i s of
[lo ]
l e d t o t h e r e s u l t presented h e r e . ) Since t h e proof
a c r i t e r i o n f o r the a.p. Cn(R)
E
Y
due t o Schwartz [ 2 2 ] , w e f i r s t c h a r a c t e r i z e
F a s a t o p o l o g i c a l s u b s p a c e o f C n ( Q I F ) . The main lemma ( 3 . 3 )
[lo]
f o r theorem 3.5 g o e s back t o [20]
uses
Y
a s w e l l a s t o P r o l l a and G u e r r e i r o
( i n t h e case of Banach s p a c e s ) . I t h a s a l s o some f u r t h e r a p p l i -
c a t i o n s which g e n e r a l i z e p a r t s o f t h e r e s u l t s o f [ 2 0 ]
and which may
h e of i n t e r e s t i n c o n n e c t i o n w i t h t h e t h e o r e m of Paley-Wiener f o r t h e elements o f
m
Cco(E,C)
'.
I n t h e l a s t s e c t i o n w e p r o v e t h a t , f o r open s u b s e t s i n c e r t a i n 1 . c . s p a c e s El
- Schwartz ill
and E 2 r e s p e c t i v e l y , t h e r e e x i s t s
t u r a l t o p o l o g i c a l isomorphism between C ~ o ( ~ l , C ~ o ( Q 2and ) ) CEo(R1 By t h e r e s u l t s on t h e a . p . o f t a t i o n of
m
Cco(Rl
x
R,)
as
Co:
(
0,) t h i s a l s o i m p l i e s
€-tensor product
m
Cco(R1)
V
and R 2 a nax
R,).
a represenm
QE C C o ( Q , ) .
While f i n i s h i n g h i s i n v e s t i g a t i o n s , t h e a u t h o r r e c e i v e d
the
p r e p r i n t 1 1 2 1 o f Colombeau, where s p a c e s o f C m - f u n c t i o n s on Schwartz b o r n o l o g i c a l v e c t o r s p a c e s a r e s t u d i e d . Colombeau h a s p o i n t e d o u t t o t h e a u t h o r t h a t any f u n c t i o n i n CZo(Q,F) i s a C n - f u n c t i o n i n S i l v a ' s l a r g e s e n s e i f t h e q u a s i - c o m p l e t e s p a c e E i s g i v e n t h e compact b o r nology. F o r F r g c h e t s p a c e s and f o r s t r o n g d u a l s o f F r g c h e t - S c h w a r t z spaces
E
b o t h n o t i o n s c o i n c i d e . T h i s shows t h a t
the
bornological
s e t t i n g i s more g e n e r a l as f a r a s t h e Schwartz p r o p e r t y o f C&(Cl)
is
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
265
concerned. However, i t i s n o t known how t o prove t h e r e s u l t s o f t h i s p a p e r w i t h b o r n o l o g i c a l methods i f
E i s t h e s t r o n g d u a l o f a Frgchet-
Monte1 s p a c e o n l y . L e t us come back t o d i s t r i b u t i o n s a g a i n : I t s h o u l d be remarked t h a t o b v i o u s l y some r e s u l t s i n t h i s a r t i c l e can a l s o b e r e g a r d e d p r o p o s i t i o n s on t h e d u a l of
m
Cco(Q)
,a
as
s p a c e which i s a n a t u r a l gen-
e r a l i z a t i o n of t h e d i s t r i b u t i o n s w i t h compact s u p p o r t . mentioned i n remark 4 . 8 w i l l be t h e subject m a t t e r of
The a
results
subsequent
paper.
ACKNOWLEDGEMENT:
The a u t h o r t h a n k s R. Aron, K.-D.
Bierstedt,
Colombeau, H . Jarchow, H . H . Keller and L . Nachbin f o r some
J . F.
helpful
d i s c u s s i o n s o r correspondence on t h e s u b j e c t o f t h e a r t i c l e .
H e also
g r a t e f u l l y acknowledges p a r t i a l f i n a n c i a l s u p p o r t from GFD and I M U .
1. PRELIMINARIES I n t h i s section we s h a l l f i x the notation, r e c a l l some definit i o n s and s t a t e some r e s u l t s which w i l l b e a p p l i e d l a t e r .
We
shall
use t h e t h e o r y o f l o c a l l y convex (1.c) s p a c e s a s i t i s p r e s e n t e d e . g . i n t h e books of Horvhth [ 1 7 1 , Kothe [19 ] and S c h a e f e r [ 2 1 1 . Throughout this a r t i c l e , a 1.c. s p a c e always means a k d H a u s d o r f f 1.c. space, b e c a u s e w e o n l y want t o d e a l w i t h real d i f f e r e n t i a b l e functions.
1.
Let
E and
subsets of
F be 1.c. s p a c e s and l e t
y
be a s y s t e m
E which c o v e r E . Then, on t h e s p a c e
of
bounded
L(E,F) of a l l con-
t i n u o u s l i n e a r maps from E i n t o F one can i n t r o d u c e t h e c o r r e s p n d i n g y-topology of uniform convergence on t h e sets i n i n g 1.c. s p a c e i s d e n o t e d by
y ; the result-
L ( E , F ) . I t i s w e l l known t h a t t h i s t c -
Y
pology d o e s n o t change i f t h e system i s e n l a r g e d i n s u c h a way
that
266
MElSE
i t i s d i r e c t e d under i n c l u s i o n and t h a t any s u b s e t o f a s e t i n y bey . W e s h a l l always assume t h a t
longs t o By
y,
, yco
y has t h e s e p r o p e r t i e s .
and y b w e d e n o t e t h e systems o f a l l
finite
dimensional bounded, a l l compact, a l l precompact and a l l boundedsulr
sets of
E
. The c o r r e s p o n d i n g s p a c e s
L (E,F) are d e n o t e d by La(E,F),
Y
L c o ( E , F ) , Lc(E,F) and
Lb(EIF). W e w r i t e
and
Ly(E,E).
2.
L (E) instead of
Y
sets which c o v e r s E and
Assume
n
L
2 2;
f o r any k w i t h
n E IN.
into F . u E e ( E n , F ) i s c a l l e d y-hypocontinuoub,
1 5 k 5 n , any
hood W of z e r o i n i n E such t h a t
c)
L y ( E , R)
E(EnlF) w e denote t h e l i n e a r space of a l l n - l i n e a r m a p
By
p i n g s from
b)
Y
L e t E and F b e 1.c. spaces, y a s y s t e m o f bounded
DEFINITION:
a)
i n s t e a d of
E'
u(Sk"
c
by
d: (E",F)
Y
E
y and any
neighbour-
F t h e r e i s a neighbourhood V o f
W e d e f i n e t h e space
n L 2
S
x
Y
S i n c e , o b v i o u s l y , any
C
(EnlF) f o r
E ( En ,F)
= iu E
V x Sn'k)
Iu
n = 1 by
L ( E , F ) and f o r
y-hypocontinuous).
is
the
case
for
Sn
Y
S E y I w e can endow
zero
W.
u E 1 (E",F) i s bounded on
any
if
E (En,F) w i t h t h e topologyof miY form convergence on t h e system y n = {Sn I S E y } As i n of
.
continuous
P,(En,F) , E C O ( E n , F ) , . . . i f
linear
mappings yco
y = yo,
, ...
write
we
. The e l e m e n t s
Xu (En,F) are c a l l e d .4qXVLLateLy continuoub n - t i n e a h mapn p i n g b . W e w r i t e d: ( E n ) i n s t e a d of X y ( E I IR)
of
.
Y
d)
u E E(En,F) i s c a l l e d bymmet/riC, i f f o r any p e r m u t a t i o n of
n e l e m e n t s and any
x = (xl
..
,.
I
xn) E E
we
71
have
287
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
L ~ ( E ~ , F:= ) {u E L~(E",F)Iu
Y
is symmetric).
The closed linear subspace
eS(En,F) of Y dowed with the induced topology.
JY(En,F) is
en-
The proof of the following lemma is an easy exercise.
3. LEMMA:
a)
Fvfi
any
Fok any
u
E
d: (En,F) t h e 6vLLowing hvLdb t h u e :
Y
S E y
the hestkiction
I
u
Sn
uni6okmty con
ib
-
.t.ifiUOUb.
b)
F v h any k , wi-th i6
4.
1 5 k 5 n, and any
S E
y,ulI&'
x
E
X
sn-k
cvntinuous.
DEFINITION:
Let E and F be 1.c. spaces and y a systemof bund-
ed subsets of E covering E . a)
b)
The 1.c. spaces Ly(E,F) I n E IN , are defined inductively by Lo(E,F) :=F and Ln+I(E,F) := LY(E,LY(E,F)). n Y Y n of linear mappings There is a unique sequence (rp I n ~
qn : Ln(E,F) * L(En,F)
Y
vn+l(u) [x,,
satisfying
. ..,xn+l~ = rpn (u(x,))
is injective for any
[ x2,.
we may
n E IN
idL(E,F)
and
. .,xn+l~.Since
pn
=
(and shall)
regard
Ln(E,F) as a linear space of n-linear mappings on En with Y values in F . C)
Let
n
E
IN be fixed. An element
u E Ln(E,F) Y
is
bymme-thic, if the corresponding n-linear mapping symmetric. We define L"~(E,F) := {u
Y
E
L"(E,F) Y
I
u
is symmetric)
called
rp" (u) is
268
MEISE
and endow t h i s l i n e a r s p a c e w i t h t h e 1.c. t o p o l o g y i n d u c e d by
Ly(ErFI v
5. PROPOSITION:
byhtem i d
06
Let
and F b e l . c . hpaceh and L e t y b e acouehing
E
bounded s u b b e t n a d
E
. Then ,the
mapping p n : Ly(ErF) +J;(E"rF)
a topulagieae ibomahphibm. The p r o o f i s by i n d u c t i o n on
PROOF:
n
. For
o b v i o u s l y t r u e . Hence, l e t u s assume t h a t morphism f o r
15 j
2 n.
t h e statement is
n =1
i s a t o p o l o g i c a l is0
VJ
-
prove t h a t pn+l : L ~ ' , ~ ( E , F )+J~S(E"+~,F)
We shall
Y
i s a t o p o l o g i c a l isomorphism. T h i s w i l l b e done i n several s t e p s .
a)
For any Let
u
Y
and a neighbourhood W o f z e r o i n
S E y
Then
1 m(Sn)
U g l w := {m E Xy(En,F)
of z e r o i n
t S ( E n , F ) . By i n d u c t i o n h y p o t h e s i s -P
t h e r e i s a neighbourhood
of
0
UgrW
C
\p"+'(u)[V
I
i.e.
Sn] c W.
x
p n + l ( u ) E gs ( E ~ + ' , F )
Y
qn+'
t i v i t y of
Then
LnS(E,F) i s mntinwus,
Y
in
u ( v ) [ S n ] C W. Since
u
.
is
such
El
But
this
symmetric,
that implies
this
shows
is b i j e c t i v e :
The i n j e c t i v i t y of
any x1
V
i s a to-
pn
Y
u :E
~p ,,u(V)
F be given.
W} i s a neighbourhood
C
p o l o g i c a l isomorphism. S i n c e n
b)
is y -hypocontinuous:
L n + l o S( E , F ) \ p n + l ( u )
E
E
is clear. L e t u s show t h e surjec-
,pn+'
p n + l . Take any
E,
U(X,)
;(X,):$+F
is i n
by
m E fS(En+l,F) and d e f i n e ,
Y
; ( X , ) [ ~ ~ . . . , X ~ + ~ I: = m ( x l f . . . r ~ ~ + ~ ) .
, since
J?(E",F)
o f z e r o in E w e have
for
c(xl)[V
x
f o r a n y neighbourhood V
Sn'l]
Thus w e h a v e d e f i n e d a mapping
=
:E
m({x,} x V
x
Sn-l).
* J S ( E n , F ) which I
Y
is
l i n e a r a n d c o n t i n u o u s , b e c a u s e f o r any neighbourhood W o f 0 i n F t h e r e i s a neighbourhood
;(V)[Sn] = m ( V
x S")
C W,
i.e.
in
V
;(V)
C
E
such
that
U i r W . By i n d u c t i o n
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
hypothesis
c)
vn+l is
u := (@')-loi s i n
26g
v"+~(u)=m.
LYn + l,s ( E , F ) and
a t o p o l o g i c a l isomorphism:
T h i s i s e a s y t o see, s i n c e
SIW
IS Ey,W neighbowhccdof 0 inF1
ES ( E n + l I F ), Y
i s a fundamental system o f neighbourhoods i n
in+'=
{u L n + l r S ( E , F ) 1 u ( S ) C (Vn)-'($J} Y S,W d e s c r i b e a fundamental system of neighbourlloods inLn+lrs(E,F)
while t h e sets
Y
if
S r u n s t h r o u g h y and
a)
the
neighbour-
0 in F.
hoods o f
6 . DEFINITION:
W runs through
L e t E and F b e 1.c. s p a c e s .
The e - p a o d u c t o b
E and F i s d e f i n e d a s
E E F :=L
e ( Fc' , E )
,
where e d e n o t e s t h e t o p o l o g y of uniform convergence on t h e F'
equicontinuous s u b s e t s of
( c f . Schwartz [ 22 I o r Bierstedt
and Meise [ 5 ] 1. b)
The mapping j : E B F + E E F , j( i s i n j e c t i v e , hence
on
E Q F, c a l l e d t h e
tion
E
v
eE
phoduct 04
c)
E
F E
of and
E
F
n
i=l
induces v i a
[211
,
j
(fi,y')*ei
i=l
a 1.c. topology
i n j e c t i v e or e - t o p o Q a g y . The comple-
E 8, F i s c a l l e d t h e i n j e c t i v e o r E-ten604
F
.
E has t h e approximation property
Grothendieck, i f
n
ei8fi)[y']:=
E' Q E
(a.p.1
i s dense i n
i n the Lc(E)
sense
of
( c f . Schaefer
111, 9 . 1 ) .
We s h a l l use t h e f o l l o w i n g r e s u l t o f Schwartz [ 2 2 1 , Ch. 1, 51, Prop. 11, i n t h e form s t a t e d i n B i e r s t e d t and Meise [ 6
7. THEOREM: A quaAi-compf?ete L . c . bpace
Banach space F t h e a t g e b a a i c 8.
teMbOa
E
hub
pkoduct
the a.p.
1. doh e v a y
E 8 F i b devise &
k and km-ApaCeA: A c o m p l e t e l y r e g u l a r t o p o l o g i c a l s p a c e
E EF.
X
is
MEISE
270
a k-Apace
(km-dpace) if f o r any t o p o l o g i c a l
space
Y
(Y
e q u i v a l e n t l y Y a completely r e g u l a r t o p o l o g i c a l s p a c e ) f :X + Y
i s continuous i f f
. By' A r h a n g e l ' s k i i
set K of
X
k
(km-spaces) a r e k
- spaces
f
1
=
a
or
IR
function
i s continuous f o r any compacts+
K
1
[ 1 ] (Blasco [ 8
- spaces
open
)
(kn-spaces)
subsets
of
again.
2. SPACES OF DIFFERENTIABLE FUNCTIONS
n times c o n t i n u -
I n t h i s s e c t i o n w e i n t r o d u c e 1 . c . s p a c e s of
o u s l y y - d i f f e r e n t i a b l e f u n c t i o n s and i n v e s t i g a t e t h e i r
topological
p r o p e r t i e s . Because o f t h e a p p l i c a t i o n s i n s e c t i o n 3 , w e are i n t e r e s t e d i n t h e completeness and t h e Schwartz
property
mainly of
such
spaces. W e begin by r e c a l l i n g some d e f i n i t i o n s .
1. DEFINITION:
L e t E and F be 1.c. s p a c e s , 51 an open subset o f E l
f a f u n c t i o n on 52 w i t h v a l u e s i n F and y a system of bounded s e t s i n E which covers E . f i s c a l l e d y-didtjehentiable i f t h e r e exists uEL(E,F) uniformly i n
t h a t for every S E y l i m E(f(a+th)- f ( a ) -u(th))=O t+o h E S ( i . e . f o r any S E y and any continuous semi-norm
q on F , t h e r e i s 0 <
I
at a p o i n t a € 0
1
SIX&
6 > 0
such t h a t f o r any
t
E
IR, with
ti 5 6 1 1
s u p q ( t ( f ( a + t h )- f ( a ) - u ( t h ) ) 2 1). he S Obviously y-
u i s uniquely determined by
d e h i v a t i v e ob
f i n a. W e w r i t e
f and a ; u i s c a l l e d
f ' ( a ) i n s t e a d of u .
t h e system of a l l bounded ( f i n i t e ) s u b s e t s of ( G i i t e a u x - ) di66ehentiable at
if
a. f
f i s y - d i f f e r e n t i a b l e a t any
E
,f
If
the y
is
is c a l l e d Fhzchet-
i s c a l l e d y-diddehentiable o n
51,
a E 52.
For G l t e a u x - d i f f e r e n t i a b l e f u n c t i o n s t h e r e e x i s t s e v e r a l g e n e r a l i z a t i o n s of t h e c l a s s i c a l mean v a l u e theorem (see e . g .
Yamamuro
SPACES OF OlFFEAENTlABLE FUNCTIONS AND THE APPROXlMATlON PROPERTY
[24I
,
27 1
1 . 3 ) . We s h a l l u s e t h e f o l l o w i n g o n e , which i s a consequence o f
t h e Hahn-Banach theorem and a r e s u l t o f c l a s s i c a l c a l c u l u s .
2. LEMMA:
l e t E and F be l . c . b p a c e b , 51 an open bub6e.t i n
a,b E R
let
.
Abbume
fitiabte at any
x E s
tained i n R g ( t ) := f ' (a
S [ a , b l : = {a t t ( b
b e buch t h a t
+
t(b
f(b)
-
-
f : S2
duhthehmohe t h a t
+
-
a ) I t E [0,11 1
F
and
E
con-
i h
Gzteaux- d;ddmen-
i h
and t h a t t h e mapping g : [ 0 , 1 ] * L a ( E , F ) , [a,bl a ) ) , i d c o n t i n u o u s . T h e n t h e doU0wing hoLh .thue: 1 f'(a
f(a) =
+
t(b
-
a))[b
-
aldt.
The f o l l o w i n g lemma i n d i c a t e s t h a t y - d i f f e r e n t i a b i l i t y
of a function
f i s a l r e a d y i m p l i e d by Gzteaux d i f f e r e n t i a b i l i t y and
a
continuity
p r o p e r t y of t h e derivative (see also K e l l e r [ 1 8 1 , 1 . 2 . 1 a n d Y a m a m u r o [24
1 , 1.4.4).
3. LEMMA:
L e t E and
F be L.c.
dpacen, 51 an open b u b b e t
f : S2 + F Gzteaux didbetentiable on
t i n u o u b , &en f PROOF:
i b
f'
16
: 52 +
Ly(E,F)
i b
con-
y-diddehentiable o n R .
L e t a be any p o i n t i n
bounded subsets of
$2.
and
E
06
S any e l e m e n t of t h e s y s t e m
S2,
y
of
E and l e t q be any c o n t i n u o u s semi-norm o n F . Py
Uleoontinuity of f ' in a, f o r
E
> 0 t h e r e e x i s t s a convex b a l a n c e d n e i g h -
bourhood U o f zero i n E s u c h t h a t
a
+
U
C 51
and s u c h t h a t f o r any
x € a + U
S i n c e S i s bounded i n E 2 we
have f o r any
, we
t with
can f i n d
0 <
I
t
I 5
6 > 0 6
with
and any
6s
C
h E S:
U.
By lemna
272
MElSE
This implies
Hence
f is y-differentiable a t a .
Let
4 . DEFINITION: E and
-
a system of bounded subsets of
y
n E mm(:=
U (
1
#
E and F be 1 . c . s p a c e s , s2 E
which
we d e f i n e t h e s p a c e o d
)
n
if
:R
-+
j E
F I f o r any
covers
t.imea
y - d i d 6 u e n t i a b L e dunctions o n R w i t h vaLuea i n
c ~ ( P , F ) :=
$ an o p e n s u b s e t o f
F
m0
cantinuouaLy
a6
with
0
~ < jn + l
) : = f ) and f o r any f . E C ( C ~ , L ~ ( E , F )(fo 1 Y
with
0
5j
on R and
A e t d 06
R
i s f . Gsteaux
3
f; = f j + l
j
E
06
. This
- differentiable
I .
Cn(B,F)
topology i s given by t h e system {pLrKrSrq 1
5
e
+
of semi-
norms, where
L
s u b s e t of
S is any e l e m e n t of y and q is any c o n t i n u o u s
norm on F ,
ill
lNo
i s endowed w i t h t h e t o p o l o g y od unidohm Y t h e dehiuatiweb up t o t h e ohdeh n a n t h e compactaub-
The v e c t o r s p a c e convehgence
For
E .
i s any i n t e g e r w i t h
and where
pLIK,s,q
0
< n
is d e f i n e d a s
1, K i s any compact
semi-
SPACES OF DIFFERENTIABLE FUNCTIONS AN0 THE APPROXIMATION PROPERTY
I n t h e sequel we s h a l l w r i t e REMARKS: a ) By lemma 3 ,
f
O ( j < n . b)
A function
f
f ( j ) instead of
is really
j
is i n
f
213
j'
y-differentiable
on
Cn(fl,F) i f f it i s of c l a s s
Y
i n t h e s e n s e of Keller 1181, 2.5.0.
for
fl
C"
Y
on
Q
The advantage o f d e f i -
n i t i o n 4 , however, w i l l become clear p r e t t y soon (see e . g . proposition 5).
c)
Obviously w e have f o r any semi-Monte1 space
L e t t h e open s u b s e t i2 a6 ,the L . c . space E be a $-space
5. PROPOSITION: and L e t
n E
mo
with
mINDD be g i v e n . Assume 0 5 j < n + 1. T h e n
Let
(fa)a
j E
PROOF:
of t h e topology o f C2,and any
E:
j with
L J ( E , F ) in c o m p l e t e doh
that
Y
any
Cn(S1,F) k d complete.
Y
be any Cauchy n e t i n
CY(fl,F), The d e f i n i t i o n
Cn(R,F) i m p l i e s t h a t f o r any compact s u b s e t K of Y 0 5 j < n + 1, ( f i j ) ( K ) i s a Carny n e t i n C(K,L;(E,F)).
L 7 ( E , F ) i s complete by h y p o t h e s i s and s i n c e Q i s a kR-space, Y f o r any j w i t h 0 5 j < n + 1 t h e r e i s g j E c ( s 2 , ~( jE , F ) 1 such that
Since
(fy)aEA
converges t o
g
j
uniformly on e v e r y compact s u b s e t of fl.
Now w e s h a l l show t h a t f o r any
d e r i v a t i v e of
g
equals
gj+l : L e t
j Then t h e r e e x i s t s an open i n t e r v a l
v
a
:t + f i j )
v
a
(a
+
j
a E Q I in
t h ) i s d e f i n e d f o r any
E C 1( I , L ; ( E , F ) )
and
with
0
and
j < n
the &teaux-
h E E
be g i v e n .
IR on which t h e
function
a E A.
v;(t)
5
Obviously
= f(J+')(a+th)[hI. a
MEISE
274
For any two 1.c. spaces X and Y the evaluation map E: L (X,Y) x X + Y ,
Y
E(U,X) = u(x) is separately continuous. Hence ( ~ 4E )A ~ towards the function
(a + th)[ h
* compact subset of I. Thus, v
v' = w. Because of
:
lim va
:=
]
I
converges
uniformly
on
every
is differentiable on I and
a+
v(t) = g (a + th), this implies j
=w(o) = g. (a + th) r 3
i.e.
lim t+o
4
(gj( a + th)
-
g.(a) - qjtl(a) th]) = 0. 3
This shows go = lim fa E Cy(Q,F). a+
6. REMARKS: a) Let us recall that any open subset Cl of a metrizable
1.c. space or a (DFM)-space is a km-space. b)
Concerning the completeness of : L (E,F) the following should be remarked: If F is complete and E
bornological, then
IN^.
If F and EL are COPTT
L~(E,F) is complete for any
plete, and
j
E
Elc equals E topologically,then
complete for any
j E INo.
PROOF:
is
Especially for any (F)-or(DFM)-
space E and any complete 1.c. space F is complete for every
L:(E,F)
, the
spa=
Li(E,F)
j E IN,.
As it was shown by Dineen [13] I Prop. 1 and Prop. 5, Cl is a
hemicompact k-space. Hence o:C proposition 5,
C:~(S~,F)
(R,F) is metrizable. By remark 6.h) and
is complete.
The following lemma will be useful in the sequel.
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
Let E be a 1 . c . opace, y a n y o t e m
8. LEMMA:
El
Y
i b
06
C"(51)
Y
Then w e d e f i n e
51 #
i s defined only f o r
* Cy(R)
A : C:(51)
by
A ( f ) :x
ob Cy(Cl).
0, w e c a n c h o o s e a +
06
E and nEINEa.
a COntiflUOUbey pao j e c t e d Zapotogicae bubopace
Since
PROOF:
bounded n u b n e t n
o p e n bubnet
E which c o n t a i n n t h e compact b e t s , R an
Then
06
276
En.
( f ' ( a ) , x ) . S i n c e any
c o n t i n u o u s l i n e a r map y o n E c o i n c i d e s w i t h i t s own derivative a n d since
o
for
for
j
y(j) =
A(f) ( j ) = 0
j 2 2, A p r o j e c t s
c"(R) o n t o Y
2, t h e c o n t i n u i t y o f
c cn(n). Y
E'
AS
A f o l l o w s from t h e e s t i -
mates :
f o r a n y compact s u b s e t K of
R.
for a n y compact s u b s e t K of
51 a n d a n y
S i m i l a r a r g u m e n t s show t h a t
n
CY ( 5 1 )
.
9. DEFINITION:
Let
E'
Y
S E y
is a t o p o l o g i c a l subspace
E be a 1.c. s p a c e . A subset K of
E is
uehy compact, i f t h e r e i s a Banach d i s c B ( i . e . a convex bounded s u b s e t B o f that
E f o r which
K is contained i n
1 0 . REMARKS:
EB
EB
called
balanced
a n d compact t h e r e .
a ) The n o t i o n o f v e r y compact sets w a s i n t r o d u c e d ( w i t h 111, 4. Def.
By a c o n s e q u e n c e of t h e Banach-Dieudonng t h e o r e m a
K of
of
i s a Banach s p a c e ) i n E s u c h
a d i f f e r e n t d e f i n i t i o n ) by d e Wilde [23],-. b)
.
.
subset
E i s v e r y compact i f € t h e r e e x i s t s a convex b a l a n c e d
278
MElSE
compact s u b s e t Q of
E such t h a t
in
K is contained
EQ
and compact t h e r e .
11. PROPOSITION: det
L e t E be a L . c . Apace i n w h i c h euehy compact
cEo(a) i d
LA v e t y compact. Then
bub-
any
a Schwaatz Apace doh
open
6ubAe.t
Cl ad
PROOF:
By a well-known c h a r a c t e r i z a t i o n o f Schwartz s p a c e s , it s u f -
E.
f i c e s t o show t h a t f o r any compact s u b s e t K o f $2
set
Qo of
E l and any
t h a t any sequence
, any compact
sub-
n E IN t h e r e i s a compact subset Q of E such m
IN i n
(fe)e
Cco (Q) w i t h sup
IK,Q
t E IN
'fe
1 5
1
c o n t a i n s a subsequence which i s Cauchy w i t h r e s p e c t t o t h e semi-norm
S i n c e t h e c l o s e d convex h u l l of a compact s e t i n E i s compact again,
K can be covered by a f i n i t e number o f compact convex
Hence, w . 0 . 1 . g . ~ we may assume t h a t
K i s convex. By h y p o t h e s i s
by remark 1 0 . b ) t h e r e i s a balanced convex compact s u b s e t Q such t h a t
K U Qo C Q
and K as w e l l as Qo a r e compact i n
Now t a k e any sequence ( f R )
m
IN i n Cco(n) w i t h s u p pml
.Ern
w i t h r e s p e c t t o t h e semi-norm
p
,
nrKIQO
f i x j w i t h 0 5 j 5 n and d e f i n e f o r any i s an
., y j )
:= f i j ) ( x ) [ y l , . .
equicontinuous s u b s e t of
t h e topology induced by
EQj + l
L e t ( a , y ) , (b, z ) E K f k j ) (x) implies
and
of
EQ ,(f,)
El
*
(1.
c o n t a i n s a subsequence which is Cauchy
I n o r d e r t o show t h a t ( f e l l
gjlt(x,yl,..
sets.
x
.
Qj
we proceed a s follows:
1
E
W e
IN, g j I e : K x QJ * lRby
. ,yj]. Then w e C ( K x Q j )I where
show t h a t fgjlel&gJ} K x Qj
be given. Then m u l t i
i s given
- l i n e a r i t y of
2??
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
The g e n e r a l mean v a l u e theorem (lemma 2 ) g i v e s
By
Sup sUPj+ll
xEK wEQ
fj1")
serve t h a t only f o r
< llb
(X)"iJ
a # b
1
I 5
P ~ + ~ , K(f , 1Q) 5 1,
t h e r e i s something to p r o v e ) :
1
b - allEQ[ l o f t( j + l ) ( a + t ( b - a ) ) [ z , -nb-all
a EQ
ldtl < llb- allE
Concerning t h e o t h e r t e r m s i n (l), the f o l l o w i n g 1
5
k
5
j (observe t h a t o n l y f o r
Thus we have shown
t h i s i m p l i e s (ob-
zk # yk
Q
holds
true
. for
t h e r e is something to prove):
218
MElSE
Hence
I .t E
{gj
JN I
i s e q u i c o n t i n u o u s on
K x QA i s a compact subset o f
i s u n i f o r m l y bounded on
p a c t s u b s e t of
C(K
x
K x QJ i n Ej'l
Q
,{gjrL]LE
K x :Q
Qi) by t h e theorem of
i t i s p o s s i b l e t o choose ( i n d u c t i v e l y )
c
K x ~j
EF' .
, andsince JN 1
Ig
j
Since
,l
le E IN 1
i s a r e l a t i v e l y com-
Arzela -Ascoli.
a subsequence o f
But t h e n (fg)e
IN
which i s a C a u c h y - s e q u e n c e w i t h r e s p e c t t o t h e semi-norm
'n ,K ,Qo' By t h e c o n s i d e r a t i o n s a t t h e b e g i n n i n g , t h e p r o o f i s now complete.
REMARK:
A s i m i l a r argument as i n t h e p r o o f
of P r o p o s i t i o n 11
used i n t h e a r t i c l e of B i e r s t e d t and Meise [ 7 ] i t w a s shown t h a t t h e s p a c e
subset
,
was
theorem 7. ( a ) ,where
H ( K ) of holomorphic germs o n a
compact
K of a m e t r i z a b l e Schwartz s p a c e i s a Schwartz s p a c e a g a i n .
1 2 . THEOREM:
Let E be
u
quabi-comp4?ete L.c. npace. Then t h e ~o&bw-
i n g ahe e q u i v a l e n t :
i d a Schwuhtz bpace.
(1) EA
compact n u b n e t
( 2 ) Euehy
( 3 ) Foh any open
bUbb&
06 E
R
06
i n v e h y compact. E,
C:,(R)
i n a Schwahtz
= C:(!J)
Apace. ( 4 ) Thehe e x i b t b i b
PROOF:
(1)
UM
open bubnet
n ( # 0) o d
E
doh
w h i c h CEo(R)
a Schwahtz b p a c e .
=)
(2). Let
K be any compact s u b s e t of
neighbourhood of z e r o i n EA a n d hence S i n c e E i s quasi-complete,
KOo
t h e topology
t h e d u a l i t y between E and E'
,
E.
Then
KO
isa
is e q u i c o n t i n u o u s i n (EA)I .
X(E',E) is c o m p a t i b l e w i t h
t h u s , KOo is equicontinuous in (EA)'=E.
BY t h e d u a l c h a r a c t e r i z a t i o n of Schwartz s p a c e s (see e.g.Horvgth [171
3 , 915, Prop. 5) , t h e r e is a compact s u b s e t Q of
is r e l a t i v e l y
compact i n (EA)1
Qoo
.
E such t h a t
KOo
S i n c e E i s quasi-complete,
QOO
i s compact by t h e theorem o f b i p o l a r s . But t h e n K , b e i n g compact
in
SPACES OF DIFFERENTIABLE FUNCTIONSAND THE APPROXIMATION PROPERTY
E , is compact i n t h e Banach s p a c e
(2) (3) (4)
-
E
Qoo (3)
by p r o p o s i t i o n 11.
(4)
trivial.
278
.
(1) by lemma 8.
Using t h e c o n c e p t of bornology and t h e n o t i o n o f S i l v a d i f -
REMARK:
f e r e n t i a b i l i t y , Colombeau 1121 g i v e s i n d e p e n d e n t p r o o f s o f p r o p o s i t i o n 5 and theorem 1 2 i n a more g e n e r a l s e t t i n g .
L e t E be any (F)-Apace 0% a n y (DFS)-bpaCe and
13. COROLLARY:
be an a k b i t k a k y open nubbet
06
E
. Then
C:o(Q)
Let Sa
= CE(Sa) i6 a S c h W z
Apace. I t i s a consequence of t h e Banach-Dieudonns theorem (see e.g.
PROOF:
Kothe [ H I , 521, 1 0 . ( 3 ) ) t h a t any compact s e t i n a F r g c h e t s p a c e
is
v e r y compact. S i n c e (DFSI-spaces c a n be r e p r e s e n t e d as compact i n j e r t i v e c o u n t a b l e i n d u c t i v e l i m i t s of Banach s p a c e s , by
Floret-Wloka
e v e r y compact subset o f a (DFS)-space is veryccmpact.
[ 1 4 1 , 525, 2 . 2 ,
I t would be i n t e r e s t i n g t o know w h e t h e r , c o n c e r n i n g nu-
14. REMARK: m
c l e a r i t y , Cco-functions behave s i m i l a r as holomorphic f u n c t i o n s complex 1.c. s p a c e s ( c f . Boland [ 9 1 ) .
on
Since nothing i n t h i s d i r e c
t i o n seems t o be known, l e t us remark t h a t f o r any open s u b s e t E =
@
n€lN
IR
the space
m
Cco(Sa)
jn : IRn
+
E
d e n o t e t h e c a n o n i c a l embedding. Then
i s an open s u b s e t o f
IRn
Sa of
is nuclear.
But t h i s r e s u l t is e s s e n t i a l l y f i n i t e d i m e n s i o n a l : F o r let
-
, hence
m
Cco(iln)
an
n
E
N
: = I -l n (Q)
is n u c l e a r f o r any n
E
IN.
Now i t i s a consequence of Yamamuro [ 2 4 L(1.6.1) ,that Czo(Q) =pmj C" (Q 1. c n co n S i n c e t h e projective l i m i t of n u c l e a r s p a c e s i s n u c l e a r , t h i s p r o v e s
the nuclearity of
m
.
Cco (Q)
280
MElSE
3 . THE ROLE OF THE APPROXIMATION PROPERTY
The aim of t h i s s e c t i o n i s t o d e r i v e a s u f f i c i e n t c o n d i t i o n f o r t h e a.p.
of
Czo
( a ) . T h i s w i l l b e done by an a p p l i c a t i o n o f theorem
1.1. T h e r e f o r e , w e f i r s t g i v e (under a p p r o p r i a t e h y p o t h e s e s ) a c h a r a c t e r i z a t i o n of t h e €-product o f
t h a t EJ
Y
Let E and F b e L . c . ApaCe6, L e t
1. THEOREM: d u b d e t d 06
Cn(il) and a quasi-canplete 1.c.space.
y
beabybtem
E which containd .the compact 6 e t d and
Let l ' n ( m .
a
i d
1 5 j < n + 2 and t h a t
km-Apace d o t
quadi-complete. Then
Cn(Q)
Y poLogicaL L i n e a t dubdpace
E
F
i b
06
bounded hdume
F ate Y t o p o L o g i c a ~ g i b o m o t p h i ct o t h e to-
Cnp(R,F)
06
Y
C n ( Q ) and
Cn(O,F) w h e t e
Y
ptecompact i n F 1 . The proof i s s i m i l a r as i n t h e f i n i t e dimensional case,
PROOF:
but
becomes more i n v o l v e d , s i n c e w e have t o d e a l w i t h t o t a l d e r i v a t i v e s . The g e n e r a l i d e a i s t h e f o l l o w i n g : Define A(x) := 6x and show t h a t t h e mapping morphism between
Cn(Q) E F
Y
%
f
+
Le(Cy(S2)A,F)
A : R
f
0
A
and
+
CY(S2);
by
i s a t o p o l o g i c a l isoCnp(il,F). T h i s
Y
will
be done i n s e v e r a l s t e p s .
a)
For
0
5
j < n
+
1 d e f i n e t h e mapping A j : Q X E J + C f s ( Q ) r by
( A J ( x , y ) , f ) := f ( 1 ) ( x ) [ y 1
By h y p o t h e s i s Blasco [ 8 A
IK
x QJ
subset
Q
1, Q
x Ej
Ej''
.
Then
A1
i s a k m - s p a c e , hence
i s a km-space.
is c o n t i n u o u s .
by
Thus, i t s u f f i c e s t o show
i s c o n t i n u o u s f o r any compact s e t K i n
of
obvious t h a t
E.
the result
of that
0 and any ampact
From t h e d e f i n i t i o n o f t h e topology o f
A J ( K x Q J ) i s an e q u i c o n t i n u o u s s u b s e t of
Cn(0) it i s
Y C Y ( Q )I .
On
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
e q u i c o n t i n u o u s subsets t h e t o p o l o g y o f
coincides with
C y (52)
weak t o p o l o g y , hence w e o n l y have t o show t h a t A J j K x $
i s continuous. L e t
f
Y
the
& q(Q);
x
- th
deriva-
f i s symmetric, by p r o p o s i t i o n 1 . 5 w e have t h e c o n t i n u i t y o f
t i v e of
= d : s ( E J ) . For S Y Y denote t h e r e s t r i c t i o n . Then p i f(1) :K
p:
:K
C n ( Q ) be a r b i t r a r y . S i n c e t h e j
E
28 1
+
LnS(E,E3)
: z:(Ej)
-+
is
ps1 : J Y ( E j )
continuous,
CB(Sj)
-+
especially
C ( Q j ) i s c o n t i n u o u s . T h i s shows t h a t
and hence t h e mapping c o n t i n u o u s on b)
y , let
E
K
X
Q’.
15 j
For ping
(x,y)
+
f‘j) (x)[y 1 = p i
Thus t h e c o n t i n u i t y o f c
i j : 52
+
n +
I K)
o
A’
is
(x,y)
i s proved.
1 one can d e f i n e a c o n t i n u o u s l i n e a r map-
LS(EJ,CY(Q))A) by Y
i J ( x ) :=
AJ(x,
- 1.
By p a r t a ) and by t h e symmetry of t h e d e r i v a t i v e it i s obvious that
AJ(x,
Cy(Q)A
.
i s a symmetric j - t i m e s
* )
L e t us prove t h a t
w e have shown, t h a t bourhood
AJ(x,
9
l i n e a r mapping from i s y-hypocontinuous:
)
into
EJ
In p a r t a)
Q j i s c o n t i n u o u s i n ( x , o ) , hence f o r any neigh-
W of zero i n
C ny ( Q ) b t h e r e i s a neighbourhood
of
zero
X > 0
such
IT
i n E such t h a t
f o r any that
u E U’.
S C XU.
If
S E y
Hence f o r
t h e symmetry o f
A 1 (x,
i s a r b i t r a r y , t h e n t h e r e is one h a s AJ(x,V
V := X-J’lU )
A 1 (x,
t h i s proves
Now l e t us show t h e c o n t i n u i t y of f o r any compact K i n pology of Cy“ ( Q )
I.
Q and any
Cy(Q), t h e set A j ( K
By t h e c o i n c i d e n c e of
S E y,
x Sj)
-
a’.
x SJ-’)
C W.
By
) E Ls(Ej,Cn(Q)i).
Y
Y
F i r s t w e observe
that
by t h e d e f i n i t i o n o f Dheto-
A j ( K x SJ) i s equicontinuous i n
X ( C y (L?)I , C y ( 5 2 )
)
and
u (Cn ( a ) I ,Cy”(f2)) Y
282
MElSE
on e q u i c o n t i n u o u s subsets of bourhood W of zero i n
Cy(R)
Cy(n)A
I
f o r any convex b a l a n c e d neigh-
t h e r e are
flr
. . . ,fm
in
C n ( Q ) such
Y
that
(1) : s2 + Ls(EJ) is c o n t i n u o u s for 1 5 k 5 m, f o r any x E K fk Y t h e r e e x i s t s a neighbourhood U of x s u c h t h a t f o r any x ' E U and
Since
any k w i t h
15k 5 m
Hence w e have for any
x' E U
and any
y
Sj
E
By o u r f i r s t o b s e r v a t i o n t h i s shows f o r any
where
q ,
d e n o t e s t h e gauge o f
W
. Since
x'
ij
U
n K
,,I
S
i s c o n t i n u o u s for any compact s u b s e t K of hence
E
E
y
was a r b i t r a r y , A'
1K
0 . But R is a k m - s p a c e ,
i s continuous.
I t is e a s y t o see t h a t f o r
0
i s c o n t i n u o u s . Hence w e have proved
5
j < n
+
1
t h e mapping
f U E CY(R,F), i f w e
can
show
283
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
A- j + l
f(j+l) = U
h E E, any
any
K := { x
+
thl
i
t
x
I 5
+
y
j < n. T o do t h i s , t a k e j w i t h 0
and an a r b i t r a r y for
t h E S2
6 1.
i s equicontinuous i n
and any
5
0
x E R
such t h a t
6 > 0
for
with
t
with
S E y
I
t
5
h
5 j
Choose
E S.
6
< n,
and
put
Then remark t h a t t h e s e t
Cy(Q)
',
s i n c e by 2 . 2 w e have f o r any f
E Cn(R)
Y
E SJ
S i n c e t h e t o p o l o g y of
Cy(il)A
c o i n c i d e s w i t h t h e weak t o p o l o g y
on
e q u i c o n t i n u o u s s u b s e t s , w e have
l i m +(Aj(x t th,y) t+o uniformly i n
- Aj(x,y) y E SJ
-
Aj+'(x, ( t h , y ) 1 ) = 0
i f t h i s holds i n
Cy(S2); u n i f o r m l y
in
Cy(S2);
in
y E S J . But t h e l a t t e r i s a consequence of the d e f i n i t i o n o f c"(R), Y
s i n c e f o r any
f E cC(il)
tends t o zero uniformly i n w e g e t by i n d u c t i o n
y E Sj
it
t t e n d s t o z e r o . From
this
284
in
MEISE
e S ( E j , F ) . Hence
( j + l )( x ) = u
fU
shown
fu
0
is Gzteaux-differentiable i n
f'j)
Y
U
,jcl ( x )
. Since
E
and
R
i s continuous, by 2 . 3 we have
u 0 Lj"
E CY(Q,F).
I n o r d e r t o show t h a t w e even have compact K i n R and any Alaoglu-Bourbaki t h a t f o r n t i v e l y compact i n Cy (a); pact i n F .
d)
x
S
E
0
,
fUE Cy(R,F),
take
any
y. Then i t f o l l o w s from t h e theorem of
5
j < n
hence
u
+ 0
1 the set
A 1 ( K x ,911 is rela-
A 1 (K x S J ) i s r e l a t i v e l y com
-
But w e have shown above t h a t
The mapping
k :F
CnP(Q,F) d e f i n e d by Y i s an i n j e c t i v e t o p o l o g i c a l homomorphism.
Y
L e t any compact K i n 51
-+
, any
S E y
For
we have
--
'l! , K
I
,
S 1
By t h e theorem of b i p o l a r s t h i s i m p l i e s
and
k(u)
:=
fU
L < n t 1 be given.
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
S i n c e any
286
u E F eC"(S2) i s weakly c o n t i n u o u s on e q u i c o n t i n u o u s sub
Y
sets, w e have f o r any c o n t i n u o u s semi-norm
q on F and
-
any
Hence t h e r e s u l t claimed under d ) i s proved.
e)
The mapping k d e f i n e d i n d ) i s s u r j e c t i v e , i . e .
a
topo-
l o g i c a l isomorphism. The s u r j e c t i v i t y of inverse
j
of
k w i l l be proved by c o n s t r u c t i n g a r i g h t -
k . Take any
f
an e a s y e x e r c i s e t o show t h a t uf : F'
u
E
+
C"(n)
f Y j ( f ) = tuf
Cfj(S2)
by
€
and any
CnP(Q,F)
Y
y'
Hence
€
we
F ' , then it is
y'
0
f
uf ( y ' ) := y '
0
f . L e t us assume f o r a moment that
E C;(Q).
can
define
E F h o l d s . Then w e can d e f i n e
,
since
isomorphism between
j : C Y ( Q , F ) + F E C " ( R ) by Y by o u r h y p o t h e s e s t r a n s p o s i t i o n i s a t o p o l o g i c a l Cn(n)
Y
i s now proved, i f we show
E
F
and
ka j = id
F
E
Cn(n).
Y
Cnp (O,F)
Y
The s u r j e c t i v i t y of
. But
k
t h i s is a consequence
286
MElSE
of the following identity which holds for any any
y'
F, any x
E
R, and
E
f E c"P(~,F): Y
Hence the proof of the theorem is complete, if we show uf ECn(R) EF. Y Let any compact subset K of 0 , any S E y , and l? < n + 1 be given. By hypothesis, the set Ll :=
is a neighbourhood of zero in FA
hence :L
vl,K, S, 1 *
i.e. uf(~y) c
2. COROLLARY: nyntem
yco
Aubbet
R
Thib
the
i d
06
.
Thus we have shown
16 t h e h y p o t h e b e b
06
0 4 aLL compact b u b b e t n
Rheohem 1 06
precompact
For any
y' EL:
in
F
,
we ham
L(F;,c~(o)).
uf
U h t
batib6ied doh t h e
E , t h e n #e have d o t a n y o p e n
E:
C u b e 6 0 ) ~a n y
(FI-npace o h a n y (DFM)-bpace E
c o m p lete 1 . c . space F , and a n y
PROOF:
6 f (1) (K)[SJ] is
j=o
n
E INm
.
,
any yuabi-
The first part of the statement follows from theorem 1, and
CEE(f2,F) = Czo(R,F). To prove this identity, let
f
E
Cn co (R,F)
be
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
g i v e n . Then f o r any
15 j
j E lN w i t h
and any compact Q i n
E
,
n
287
+ 1, any compact K i n
SZ,
we
by p a r t a ) o f t h e proof o f theorem 1,
have
Now o b s e r v e t h a t t h e e v a l u a t i o n map A : C Q J I F ) X Q j -+F, A ( g , x ) : = g ( x ) ,
i s c o n t i n u o u s . Hence
B :K
X
Qj
F , B(x Y ) = A ( f ( j )
-+
(XI ,y) ,
i s con-
t i n u o u s and t h u s
i s compact i n F . The second p a r t o f t h e s t a t e m e n t f o l l o w s from remark 2 . 6 .
REMARK:
C o r o l l a r y 2 g e n e r a l i z e s a r e s u l t o f Bombal Gord6nand Conzaez
Llavona [lo 1 , who c h a r a c t e r i z e d f o r Banach s p a c e s E of
, Aron
[2
C:o(E)
E
f o r Banach s p a c e s E . Also
F
1 gave a somewhat d i f f e r e n t description
Cb(E) E F .
Now w e come t o t h e main lemma for many of t h e r e s u l t s p r e s e n t e d i n t h e s e q u e l . For Banach s p a c e s E i t goes back t o Bombal Gord6n and Gonzslez Llavona 1101 a s w e l l a s t o P r o l l a and G u e r r e i r o [ 2 0 1 .
3 . LEMMA:
Let
be a quasi-compeete bahheLLed L . c . n p a c e w i t h
E
the
~ o L L o w i n g phopehty: (CFA) : Foh a n y compact n u b n e t EK
j,
K i n
w i t h a . p . and a c o n t i n u o u n L i n e a h : EK
-+
E duch t h a t K C jK(EK)
Fuhthehmote l e t SZ be an open n u b b e t and L e t
n
E
E t h e h e e x i n t b an
INm
and
f E CEo(n,F)
06
injective
and j , l ( K )
E , Let
be g i v e n .
F
(P)-Apace
mapping
.& compact i n
8.
b e a nohmed bpace,
Then d o h any
compact
288
MElSE
06
R,
> 0 thehe i n
u
KO
hubnet E
,that
f0 u
PROOF:
E
Put
a n y compact E
E' 8 E
Qo
i n
E
,
any
L
n
and a n o p e n neighbouhhood
+ w
1, and
any
K,
nuch
06
n Cco(wlF) and h u c h t h a t
K := KO
U
and choose -according to (CFA)-an(F)-space
Qo
EK with a.p. for which there is a continuous embedding jK Assume that the topology of EK is given
by
can find w m
E
(EK)' 8 EK
EK
the increasing
of semi-norms. Since EK has the a.p. I for any
(qs)s
:
+
E.
system
m E lN
we
such that
By the quasi-completeness of EK and E it follows from the injectiK K K vity of 1, that tjK(E') is A ( (E ) ',E )-dense in (E ) I . Hence by um E E ' 8 E K C E '
approximating wm appropriately we can find
8
E
such that
Consequently there is a sequence for any
E' 8 EK
such
that
s E
lim m+m
sup qs(um(x) xEK
luation map
A
K
: C(K,E )
x
K
-+
- x)
=
0,
in C(K,EK )
.
Since the eva K E , A(f ,x) := f (x), is continuous, the
m restricted to K tends to
i.e. u
set
in
idK
269
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
i s compact i n E K r h e n c e a l s o compact i n E. S i n c e K K t h e same arguments show t h a t f o r any
s e t of
Ls
:= KO U
0
i s a c o m p a c t sub-
s E IN
t h e set
um(Ko)
U
m>s
i s compact i n EK and E .
Wo b e a convex b a l a n c e d neighbourhood of z e r o
Now l e t f o r which
+
KO
Wo
C
KO C K , t h e r e e x i s t s
n. s
l i m um = i d K i n
Since
Lo := Ls
J
and
E
since
m*m 0
E IN
s u c h t h a t f o r a n y s ? s o a n d a n y x E KO
us(x)
Put
C(K,E)
in
-
x
E
wo.
then it follows
0
LoCKo+WoCR.
S i n c e Lo tion
is a compact subset o f
0 2 1 ( 1 t h e func-
f ( j ) : 51 + L A ~ ( E , F )i s c o n t i n u o u s , t h e r e e x i s t s a convex
a n c e d neighbourhood any
51 and s i n c e f o r
j with
U of z e r o i n
0 < j 5 l , any
x
E
E with
Lo
Lo
and a n y
+
bal-
U C s2 s u c h t h a t
z E E
with
for
x-zEU
t h e f o l l o w i n g estimate h o l d s
For
1 5 j 5 !k t h e s e t
f (j) (Lo)
i s compact and hence
bounded
in
= L c o ( E r L f ~ l ( E r F1). E i s b a r e l l e d by h y p o t h e s i s , hence fJ(Lol j- 1 i s e q u i c o n t i n u o u s i n Lco(E,Lco (E,F)). T h i s i m p l i e s t h a t t h e r e i s a
L:O(E,F)
MEISE
280
neighbourhood W and any
j
of z e r o i n
y ' E Lj-l
E such t h a t f o r any
f o r any
y = (ylI...,yj)
o n e of t h e yk i s i n The s e t
Now w e d e f i n e
W
e
x
El, where
E
Lo,any y1 E W
j
,
t h i s means t h a t we have
j
- 1 of
t h e yk a r e i n
L and
j '
-1 ( n ( W . j, j=1 J
hence t h e r e e x i s t s
E
w e have
f (1) (x) i s symmetric f o r any
Since
x
fl
U) i s a neighbourhood of z e r o i n
s E IN w i t h
s
our construction we get
u(x)
E KO
,
2 s0 s u c h t h a t
and o b s e r v e t h a t b y t h e c h o i c e of
u := us
EK
+
U C B
f o r any
x
E
s
and by
Ko(from now
on l e t us omit t h e map j, l i . e . w e r e g a r d u as mapping from E i n t o E). Then t h e s e t w := u-1 (s1) i s an open neighbourhood of KO and on w w e c a n d e f i n e t h e mapping
is e a s y t o see t h a t with
j
f
0
n t 1 and any
f
0
u : w
+
F. By o u r d e f i n i t i o n 2 . 4
u E Cgo(w,F) and t h a t f o r any y E Ej
x
E
w,
any
it j
t h e following holds
I n o r d e r t o prove t h e d e s i r e d estimate, w e o b s e r v e f i r s t t h a t we have
u(Ko) (1):
C
Lo, and t h a t f o r any
x
E
KO
,u(x) -
x E U. Hence w e g e t fran
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
Then w e o b s e r v e t h a t f o r any
x
hence f o r any
KO
E
j
and any
with y
E
291
1 5 j 5.e w e have u(Q,) Q:
1
C
L
1’
w e g e t from (11,( 2 ) and ( 3 )
< (e+1).L + 1 -
By ( 4 ) and (S), t h e proof of t h e lemma i s complete.
4.
L e t us r e c a l l from B i e r s t e d t and Meise 1 6
REMARK:
d u c t i v e i n j e c t i v e system
s u b s e t of
Ea
~
o A f 1 . c . s p a c e s Ea
(CFA)
. Then
i t i s obvious t h a t any 1 . c . s p a c e E which c a n b e
E A of
lEa I
. Hence i)
is called
E = ind E
r e p r e s e n t e d a s an i n d u c t i v e l i m i t o f a compactly r e g u l a r system
in-
i s Hausdorff And i f f o r any acanpact a+ a E t h e r e e x i s t s a E A such t h a t K i s a l r e a d y a ccnpact
campactLy heguLah, i f s u b s e t K of
{ E a I*j u BIa
1 t h a t an
( F ) - s p a c e s Eu w i t h a.p.
has
inductive
the
property
i n any of t h e f o l l o w i n g c a s e s E h a s (CFA):
E i s a (F)-space with a.p.
i i ) E = i n d En I where { E n l j n m } i s a s t r i c t i n d u c t i v e n + of ( F ) - s p a c e s En w i t h a . p . i i i ) E = in$ En
n
I
system
where { E n l j n m } i s a compact i n j e c t i v e induc-
t i v e system of
( F ) - s p a c e s En w i t h a . p . For b r e v i t y w shall
c a l l any s p a c e o f t h i s t y p e (DFSA)-space. Using a t r i c k which g o e s back t o Aron and S c h o t t e n l o h e r [ 4 ],we can now prove t h e d e s i r e d r e s u l t on t h e a . p .
of
Czo(Sa).
MElSE
232
5. THEOREM:
L e t E be an in Lemma 3 and
and
y = yco
n E INm aLl t h e hypothedeb
a t e d a t L d 6 i e d . Then
PROOF:
in
of
n Cco(Q)
identify
c a n b e proved by showing t h a t C z o ( Q ) 8 F is dense
Czo(R)
E
F
f o r any Banach s p a c e F.
C:o(Q)
E
F
in
Q,
ma 3 , t h e r e e x i s t s such t h a t Let
f
0
us define
fo E Cao(Qo,F)
E , any
u
E
e
C g o ( Q , F ) , a n y compact subset KO o f
+ I, and
Then
any
Eo : = I m u,
,
no:
=
Q
and by a c l a s s i c a l
gE F.
~
Q:
R,
be g i v e n . By l e m -
of
w
KO
I
Ro
.
dimensional)
Then result
S i n c e i t w a s shown i n t h e proof of lemma
h E CZo(Eo) 8 F
Furthermore w e have f o r any E
> 0
and E ~ f o := f
(finite
CEO (Eo)
i s dense
in
Czo(R)
3
,
such t h a t
g : = hou E C a o ( E ) 8 F, and f o r any
Y
E
E ’ 8 E and an open neighbourhood
u(Ko) C Q n Eo = Qo, and s i n c e
there exists
any
f E
u E CZo(wlF) s a t i s f i e s t h e estimates g i v e n i n lemma 3 .
CZo(QotF) = Ccn0(Q) that
that
C Z o ( Q t F ) f o r any Banach s p a c e F .
To do t h i s , l e t any any compact
By c o r o l l a r y 2 w e mayand shall
CZo(R,F). Hence w e o n l y have t o show
and
i s dense i n
Czo(Q) 8 F
6 a h a n y open d u b n e t R a 6 E.
had t h e a . p .
Cgo(Q)
c0(Q)
theohem 1 o n E and
06
i s quasi-complete by h y p o t h e s i s . Hence, by theorem 1.7
Cgo(Q)
t h e a.p.
adburnt duathehmohe t h a t doh
x E KO
,
any
x
E KO
j with
15 j
5 L
and
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
Hence w e have shown of
mm
(f
-
5
g)
2s
,
which proves t h e d e n s i t y
a F in c ~ ~ ( . Q , F ) .
c:~(E)
A l l t h e hypotheses of theorem 6 a r e s a t i s f i e d
6 . REMARK:
n E
K~ ,Q,
293
s2 of
and any open s u b s e t
E,
if
E is either
w i t h a . p . o r a (DFSA)-space. T h i s follows from 2 . 5 ,
for
any
an (F) - s p a c e
2 . 6 andremark 4 .
We s h a l l show now t h a t f o r Frgchet spaces E w i t h a . p . t h i s r e s u l t i s optimal.
7. THEOREM: a)
Fax a F h e c h e t Apace E t h e doLLowing axe e q u i u a e e n t :
C:o(Sl)
Q
# $
has t h e a . p . 06
that
c)
E
n E INw
and a n y o p e n n u b s e t
E-
Thexe exidt
b)
d o h any
n E INm and a n open n u b b e t
czo(a)
had
R # pl
ad
E
nuch
t h e a.p.
had t h e a . p .
(a) * (b): t r i v i a l
PROOF:
(b)
=.
( c ) : By 2 . 8 ,
ELo = E ’ i s a continuously p r o j e c t e d topo-
l o g i c a l l i n e a r subspace of Frgchet space E t h e a . p . of
C
C : o( . Q ) ,
hence
EA has t h e a.p.
But f o r a
EA i s e q u i v a l e n t t o t h e a . p . of Elhence
E has t h e a . p .
( c) * ( a ) : This is clear according t o t h e remark 6.
REMARK:
For Banach spaces E theorem 7 was shown by Bombal
Gorddn
294
MEISE
and Gonzslez Llavona [lo] f o r
51 = E . Again f o r Banach s p a c e s
s l i g h t l y d i f f e r e n t version (using
[ 201 and a l s o by Aron [ 3
topology
.
I
T h e h e C X i 4 t A a n (FS)-npace
8. COROLLARY:
06
the
a
C z o (51) ) of theorem 7 w a s p r e s e n t e d by P r o l l a andGuerreiro
i n d u c e d by
t h e a . p . go& a n y
n o t have
Cf: ( 0 ) endowed w i t h
E
E huch t h a t
doeA
Czo(51)
mm a n d a n y n o n - e m p t y o p e n nubnet
n E
R
E.
T h i s i s a consequence of theorem 7 and t h e e x i s t e n c e of (FS)-
PROOF:
s p a c e s w i t h o u t a . p . The e x i s t e n c e of s u c h (FS) - s p a c e
follows
from
E n f l o ' s c o u n t e r e x a m p l e , a s Hogbe-Nlend p r o v e d i n [ 1 6 1 . Because of lemma 3 , t h e method a p p l i e d i n t h e proof o f theorem 5 c a n be used a l s o t o d e r i v e some f u r t h e r d e n s i t y r e s u l t s
just
by
" l i f t i n g " d e n s i t y r e l a t i o n s known i n t h e f i n i t e d i m e n s i o n a l case. Bef o r e s t a t i n g them l e t u s r e c a l l t h a t a c o n t i n u o u s n-homogeneous p o l r nomial
p on E i s c a l l e d Ainite, i f t h e r e
exist
y i ,...,y;
E
E'
such t h a t n p(X) =
By
n
j =1
f o r any
(y;,X)
x E E.
P f ( E ) w e d e n o t e t h e l i n e a r h u l l o f a l l c o n t i n u o u s n-homogeneous
p o l y n o m i a l s on E
,
9. THEOREM:
E be a q u a A i - c o m p k k t e b a h a L t e d
Let
(CFii). Then doh 0(#
0)
PROOF:
06
E
n
any
E
n E
t h e space
L e t any
p a c t s u b s e t Q of
f
INo.
I t i s e a s y t o see t h a t
and 1 . c . Pf(E) @ F
,
E CZo(QIF)
E , any
m
Pf (E) C C c o ( E l .
L.c.
pace F a n d a n y
i n dense i n
Apace open
With
oubaet
Czo(51,F).
a n y compact s u b s e t K of
51 , a n y can-
1 < n +1, any c o n t i n u o u s seml-norm
q onF,
296
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
and
> 0
E
be g i v e n . W e s h a l l show t h a t t h e r e e x i s t s
g
E Pf(E)
€3 E
with
Let
F
9 and l e t
d e n o t e t h e c o m p l e t i o n o f t h e c a n o n i c a l normed s p a c e
F/ker q
d e n o t e t h e c a n o n i c a l c o n t i n u o u s l i n e a r map.Since q II 0 f E Cn ( Q I F ) , a c c o r d i n g t o lemma 3 t h e r e e x i s t s u E E' 8 E s u c h co 9 that a : F
+
F
Now we p r o c e e d as i n t h e proof of theorem 5 and d e f i n e
no :=
Ci n Eo and f o : (IT
0
f ) ICio.Then f o E C,"o(Qo,F
S i n c e t h e p o l y n o m i a l s o n Eo are d e n s e i n in F
q i
and s i n c e
ho
E
)
= CEo(Qo)
gE
Fq.
CEo(Qo) ;since II(F) is dense
no ( t h i s was shown
u(K) i s c o n t a i n e d i n
p r o o f of lemma 3 ) , t h e r e e x i s t s
4
:= I m u I
Eo
P(Eo) 8
IT
(F) =
Pf (Eo)
in @
IT (
the F)
such t h a t
Assume t h a t
i=ll...,m.
ho = Then
m -..
Z pi 8 r ( y i ) , where i=1 h :=
Z piOu8yi
i =1
pi E Pf (Eo) and
is i n
Pf(E) 8 F
yi E F f o r and
as
in
t h e p r o o f of theorem 5 i t f o l l o w s
PL,KIQ,q
Hence we have shown t h a t
(f
-
h) 5 2~
Pf (E) €3 F
.
is d e n s e i n
Czo(Q,F).
The following c o r o l l a r y i s a n immediate consequence of theorem 9.
286
MElSE
10. COROLLARY:
Let E be a q u a b i - c o m p l e t e b a w i e l l e d l . c . npace w i t h
(CFA). T h e n
any
n ( # 0 ) 06
604
.the a p a c e
E
m m , any
n E
l . c . b p a c e F , and a n y o p e n n u b a e t
63 F
C:o(E)
CEo(n,F).
dense i n
i b
Looking a t theorem 5 and c o r o l l a r y 1 0 and t h e i r p r o o f
in
f i n i t e d i m e n s i o n a l c a s e one h a s t h e i m p r e s s i o n t h a t c o n d i t i o n ( o r more o r less t h e a . p . )
t o g e t h e r w i t h f i n i t e dimensional
the (CFA)
results
c a n b e u s e d i n s t e a d o f C m - f u n c t i o n s w i t h compact s u p p o r t . T h e f o l l o w i n g theorem i s o f t h e s a m e n a t u r e . B e f o r e w e s t a t e i t , l e t us remark E be any 1.c. space
t h a t a n e a s y c a l c u l a t i o n shows t h e f o l l o w i n g : L e t and l e t
d e n o t e i t s ( c o n t i n u o u s ) d u a l . For any system y o f bounded
E'
subsets of
( c o v e r i n g E ) and any
E
m
C y ( E ) . Using t h i s and t h e c l a s s i c a l theorem
belongs to
Wiener-Schwartz
11. THEOREM:
0) 0 6
denbe
ifl
E
Paley
L e t E b e a q u a b i - c o m p l e t e b a a a e l l e d l . c . bpace n
a n y 1 . c . b p a c e F , and a n y o p e n
E INm,
t h e L in ea h hue1
06
the net
Ie,
-
*
with bubbet
f I y E E', f E F)
LA
Cgo(Q,F).
4 . A KERNEL THEOREM FOR FUNCTIONS OF CLASS
CEO
I n t h i s s e c t i o n w e s h a l l show ( u n d e r a p p r o p r i a t e t h a t any f u n c t i o n s i n m
of
t h e proof o f theorem 9 a l s o g i v e s
(CFA). T h e n d o h a n y
fi(#
y E E', t h e f u n c t i o n
m
Cco(Ql
x
hypotheses)
Q 2 ) c a n b e r e g a r d e d as a n e l e m e n t o f
m
C c o ( Q l , C c o ( ~ 2 ) ) and v i c e v e r s a . Using theorem 3 . 5 t h i s a l s o
a tensor product representation f o r
m
Cco(Ql
x Q,)
.
B e f o r e w e c a n prove
o u r r e s u l t w e need s e v e r a l lemmas. The f i r s t lemma i s consequence o f d e f i n i t i o n 2 . 4 .
implies
an
immediate
SPACESOF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
1. LEMMA: 06
Let E,F and
b e L . c . b p a c e b , l e t R be an o p e n
G
y b e a covetring b y b t e m
E l leX
297
0 6 bounded b u b b e t
dubbet
let
E , and
06
u E L(F,G) b e g i v e n .
any
a)
Foh
b)
Fotr a n y
f E Cm(51,F) t h e d u n c t i o f l
Y
f E Cm(R,F) a n d a n y
belongs t o c)
2.
06
subbet
x1
1e.t
LEMMA:
Ei
PROOF:
t o p o t o g i c a l bubbpace
and
doh i =1,2.
F
The mapping
m
+
tonuous and j - l i n e a r ,
El
(ii)* :
( i d ) * ( m ) [ x ]= m ( [ i d ( x ) ] ) .
t h e n by lemma 1 . b )
,
j
f o r any
c o n t i n u o u s l i n e a r map
1.5,and l
(ii)*
( i a )*
o
f'j)
any j E IN
0
6
be an
open
,F)
by
g E C(R1,Cco(Q2,F)1 .
i2
d e f i n e d by
ii
: E';
IN. Thus
eEo( (El x
(0,x2)is
=
(El x E 2 ) j i s con-
+
gives rise
ia
E2)JrF)
( x2 )
to
a
eEo(E);,F), d e f i n e d
-+
I f now f i s a n y e l e m e n t of C~o(511xR2rF)r m
m l.c), f ( 1 ) is i n C c o ( R l ~ ~ 2 , ~ ~ o ( (E2)jrF). E1~ m
(EJrF)), (i$*o f ( j ) i s in Cco (511X 512' Ls co 2
L3 (E ,F)). L e t u s d e n o t e t h e f u n c t i o n 512' co 2 m R2,F) t h a t f o r Then it f o l l o w s from f E Cco(R1x
Cco(nl
f (1) by
Cco(R,G).
m
x E2
Then i t f o l l o w s from lemma 1.a) that hence
m
x R2
f E Cco(Rl
T h e n dotr any
i2: E 2
b e l o n g n to
.then
F,
06
be 1 . c . hpaceb and l e L Qi
o b v i o u s l y l i n e a r and c o n t i n u o u s , hence
by
f(1)
Y
1 , o n e de,4inen a 6 u n c X i o n
f (xlf
+
Y
E2
El
t h e dunction
E INo
f E Czo(RrF) m i t h f ( R ) C G
any
Y
C m ( R , L j (E,F)1 .
G i b a closed lineah
76
j
Y
beloflgb to C m ( R r G ) .
uo f
gj
.
and any x1 E
R1 t h e f u n c t i o n
gj(xlr
) :
R2
-+
Lio(E2rF)
i s G z t e a u x - d i f f e r e n t i a b l e and t h a t i t s Gsteaux-derivative is g j + l ( y , * 1. T h i s p r o v e s t h a t f o r any m
x1 E Q1
Cco(512,F) , hence t h e f u n c t i o n
t h e function
g : nl
f(xlr
) belongs
Cco(n2,F) , g(Xl) = f 00
-+
to
( X l l o ) Can
be d e f i n e d . I n o r d e r t o show c o n t i n u i t y o f E
> 0 , and any c o n t i n u o u s
be given. Since
g
j
g on
semi-norm on
R1
, let
any
x1
6
61,
any
m
Cco(S22rF) of the f o m p
l r 5 r Q 2 r ~
i s uniformly c o n t i n u o u s on {x,} x K 2 f o r any
j,
MElSE
258
t h e r e e x i s t s a neighbourhood f o r any ( x l , x 2 ) any j w i t h
(where
0
{xll
E
and any ( h l , h 2 )
d e n o t e s t h e semi-norm
I
Pj,Q2,s
E
V1
El
w e have
V2
x
such t h a t
x E2
for
hl
E
u
+
s u p . q(u(y)) on L20(E,F)).
FQ;
V1
g i s continuous.
3 . PROPOSITION: 604
of zero i n
x V2
5 j5 l
T n i s i m p l i e s f o r any
hence
K2
x
V1
Which
and L e t
i =1,2
Fok
(Ei)A
Let
C o m p L e t e and w h i c h eQllaeA (Ei)AA
i d
be a n o p e n h u b b e t a d
sli
b e a quabi-compLete L . c . bpuce
Ei
t#pOfOgiCUk?Ly,
Abdume 6uhthekmohe t h a t
Ei.
E2
in
a k I R - b p a c e . Then t h e h e e X i b t h a continuoub Lineah a n d i n j e c t i v e map 03
A : Cco(Ql 6oh
any
PROOF:
tion
x
f
E
Q2)
m
+
m
Co(Q1,Cco(~Z)),
m
Cco(nl
x
dediMed b y
-+
f
(xl,
: x1
*
f(X1,
1
a,)
t h e func
-
Q2).
L e t u s show f i r s t t h a t for any
g : x1
A(f)
)
belongs to
m
Cco
f E CEo(fil
x
(Ql,c;o(i22)1 *
L e t il d e n o t e t h e l i n e a r c o n t i n u o u s mapping i l : E 1 El x E2, m i l ( x l ) = (xl,O) a n d l e t f E Cco(Ql x Q,) b e g i v e n . As i n t h e p r o o f -+
of lemma 2 one shows t h a t f o r any
j E IN
t h e mapping
~p :=
j
( i i ) * o f(1)
SPACES OF OIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
m
belongs t o g j : x1
+
Cco ( i l l
x
R2,L~o(El,IR1 ) .
29%
Hence, by lemma 2 t h e
mapping
i s i n C(Rl,C~o(R2,L~o(E1,1R))). Now o b s e r v e
9 . (xl,
3
i t f o l l o w s from 3.2
by o u r h y p o t h e s e s
and g e n e r a l
results
that
on t h e
€ - p r o d u c t t h a t w e have n a t u r a l isomorphisms
Using t h i s isomorphism, w e g e t from f o r any
j E IN
. Obviously
g.7 the napping
g. E C(Ql,L~o(E1,C~o(Q2)) 7
go = A ( f ) , and w e s h a l l p r o v e
now t h a t
i s t h e G s t e a u x - d e r i v a t i v e of g I n o r d e r t o do t h i s , gj+1 j ’ (k) = (‘j * 0 f f i r s t remark t h a t f o r any k E IN w e have
l e t us and
5)
‘Oj
that
f o l l o w s f r o m t h e proof of lemma 2. Hence w e g e t
Now l e t
R2
x1 E Q1
hl E E l ,
any compact Q2 o f
any
L
E M
, any
E 2 , any compact s u b s e t Q1 o f
b e g i v e n . W e have t o p r o v e t h a t t h e r e e x i s t s any
t
with
0 <
I
t
I 5
compact subset
6 > 0,
6
By (1) and 1 . 5 w e have t o e s t i m a t e f o r
0
5
k
5 R
El
K2 o f
and
such t h a t
E
> O for
MEISE
300
-t ( f ( j + k )(xl
+
-
thl,x2)
f
(X1'X2)
f ( j + k + l ) i s c o n t i n u o u s on
Since
R~
x
a2 ,
-
it i s uniformly continu-
ous on a s u i t a b l e neighbourhood o f t h e compact set
uniform c o n t i n u i t y o f
Cx,)
element o f
.
By
g = go
isan
.
m
m
K2
f ( j + k + l ) a n d ( 3 ) i t i s clear t h a t t h e r e exists
s a t i s f y i n g ( 2 ) . C o n s e q u e n t l y w e h a v e shown t h a t
6 > 0
x
Cco (Ql,Cco (Q,)
L i n e a r i t y and i n j e c t i v i t y o f
A are o b v i o u s . C o n t i n u i t y
of
A
follows i m m e d i a t e l y from (1) a n d t h e d e f i n i t i o n o f t h e c o r r e s p o n d i n g topologies.
Now w e want t o p r o v e t h a t A i s s u r j e c t i v e i f w e impose
some
f u r t h e r conditions.
4 . LEMMA:
subset
06
Fon Ei.
i =1,2
El
Asbume t h a t
'Let g b e a a n y 6unc;tion i n a)
Fon a n y ( j, k )
b e a L . c . bpace and l e t
L e t Ei
E
x m
E2 k
i b
604
be an open
any ( j , k )
E
E!
k E2
IN 2
.
cco(~l,cco(~2)). m
IN2 t h e mapping f ( j r k :Ql )
dehilzed b y f ( j t k ) (x1,xi,y1,y2)
is c o n t i n u o u s .
a km-6pace
Ri
.
x
Q2
X
x
+
IR
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
331
a ) Observe t h a t f o r any open s u b s e t R of a 1.c. s p a c e E l a n y
PROOF:
1 . c . s p a c e F and any
i s c o n t i n u o u s on
K
x
f
QJ
E
m
Cco(QIF)
the function
f o r any compact s u b s e t
K
R and
of
any
compact s e t Q i n E and has v a l u e s i n F . Hence f o r any compact s u b s e t K1
of
Rl
and any compact s u b s e t Q1
of
Ell
the function
i s l i n e a r and c o n t i n u o u s f o r any compact s e t K 2 i n p a c t s e t Q 2 i n E2 , w e g e t from lemma 1.b) t h a t b e l o n g s t o C(K1 x QllC(K2 x Q,k)) = C(K1 x Q! x K2 x
b)
x
-
) [
x K2 x
Q:'
x
Q,). k
is a kR-space
The s e c o n d a s s e r t i o n i s a consequence o f t h e f o l l o w i n g con-
siderations:
=: A ( t )
pk((g(j)(
d;) =C(K1
T h i s p r o v e s t h e c o n t i n u i t y o f f ( j r k )I s i n c e El1' f o r any ( j , k ) E IN 2
.
R 2 and any com-
+
B(t).
302
MElSE
uniformly i n
y1 E Qi
By lemma 2.2
=
k y 2 E Q,
and
.
we g e t
lo
1
f ( j r k + ' )(x,
+ t h l r x 2 + T t h Z r y l r ( h 2 , y 2 ) )dT
f ( J r k + l ) i s u n i f o r m l y c o n t i n u o u s i n a neighbourhood of t h e compact set {xl} x {x,) x Qi* x Q2k hence w e also have I t f o l l o w s from a ) t h a t
uniformly i n
5. THEOREM:
equal6
Foh
(Ei)AA
Then t h e mapping ib
i =1,2
a topological
l e t Ei
E!
x E:
m
A : Cco(Ql
ment o f
m
C c o ( a l f C ~ o ( ~ 2 ))
any ( j , k ) A(f) : x l + f
E
.
2 D l
(Xlf*)r
Q1
A i s s u r j e c t i v e . L e t g be any e l e
-
1 . By lemma 4 t h e f u n c t i o n f : (x,,x2) +g(xl) (x,)
Cco(Ql,C~,(Q,)
is obvious t h a t
+
doh
Ei.
06
idomohphibm.
m
i s c o n t i n u o u s on
be an open bubbet
i b a km-hpace
x "2)
F i r s t l e t us show t h a t
PROOF:
be a q u a b i - c o m p l e t e l.c.bpace w h i c h
t o p o l o g i c a l t y and l e t Qi
buathekmoke t h a t
Addume
k y2 E Q 2 .
and
y1 E Q:
x Q,.
W e s h a l l prove
A ( f ) = g, hence
I n order t o prove
m
m
f E CCo(al
x
a,).
Then it
A is surjective.
f E Cc0(R1 x
a,)
let usremark t h e following:
SPACES OF DIFFERENTIABLE FUNCTIONS AND THE APPROXIMATION PROPERTY
and a s u r j e c t i o n u
Let (j,k) E
:
{I,...f j j
U
303
{l'f...fk'}+.{lf,..,j+k~
b e g i v e n . Then w e d e f i n e a c o n t i n u o u s l i n e a r map ru :(El x E 2 ) j + k
+
E
by
= ( ( e l , u (1)'
-
r e l (, j~ )
-(j,k)
By lemma 4.a) t h e f u n c t i o n Ql
X
R2
x
(El
any (x1,x2)
€
X
R1
:= f ( I , k ) o r
fU
-f a(j,k) ( x l , x 2 , * )
E2)j+k and x
' (e20 (1')'. . - 'e2u ( k ' )
X
E2,1R),
is c o n t i n u o u s . Using t h e mappings duction t h a t
f
m
belongs t o
Cco(R1
for
is ( j + k ) - l i n e a r
-f a(I 'k)
R2. Because of t h e c o n t i n u i t y o f c]+'(El o
c o n t i n u o u s on
is
U
t h e map-
d e f i n e d by
f u( J ' k ) I t is e a s y t o p r o v e by i n x
Q,).
Let
us
show
that
f
is
Gsteaux-differentiable:
u
Define
u2
:
0
U (1')
+
{l}
*
1 '
11)
0
U
{l}
by
ul(l)
= 1
and
define
~ ~ ( 1 =' 1. ) Thenwe g e t from 4 . b ) t h a t f o r
by
x = (x1,x2) E Q1 x R2
+
and
h = ( h l , h 2 ) E El
E2
x
1
+
f ; O r 1 ) ( x ) ) [ h ] , and f E Cco(R1 x R,) ul 2 by lemma 2.3. From t h i s and l e m a 4 . b ) w e g e t by i n d u c t i o n t h a t f o r
Hence
f'(x)[h] = (f(l'')(x)
any 1 E W t h e f u n c t i o n
f
be r e p r e s e n t e d as a sum of
k in
INo w i t h
j +k =
L
is i n
L
Cco(Rl
x Q,)
and t h a t
f ('1
can
f, ( J r k ) where t h e sum r u n s o v e r a l l j and and o v e r c e r t a i n
(J
.
This proves
that
MEISE
f E
cco (a, m
x
Q,).
Hence w e have shown t h a t
A : Cm c o ( ~ xl o 2 )
b i j e c t i v e . From t h e r e p r e s e n t a t i o n o f (A'l ( g ) )
follows t h a t A - l
m
+ c ~ ~ ( Q ~ , ci s~ ~ ( ~ ~ i n d i c a t e d above i t
i s c o n t i n u o u s . Then A i s a t o p o l o g i c a l isomorphism
by p r o p o s i t i o n 3 .
REMARK:
R e s u l t s of t h e same t y p e as i n theorem 5 a r e also g i v e n
in
t h e l e c t u r e n o t e s of F r o h l i c h e r a n d Bucher [151 ( w i t h a d i f f e r e n t defin i t i o n o f d i f f e r e n t i a b i l i t y ) and i n Colombeau [111I [ 1 2 ] . Itseems t o b e i m p o s s i b l e t o g e t t h e r e s u l t on (DFM)-spaces g i v e n below by
bor-
n o l o g i c a l methods. Concluding t h i s s e c t i o n , l e t us combine theorem 5 and some
of
t h e r e s u l t s i n s e c t i o n 3 . Then w e g e t
6 . THEOREM:
Let El
and E 2 b e e i t h e k (F)-Apactd o h (DFM)-bpacesand
L e t Oi be a n o p e n d u b n e t a d Ei
doh
i =1,2.
Then we h a v e t h e
dot-
bowing t o pob a g i c a l 16 a ma h p hid m d
7 . THEOREM: Ei
.
Foh
i =1,2, b e t
be an open s u b s e t
Cli
06
t h e L . c . space
Assume t h a t e i t h e n . 1)
El
2)
El
and E 2 and
E2
ahe ( F ) - d p a c e n , o n e
06
a h e (DFM) - s p a c e s , o n e
w h i c h had a . p . ,
05
ah
wkich A a ( D F S A ) -npace.
T h e n t h e d o l l o w i n g hold4
8. REMARK:
The d u a l of
CEo(Sa)
forms a n a t u r a l g e n e r a l i z a t i o n of the
s p a c e of d i s t r i b u t i o n s w i t h compact s u p p o r t t o i n f i n i t e
dimensions.
SPACES OF DIFFERENTIABLE FUNCTIONS ANDTHE APPROXIMATION PROPERTY
306
It is obvious that many of the results of this article can regarded as results on the dual of
also
be
m
Cco(Q). E.g. theorem 3.10 is of
importance in connection with the theorem of Paley -Wiener -Schwartz (in order to see this one has to extend several results
to
complex
valued functions on R , then (for certain 1.c. spaces E) one can define the Fourier-Laplace transform of any
m
T E Cco(Q,fl!)'
morphic function on the complexification of EA growth condition). Theorem 6 can be used
* : Cmco (E)'
x
CZo(E) '
-+
to
,
as a
holo-
satisfying acertain
define
a
convolution
Czo(E) I . The precise formulation of the results
just mentioned will be contained in a subsequent paper.
REFERENCES
[
11
A. ARHANGEL'SKII, Bicompact sets and the topology Soviet Math. (Doklady) 4 (1963),, 561 - 564.
of
spaces,
[ 21
R. ARON, Compact polynomials and compact differentiable mappings between Banach spaces, in "Si?minaihe P i t h h e L e h f l g ( A n a l y b e ) Annee 1974/75", Springer Lecture Notes Math. 524 (1976), p. 213-222.
[ 31
R. ARON, Approximation of differentiable functions on a Banach space, in 'I 1 n d i n i t e d i m e n d i o n a l hoComo/rphq a n d appfic&ovl~': North-Holland Mathematics Studies (19771, p. 1-17.
[ 41
R.
[ 51
K.-D. BIERSTEDT and R. MEISE, Lokalkonvexe Unterraume in topologischen Vektorramen und das c-Produkt,manuscripta math. 8 (1973)I 143 -172.
[ 61
K.-D. BIERSTEDT and R. MEISE, Bemerkung uber die Approximationseigenschaft lokalkonvexer Funktionenrame, Math. Ann. 209 (19741, 99 -107.
ARON and M. SCHOTTENLOHER, Compact holomorphic mappings on Banach spaces and the approximation property, J. Functional Analysis 21 (19761, 7-30.
MElSE
306
[ 71
X.-D.
BIERSTEDT and R. MEISE, N u c l e a r i t y and t h e Schwartz prope r t y i n t h e t h e o r y of holomorphic f u n c t i o n s on
metrizable
l o c a l l y convex s p a c e s , i n " l n d i n i t e dimenbionaL hoLomohphy
and a p p L i c a t i o n d " , North-Holland Mathematics S t u d i e s (1977), p. 9 3 - 1 2 9 . 81
J. L. BLASCO, Two p r o b l e m s on k m - s p a c e s , Math. Sci. Hung.
t o appear
in
Acta
[ 91
P. L. BOLAND, An example of a n u c l e a r s p a c e i n i n f i n i t e dimens i o n a l holomorphy, Ark. Mat. 1 5 ( 1 9 7 7 ) , 87 - 9 1 .
1101
F. BOMBAL GORDON and J. L. GoNZaEZ UAVONA, La p r o p i e d a d
de
aproximacidn en e s p a c i o s de funciones diferenciables,Revis-
t a Acad. C i . Madrid 70 ( 1 9 7 6 1 , 7 2 7 - 7 4 1 . [ l l ] J. F. COLOMBEAU, Uiddekentiation
e t b o k n o l a g i e , t h S s e , Bordeaux
1973.
[12]
J. F. COLOMBEAU, S p a c e s of Cm-mappings i n i n f i n i t e l y many
di-
mensions and a p p l i c a t i o n s , p r e p r i n t Bordeaux 1 9 7 7 . [131
S. DINEEN, Holomorphic f u n c t i o n s on s t r o n g d u a l s of Fr6chetMonte1 spaces , i n " I n d i n i t e d i m e n d i o n a l holomokphy and app t i c a t i o n d t'
[14]
,
North-Holland Mathematics Studies (1977),147-166.
K. FLORET a n d J . WLOKA, Eindiihtung i n die Thgohie d e n LokaLkonwexen
Raume, S p r i n g e r L e c t u r e Notes i n Math. 56 ( 1 9 6 8 ) . [15]
A. FROLICHER a n d W. BUCHER, CaLcuLud i n wectoh dpaced nohm, S p r i n g e r Lecture Notes i n Math. 30 ( 1 9 6 6 ) .
[161
H.
without
HOGBE-NLEND, L e s e s p a c e s de F r 6 c h e t - S c h w a r t z e t l a p r o p r i e t e d ' a p p r o x i r n a t i o n , C.R.
Acad. S c i . P a r i s A 275(1972) ,1073-1075.
[171
J. HORVhTH, T o p o L o g i c a e v e c t o h b p a c e d and d i b t h i b u t i o n b 1,Readi n g , Mass, Addison Wesley 1965.
[18]
H.
PiddehcntiaL cak?cutub i n eocaeCy c o n v e x S p r i n g e r L e c t u r e Notes i n Math. 417 ( 1 9 7 4 ) .
H . KELLER,
bpaced,
SPACES OF DIFFERENTIABLE FUNCTIONS ANOTHE APPROXIMATION PROPERTY
[191
G . KOTHE,
T a p o L o g i c a L v e c t v t r h p a C e b I, Springer
307
Grundlehren
der Math. 159 (1969). [20] J. B. PROLLA and C. S. GUERREIRO, An extension of Nachbin's theorem to differentiable functions on Banach spaces with the approximation property, Ark. Mat. 14 (19761, 251 - 258. [21] H. H. SCHAEFER, T o p o L o g i c a L v e c t o h dpaces, Springer 1970. [221
L. SCHWARTZ, Theorie des distributions 5 valeurs I, Ann. Inst. Fourier 7 (19571, 1-142.
[ 231
M. DE WILDE, R6seaux dans les espaces lin6aires 2 semi-normes, Mgmoires SOC. Royale Sc. Lisge, 5e sGrie,l8, 2 (1969).
[24I
S. Y A W U R O , Uia6ekentiaL
CdCU&Uh
vectorielles
i n t o p o t a g i c a L fitzeah
Springer Lecture Notes in Math. 374 (1974).
hpaceb,
This Page Intentionally Left Blank
Approximation Theory and Functional A n a l y s i s J.B. Prol2a ( e d . ) 0North-HoZland Publishing Compmzy, 1979
A LOOK AT APPROXIMATION THEORY
LEOPOLDO NACHBIN I n s t i t u t o de Matemgtica U n i v e r s i d a d e F e d e r a l do Rio de J a n e i r o 20.000 R i o de J a n e i r o RJ ZC-32 Brazil Department of Mathematics U n i v e r s i t y of R o c h e s t e r R o c h e s t e r NY 14627 USA
1, INTRODUCTION I would l i k e t o d e s c r i b e v e r y b r i e f l y how I w a s l e d t o
become
s e r i o u s l y i n t e r e s t e d i n Approximation Theory, t h a t i s , t o i n d i c a t e t h e m o t i v a t i o n t h a t I had i n my mind. T h i s f i e l d h a s d e v e l o p e d i n B r a z i l i n t h e p a s t t e n y e a r s or so, t h a n k s a l s o t o t h e work of S i l v i o Machado,
Joao Bosco
Prolla
and
Guido Z a p a t a , as w e l l as t h e r e s e a r c h s c h o o l t h a t t h e y formed. I f I had t o r e d u c e b i b l i o g r a p h i c a l r e f e r e n c e s t o a b a r e
mini-
mum, i n what c o n c e r n s t h e work of t h e B r a z i l i a n s c h o o l i n Approximat i o n Theory and i t s r e l a t i o n s h i p t o t h e r e s e a r c h o f o t h e r g r o u p s ,
I
would q u o t e my monograph Element6 ad A p p t o x i m a t i a n T h e o h y ( 1 9 6 7 ) , as w e l l as P r o l l a ' s monograph Apphoximation (1977) (see [ 3 4 1 up-to-date
,
0 4 Vectoh Vatued
[ 5 4 1 ) . However, t h e b i b l i o g r a p h y
at
the
Funciionb end
is
and complete w i t h r e s p e c t t o t h e work by Machado, P r o l l a ,
Zapata and m y s e l f .
I t i s extremely incomplete o th er wis e.
emphasize t h e f o l l o w i n g aspects:
Let
me
310
NACHBIN
1)
I s h a l l r e s t r i c t myself
h e r e t o t h e r e a l v a l u e d c a s e . The
v e c t o r v a l u e d c a s e was t r e a t e d i n a d e s i r a b l e d e g r e e
of
generality
(see also
through v e c t o r f i b r a t i o n s by Machado [ 1 6 ] and P r o l l a [ 4 0 ] [35 1
I361 1 * 2)
I n t h e complex c a s e ,
Bishop and W e i e r s t r a s s - S t o n e 3)
I p o i n t o u t t h e work by Machado on the
theorems [ 181
.
W e c a l l a t t e n t i o n t o t h e work by Zapaka on Mergelyan's the-
orem and q u a s i - a n a l y t i c classes [ 65 ] (see a l s o [ 541 ) 4)
.
(See a l s o [ 541 )
.
Weighted approximation i n t h e c o n t i n u o u s l y
differentiable
c a s e was s t u d i e d by Zapata [631 , [ 6 4 1 . 5)
A d e n s i t y theorem f o r polynomial a l g e b r a s of
continuously
d i f f e r e n t i a b l e mappings i n i n f i n i t e dimensions and i t s
relationship
t o t h e Banach-Grothendieck
approximation p r o p e r t y was i n v e s t i g a t e d by
P r o l l a and G u e r r e i r o I 5 3 I (see a l s o [ 38 1 ) . 6)
Nonarchimedean Approximation Theory h a s
P r o l l a [ 561,
and C a r n e i r o [ 7 1
,
[ 8
1
.
been
sthdied
by
2 . APPROXIMATION OF CONTINUOUSLY DIFFERENTIABLE MAPPINGS
I n 1 9 4 7 , M a r s h a l l S t o n e came from t h e U n i v e r s i t y o f Chicago t o l e c t u r e a t t h e U n i v e r s i d a d e F e d e r a l do R i o de J a n e i r o (known t h e n as U n i v e r s i d a d e do B r a s i l ) f o r t h r e e months.
He
offered
a
beautiful
c o u r s e on "Rings of Continuous F u n c t i o n s " . Among o t h e r t h i n g s ,
he
t a l k e d a b o u t h i s c e l e b r a t e d p a p e r A GenehaLized W C i e & A t h U A A A p p h o x i -
m a t i o n Theohem which he had j u s t w r i t t e n . I t was p u b l i s h e d n e x t y e a r i n volume 21 (1948) of Mathematics Magazine. T h i s is a good
example
of an a r t i c l e t h a t became famous i n s p i t e of t h e f a c t
is
that
was
p u b l i s h e d i n an o b s c u r e j o u r n a l . S t o n e ' s c o u r s e d e a l t w i t h c o n t i n u o u s f u n c t i o n s , and was
going
t o have a l a s t i n g i n f l u e n c e on m e . I t was d u r i n g a n d shortly a f t e r i t t h a t , i n 1948, I t h o u g h t of and proved, b u t d i d n o t
gublish
then,
A LOOK AT APPROXIMATION THEORY
31 1
I will
w h a t I c a l l e d the W e i e r s t r a s s - S t o n e theorem f o r modules [ 3 4 ] .
come back t o t h i s a s p e c t i n a b r i e f w h i l e . The r e a s o n I d i d n o t publ i s h r i g h t aw'ay t h a t r e s u l t f o r modules w a s t h i s . I t took u n t i l 1960
to r e a l i z e
- 1961,
years
me
w h i l e I v i s i t e d B r a n d e i s U n i v e r s i t y f o r four months,
th!, i n t e r e s t f o r Approximation Theory o f
modules i n p l a c e
o f a l g e b r a s , and t o g e t s t a r t e d i n w e i g h t e d a p p r o x i m a t i o n p r o p e r f o r continuous functions. I n 1948, I went t o t h e U n i v e r s i t y of Chicago v i s i t during 1948-1950,
for
a two
a t t h e i n v i t a t i o n o f S t o n e . While t h e r e ,
had an a p p o r t u n i t y , i n 1 9 4 9 , o f p r e s e n t i n g a t And& Weil's
I
seminar
the t h e n r e c e n t a r t i c l e "On i d e a l s of d i f f e r e n t i a b l e f u n c t i o n s "
Hassler
year
by
Whitney, j u s t p u b l i s h e d i n volume 70 (1948) of t h e American
J o u r n a l of Mathematics. A f t e r my l e c t u r e ,
I r v i n g Segal
asked
me:
how a b o u t a s i m i l a r r e s u l t f o r a l g e b r a s of c o n t i n u o u s l y differentiable f u n c t i o n s , a l o n g the l i n e s o f t h e W e i e r s t r a s s - S t o n e t h e o r e m ? I n o t h e r
words, t h e problem w a s t o describe t h e c l o s u r e of
a subalgebra
continuously d i f f e r e n t i a b l e functions , or e q u i v a l e n t l y , to
of
describe
t h e closed subalgebras of c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s ,
in
t h e s p i r i t of t h e W e i e r s t r a s s - S t o n e theorem. To t h e b e s t of my knowl e d g e , t h i s problem h a s remained open so f a r : see below f o r t h e con-
jecture t h a t I h a v e i n mind i n t h i s r e s p e c t . Pressed by S e g a l ' s q u e s t i o n , I s t u d i e d i m m e d i a t e l y i n 1949 [ 231
t h e noteworthy case of d e n s e s u b a l g e b r a s , t o o b t a i n t h e f o l l o w i n g r e s u l t . L e t E be a r e a l m - d i f f e r e n t i a b l e ( m = 0 , 1 , f i n i t e dimension. Denote by
...,-)
Cm(E) t h e a l g e b r a o f
m - d i f f e r e n t i a b l e real f u n c t i o n s o n E ,
all
manifold
continuously
endowed w i t h t h e t o p o l o g y
o f uniform convergence on t h e compact subsets of and a l l t h e i r d i f f e r e n t i a l s up t o o r d e r m .
of
T~
E of suchfunctions
312
NACHBIN
(Nl)
F o h euehg
x
E
E, thehe i d
(N2)
F a t euehg
x
E
E,y
E
E
E
f E A
duck t h a t f ( x ) #O.
# y , ,?%thehe0
E, x
f E A nuch t h a t
f(x) # f(y).
F a h evehy
(N3)
x
x , thehe
at
ib
t # 0
and evehy t a n g e n t u e c t o h f
E A
t o E
buck t h a t
These c o n d i t i o n s do n o t depend on m . The case
excluded
m=O
by t h e above r e s u l t i s c o v e r e d by t h e W e i e r s t r a s s - S t o n e
theorem.
Coming back t o t h e q u e s t i o n S e g a l a s k e d m e i n 1 9 4 9 , b i t by b i t I was l e d t o f o r m u l a t e t h e f o l l o w i n g c o n j e c t u r e . I f i t i s t r u e ,
the
Whitney i d e a l theorem a n d t h e above d e n s i t y theorem are subsumed
by
i t . For t h e s a k e o f s i m p l i c i t y o f terminology and n o t a t i o n o n l y , l e t
15 m
us assume t h a t of
f o r some
lRn
n = 1,2,.
16
belongb t o t h e
CeObUhe
modueo
jD"g(x)
06
-
and evehy
U/A
Daf(x) 1 c
oadeh a t
mobt
Notice t h a t
E
doh
A
any
f
K ad
E
U
doh
T~
x E K
and
QO
equivalence are
Cm(U) t h e n f
i d (and ahJay4
contained i n
> 0, theke i b
E
m=
A.
LA a d u b a l g e b k a a d
Cm(U)
in
doh e u e h g compact h u b b e t
ceabb
D"
A
consider the
U, a c c o r d i n g t o which x , y E U
f E C m ( U ) and 06
subset
t o a r b i t r a r y E and t o
C"(U),
f(x) = f(y) for a l l
CONJECTURE 2 :
i s a nonvoid open
E=U
.., e x t e n s i o n
d e f i n e d by A on
U/A
e q u i v a l e n t when
id),
and t h a t
A is a subalgebra of
being easy. I f relation
m
g
any
dome
E
A
pahtiae
OMLy
equivalence buch
that
dehiuadiue
epual t o m.
f belongs to t h e c l o s u r e of
A in
Cm(U)
for
T~
when t h e above c o n d i t i o n h o l d s t r u e f o r e v e r y compact s u b s e t K o f U, by d e f i n i t i o n , n o t j u s t f o r t h o s e K c o n t a i n e d i n
some
equivalence
313
A LOOK AT APPROXIMATION THEORY
class modulo
The above c o n j e c t u r e i s a n a s p e c t of what I called
U /A.
LacaLizabiLity (see below too). If t r u e , t h e above c o n j e c t u r e h a s a n a t u r a l e x t e n s i o n t o modules i n p l a c e of a l g e b r a s . T h e r e i s a more n a i v e c o n j e c t u r e , which i s e a s i l y s e e n
to
f a l s e . W e m i g h t i n d e e d c o n j e c t u r e t h a t e v e r y s u b a l g e b r a A of which i s c l o s e d f o r
p l e convergence a t p o i n t s o f
t i a l s up t o order m. F o r
E of f u n c t i o n s and a l l t h e i r d i f f e r e n -
m = 0, t h i s i s i n d e e d t h e case; as a matter
of f a c t , t h e s t a t e m e n t t h a t g e b r a s of
Co(E)
Cm(E)
of s i m -
i s a l s o c l o s e d f o r t h e t o p o l o g y rms
T~
be
and
T~
= C(E) i s easily
have t o same c l o s e d s u b a l -
T~~
seen
to
be
to
equivalent
the
W e i e r s t r a s s - S t o n e theorem.
Lef: A be f:he n u b a t g e b f i a 0 6
EXAMPLE 3:
f(l/k) = f(0)
n u c h bha-t
Then A i d c t o b e d
60fi
60k
- c ~
a&!
k=1,2,
C1(n) a 6 a t e
... and
f
E
m
tnofieaweh Z,=,f'(l/n)/n
but it in n o 2 c t o n e d
doh
C1 ( R ) 2
=O.
71s.
A few y e a r s a g o , I a s k e d J a i m e Lesmesthe q u e s t i o n o f e x t e n d i n g
t h e above Theorem 1 t o i n f i n i t e d i m e n s i o n s . I a l s o d i d
raise
q u e s t i o n d u r i n g a l e c t u r e I gave a t Madrid, where Jos6 L l a v o n a
that got
i n t e r e s t e d i n i t . Recent work a l o n g t h i s l i n e w a s done by L e s m e s [13] and P r o l l a [ 4 9 ]
,
[ 531 i n B r a z i l , and by Llavona [14 I
,
[15]
i n Spain.
W e now summarize t h a t a s p e c t v e r y s u c c i n t l y , a l o n g t h e l i n e s o f 1381.
L e t E l F be Hausdorff r e a l l o c a l l y convex s p a c e s , E # 0 , U a nonvoid open s u b s e t o f
m
and
E
t h e v e c t o r s p a c e of a l l mappings
=
1,2,.
f :U
+
F
..
,m.
W e d e n o t e by
FfO,
?(U;F)
t h a t a r e c o n t i n u o u s l y m-
d i f f e r e n t i a b l e i n t h e following sense: 1)
f
is f i n i t e l y m-differentiable;
dimensional v e c t o r subspace
S
w e assume t h a t the r e s t r i c t i o n
of
E with
f ] (U
13 S )
t h a t is, f o r every f i n i t e
S # 0
+
fa,(
k
E;F)
U n S nonvoid,
is m - d i f f e r e n t i a b l e i n t h e
c l a s s i c a l s e n s e . Thus we h a v e the d i f f e r e n t i a l s dkf : U
and
NACHBIN
314
for
k =0,1,.
. ., k
5 n,
k
w i t h v a l u e s i n t h e v e c t o r s p a c e gas ( E;F) o f
Ek t o
a l l symmetric k - l i n e a r mappings of 2)
F.
The mapping
i s c o n t i n u o u s of e v e r y
longs t o t h e v e c t o r subspace k - l i n e a r mappings of
We endow
.,
k = 0,1,.
Ek
to
k 5 m . I n p a r t i c u l a r , d k f ( x ) be-
k ~ ; of ~ ) all Is
c o n t i n u o u s symmetric
F.
Cm(U;F) w i t h t h e t o p o l o g y
T~
of seninorms depending on t h e p a r a m e t e r s k ,
fc
d e f i n e d by t h e
4, K, L
a c o n t i n u o u s seminorm on
b e i n g nonvoid compact subsets of
family
F and K , L
respectively.
U, E
W e s h a l l u s e t h e n o t i o n of polynomial a l g e b r a ; see t h e convent i o n on page 6 3 , [ 5 4 1 .
THEOREM 4 :
Let
m 2 1 and A be a p o L y n a m i a l b u b a t g e b t a
SUppObe t h a t t h e t e i n a b u b n e t G c o n t i n u o u n L i n e a z endomohphibmh duch
06
06
t h e v e c t a h Apace
E'
?(U;F).
06
06 all
@ E
E w i t h d i n i t e dimenbionaL h a g u ,
that: 1)
T h e i d e n t i t y mapping
IE
belong4 t o t h e clonute 0 6 G doh
t h e compact-open t o p o l o g y an t h e v e c t o h npace a L L c o n t i n u o u n L i n e a h endomohphibmb 2)
Fon evetry that
06
and e v e h y
f
E
h t h i c t i o n ( f o J) IV = f o (J I V)
A,
; E)
06
U
06
E.
J E G, e v e t y n o n v a i d open nubbet V
J(V) c U
E(E
it 6 o l l o w h t h a t t h e
belongn t a t h e
nuch ze-
ctonuhe i n
316
A LOOK A T APPROXIMATION THEORY
Cm(U;F)
Then A i n d e n s e i n (Nl)
Foh e v e h y
x
(N2)
Fah evehy
x E U, y
that (N3)
6
in
f
E
A
nuch
thehe i b
f
E A
buch
x # y, t h e h e
U,
E
h u c h t h a t f ( x ) # 0.
f E A
U, t h e h e in
f ( x ) # f(y)
Foh e V e h y
don .rm i d and ondy id:
.
x E U, t E E , t # 0,
that
aa ft ( X I If
E
= d f ( x ) (t)
# 0.
i s f i n i t e d i m e n s i o n a l , c o n d i t i o n s 1) and 2 ) of Theorem 4
are s a t i s f i e d by G r e d u c e d t o
IE. Hence Theorem 4 i m p l i e s
Theorem
1. C o n d i t i o n 1) o f Theorem 4 i m p l i e s t h a t Grothendieck approximation p r o p e r t y , t h a t i s , closure of
E' 8 E
in
L(E;E)
E
has
the
belongs
IE
f o r t h e compact-open
Banachto
topology.
the Thus
Theorem 4 leads t o t h e f o l l o w i n g c o n j e c t u r e :
CONJECTURE
5:
F a h evetry
given
E,
t h e 60tLaiuing c o n d i t i o n n a h e equi-
vatent: Foh a h b i t h a h y
U, F , m 1. 1, t h e n e v e h y poLynamiaL
adgebha A i n d e n h e i n
o n l y id) A E
hatihdieh
Cm(U;F) d o h
(Nl)
,
-rm id ( a n d
nubaLwayn
(N2) , ( N 3 ) .
hab t h e Ranach - G h o t h e n d i e c k a p p h o x i m a t i o n p h o p e h t y .
I t i s known t h a t ( C 1 ) i m p l i e s ( C 2 ) . The c o n j e c t u r e i m p l i e s ( C l ) i s an a t t e m p t t o improve Theorem 4 .
that
(C2)
316
NACHBIN
I n t h e d i r e c t i o n of r e s e a r c h t h a t I j u s t mentioned,
there
is
more g e n e r a l l y t h e q u e s t i o n of s t u d y i n g Approximation Theory f o r a l g e b r a s o r modules of c o n t i n u o u s l y d i f f e r e n t i a b l e v e c t o r v a l u e d mappi n g s by u s i n g w e i g h t s . T h i s q u e s t i o n however i s s t i l l wide open,
in
s p i t e o f t h e a v a i l a b l e r e s u l t s . See t h e n e x t s e c t i o n f o r t h e c o n t i n uous c a s e .
3 . WEIGHTED APPROXIMATION FOR MODULES AND ALGEBRAS OF CONTINUOUS F"C-
TIONS L e t m e t a l k now a b o u t t h e W e i e s t r a s s - S t o n e theorem f o r m o d u l e s , how i t l e d m e t o t h e B e r n s t e i n a p p r o x i m a t i o n problem and what I t h e n c a l l e d t h e w e i g h t e d a p p r o x i m a t i o n problem ( o r t h e B e r n s t e i n - N a c h b i n a p p r o x i m a t i o n problem, a c c o r d i n g t o a more r e c e n t terminology by other authors). Let
E be a c o m p l e t e l y r e g u l a r t o p o l o g i c a l s p a c e , and C ( E ) de-
n o t e t h e a l g e b r a o f a l l c o n t i n u o u s r e a l f u n c t i o n s on E endowed w i t h t h e compact-open t o p o l o g y . T h e r e i s t h e i d e a l theorem f o r I f I i s an i d e a l i n
C ( E ) and
b e l o n g s t o t h e c l o s u r e of
-1
1
I
I-'(O)
in
as
C(E) which reads =
nf
,
I f-'(O)
C ( E ) i f and only i f
then
follows. f
E
f vanishes
C (E) on
(0). More g e n e r a l l y , t h e r e i s t h e W e i e s t r a s s - S t o n e
a l g e b r a A of
C ( E ) which r e a d s as f o l l o w s . L e t
l e n c e r e l a t i o n on E d e f i n e d by for every
f E A. C o n s i d e r
x1
A in
E / A
be t h e e q u i v a -
i f xl, x2 E E and f ( 5 ) = f ( x 2 )
n f € A f"(0)
A-'(O)=
o f e q u i v a l e n c e c l a s s e s modulo E / A belongs to t h e c l o s u r e of
- x2
theorem f o r a sub-
which e i t h e r is one
o r e l s e is v o i d . Then
C ( E ) i f and only i f
on e v e r y e q u i v a l e n c e class modulo E / A -1 A (0) i s nonvoid.
and f
f
f E C(E) i s constant
v a n i s h e s on A-'(O)
if
I n t h e i d e a l theorem, we have a module I over the algebra A = C ( E ) .
A LOOK AT APPROXIMATION THEORY
31 7
I n t h e W e i e r s t r a s s - S t o n e theorem, w e have a module bra
A
+
A over t h e alge-
IR g e n e r a t e d b y A and a l l c o n s t a n t r e a l f u n c t i o n s on
w e c o n s i d e r e d j u s t a v e c t o r subspace W of
C(E)
,
w e would
If
E.
have
a
module W o v e r t h e a l g e b r a A of a l l c o n s t a n t r e a l f u n c t i o n s on E . I n t h e succession of these t h r e e c a s e s , t h e algebra of m u l t i p l i e r s vari e s from t h e l a r g e s t t o t h e s m a l l e s t p o s s i b i l i t y c o n t a i n i n g t h e unit. More g e n e r a l l y , l e t A be a s u b a l g e b r a o f
C ( E ) whichwe mayrylw
w
assume t o c o n t a i n t h e u n i t w i t h o u t loss o f g e n e r a l i t y , and l e t a v e c t o r subspace o f
C ( E ) which i s a module o v e r
be
A so t h a t A W C W .
The W e i e r s t r a s s - S t o n e theorem f o r modules r e a d s a s f o l l o w s . 1 n t r o d u c e as b e f o r e t h e e q u i v a l e n c e r e l a t i o n E/A to t h e closure of
s e t K of E
E
W in
on E .
Then
f E C(E) belongs
C ( E ) i f , a n d o n l y i f , f o r e v e r y compact sub-
c o n t a i n e d i n some e q u i v a l e n c e c l a s s modulo E/A and every
> 0, there is
g E W
such t h a t
Ig(x)
-
f ( x )1 <
E
for every x E K.
T h i s i s an aspect of what l a t e r I c a l l e d " l o c a l i z a b i l i t y " (see below). I t i s known t h a t a t o p o l o g i c a l v e c t o r s p a c e
s e n t a t i o n by c o n t i n u o u s r e a l f u n c t i o n s , t h a t i s
W h a s some r e p r e -
W
i s isomorphic and
homeomorphic t o a t o p o l o g i c a l v e c t o r subspace W o f some and o n l y i f
W
i s a Hausdorff l o c a l l y convex s p a c e . Thus t h e f o l l o w -
i n g r e p r e s e n t a t i o n t h e o r y q u e s t i o n a r i s e s n a t u r a l l y . Given l o g i c a l v e c t o r s p a c e W and an a l g e b r a where
if
C(E),
p l i c t l y , w e want t o know when w e can f i n d
A, W
T E A
and
x
E
* A
im: W
*
W
W.
n e v e r p u b l i s h e d . L e t W be a t o p o l o g i c a l v e c t o r s p a c e , and
t i t y o p e r a t o r of
ex-
so t h a t im[ T ( x ) ] =
W e have t h e f o l l o w i n g t h r e e r e s u l t s t h a t I proved i n
a l g e b r a o f l i n e a r o p e r a t o r s of
More
i n some C(E) a s above,
a s u r j e c t i v e v e c t o r s p a c e isomorphism and homeomorphism ia : A
W,
W , when d o e s t h e p a i r A ,
W have some r e p r e s e n t a t i o n by c o n t i n u o u s r e a l f u n c t i o n s ?
i a ( T ) i m ( x ) f o r every
topo-
A o f l i n e a r o p e r a t o r s of
A c o n t a i n s t h e i d e n t i t y o p e r a t o r of
and a s u r j e c t i v e a l g e b r a isomorphism
a
W . Assume t h a t
1956,
A
but
be an
A c o n t a i n s t h e iden-
W and i s commutative ( w i t h o u t commutativity o f
A
318
NACHEIN
w e would below r e p l a c e i d e a l s by l e f t i d e a l s i n For every i d e a l s u b s p a c e of
A
J in
W spanned by t h e
t h a t a subset
X of
,
T(x) with
T E J
and
x E W.
W e say
W i s A-convex if X i s convex and X = f \ J ( X A.
We
say t h a t
+ JW) W
is
A i n case t h e A-convex neighborhoods o f 0 i n W
l o c a l l y convex under
form a b a s i s o f neighborhoods a t
0.
This implies l o c a l convexity i n
A i s reduced t o t h e s c a l a r o p e r a t o r s
t h e u s u a l s e n s e , of c o u r s e . I f of
JW d e n o t e t h e A - i n v a r i a n t vector
let
J i n A o f codimension 1 i n
for all ideals
A).
W , t h e n A-convexity and l o c a l c o n v e x i t y under
A reduce t o
con-
v e x i t y and l o c a l c o n v e x i t y i n t h e u s u a l s e n s e . The above d e f i n i t i o n s a r e subsumed by
$ 3 , [241.
A linear operator
neighborhoods
V of
0
T on
in
W,
W i s s a i d t o be " d i r e c t e d "
when
the
f o r e a c h of which t h e r e i s X = h ( V ) > 0
T(V) C X V , form a b a s i s o f neighborhoods a t 0 ; i n equiva-
such t h a t
l e n t t e r m s , when c o r r e s p o n d i n g t o e v e r y neighborhood U of
0 in
W
w e may f i n d a n o t h e r neighborhood V o f 0 i n W and E > 0 such t h a t k k Urn T ( E V) C U. More g e n e r a l l y , t h e members o f a c o l l e c t i o n C o f k =O l i n e a r o p e r a t o r s on W are s a i d t o be " s i m i l a r l y d i r e c t e d " i f the neighborhoods such t h a t
a t 0.
V of
T(V) C
0 in
W
,
f o r e a c h o f which there i s X = A ( V , T ) > 0 T E C , form a b a s i s of neighborhoods
X V f o r every
D i r e c t e d n e s s o f a l i n e a r o p e r a t o r i m p l i e s i t s c o n t i n u i t y . Both
d i r e c t e d n e s s and s i m i l a r d i r e c t e d n e s s r e d u c e t o continuity when a normed s p a c e . These c o n c e p t s a r i s e o n l y i n t r e a t i n g
more
t o p o l o g i c a l v e c t o r s p a c e s . Thus t h e h y p o t h e s i s i n Theorem t h a t the operators i n i s f i e d when
THEOREM 6:
W is
general 6
below
A be s i m i l a r l y d i r e c t e d is a u t o m a t i c a l l y sat-
W i s a normed s p a c e .
The p a i h A , W ha6
b0Me
h e p h e s e n t a t i o n b y continuous A e a L
a Haubdoadd s p a c e lukich 4~ lady convex
dunc.tionb id and o n L y i6 W
i6
undeh A ,
i n A a t e bimieahdy d i t e c t e d .
and the
0pehU.tOth
A WOK AT APPROXIMATION THEORY
76 the pait
THEOREM 7: h u e
6uncXionn and
undex A ,
S
i 4
A , W han 40me h e p h e n e n t a t i o n b y
06
a wectoh oubnpuce
t h e n t h e q u o t i e n t paih
A / S , WIS
16 t h e p a i h
areal d u n c t i o n b , t h e n 16
dea W
A, W
bpeC.tkae
cont i nuoun
which in
inuahiant
aeptenentation
S i 4 cloaed i n
had n u m e h e p h e d e n t a t i o n
W.
by cant i nuoun
nynt hebi d hoedn i n t h e doLl!owing
S i n a cLoned p h o p e h v e c t o h bubnpuce
A,
W
hab dome
b y Cona%tUOUb heal! 6unct i onn i6 a n d o n l y id
THEOREM 8:
319
t h e n S in t h e intehnection
06
06
W which
in
inwahiant un-
a l l ! C t 0 4 e d w e c t o h n u b 4 p a c ~0 6
w h ich a x e i n v a a i a n t undeh A , have cadimenhion o n e i n W and c o n - .
tain S. The p a s s i n g t o a q u o t i e n t s t a t e m e n t of Theorem 7 i m p l i e s
spec;
t r a l s y n t h e s i s i n Theorem 8 , which may be viewed a s an a b s t r a c t v e r s i o n o f the W e i e r s t r a s s - S t o n e theorem f o r modules. L e t u s a l s o p o i n t
,
then
Theorem 8 becomes t h e f o l l o w i n g s t a t e m e n t . Every c l o s e d p r o p e r
vec-
o u t t h a t , when
tor subspace
A i s reduced t o t h e scalar o p e r a t o r s
S o f a l o c a l l y convex s p a c e
a l l c l o s e d vector s u b s p a c e s o f and c o n t a i n
W
of
W
is t h e i n t e r s e c t i o n
of
W which have codimension one i n
S . As i t i s c l a s s i c a l , such a s t a t e m e n t
is
W
equivalent
t o t h e Hahn-Banach theorem. Thus Theorem 8 may be looked upon
as
a
g e n e r a l i z a t i o n of b o t h t h e W e i e r s t r a s s - S t o n e theorem f o r modules a n d t h e Hahn-Banach theorem f o r l o c a l l y convex s p a c e s . We may t h e n ask t h e f o l l o w i n g n a t u r a l q u e s t i o n . To what e x t e n t
the c o n d i t i o n o f t h e o p e r a t o r s i n
A b e i n g s i m i l a r l y d i r e c t e d i s mu-
c i a l f o r the v a l i d i t y o f Theorem 6, o r Theorem 7 , or Theorem 8 ? Lo-
c a l c o n v e x i t y under
A
i s n o t superfluous.
In fact,
r e d u c e d t o t h e scalars o p e r a t o r s o f
W , t h e n i t may
e v e r y c l o s e d p r o p e r v e c t o r subspace
S of
s l l closed vector subspaces of
and c o n t a i n
S,
letting be
A
false
be that
is the intersection
of
W which have condimension one i n
W
W
i n case W i s n o t assumed t o be l o c a l l y convex.
The
NACHBIN
320
answer t o t h e above n a t u r a l q u e s t i o n i s no. The example t h a t I found i n 1957 l e d m e t o t h e c l a s s i c a l B e r n s t e i n a p p r o x i m a t i o n problem, a s 1 s h a l l describe next.
EXAMPLE 9 :
t i o n s on
Let
R
W be t h e F r g c h e t s p a c e o f a l l c o n t i n u o u s r e a l f u n c -
t h a t are r a p i d l y d e c r e a s i n g a t i n f i n i t y . C a l l
t h e a l g e b r a o f a l l r e a l p o l y n o m i a l s on
a
R . Every
E
A = P (33)
is
that
C(lR)
s l o w l y i n c r e a s i n g a t i n f i n i t y g i v e s r i s e t o t h e c o n t i n u o u s l i n e a r opTa : f E W
erator
+
af
E W
which i s d i r e c t e d i f and only a is bounded.
Thus A may be v i e w e d . a s a commutative a l g e b r a operators of
of
continuous l i n e a r
W c o n t a i n i n g the i d e n t i t y o p e r a t o r o f
W , b u t each such
o p e r a t o r i s d i r e c t e d i f and o n l y i f t h e c o r r e s p o n d i n g p o l y n o m i a l
is
c o n s t a n t . I t i s c l e a r t h a t W i s l o c a l l y convex u n d e r A .
is
w
some
E W
v a n i s h i n g nowhere
i n lR s u c h t h a t
W ( t h i s i s e a s i l y seen t o be e q u i v a l e n t
v a n i s h i n g nowhere i n of
BAP
-2
or BA P
t o e x i s t e n c e o f some
W which i s i n v a r i a n t u n d e r
lR, i t can be shown t h a t A w
any c l o s e d v e c t o r s u b s p a c e o f condimension o n e i n W .
in
b e l o w ) . Then t h e c l o s u r e
p r o p e r v e c t o r subspace o f never vanishes i n
i s n o t dense i n
Aw
w
E
W
t h a t i s n o t a f u n d a m e n t a l w e i g h t i n the sense
R
-1
There
W is a closed
Since
w
i s n o t contained
in
A.
W which i s i n v a r i a n t under
A, having
Thus Theorem 8 d o e s n o t h o l d i n t h i s case due
t o l a c k o f d i r e c t e d n e s s . A f o r t i o r i Theorem 7 a n d Theorem 6
do n o t
h o l d i n t h i s c a s e f o r t h e same r e a s o n . This counterexample l e a d s us t o t h e
CLUbbiCae
&MnAZeh a p p o x i -
m a t i o n p t o b L e m , u s u a l l y f o r m u l a t e d i n t h e f o l l o w i n g t w o forms, where P(lRn)
i s t h e a l g e b r a o f a l l r e a l p o l y n o m i a l s on IRn B AP
and
- 1.
Let
v : IRn
+
IR,
b e an upper s e m i c o n t i n u o u s " w e i g h t "
Cvm(lRn) be t h e v e c t o r s p a c e o f a l l
tends to
... .
f o r n = 1,2 ,
f E C(IEln)
such
that
0 a t i n f i n i t y , seminormed by II f Ilv = s u p { v ( x ) If ( x ) ; x EW
Assume t h a t
vf n
v i s r a p i d l y d e c r e a s i n g a t i n f i n i t y , t h a t is P(Rn) CCv,(*).
1.
A LOOK AT APPROXIMATION THEORY
When i s
dense i n
P(IRn)
321
Cvw(lRn) ? W e t h e n s a y t h a t
mentaL w e i g h t . W e s h a l l d e n o t e by R n
v is a
dunda-
t h e s e t o f a l l s u c h fundamental
w e i g h t s i n t h e s e n s e of B e r n s t e i n . F o r t e c h n i c a l r e a s o n s w e a l s o i n -
rn
troduce the set
rn
Clearly
BAP
i n g to
0
C
f o r a l l k > 0.
Rn
E
This inclusion i s proper.
Rn.
- 2.
vk
o f a l l such v such t h a t
Let
Cw(lRn)
be t h e Banach s p a c e o f a l l
a t i n f i n i t y , normed by
the s p e c i a l case of
E
C(#)tend-
Ilfll= s u p { i f ( x ) I ; x E lRn 1 ;
Cvm(lRn) when
w
v = l . Assume t h a t
rapidly decreasing a t i n f i n i t y , t h a t i s
w a w e i g h t . When i s
f
P(IRn) w dense i n
P(lRn) w
it
is
E C(IRn)
is
and c a l l
Cm(IRn),
C
Cw(lRn) ? W e t h e n s a y
that
w
is a 6undarnentaL w e i g h t . If
w
E
C(IR")
is rapidly decreasing a t i n f i n i t y , then w i s a
f u n d a m e n t a l w e i g h t i n the s e n s e of v a n i s h e s on B AP
- 1. H o w e v e r
v a n i s h on that
and
IRn
IRn
B A P -1
I wI
BAP- 2
i f and o n l y i f
is a fundamental weight i n t h e
a fundamental w e i g h t v i n t h e s e n s e of
a n d may f a i l t o be c o n t i n u o u s .
is
It
B AP
sense
of
-1
my
€3 A P
i n t h i s sense
i s a b e t t e r way o f l o o k i n g a t t h e c o n c e p t
m e n t a l w e i g h t s i n t h e s e n s e of B e r n s t e i n t h a n
never
w
of
funda-
- 2.
The f o l l o w i n g a r e t h e s i m p l e s t c r i t e r i a f o r a n upper s e m i c o n tinuous function
v : IR
+
IR+
t o belong to
rl ,
thus t o
R1 ,
by
i n c r e a s i n g d e g r e e of g e n e r a l i t y : BOUNDED CASE: ANALYTIC CASE:
v
hub a b o u n d e d buppoht.
Thehe ahe
C > 0
and
c > 0 dvh w h i c h , doh any x E IR,
we have
QUASI-ANALYTIC CASE:
We h a v e
1 z;=l -
VM,
=
+
-
whehe,
{oh
NACHBIN
322
m = O,l,...,
In
we b e t
B A P - 1, t h e s u b a l g e b r a
Cvm(IRn), and we have t h e weight BAP
- 2,
i s contained i n
C(IRn)
Cvm(IRn). I n
v i n the definition of
t h e submodule P ( I R n ) w o v e r t h e s u b a l g e b r a
is contained i n of
of
P(IRn)
of
P(IRn)
C (IR")
c,(IRn), and w e have t h e w e i g h t w i n t h e d e f i n i t i o n
P(EP)W. Thus
was l e d
I
t o t h e following general
formulation
of t h e
weighted a p p h o x i m a t i o n phobLem. The v i e w p o i n t t h u s adopted embraces the
Weierstrass
- Stone
theorem f o r modules, t h u s f o r a l g e b r a s ,
B e r n s t e i n approximation problem. A c t u a l l y , it i s guided by
and t h e the
idea
of e x t e n d i n g t h e c l a s s i c a l B e r n s t e i n approximation problem i n t h e same s t y l e t h a t the Weierstrass
- Stone
theorem g e n e r a l i z e s
W e i e r s t r a s s theorem (see [ 3 4 ] f o r d e t a i l s )
.
t h e classical
L e t V be a s e t of upper semicontinuous p o s i t i v e r e a l f u n c t i o n s
on a completely r e g u l a r t o p o l o g i c a l s p a c e E.
d i m c t e d i n t h e s e n s e t h a t , i f vl, v 2 v1 5 X v and
such t h a t
v2
E V,
and any
v E V
E
f
+
is
V
i s called
f E C ( E ) such t h a t ,
a for
> 0 , t h e c l o s e d s u b s e t CxEE; v ( x ) - i f ( x ) l L E I
i s compact, w i l l be denoted by seminorm
V
t h e r e a r e h > 0 and v E V
5 X v. Each element of
w e i g h t . The v e c t o r subspace of C ( E ) o f a l l any
W e assume t h a t
CVm(E).
It f l l v = sup I v ( x )
0
Each
If ( x ) 1 ; x E E
n a t u r a l topology on t h e w e i g h t e d d p a c e
CV,(E)
v
determines a
E V
on
the
CVm(E).
is defined
by
the
f a m i l y of a l l such seminorms. Let
A
C
C ( E ) be a s u b a l g e b r a c o n t a i n i n g t h e u n i t , and W
be a v e c t o r subspace. A s s u m e t h a t W i s a module o v e r A W C W.
A
,
C
CVm(E)
that
is
The w e i g h t e d a p p h u x i m a t i o n pAObeem c o n s i s t s of a s k i n g f o r a
d e s c r i p t i o n of t h e c l o s u r e of
W in
CVm(E) under such c i r c u m s t a n c e s
We s a y t h a t W i s LocaLizabLe undefi A i n Wm(E)when the following
A LOOK AT APPROXIMATION THEORY
condition holds true: i f of
W
in
CVm(E) i f
f(x)1 <
-
(w(x)
belongs t o the closure
f
( a n d always o n l y i f ) , f o r any
and any e q u i v a l e n c e class v(x)
then
f E CV,(E),
323
E
X modulo
f o r any
E
w
there is
E/A,
x
v
V,
any
E W
E >
0
such t h a t
The n t h i c t w e i g h t e d appaoxi-
E X.
mation phab-tern c o n s i s t s of a s k i n g f o r n e c e s s a r y and s u f f i c i e n t c o n d i tions i n order t h a t W e d e n o t e by
W b e l o c a l i z a b l e under G ( A ) a s u b s e t of
A as a n a l g e b r a w i t h u n i t ,
W e a l s o introduce a subset W a s a module o f
t h a t i s , such t h a t t h e s u b a l g e b r a
G(W) of
f o r t h e t o p o l o g y of
A
of
A
C(E)
.
W which t o p o l o g i c a l l y g e n e r a t e s
t h a t i s , the submodule over A of
A,
G(W) i s dense i n W
by
CV,(E).
A which t o p o l o g i c a l l y g e n e r a t e s
G ( A ) and one i s d e n s e i n
g e n e r a t e d by
A in
f o r t h e topology of
W
generated
CVm(E).
A b a s i c r e s u l t i s t h e n t h e f o l l o w i n g one.
THEOREM 10:
w
E
doh
G(w),
any
Addume
thehe id
x
E E.
t h a t , 604
Y
E
rl
v
eUChg
E
V,
euehy
a
E G(A)
and e u e h q
nuch t h a t
Then W i n locaLiza6Le undeh A i n
CVm(E).
W e may combine Theorem 10 w i t h t h e i n d i c a t e d c r i t e r i a f o r memb e r s h i p of
rl.
COROLLARY 11: evekg
L e t u s c o n s i d e r e x p l i c i t l y the a n a l y t i c case.
Anbume t h a t , d o h e v e h y
w E G(W), t h e t e a t e
6 o h any
x
E
E.
Then W
i b
C > 0
and
v
E
V,
c > 0
evehy
a E G(A)
and
nuch t h a t
LocaLizabLe undeh A i n
CV=(E).
A s a p a r t i c u l a r c a s e o f t h e above r e s u l t s f o r modules,
w e have
324
NACHBIN
t h e f o l l o w i n g o n e s f o r a l g e b r a s . For s i m p l i c i t y s a k e , assume t h a t
i s s t r i c t l y p o s i t i v e , t h a t is, f o r every that
v ( x ) > 0 . L e t A be c o n t a i n e d i n
,
there is v E V
su&
We s a y t h a t A i s
lo-
E E
CV,(E).
C V m ( E ) when t h e f o l l o w i n g c o n d i t i o n
calizabte i n f E CV,(E)
x
then
always o n l y i f )
belongs t o t h e c l o s u r e o f
f f
holds
A in
true:
CV-(E)
if
is c o n s t a n t on e v e r y e q u i v a l e n c e class mdulo
W e d e n o t e by
G ( A ) a s u b s e t of
V
A which t o p o l o g i c a l l y
if (and E/A.
generates
A as an a l g e b r a w i t h u n i t , t h a t i s such t h a t t h e s u b a l g e b r a o f A g e n -
e r a t e d by
G ( A ) and one i s d e n s e i n
A
f o r t h e topology of
CVm(E).
The p a r t i c u l a r c a s e i s t h e n t h e f o l l o w i n g one.
W e may combine Theorem 12 w i t h t h e i n d i c a t e d c r i t e r i a f o r membership of
rl.
COROLLARY 1 3 : ahe
C > 0
d o h any
x
and
E E.
W e quote
L e t us c o n s i d e r e x p l i c t l y t h e a n a l y t i c case.
Andume t h a t , 6 0 4 e v e h y
c > 0
buch
Then A [34]
,
i b
v
E
V and evehy a E G ( A ) , t h e t r e
that
localizable i n C V m ( E ) .
[37] for additional details.
A LOOK AT APPROXIMATION THEORY
325
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t h e B e r n s t e i n problem
over f i n i t e dimensional
vector
s p a c e s , Topology 3 ( 1 9 6 4 ) , s u p p l . 1, 1 2 5 - 1 3 0 . [31 ]
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-
complex cases,
302.
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de
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S c o t t a n d Foresman, USA.
E l e m e n t 4 a 6 apphoximatian Rheahy (1967) , Van N o s t r a n d .
R e p r i n t e d ( 1 9 7 6 ) , K r i e g e r , USA. 135 ]
L . NACHBIN, J . B . PROLLA a n d S. MACHADO, Weighted a p p r o x i m a t i o n ,
v e c t o r f i b r a t i o n s and a l g e b r a s o f o p e r a t o r s , J o u r n a l de
Mathgmatiques P u r e s e t A p p l i q u g e s 5 0 ( 1 9 7 1 ) , 2 9 9 [36 ]
L . NACHBIN, J . B . PROLLA and S . MACHADO,
- 323.
Concerning weighted
approximation, v e c t o r f i b r a t i o n s and a l g e b r a s of
opera-
t o r s , J o u r n a l o f Approximation Theory 6 ( 1 9 7 2 1 , 80 - 8 9 . [371
L.
N A C H B I N , On t h e p r i o r i t y o f a l g e b r a s o f c o n t i n u o u s f u n c t i o n s
i n w e i g h t e d a p p r o x i m a t i o n , Symposia Mathematica 17(1976),
169 [ 38
I
- 183.
L. NACHBIN , S u r l a d e n s i t 6 d e s s o u s - a l g g b r e s p o l y n o m i a l e s
d'ap-
p l i c a t i o n s c o n t i n h e n t d i f fgrentiables ,Seminaihe Piehhe
LeLung e t Henhi S k a d a (Andyde) , 1976/77,
Springer Verlag
L e c t u r e Notes i n Mathematics, t o appear. [39]
J . B.
PROLLA,
Vectah
dibhatiann
and
aLgebhan a d o p e h a t o f i b ,
P u b l i c a t i o n s du S g m i n a i r e d ' A n a l y s e Moderne, U n i v e r s i t g de S h e r b r o o k e (1968/69)
,
Canada.
A LOOK AT APPROXIMATION THEORY
[40]
J. B.
329
PROLLA, Aproximaqiio p o n d e r a d a e S l g e b r a s
de operadores,
A n a i s d a Academia B r a s i l e i r a d e C i g n c i a s 43(1971), 23
[41]
L , B.
PROLLA, The w e i g h t e d Dieudonn6 t h e o r e m
for
- 36.
density
in
t e n s o r p r o d u c t s , I n d a g a t i o n e s Ebthermticae 33(1971), 170-175. 142
I
J. €3. PROLLA, Weighted s p a c e s o f v e c t o r - v a l u e d
c o n t i n u o u s func-
t i o n s , A n n a l i d i Matematica P u r a e d A p p l i c a t a 145 [43 ]
J . B.
- 158.
PROLLA, B i s h o p ' s g e n e r a l i z e d S t o n e - W e i e r s t r a s s f o r weighted s p a c e s , Mathematische 283
[44 1
89 (1971),
- 289.
theorem
Annalen 1 9 1 (1971) ,
J . B . PROLLA, Weighted a p p r o x i m a t i o n o f c o n t i n u o u s
functions,
B u l l e t i n o f t h e American M a t h e m a t i c a l S o c i e t y 7 7 ( 1 9 7 1 ) , 1021-1024.
[45 I
J . B.
PROLLA, Weighted a p p r o x i m a t i o n a n d s l i c e p r o d u c t s of iscdu-
l e s o f c o n t i n u o u s f u n c t i o n s , A n n a l i d e l l a S c u o l a Nomle S u p e r i o r e d i P i s a 2 6 ( 1 9 7 2 ) , 5 6 3 571.
-
[46 1
J . B . PROLLA a n d S . MACHADO, W e i g h t e d G r o t h e n d i e c k
subspaces,
T r a n s a c t i o n s o f t h e American M a t h e m a t i c a l S o c i e t y (1973) [471
J. B.
,
247
- 258.
186
PROLLA, Modules od c o n t i n u o u s f u n c t i o n s , i n
Functional A n a e y b i n and A p p l i c a t i a n n ( E d i t o r : L. N a c h b i n ) ,S p r i n g e r
V e r l a g L e c t u r e N o t e s i n M a t h e m a t i c s 384 (19741, 123- 128. [48]
J . B.
PROLLA, Then c o n d e 4 e n c i a b n a b h e t e o h i a
de aphoximacion,
P u b l i c a c i o n e s d e l D e p a r t a m e n t o de E c u a c i o n e s F u n c i o n a
-
lest U n i v e r s i d a d d e S e v i l l a ( 1 9 7 4 1 , S p a i n . (491
J. B.
PROLLA, On p o l y n o m i a l algebras o f c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s , R e n d i c o n t i d e l l a Accademia N a z i o n a l e d e i L i n c e i 57 ( 1 9 7 4 ) , 481
[501
J . B.
- 486.
PROLLA, On o p e r a t o r i n d u c e d t o p o l o g i e s , i n A n a l y n e
tionnelee e t Applications 225 - 2 3 2 ,
Hermann, P a r i s .
( E d i t e u r : L. N a c h b i n )
Fonc-
(1975) ,
NACHEIN
330
[51]
J.
1521
J . B. PROLLA,
B. PROLLA,
Dense approximation f o r polynomial algebras, Bonner
Mathematische S c h r i f t e n 8 1 (1975) , 115
- 123.
Aphaximacizn en litgebhab p a l i n o m i c a b de duncianu dibehenciabled,Publicaciones d e l Departamento de An6l i s i s Matemstico, U n i v e r s i d a d de S a n t i a g o de Compostela (19751, S p a i n .
1531
[54
1
J . B. PROLLA and C . S . GUERREIRO, An e x t e n s i o n
of Nachbin's theorem t o d i f f e r e n t i a b l e f u n c t i o n s on Banach s p a c e s w i t h t h e approximation p r o p e r t y , Arkiv f o r Matematik 1 4 (19761, 251 - 258.
J. B . PROLLA, Apphoximation o d u e c t o t - v a t u e d d u n c t i o n n ,
de Matemitica 6 1 ( 1 9 7 7 ) [55]
J. B.
,
Notas
North-Holland.
PROUA, The approximation p r o p e r t y f o r Nachbin s p a c e s , i n Appaaximation Theohy and F u n c t i o n a l A n a l y b i b ( E d i t o r : J. B. P r o l l a )
, Notas
de Matem6tica (19791, North - H o l l a n d ,
t o appear. [56 1
J. B.
PROLLA, Non-archimedean f u n c t i o n s p a c e s , i n Lineah Spaces
a n d Appkoximation ( E d i t o r s : P. L. B u t z e r and B.Sz-Nagy)
,
I n t e r n a t i o n a l S e r i e s i n Numerical Mathematics 40 (1978) , 1 0 1 - 1 1 7 , B i r k h a u s e r Verlag B a s e l , S w i t z e r l a n d . [57]
J. B. PROLLA and S. MACHADO, S u r l ' a p p r o x i m a t i o n polynomialeen
dimension i n f i n i e , Acted d e l a VT Rzunion du Ghoupernent d e Mathematiciend d ' Exphehbion L a t h e , Palrna de M a l l o r c a 1977, S p a i n , t o a p p e a r . [581
W.
H.
SUMMERS, Weighted bpaceb and w e i g h t e d a p p h o x i m a t i o n , PUb l i c a t i o n s du S i m i n a i r e d'Analyse Moderne , U n i v e r s i t S de Sherbrooke ( 1 9 7 0 ) , Canada.
[59 1
W.
H.
SUMMERS, The bounded case of t h e weighted
approximation problem, i n FunctionaL Analydid and A p p t i c a t i o n b (Editor: L. Nachbin) , S p r i n g e r V e r l a g L e c t u r e Notes i n Mathematics
384 ( 1 9 7 4 ) , 1 7 7 - 183.
331
A LOOK AT APPROXIMATION THEORY
160 ]
W.
H. SUMMERS, Weighted a p p r o x i m a t i o n f o r modules o f c o n t i n u o u s f u n c t i o n s 11, i n Anak?yne F a n c t i o n n e L l e ( E d i t e u r : L . Nachbin) ( 1 9 7 5 ) , 2 7 7
[61]
G.
I . ZAPATA, A p t o x i m a C Z a
- 283,
et
Appticationd
Hermann, P a r i s .
p o n d e a a d a paka d u n ~ o e b &&?hen&&&,
M o n o g r a f i a s do C e n t r o B r a s i l e i r o de P e s q u i s a s F i s i c a s 30
(1971) , B r a s i l .
[62]
G.
I . ZAPATA, S u r le problsme d e B e r n s t e i n e t l e s a l g s b r e s
de
f o n c t i o n s c o n t i n h e n t d i f f g r e n t i a b l e s , Comptes Rendusde
1'Acadgmie des S c i e n c e s de P a r i s 274 (1972) , 70 [631
G.
- 72.
I . ZAPATA, B e r n s t e i n a p p r o x i m a t i o n problem f o r differentiable
f u n c t i o n s and q u a s i - a n a l y t i c weights, T r a n s a c t i o n s
of
t h e American M a t h e m a t i c a l S o c i e t y 182 (19731, 503-509. [64]
G.
I . ZAPATA, Approximation f o r w e i g h t e d a l g e b r a s o f d i f f e r e n -
t i a b l e s f u n c t i o n s , B o l l e t t i n o d e l l a Unione I t a l i a n a 9 ( 1 9 7 4 ) , 32 [651
G. I . ZAPATA, Weighted a p p r o x i m a t i o n , Mergelyan t h e o r e m and q u a s i - a n a l y t i c w e i g h t s , A r k i v f o r Matematik 1 3 ( 1 9 7 5 ) ,
255 [66 ]
Matematica
- 43.
G.
- 262.
I . ZAPATA, Fundamental seminorms, i n A p p a o x i m a t i o n Theoty and
F u n c t i o n a l Anadgdid
( E d i t o r : J. B . P r o l l a ) ,
Matemztica (1979) , N o r t h - H o l l a n d ,
t o appear.
Notas
de
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. ProZla led. I @North-HoZZand PubZishing Company, 1979
BANACH ALGEBRAS OVER VALUED FIELDS
LAWRENCE N A R I C I
St. John's University
Jamaica, N e w York, 11439, USA and EDWARD BECKENSTEIN
S t . John ' s U n i v e r s i t y S t a t e n I s l a n d , N e w York 1 0 3 0 1 , USA
ABSTRACT By " G e l f a n d t h e o r y " h e r e is meant t h e s t u d y o f t h e c o n s e q u e n c e s o f t o p o l o g i z i n g t h e maximal i d e a l s of a Banach algebra.
The
i s most r i c h when t h e u n d e r l y i n g f i e l d i s t h a t of t h e complex
R o r some o t h e r v a l u e d
b e r s . I f t h e u n d e r l y i n g f i e l d is
theory num-
field,
a
t h e o r y c a n s t i l l b e d e v e l o p e d however and t h a t i s d i s c u s s e d here. F i r s t t h e G e l f a n d t h e o r y for complex Banach a l g e b r a s i s reviewed
briefly;
t h e n t h e a n a l o g o u s t h e o r y f o r t h e case when t h e f i e l d c a r r i e s a nonarchimedean r e a l - v a l u e d v a l u a t i o n i s p r e s e n t e d . I n t h e c o u r s e o f t h e
l a t t e r d i s c u s s i o n , a S t o n e - W e i e r s t r a s s t h e o r e m is needed. I n t h e l a s t p a r t of t h e p a p e r some versions of t h e S t o n e - W e i e r s t r a s s
theoremwhich
h o l d i n a l g e b r a s of c o n t i n u o u s f u n c t i o n s over f i e l d s w i t h n o n a r c h i medean v a l u a t i o n are d i s c u s s e d .
1. CLASSICAL GELFAND THEORY.
If
I,
G i s an open s u b s e t o f
a t o p o l o g i c a l vector s p a c e , a map
[9
1
C , t h e complex numbers,
x:G + X
333
and
X
is
is a n a l y X h i n G i f t h e
NARlCl and BECKENSTEIN
334
d i f f e r e n c e q u o t i e n t has a l i m i t a t each p o i n t i n
G.
For t h e v e c t o r - v a l u e d v e r s i o n o f L i o u v i l l e ' s theorem t o
hold,
t h e v e c t o r s p a c e must have a good s u p p l y o f c o n t i n u o u s l i n e a r f u n c t i o n a l s . The d u a l s p a c e X' must be t o t a l i n t h e s e n s e t h a t i f e v e r y vanishes a t
f E X'
x , then x must be
1.1. LIOWILLE'S THEOREM:
0.
16 X LA a TVS
and
X'
i A
totae then
, p.211).
([l1
x : & + X in e n t i h e and b o u n d e d , t h e n x mu4.t be c o n h t a n t .
id
F o r t h e remainder o f t h e r e s u l t s i n t h i s s e c t i o n w e a s s u m e t h a t X is a complex commutative Banach a l g e b r a w i t h i d e n t i t y e
A complex number
A
v e r t i b l e . The s e t
06
i s a hegueah p a i n t
if
x E X
x
-
(11 e 11 =1). X e is i n
-
p ( x ) of r e g u l a r p o i n t s o f
over t h e r e s o l v e n t map
rx : p ( x )
X,X
+
x i s an open set.More-1 ( x -Xe) i s a n a l y t i c , "11,
+
p. 2 0 8 ) . An i m p o r t a n t consequence o f t h e s e r e s u l t s i s :
1.2.
GELFAND - M A Z U R THEOREM ( [ i ] , p.
bpecthum
u(x)
06
212):
x, t h o b e compgex numbetrn
( a ) F o t evehy A
doh
which
x E
x,
the
-
he
ib
x
n o t i n v m t i b e e , i b n o t empzy. ( b ) 16 X i n
a d i v i n i a n a l g e b t a [ a l l nanzeto elementn have i n -
v e t b e d ) t h e n X i6 i n o m o t p h i c and i n o m e t h i c t o
PROOF:
(b) Since
a(x) #
@, x
Since X i s a d i v i s i o n algebra,
-
Xe x
-
&.
i s n o t i n v e r t i b l e for sore A Xe
must be
E
Q.
0 , i . e . x = Xe.
The proof of p a r t ( a ) depends h e a v i l y on t h e L i o u v i l l e theorem. Consequently one would s u s p e c t t h a t t h i s r e s u l t would
not
transfer
e a s i l y t o Banach a l g e b r a s o v e r o t h e r f i e l d s , and i n d e e d t h i s i s c a s e . Even i n r e a l Banach a l g e b r a s t h e r e may be e l e m e n t s w i t h
the empty
spectrum. A s l o n g as t h e u n d e r l y i n g f i e l d i s Q , however, wecan obtain v e r s i o n s o f t h e above r e s u l t f o r l o c a l l y convex Hausdorff and l o c a l l y m-convex a l g e b r a s ( [ lI
,
p . 212
- 3).
algebras
The o n l y change that
336
BANACHALGEBRASOVERVALUEDFIELDS
o c c u r s i s t h a t t h e " i s o m e t r y " of p a r t ( b ) i s r e p l a c e d by "homeomor phism". For a t i m e i t was wondered ( [ 6 1 ) l o g i c a l d i v i s i o n a l g e b r a s o t h e r t h a n Q.
i f t h e r e w e r e complex top.
1 ,[ 1, p. 2141)
Williamson "12
showed t h a t t h e r e were by p r o v i d i n g an a l g e b r a i c a l l y c o m p a t i b l e
t with
Q(t) of r a t i o n a l f u n c t i o n s i n
pology f o r t h e f i e l d
-
to-
complex
coefficients. ( b ) above i s t h a t
An i m p o r t a n t consequence o f maximal i d e a l M of
X
. We
(I:
for any
d e n o t e t h e c o s e t (complex number) x + M by
x(M). I t now becomes p o s s i b l e t o view on t h e s p a c e M o f maximal i d e a l s of function
is
X/M
2 which s e n d s M i n t o
X as a collectionoffunctions X
.
We a s s o c i a t e x
x(M).Once M h a s been
X as a c o l l e c t i o n
it becomes p o s s i b l e t o view
of
X with
the
topologized,
c o n t i n u o u s func-
t i o n s mapping M i n t o Q. Among o t h e r t h i n g s , even w i t h o u t
endowing
M w i t h a t o p o l o g y , i t now f o l l o w s t h a t
a(x)
1.3
= %(MI.
I n a l g e b r a s o f c o n t i n u o u s o r a n a l y t i c f u n c t i o n s ( [ 1 1 , p.202-3) c h a r a c t e r i z a t i o n s such as 1 . 3 are t h e r u l e f o r d e s c r i b i n g
spectra,
i . e . , t h e spectrum of a f u n c t i o n x i s i t s r a n g e . We endow M w i t h t h e weakest t o p o l o g y which w i l l make each t h e maps
Z
c o n t i n u o u s and c a l l t h i s t h e Gelband t o p o l o g y .
of then
M
becomes a compact Hausdorff s p a c e .
i s a Banach a l g e b r a w i t h i n v o l u t i o n s a t i s f y i n g t h e
A B*-algebra
condition
II x* x 11 = I1 x 11 2 . The c e l e b r a t e d r e p r e s e n t a t i o n theorem
of
Gelfand and Naimark states:
1.4. REPRESENTATION OF B*-ALGEBRAS ( 1 1
algebta, t h e n X
06
CVfltiflUVUb
maximal i d e a &
i b
1
, p.
259f.
:
iboaethically i ~ o m a ~ p h ti oc t h e algebha
c o m p l e x - v a l u e d dunctiond on t h e compact 06
16 X i b a
X with
bUp
nahm ( a n d p V i n t w i b e
bpaCC
VpehUtiVnb).
C(M
U
B*I
Q)
04
NARlCl and BECKENSTEIN
336
2.
GELFAND THEORY OVER VALUED F I E L D S
Here w e assume t h a t
X i s a commutative Banach
algebra
with
i d e n t i t y o v e r a f i e l d F where t h e norm on X and t h e v a l u a t i o n on F each s a t i s f y t h e s t r o n g IIx + y 11 5 max (Ilx 11 t y a r e t h a t if
,
("nonarchimedean")
II y I1 )
. Among
IIxII ZIlyll , t h e n
triangle
inequality:
t h e consequences of t h i s i n e q u a l i IIx+ yII =max (Ilxll
, IIy 1 1 )
and t h a t
e v e r y p o i n t i n a s p h e r e i s a c e n t e r . A l l norms and v a l u a t i o n s areassumed r e a l - v a l u e d .
A d e t a i l e d d i s c u s s i o n of such normed s p a c e s
a l g e b r a s can be found i n
[lo ] ,
and
such s p a c e s b e i n g c a l l e d n o m c k i m e d u n
hpaces, The c r i t i c a l r e s u l t ( ( 1 . 2 ) ) t h a t
each
e l e m e n t have
nonempty
spectrum f a i l s t o h o l d f o r nonarchimedean a l g e b r a s . There may b e e l e ments w i t h empty spectrum ( [ l o ] , p. 1 0 5 ) . The w o r s t consequence t h i s is t h a t w e c a n n o t s a y t h a t
X.
X/M
X/M
i s merely a s u p e r f i e l d of
F.
of
i s F f o r e a c h maximal i d e a l of If we hypothesize
separately
t h a t e a c h element have nonempty spectrum t h e n , e x a c t l y a s i n p r o o f o f (1,2) ( b ) , d i v i s i o n a l g e b r a s are i s o m e t r i c a l l y i s o m o r p h i c t o t h e
d e r l y i n g f i e l d . We d e f i n e a Geldand atgebha t o be
a
commutative Banach a l g e b r a X w i t h i d e n t i t y such t h a t each maximal i d e a l
M of
un-
nonarchimedean X/M = F
for
X.
Although w e c a n n o t show t h a t each e l e m e n t h a s n o n e m p t y s p e c t r u m i n an a r b i t r a r y nonarchimedean Banach a l g e b r a , w e c a n show f o r any x that
u ( x ) i s c l o s e d and bounded, t h e proof b e i n g a b o u t t h e same
f o r t h e complex c a s e
([lo] ,
p . 114). Thus i f
e a c h e l e m e n t h a s compact spectrum. A l s o ( c f . true that
u ( x ) = G(M)
F
is locally
(1.3) 1 i t i s
as
compact, generally
fl F.
I n an a t t e m p t t o d u p l i c a t e t h e complex Gelfand t h e o r y , w e wish t o i n t r o d u c e a t o p o l o g y t o t h e maximal i d e a l s . Two main c h o i c e s
are
a v a i l a b l e : R e s t r i c t c o n s i d e r a t i o n of what e l e m e n t s x are t o b e chosen
or c o n s i d e r o n l y c e r t a i n maximal i d e a l s . More s p e c i f i c a l l y we consider
([lo] , p . 1 1 7 f . l :
337
BANACH ALGEBRASOVER VALUED FIELDS
2.1.
THE GELFAND SUBALGEBRA
maximal i d e a l M I x(M)
2.2.
9: or
F
E
THE GELFAN'D IDEALS
f o r every
X
x
Those
E
such t h a t f o r
X
every
Those maximal i d e a l s M such that x(M) E F
:
Mg
x.
I n t h e f i r s t c a s e w e r e t a i n a l l t h e M ' s ; i n t h e s e c o n d , a l l the x's.
I t now f o l l o w s t h a t ( a ) f o r each
M E M, M 0 X
9 = F); ( b ) X = X
(i.e.l X / M n X
i s a Gelfand
iff M = M 4 9 g (X i s a Gelfand a l g e b r a i f f e a c h maximal i d e a l i s a Gelfand i d e a l or
(maximal) i d e a l i n X
9
9
X c o i n c i d e s w i t h i t s Gelfand s u b a l g e b r a ) ;
gebra o f
(c) X
g
is a closed subal
-
X.
W e may now c o n s i d e r t h e f o l l o w i n g t o p o l o g i e s .
Define t h e w c a k G e L d a n d Z o p o L o g y
2.3. THE WEAK TOPOLOGY: weakest topology f o r
M such t h a t each
i n d u c e s t h e weak Gelfand t o p o l o g y on
2.4.
x
E
X
4
t o be t h e
i s continuous.
Mg'
Define t h e s t t o n g GcLdand t o p o l o g y
THE STRONG TOPOLOGY:
t h e weakest t o p o l o g y f o r M
g
This
t o be
such t h a t e v e r y X E X i s c o n t i n u o u s . T h i s is
c l e a r l y s t r o n g e r t h a n t h e weak Gelfand t o p o l o g y .
REMARKS: M
9
( a ) S t r o n g t o p o l o g i e s y i e l d s p a c e s w i t h more s t r u c t u r e . (b)
i s g e n e r a l l y n o t b i g enough t o y i e l d i n f o r m a t i o n a b o u t
t h e Gelfand i d e a l s
M
of
X
X whereas
are r i c h enough t o h e l p d e s c r i b e
X
g' a ( x ) = B ( M ) . ( c ) These t o p o l o g i e s a r e unigg 9' a r e complete. Thus M o r M i s compact f o r m i t i e s and M and M g 99 9 gg i f f t h e y are t o t a l l y bounded.
e.g.
if
x
E
X
99
g
then
The l a s t remark h e l p s t o o b t a i n t h e f o l l o w i n g compactness sult.
re-
NARlCl and BECKENSTEIN
338
2.5.
COMPACTNESS
and
Mgg
([lo],
7 6 F in eocaU?y c o m p a c t t h e n
p. 124):
ahe n t h u n g e y c o m p a c t . Convehnek?y i d
Mgg
11X
A4
oh
h t h o n g l y compact, t h e n e i t h e t F in L o c a l L y compact o h t h e 06
any element i n X
2.6.
9
i b
in 0 - d i m e n n i a n a L and each
ad t h e npaeen
Each
06
npecthum
t h e Geldand t o p d O g i 5 5
Mg' Mgg'
M
1 3
t o p o C o g y i b t o t a L C y d i b c o n n e c t e d and Haundoh66.
([lo],
in
9
n o n e m p t y , compact, and nowhehe d e n b e .
DISCONNECTEDNESS ([lo], p . 1 2 5 ) :
2.7. SEPARATION
Mg
p. 1 2 6 ) :
X
4
i n fie nfivng
T h e d o L l o ~ i n 9n t a t e m e n . t n a t e e q u h a -
Cent. (a) The (weak1 Geldand t a p o l o g y on M ( b ) T h e dunctionn dhom
X
(c) The dunctiono dhom X ( d ) The map M + M n X
g
in
g g
i n Haundohdd.
nepahate p a i n t n .
nepahate pointb
hth0Mgly.
1-1.
Maximal i d e a l s must a l w a y s be o f codimension 1. C o n v e r s e l y , i n
I , Gleason p r o v e d t h a t a l i n e a r s u b s p a c e o f codimension
1
in a
complex commutative Banach a l g e b r a w i t h i d e n t i t y i s a maximal
ideal
I 5
i f f i t c o n s i s t s o f s i n g u l a r e l e m e n t s . Hence,in a nonarchimedean Banach a l g e b r a , one might c o n s i d e r t h e q u e s t i o n : I f
M i s a l i n e a r subspace
o f codimension 1 c o n s i s t i n g s o l e l y o f s i n g u l a r e l e m e n t s , must M b e a G e l f a n d i d e a l ? The f a c t t h a t G l e a s o n ' s
argument uses d e e p theorems
from complex v a r i a b l e t h e o r y g i v e s warning t h a t
the
nonarchimedean
q u e s t i o n c o u l d be d i f f i c u l t . In [ 2 ]
t h e a u t h o r s c o n s i d e r e d G l e a s o n ' s q u e s t i o n i n t h e topo-
l o g i c a l a l g e b r a (endowed w i t h t h e compact-open t o p o l o g y ) C (T,F) c o n t i n u o u s f u n c t i o n from a t o p o l o g i c a l s p a c e T i n t o
a
of
topological
f i e l d F . I t i s shown t h e r e t h a t G l e a s o n ' s r e s u l t i s t r u e i f F i s t h e f i e l d of complex numbers, f a l s e i f
F i s t h e reals, a n d t r u e i f F i s
a n u l t r a r e g u l a r f i e l d c o n t a i n i n g a t l e a s t t h r e e p o i n t s under a n y t h e following conditions.
of
BANACH ALGEBRAS OVER VALUED FIELDS
339
1. F
i s n o t a l g e b r a i c a l l y closed.
2. F
p o s s e s s e s a s e q u e n c e o f d i s t i n c t e l e m e n t s converging to 0.
3. F
i s d i s c r e t e l y valued.
4 . The t o p o l o g y of
i s g i v e n by a v a l u a t i o n .
F
is ultranormal.
5. T
2
W e s a y t h a t a Gelfand a l g e b r a i s fiegulah i f t h e f u n c t i o n s s e p a r a t e p o i n t s and closed subsets of
2.8.
REGULAR:
X
i d
M strongly.
f i e g u l a f i i d 6 t h e I w t a k l Geldand t o p o l o g y
c o i n c i d e n w i t h t h e h u l l - k e f i n e l t o p o l o g y on M .
( I10
1
,
on
M
p. 1 3 5 ) .
I n t h e complex case, X i s r e g u l a r i f f t h e h u l l - k a r n e l t o p o l o g y
i s Hausdorff and t h e p r o o f r e l i e s h e a v i l y on t h e compactness o f M i n t h e Gelfand t o p o l o g y . By c!ioosing nonarchimedean a l g e b r a s i n which M
i s n o t compact, one o b t a i n s c o u n t e r e x a m p l e s t o ' i f t h e X is regular".
topology is Hausdorff, t h e n
U be the u n i t b a l l i n
Let
each maximal i d e a l M 1 . that
U C W. If
hull - k e r n e l
X and l e t
S i n c e II x(M) 11'
U = W, w e c a l l
II x I1
W = { x I Ilx(M)II
f o r every M
I
5
it i s clear
X a v*-aLgebaa.As w i l l ba seen shortly,
t h e V*-algebras are t h e nonarchimedean a n a l o g s o f B*-algebras (2.10)).
for
1
I t i s e a s y t o v e r i f y ([lo 1
I
(
see
p . 1 4 8 ) t h a t V*-algebras m u s t be
semisimple.
2.9.
16
T i n a 0 - d i m e n n i o n a e compact Haubdofid6 Apace and F in com-
p l e t e t h e n T in homeomofiphic t o t h e n p a c e M a d maximal C(T,F) undeh t h e map
t
+
Mt
S
i d
06
= { x E C(T,F) I x ( t ) = 0 ) urhefl \#i c~hhieA
t h e GeLdand t o p o l o g y . A C A ~ ,C ( T , F ) L A a V * - a t g e b k a
a d d i t i o n , id
idcaln
( [ l o ] , 9. 1 5 4 ) . I n
0 - d i m e n h i a n a l , compact and Haubdok.d6 t h e n S 0 ho-
meomohphic t o T id6 C(S,F)
i b
ibomofiphic t o
C(T,F).
As a f i r s t r e p r e s e n t a t i o n t h e o r e m w e have 2 . 1 0 . ( [ 1 0 ] , ~ . 164)
16
Xg i b a V*-Gd6and
aegebfia and
Mg
in compact
NARlCl and BECKENSTEIN
340
then X
9
in i n o m e t h i c a L L y i n o m o h p h i c t o
t o w n 2ha.t id X
C(Mg
I
dhom w h i c h it
p),
a V*-Gebaand a t g e b t a i n w h i c h
X id
in idorne-th.icaley i n o m o h p h i c t o
bl
601-
i d compact t h e n
C(M,F).
F o r t h e p r o o f of ( 2 . 1 0 ) one n e e d s a version of
a Stone-Weierstarss
t h e o r e m f o r a l g e b r a s of c o n t i n u o u s f u n c t i o n s which t a k e v a l u e s i n
a
nonarchimedean v a l u e d f i e l d . Such t h e o r e m s a r e t h e s u b j e c t o f t h e n e x t and l a s t p a r t o f t h e p a p e r .
3 . STONE-WEIERSTRASS THEOREMS
F d e n o t e s a f i e l d w i t h nonarchimedean v a l u a t i o n . G e n e r a l i z i n g a r e s u l t of Dieudonn6 ([ 4 ] ) , K a p l a n s k y ( [ 7 1 1
ob-
t a i n e d t h e f o l l o w i n g a n a l o g of t h e c l a s s i c a l S t o n e - W e i e r s t r a s s t h e o -
rem.
3.1.
KAPLANSKY-STONE-WEIERSTRASS
THEOREM:
([ 7
1,
i n a compact Haundohdd n p a c e and Y a nubaegebha & a t e 4 p o i n t s and COntainb C O n b . t U M f b t h e n
[
10, p . 162 ]
06
'Id T
:
C ( T , F ) w h i c h nepa-
Y i n dende i n
C(T,F).
An immediate c o n s e q u e n c e o f t h i s is
3.2. and
([
71,
[ 1 0 , p.
1631 1:
Y a bubatgebha
05
16 T i n a LocaLLy c o m p a c t Haundohbd n p a c e
C-(T,F)-continuoud
buncfionb which vanidh a t
i n d i n i t y - w h i c h n e p a t a t e n p a i n t d and c o n t a i n d conn.tan;tA then Y i n denbe in
Ca(T,F). A s h a s b e e n o b s e r v e d b y Nachbin
( [ a 1 ) , i t is
n o t r e a l l y neces-
s a r y t o c o n s i d e r s u b a l g e b r a s Y f o r S t o n e - W e i e r s t r a s s t y p e theorems: sub-modules s u f f i c e . T o q u o t e just one o f many p o s s i b l e i l l u s t r a t i o n s of t h i s
viewpoint ([ 3 ]
I
f o r example) w e h a v e t h e f o l l o w i n g r e s u l t of
P r o l l a ' s. 3.3.
( [ 111
,
Cor. 2.5):
Le-t T be a compact Haubdohd6
dpace,
X
a
341
EANACH ALGEBRASOVER VALUED FIELDS
n o n a h c h i m e d e a n nohmed b p a c e o v e h
F
whehLe A i n a n e p a h a t i n g n u b a e g e b h a Then i d
denne i n
and
Id a n A-bubmoduLe
06
W i d denne i n C(T,X)
06
C(T,x),
C(T,F). d o h eaclz
t i n T, V 7 ( t ) ={w(t)lwEW}
X.
REFERENCES
[ 11
E. BECKENSTEIN, L. N A R I C I a n d C .
SUFFEL, T o p o L o g i c a L A l g e b h a b ,
North-Holland P u b l i s h i n g Co., [ 21
E. BECKENSTEIN, L. NARICI,
C.
Amsterdam, 1977.
SUFFEL and S . WARNER,
Maximal
ideals i n algebras of c o n t i n u o u s f u n c t i o n s , J. Anal. Math. 31(1977) , 293
[
31
[ 41
- 297.
R. C . BUCK, A p p r o x i m a t i o n p r o p e r t i e s of vector - v a l u e d
t i o n s , P a c i f i c J. Math. 5 3 ( 1 9 7 4 ) , 85
J. DIEUDONNE, S u r l e s f o n c t i o n s c o n t i n u e s p - a d i q u e s , Math. 6 8 ( 1 9 4 4 ) , 79
51
[ 61
A.
- 95.
func-
Bull.Sci.
GLEASON, A c h a r a c t e r i z a t i o n of maximal i d e a l s , J.Anal. Math.,
vol. 1 9 ( 1 9 6 7 ) , 1 7 1
- 172.
I . KAPLANSKY, T o p o l o g i c a l r i n g s , B u l l . Amer. M a t h . SOC. 45(1948) 809
[ 71
- 94.
- 826.
I. KAPLANSKY, T h e Weierstrass t h e o r e m i n f i e l d s w i t h valuations, P r o c . Amer. Math. SOC. 1 ( 1 9 5 0 ) , 356 - 3 5 7 .
[ 81
L. NACHBIN, A p p h o x i m a t i o n T h e o h y ,
van Nostrand, P r i n c e t o n , l 9 6 7 .
R e p r i n t e d by Krieger P u b l i s h i n g C o . , n u e , H u n t i n g t o n , N. 91
6 4 5 New Y o r k
Ave-
Y., 1 9 7 6 .
M. NAIMARK,.Nohmk?d Ringd, N o r d h o f f , G r o n i n g e n , T h e N e t h e r l a n d s , 1964.
[lo 1
L. N A R I C I , E . BECKENSTEIN a n d G . BACHIVYW, FuncfhnrLt Aaa.tydi.6 and V a L u a t i o n T h e o h g , Marcel D e k k e r , N e w Y o r k , 1 9 7 1 .
NARlCl and BECKENSTEIN
342
[ll]
J . B.
PROLLA, Nonarchimedean f u n c t i o n spaces. T o a p p e a r
Birkhauser Verlag, Basel-Stuttgart,
[12]
J. H.
in:
Lineah S p a c u and A p p h a x i m a t i o n ( P r o c . Conf .,Oberwolfach, 1 9 7 7 ; E d s . P . L. B u t z e r a n d €3. S z . - N a g y ) , I S N M v o l . 4 0 , 1978.
WILLIAMSON, On t o p o l o g i s i n g t h e f i e l d C ( t ) Math. SOC. 5 ( 1 9 5 4 ) , 729 - 734.
,
Proc.
Amer.
Approdmation Theorg and Functional AnaZysds J.B. Prolla l e d . ) 0iVort.h-Holland PA Zishing Company, 1979
APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS
PHILIPPE NOVERRAZ U n i v e r s i t g d e Nancy I Ma t h g m a t i q u e s 5 4 0 3 7 NANCY CEDEX, F r a n c e
If
U
i s a n open and c o n n e c t e d s u b s e t o f
a , an a p p l i c a t i o n
s i o n a l l o c a l l y convex v e c t o r s p a c e E o n (resp. [ -
a,
+ 1
an i n f i n i t e
dimen
-
f :U+a
i s s a i d t o b e h o t o m o h p h i c ( r e s p . p.&.U&ubhahmonLc) i f
)
a)
f
b)
t h e r e s t r i c t i o n of
i s continuous ( r e s p . upper semicontinuous)
f
t o any f i n i t e d i m e n s i o n a l
subspace
i s holomorphic ( r e s p . plurisubharmonic). L e t us d e n o t e by
(resp. P ( U )
H(U)
,
P,(U))
t h e s e t o f holomor-
p h i c ( r e s p . p l u r i s u b h a r m o n i c , p l u r i s u b h a r m o n i c a n d c o n t i n u o u s ) funct i o n s on
U.
If
K i s a compact , s u b s e t of
= Ix E
(U)
In
an,
n
2
2,
1) Any v i n
u,
U
,
l e t u s d e n o t e by
v ( x ) 5 s u p v , wv E P ( U ) ) . K
t h e f o l l o w i n g r e s u l t s are w e l l known ( 3 ) : P(U) i s t h e p o i n t w i s e d e c r e a s i n g l i m i t
of
p l u r i s u b h a r m o n i c f u n c t i o n s i n a s t r i c t l y smaller o p e n
2)
(ie
U'
of
If
U i s pseudo-convex
compact
U
K of
Cm
set
d(U', C U ) > 0).
U)
then
(ie Kp(u)
343
Kp(U)
-
i s compact i n
Kpc(U)
.
U
f o r any
344
NOVERRAZ
If
U i s pseudo-convex,
compact s u b s e t of al,
..., a j
Iv
If
then f o r v i n
U there e x i s t
fll..
Pc(U)
. , f 7.
,
in
E >
0 and K
and
H(U)
p o s i t i v e numbers such t h a t
-
K = KH(U)
sup ai l o g I f i
i
i s compact i--a pseudo-convex open set
U,
then any holomorphic f u n c t i o n i n a neighborhoodof Kcan be a p p r o x i m a t d u n i f o r m l y on K by elements If
u
H(U).
-
A
and U' are pseudo-convex, U C U' t h e n K H ( U ) , = K H ( " , )
f o r any compact s u b s e t of in
of
U i f and o n l y i f
H(U') i s d e n s e
H(U) f o r t h e compact open topology.
P r o p e r t i e s 31, 4 ) and 5 ) have been g e n e r a l i s e d t o l a r g e r c l a s ses of l o c a l l y convex s p a c e s w i t h Schauder b a s i s i n c l u d i n g
Banach
spaces ( 6 ) .
8, c o n d i t i o n
W e s h a l l i n v e s t i g a t e c o n d i t i o n s 1 and 2 . I n
i s o b t a i n e d by r e g u l a r i s a t i o n ( i e c o n v o l u t i o n ) of
se-
v by a D i r a c
quencer so it i s n a t u r a l t o c o n s i d e r s o m e measure.
1)
For t h e sake
of
s i m p l i c i t y w e s h a l l c o n s i d e r h e r e only ( i n f i n i t e dimensional) Banach spaces and Gaussian measures f o l l o w i n g Gross ( 5 ) . I t i s w e l l known t h a t i n a Banach s p a c e E there are no
s t i t u t e t o t h e Lebesgue measure t h a t means t h e r e does n o t e x i s t
sub-
a
measure i n v a r i a n t by t r a n s l a t i o n s o r r o t a t i o n s . A Gaussian measure l.~ on E can be c h a r a c t e r i z e d as follows: there e x i s t s an H i l b e r t space H
v
d e n s e l y and c o n t i n u o u s l y imbeded i n E such t h a t
u
t h e c y l i n d r i c a l Gauss measure on t h e c y l i n d r i c a l s e t s of
arises H
1-I
.
from The
t r i p l e t ( H p , i , E ) is c a l l e d an a b s t r a c t Wiener space. The f o l l o w i n g p r o p e r t i e s hold:
1)
L e t be
T in
P(E,E), i f
and i s u n i t a r y t h e n
p
T restricted t o H
i s i n v a r i a n t by
T
i s i n P(H H ) I-r P I !J ( i e pT-' = 11).
APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS
L e t be
2)
clx(A)
= p(x
+
,A
A)
346
u
Bore1 i n E l t h e n
and p x are
e i t h e r e q u i v a l e n t o r o r t h o g o n a l , t h e y a r e e q u i v a l e n t if and only i f
x belongs t o
H
P
.
W e have t h e f o l l o w i n g Lemma:
16
LEMMA 1:
i n a Gaubbian meabutre on E and
p
hatrmonic 6uncLion i n an open n u b s e t U
doh
E we h a v e
Suppose t h a t v i s bounded from above i n t h e b a l l B ( x , r ) , t h e
mapping
x
+
eiex
- invariant,
but
a pLuhisub-
r bmale enough.
PROOF:
Te
06
i d v in
V(X)
induce& a
u n i t a r y mapping T e on H
P
I
so
u
is
and w e have
5
v(x
+
y e i e )do
.
The r e s u l t f o l l o w s from
Fubini
t h e o r em. L e t us note
P R O P O S I T I O N 1: 1)
A(v,x
A(x,v r )
p(r)A
2)
v(x) = l i m r =O
3)
A(x,v,r)
LA apLutribubhatrmonic d u n c t i o n 06
a c nwex and inctreasing dunc-tion
i n in6initely
any x i n E t h e f u n c t i o n y = 0.
and
0 6 Log r .
A(x,V,r).
L e t us r e c a l l t h a t a f u n c t i o n
entiable a t
x
y
+
cp
H
u - di66etrenZiable. 9 is H -differentiable
u
( x + y ) , d e f i n e d on
Hcl
I
if
for
is differ
-
NOVERRAZ
346
PROOF:
1) i s a consequence of t h e f a c t t h a t p l u r i s u b h a r m o n i c
func-
t i o n s depending o n l y from I1 x I) are l o g a r i t h m i c a l l y convex. S i n c e v i s upper s e m i c o n t i n u o u s , f o r any
2)
5 v(x) +
v ( x + y)
w e have
II y II 5 r X f E hence
for
E
> 0
E
Is a consequence of a r e s u l t of Gross (5).
3)
L e t us n o t i c e t h a t , u n l e s s
v i s continuous, A(v,x,r)
i n g e n e r a l a continuous f u n c t i o n of
is
not
x.
A s a consequence of 2 ) and 3 ) w e have:
A p l u t i b u b h a h m a n i c dunc-tion v
PROPOSITION 2:
wine l i m i t
a nequence
06
06
i-6
L a c a l l y ,the p o i n t -
i n , 5 i n i t d y H - d i d , 5 e h e n t i a b l e @~L5ubha/unonic
iuncztio nA . T h e r e i s a n o t h e r way t o a p p r o x i m a t e bounded f u n c t i o n s : l e t p be a Gaussian measure o f p a r a m e t e r
vt{ll x 11 2
c1
> 01
+
f u n c t i o n Ptf ( x ) = f Ptf
0
i,
if f (x
t
+
+
t > 0 , then
t h e n Gross ( 5 ) h a s proved t h a t
0
1.1
/f(x)
-
f
is uniformly continuous
f u n i f o r m l y on E .
tends t o
For
the
y)ut(dy) is i n f i n i t e l y H -differentiable if
i s bounded and m e a s u r a b l e . Moreover i f
PROOF:
t and
= 1
vt(E)
E
f(y)I 5
< 0, t h e r e i s E
.
<
2 E
n
such t h a t
if
Ix
- yi 5
t < t E .
rl
i m p 1i e s
APPROXIMATION
If
f
OF PLURISUBHARMONIC FUNCTIONS
347
i s only continuous, t h e n t h e convergence of
Ptf
to
f
i s u n i f o r m on e v e r y compact s u b s e t . I t is a l s o w e l l known (1) t h a t t h e r e e x i s t
ceding r e s u l t gives a
separable
f u n c t i o n s are n o t
Banach s p a c e s s u c h t h a t t h e bounded and 'C i n t h e space o f uniformly
several
dense
c o n t i n u o u s and bounded f u n c t i o n s . T h e p r e
uniform a p p r o x i m a t i o n by H-inf i n i t e l y differen-
t i a b l e f u n c t i o n . F o r p l u r i s u b h a r m o n i c f u n c t i o n s t h i s k i n d of approxim a t i o n g i v e s more o r less t h e same r e s u l t as p r o p o s i t i o n 2 . Now w e s h a l l s t a t e t h e f o l l o w i n g p r o p o s i t i o n :
PROPOSITION 3:
Let
U
be a pheudo-convex open b u b b e t o d aBanach bpace
v be a pLuhinubhahmonic
E and L e t
pointwide l i m i t
06 a
d e c h e a n i n g oequence a d con,Chow ( i n
p ~ u h i n u b h a h m o n i c 6unct i onh i n U
Let
COROLLARY: E,
60%
v
6unctian on U , t h e n
i n
the
duct L i p b C c k i t Z )
.
U b e a pbeudo-convex o p e n b u b n e t 0 6 a Bunach
Apace
then
K ad
any compact Aubbet
U.
F o r t h e p r o o f w e s h a l l follow a n u n p u b l i s e d p a p e r o f
C.Herves
and M. E s t e v e z ( 2 ) . They f i r s t g e n e r a l i z e i n t h e Banach case a n i d e a of
( 3 ) : L e t f be a l o w e r s e m i - c o n t i n u o u s f u n c t i o n bounded from
above,
t h e n for any i n t e g e r k d e f i n e
f
k
( x ) = i n f [ kll x
I t i s e a s y t o show t h a t
Moreover
f
Y
fk-l- < fk ( f
-
yll
and
+
f ( y )1
.
1 f k ( x ) - f k ( y ) I -<
i s t h e p o i n t w i s e l i m i t of t h e s e q u e n c e
fk
.
k I1 x
-
yll.
NOVERRAZ
348
If U
U i s pseudo-convex and v i s a p l u r i s u b h a r m o n i c f u n c t i o n i n
w e t a k e t h i s approximation sequence of t h e f u n c t i o n f d e f i n e d by
e x p ( - v)
i n U and z e r o o u t s i d e
and i f w e c o n s i d e r t h e norm
U
If we s t a t e
+ Iw 1
kII z 1 I
on
E x 4
, we
subharmonic i n
U,
moreover
v = l i m [-log f k 1 k
proved.
xo
B
cp(u)
.
there i s v i n
P ( U ) such t h a t
u(xo) > a
P r o p o s i t i o n 3 i m p l i e s t h a t t h e r e i s a d e c r e a s i n g sequence
> sup
K
(vn)
v. in
, hence: K C { X E U, v ( x ) < a ) =
U
n
{x E U, vn(x) < a ] .
vn+l < v
S i n c e K i s compact and
t h e r e i s an index
p
that
We have v (x 1
P
hence
is
Proposition 3
It i s s u f f i c i e n t toprove t h a t
PROOF OF THE COROLLARY:
Pc (U)
have
i s a pseudo-convex domain it follows t h a t -log fk i s p l u r i -
Since
If
.
O
2 v(xo)
xo does n o t belong t o
*
>
c1
sup v K P The c o r o l l a r y i s proved.
such
APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS
340
REFERENCES
11
R. BONIC and J. FRAMPTON, Smooth functions on Banach manifold, J. Math. Mech. 15(1966) , 877 - 898.
2
1
M. ESTEVEZ and C. HERVES, Sur une proprigts de l'enveloppeplurisousharmonique dans les espaces normds, preprint.
3
1
L. HORMANDER, An i n t h o d u c t i o n t o campeex a n a l y d i b , VanNostrand 1966.
14 1
J. P. FERRIER and N. SIBONY, Approximation ponddrde sur une sous-vari6tG totalement r6elle de an, Ann. Inst.Fourier 26 (1976)I 101 - 115.
5 1
H. H. KUO, Gaudhian meabuhe i n Banach d p a c e b , Springer Lecture Notes 464.
61
Ph. NOVERRAZ, Approximation of holomorphic or plurisubharmonic functions in certain Banach spaces. Phoc. on ' I n d i n i A e Dimen&Lonad Holomohphy, Springer Lecture Notes 364,p. 178-185.
1
Ph. NOVERRAZ, P n e u d o - c o n v e x i t e , c o n v e x i t e p o l y n o m i a l e etdomainen d ' h o l o m o h p h i e , North-Holland Publishing Cn., Amsterdam, 1972.
[
[ 7
This Page Intentionally Left Blank
Apprositnation Theory and Functional Analysis J.B. Prolla ( e d . ) @North-Holland PubZishi& Company, 1979
THE APPROXIMATION PROPERTY FOR CERTAIN SPACES OF HOLOMORF'HIC MAPPINGS
OTILIA T. WIERMANN PAQUES Instituto de Matemgtica Universidade Estadual de Campinas Campinas, SP, Brazil
50. INTRODUCTION
If E and F are locally convex complex Hausdorff spaces,
let
JCS(E;F) be the vector space of all Silva-holomorphic mappings from E to F. (See Definition 1.13 below). In section 1, after the preliminary definitions, we study
the
S (E;F) endowed with the topology of uniform convergence strict (see Definition 1.21) compact subsets of E .
on
In section 2, we prove that for a quasi-complete space E
the
space
JC
following properties are equivalent:
(a) E has the S-approximation property (see Definition 1.31); (b) JCS(E;C) 8 F
is
cam dense
in the space JCS(E;F), for every
locally convex space F ;
(c)
JCs
(E;C) with the topology
,
' c ~ has the approximation
prop-
erty. For Banach spaces, Aron and Schottenloher [ 2 1 and have some results about this for the space ( X ( E ; C ) ,
Aron
T~),
[ 1
1
(where
( J C ( E ; C ) , T ~ ) denotes the vector space of all holomorphic mappings fmm
E to C, endowed with the topology compact-open
I wish to
thank Prof. Mgrio C. Matos for
351
T ~ ) .
his
guidance
and
362
PAQUES
encouragement d u r i n g t h e p r e p a r a t i o n of t h i s p a p e r .
91.
SILVA-HOLOMORPHIC MAPPINGS.
I n t h i s p a p e r E and F a r e always l o c a l l y
convex
Hausdorff s p a c e s and U i s a non-void open s u b s e t of
complex
E . BE w i l l de-
n o t e t h e s e t of a l l c l o s e d a b s o l u t e l y convex bounded s u b s e t s of If
B
E
BE
,
EB
i s t h e v e c t o r subspace of
normed by t h e Minkowsky f u n c t i o n a l
pB
E generated
by
determined by B .
E.
and
B
cs(E)
is
t h e s e t of n o n - t r i v i a l c o n t i n u o u s seminorms on E .
1.1 DEFINITION:
n = l,21...;
Let
Eb(nE;F) w i l l d e n o t e t h e
space of a l l n - l i n e a r mappings from
... x E
= E x
E"
En,
F, which a r e bounded on bounded s u b s e t s o f
vector
( n times)
to
endowed w i t h t h e 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 a l l seminorms of t h e form:
where
T E fb(nE;F)
w e d e n o t e II TI1
for a l l
Blr *
xi E EBi
a
I
B1,.. *
rBnr B
.,Bn =
E BE
11
.
b
Notice t h a t
n
EbS( E;F) t h e v e c t o r
Eb(nE;F) of a l l such T t h a t a r e symmetric. For (OE;F) = Lbs(OE;F) = F
1.2 PROPOSITION:
...= Bn = Bl
Lb(nE;F) a r e c a l l e d S i l v a - b o u n d e d @ - b o u n d e d )
n - t i n e a h mappingb. W e w i l l d e n o t e by
f
E c s ( F ) . If B1=
i = lr...ln.
The e l e m e n t s of
of
B
and
and
11 T 11
BIB
= B(T)
n = O , we define
, forevery
T E Lb(OE;F).
16 F i b a c o m p l e t e t o c a t t y c o n v e x b p a c e ,
.in c o m p l e t e . F o h evehy
subspace
E~(%;F)
F , Ebs(nE;F) i n a c t o n e d u e c t o h bubbpace
06
'
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
1 . 3 DEFINITION:
For
n Ts E Ebbs( E;F)
Sn
.
(xl,..
=
id and o n L y .id
-
Z
u
n!
E
06
Let
Eb(nE;F)
P =
5.
T (x,
E E.
and
. ..,x) where
Txo = T . A mapping
To d e n o t e t h a t
n
IN. I f
E
x is r e p e a t e d P :E
+
T = TS
n=1,2, n
PI
we
Pb(nE;F) d e n o t e s t h e v e c t o r s p a c e o f a l l S i l v a - b o u n d e d
l o c a l l y convex topology
P
E
Pb(nE;F),
T~
B E BE
On
times.
s u c h t h a t P(x) =Txn,
corresponds to
F.
...,
i s a Silua-bounded
F
T E Jb("E;F)
T
mogeneous p o l y n o m i a l s from E t o
where
aeon-
Ebs( E ; F ) . Fuhthehmote
ont o
n-homogerztoun poLynomiaL i f t h e r e i s x
n.
n
T 6 Lb(nE;F), x 6 E
to denote
n = 0 , we define
f o r every
sn
T E Lbs(n~;~),
1 . 5 DEFINITION: Tx"
symmetrization
its
T : L b ( n ~ ; ~ +) T~ E L b s ( % ; ~ )
T h e mapping
tinuoun phojection
If
,Xn)
i s t h e s y m m e t r i c g r o u p of d e g r e e
1 . 4 PROPOSITION:
we w r i t e
define
by
Ts
where
T E Lb(nE;F), w e
363
we consider
Pb(nE;F)
write n-hothe
g e n e r a t e d b y a l l seminorms o f t h e form:
and
8
E
cs(F).
Notice t h a t
1.6 PROPOSITION: toh
The mapping
T
E
Ebs(nE;F)
Apace inomokphibm and a homeamohphinm
06
+
?'
the
E
Pb(nE;F), h a uec-
dihbt
o n t o t h e 6eCOnd
PAOUES
364
b p a c e . Moheaveh
5n
1 . 7 REMARK: (Nachbin [ 9
i s t h e best u n i v e r s a l c o n s t a n t o c c u r r i n g
1 1. 1 6 F i n a comp.i?e-te i?ocal.i?y c o n v e x Apace,
1 . 8 PROPOSITION:
1 . 9 DEFINITION:
(k =O,.
P :E
.. I n )
A
+
F
S i l v a - b o u n d c d poLynorniaL P 6hom f o r which t h e r e are
such t h a t
Pb(%;F)
.
n =Oil
i n complete d o h all
mapping
(1).
in
P = P
0
+
.. .
n = 0,1,.
.
+ Pn
..
I
E
to Pk
E
F
Pb
is
a
k
( E;F)
W e w i l l denote by Pb(E;F)
t h e vector s p a c e of a l l S i l v a - b o u n d e d p o l y n o m i a l s f r o m E t o F.
1 . 1 0 PROPOSITION:
way
06
P = Po
waiting
(k = 0 , . . . , n )
16
P E P b ( E ; F ) , P # 0, t h e h e in O M e and a n L y # M e
+
... + Pn,
with
n =0,1,.
.. ,
k
Pk E Pb( E;F)
a ~ d Pn # 0.
1.11 DEFINITION:
i s a series i n
where
An E Lbs(
where
Pn
A dohmai? p o w e h x E E
n
Aehieb
dham E t o F about
5
E E
of t h e f o r m
E;F) ( n = O , l , . . . ) ;
=in E Pb(nE;F)
or of t h e form
(n=O,l,. ..I.
c o e f f i c i e n t s of t h e power series.
An
and
Pn
are c a l l e d
the
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
m
lim
f3( Z
n =o
m+m
6va aLb
x
f
5 +
pBB,
t h e n f 3 ( P n ( t ) )= O ,
1 . 1 3 DEFINITION: A mapping f : U
( S - h o L o m o h p h i c ) on Pm
f
there is
pB > 0 ,
. .)
such t h a t f o r a l l
5 +
satisfying
f(x) =
m
c
Pn(x
n =o
uniformly w i t h r e s p e c t t o
f3 on
t h e n unique a t e v e r y p o i n t
5
tions
1 -m m! 6 f(5)
n=0,1,
Soh
= Pm'
pBB
-
t E E.
C
f3
E
5
E U,
there are
c s ( F ) and
B
f
EE
such t h a t
U,
5)l
5 +
pBB.
The s e q u e n c e
E E l by Lemma 1 . 1 2 .
m! 6 m f ( 5 )
..., a n d
i s s a i d t o be S i L u a - h a l a m a h p h i c
F
+
= 0,
i f , corresponding t o every
U
Pb(%;F) (m = 0 , 1 , .
- 5))
P,(x
355
=
im -
A ~ i,f
is
m
(PnInZo
We s e t t h e
nota-
pmt f o r ri=a,l
,....
The n o t a t i o n m
f(x) =
z
- s m f W( x -
m=o m!
i n d i c a t e s t h e T a y l o r series o f
f
at
5)
I
s p a c e o f a l l S-holomorphic mappings from U t o
1.14 REMARK:
vector
5. JfS(U;F) d e n o t e s t h e
The above c o n c e p t o f holomorphy
F.
was
introduced
by
S e b a s t i z o e S i l v a 1 1 6 1 . W e w i l l d e n o t e by X(U;F) t h e v e c t o r s p a c e of a l l holomorphic mappings from U t o t h i s s p a c e see Nachbin [ 8 ]
a mapping tinuous.
f E JCs(U;F)
and
[ 9 ]
F . F o r some b a s i c p r o p e r t i e s o f
and Noverraz 110 1
i s holomorphic i f and o n l y i f ,
. Notice f
JCS(U;F) = JC(U;F), f o r e v e r y open non-void s u b s e t
is U
that con-
of
a
PAQUES
366
seminormed or a S i l v a s p a c e E and f o r e v e r y l o c a l l y convex Hauscborff space
In general,
F.
1.16 COROLLARY:
that
(1
-
f :U
A mapping
holomorphically
Let
A) 5 + Ax
f
E
5
JCs(U;F),
LA S - h o e o m o h p h i c i6 a n d o n l y
F
+
U 17 EB
i n holomofiphic o n
flUrlEB
a
is
E
(Matos [ 6 ] 1 .
bornological space.
1 . 1 5 PROPOSITION:
x S ( U ; F ) = X(U;F) i f
,
doh evehy
x
E U,
X
U, doh e v e h y
E
t!
E U
, 1
A
f ( ( l - A ) E + Ax) A - 1
B E BE
and
p > 1
I
p T h .e n
~
.
be d u c h
dA
IxI=p
dofi
n=0,1,
...
1 . 1 8 COROLLARY: (Cauchy i n e q u a l i t i e s ) :
5
E U
~ O J L
and
p > 0
n = 0,1,.
pB C U.
f E JC,(U;F), BEcs(F), B E
Then
..
1.19 DEFINITION:
A mapping
f :U
holomofiphic i f f o r every
$ E
d u a l of
$ of
F)
5 +
be nuch t h a t
Led
the function
+
F
is said t o be
F' (where F '
weakLy
denotes t h e
i s Silva-holomorphic.
SLLva-
topological
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
L e t F be a Apace w i t h t h e p h o p e h t y t h a t id K LA
1 . 2 0 PROPOSITION:
a compact b u b n e t
367
06
F , t h e n t h e cLoned a b d a L u t e L y convex
K, r ( K ) , i n a c o m p a c t
06
dubbet
F . Then
f :U
-+
huLt
in
F
176
weahLy
u
S i l v a - h o t o m o h p h i c mapping id and o n t y id f i b S i t v a - h o t o m o h p h i c . The p r o o f o f t h i s p r o p o s i t i o n f o l l o w s from P r o p o s i t i o n 1 . 1 5 a n d
.
Nachbin [ 8 ]
1 . 2 1 DEFINITION: A subset K of E i s s a i d t o be a A t h i c t compact set i f there is pact i n
.
EB
B E BE
such t h a t
K
i s contained i n
E i s normed, o r F r s c h e t ( o r
If
EB
and
,
then
LF )
is
com-
K
E is
s t r i c t compact i f a n d o n l y i f i t i s compact i n E . W e w i l l d e n o t e by
u n i f o r m c o n v e r g e n c e on t h e s t r i c t compact s u b s e t s o f
of
1 . 2 2 PROPOSITION: (JCS(U;F),
PROOF:
T
T
~
)
i d
F id a compLeXe t o c a t l y convex
16
B
E
i s t h e compact
~
i s complete, f o r
thex
bpace,
complete.
(falUnEB
BE
JCs(U;F)
U.
L e t ( f a ) a E I be a Cauchy n e t i n (JCS(U;F) , T ~ )a n d
Then if (
t h e l o c a l l y convex t o p o l o g y on
T~
-
1a E I
13' E c s ( F )
i s a Cauchy n e t i n ( X ( U
open t o p o l o g y ) . W e know t h a t
11 E B ; F ) , ~ C
( X ( U 17 E B ; F ) , ~ O )
F c o m p l e t e . Using t h i s f a c t , i t i s e a s y t o see t h a t
there i s
f E JCS(U;F) s u c h t h a t ( f a ) a E I
(KS(U;F) ,
T ~ ) .
converges
to
f
We now d e f i n e t h e n o t i o n o f S i l v a - h o l o m o r p h i c mapping of
on
com-
p a c t t y p e , which w i l l b e n e e d e d i n t h e n e x t s e c t i o n .
1 . 2 3 DEFINITION:
For
q E E*,
l i n e a r mappings from E
of E, x E E (Pi
and +
E E*,
q(x)
b E F
,
we
b
F
by
i = 1,.
E
. ., n ,
to
where
@,
which a r e bounded o n bounded
denote 7
9
E* d e n o t e s t h e s p a c e
the
S -bounded
b E Lb(nE;F).
n E IN a n d
More
b E F', w e d e n o t e
linear
of
subsets mapping
generally, the
all
S
for
- bounded
358
PAQUES
n - l i n e a r mapping
The v e c t o r s u b s p a c e o f form Lplx
... xPn
ebf ("E;F).
Lb(nE;F)
b , Pi
E
g e n e r a t e d by a l l e l e m e n t s o f
... ,n, a n d
E*, i =1,
W e d e f i n e t h e v e c t o r subspace
be t h e c l o s u r e o f
fbf(nE;F) i n
complete space t h e n
x b f s P ~ ; ~= ) For n = O
1 . 2 5 DEFINITION:
A E Lb(nE;F)
compact Xype i f a n d o n l y i f P E E*,
b
E
Pn
*
n
P b f ( E;F) i n
x
1 . 2 6 PROPOSITION:
-
S i l v a -bounded
~ ( xn) b E F
by
Pb(nE;F) g e n e r a t e d by
all
E
-+
Pbc(nE;F) o f
P b ( n E ; F ) . The t o p o l o g y o n
be t h e i n d u c e d t o p o l o g y by Pbc(nE;F)
A E lbc(nE;F).
cp E E*, b E F
b,
W e d e f i n e t h e v e c t o r subspace
then
define
.
F, w e d e n o t e t h e
b E Pb("E;F). The v e c t o r s u b s p a c e o f
e l e m e n t s of t h e form
of
We
i s s a i d t o b e a S i L v a - bounded n-fineah
n-homogeneous p o l y n o m i a l g i v e n by *
space.
is a
F
n n f b f s ( E;F) = Pbcs( E ; F ) .
Analogously, f o r
Pn
L b ( n E ; F ) . Hence, i f
we define a l l these spaces a s F
06
X b ( n ~ ; ~ ) , to
of
The t o p o l o g y on Lbc(nE;F)
i s a complete
lbc(nE;F)
by
L ("E;F) ~ ~ n L ~ ~ ( % ; Fand ) L ~ ( % ; F )= L ~ ( % ; F ) n E ~ ~ ( ~ E ; F ) .
1 . 2 4 PROPOSITION:
mapping
Lbc("E;F)
Lb?E;F).
w i l l a l w a y s be t h e i n d u c e d t o p o l o g y by
b E F, i s denoted
the
i s d e n o t e d by
Pbf(%:F).
Pb(nE;F) to be the closure
will
Pbc(nE;F)
P b ( n E ; F ) . Hence, i f
always
F i s a complet s w e
is a c o m p l e t e s p a c e .
The n a t u h a L mapping
i n d u c e 6 a topoLogicd and
T E Lbs("E;F)
aLgeblraic i n o m o t p h i s m b e t w e e n
+
?
E
Pb(nE;F)
Pbcs( nE;F) and
.
pbC PE;F) 1 . 2 7 DEFINITION:
P E Pb(nE;F)
i s s a i d to be a Sieva-bounded n-homogeneoud
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
p o l y n o m i a l a 4 compact t y p e i f and o n l y i f 1 . 2 8 DEFINITION:
P
E
369
Pbc(nE;F).
XSc(U;F) b e t h e v e c t o r s u b s p a c e o f
Let
o f a l l S i l v a - h o l o m o r p h i c mappings f : U
+
F, such t h a t f o r
Ks(U;F) each x E U
n E IN, 1 ^6 nf ( x ) E P b c ( n E ; F ) . An e l e m e n t f E JCSc(U;F) will n! b e c a l l e d a S i Q v a - h o l u m o t p h i c m a p p i n g 0 6 c o m p a c R .type 0 6 u i n t o F . and
A main t o o l o f t h i s p a p e r i s the n o t i o n o f € - p r o d u c t
by S c h w a r t z [141
which w e want t o r e v i e w .
1 . 2 9 DEFINITION:
Given two l o c a l l y convex H a u s d o r f f s p a c e s
F , w e d e n o t e by
FA
E
and
E
F endowed w i t h t h e t o p o l o g y o f uni-
t h e dual of
form c o n v e r g e n c e on a l l b a l a n c e d convex compact s u b s e t s of E
introduced
F, a n d by
F = LE(FA,E) t h e s p a c e of a l l l i n e a r c o n t i n u o u s maps from
to
Fi
E , endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e o f a l l e q u i c o n
t i n u o u s subsets o f seminorms
F'
. The
gE(F&,E) is g e n e r a t e d b y t h e
t o p o l o g y on
d e f i n e d by:
p ~a
c s ( F ) and
1 . 3 0 DEFINITION:
a
E
cs(E). W e have t h a t
A l o c a l l y convex Hausdorff s p a c e E
a
the a p p h o x i m a t i o n p h o p e t t y , i f f o r every and e v e r y b a l a n c e d convex compact s u b s e t such t h a t
a(T(x)
1 . 3 1 DEFINITION:
-
x ) < E, f o r a l l
EB
E,
there is
and g i v e n
for all
x E K.
E
E
of
K
cs(E)
,
EEF %FEE.
i s s a i d tohave every
E
> 0,
E , there is T E E' 8 E,
x E K.
A l o c a l l y convex H a u s d o r f f s p a c e E i s s a i d t o h a v e
t h e S - a p p t o x i m a t i o n p h o p e t t y (S.a.p.1 set K o f
-
B
€
BE
> 0, t h e r e i s
,
i f g i v e n a s t r i c t compact sub-
such t h a t T E E* 8 E ,
K
C
EB
a n d i s compact i n
such t h a t
%(T(x)- x)
<E,
PAOUES
360
1 . 3 2 REMARK:
If E h a s t h e
S.a.p.,
and a l l compact subsets
= E*,
E'
are s t r i c t , t h e n E h a s t h e a p p r o x i m a t i o n p r o p e r t y . Hence,
of
E
E
i s a normed s p a c e , o r F r s c h e t , o f
E
h a s t h e approximation p r o p e r t y . I f
sequence
L F , which h a s t h e S . a . p . ,
then
is a n i n d u c t i v e l i m i t
of a
E
,
of Banach s p a c e s En
(En);=o
.
property, then E has the S.a.p.
a)
which h a v e t h e approximation
Enflo i n [ 3 1
a Banach s p a c e which d o e s n o t h a v e t h e
1.33 PROPOSITION:
.
S.a.p.
g i v e s an example of
(Schwartz [14 ] :
L e t E and F b e L o c a l l y c o n v e x Haundoh66 F
E
(&2n4oh p h o d u c t
E-topoLogy) b)
i d
06
and
E
i6
E 8 F
A
d)
E EF.
E EF ,
a l e Eanach dpaced
F.
E B F
i 4
denae i n
E EF,
F.
1 6 E and
F
and E
F ha4 t h e a p p h o x i m a t i o n phopehty, t h e n
otl
doh
q u a n i - c o m p l e t e s p a c e . Then E haa t h e a p p h 0 ~ -
m a t i o n p h o p e h t y id and o n l y i6 do&
the
ha4 t h e apphoximation
i 4 denhe i n
a t e t o c a L t y c o n v e x t i a u ~ d o h d dbpacen
Then
with
a t o p o l o g i c a l ! vecdotr 4 u 6 4 p a c e 0 6
p h o p e h t y i6 and onLy
Let E b e
npacen.
endowed
F,
A L o c a t l y c o n v e x Hauadoh&,j n p a c e E
c)
if
axe t o c a l t y c o n v e x Haundof~66c o m p l e t e bpaceb,
identical t o
F. (E
dE F
denotes t h e cornpLdon
E kE F E
BE F).
W e b e g i n o u r s t u d y w i t h an e x a m i n a t i o n of t h e closure of
the
i 4
E
E
xs (U;C).
52. THE APPROXIPZATION PROPERTY FOR
tensor product
XS(U;C) 0 F
2 . 1 THEOREM:
L e t E h a v e ,the
4u64et 0 6
Then
E.
06
i n (XS(U;F), T ~ ) .
and L e t U b e a baLanced
open
~ ~ - d e n 4i ne
doh
ev5'Lg
be a s t r i c compact s e t . By h y p o t h e s i s t h e r e
is
JCsc(U;C)
S.a.p. 0 F
16
JCS(U;F),
.tocalLg c o n v e x 4pace F. PROOF:
Let
K C U
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
B E BE
such t h a t
every all
Let
x E K.
t h a t there is
EB
U
C
t h e r e is
> 0,
E
K
and i s compact i n
T E E* Q E
f E JCS(U;F)
,E
> 0
such t h a t
EB
,
361
so t h a t
pB(T(x)
-
x) <
for for
E,
B E c s ( F ) . We f i r s t
and
show
5
6 > 0 , 6
i s t h e complement o f
d i s t ( K , C E B ( U 87 EB) ) (where C (U n EB) EB EB U n EB i n E B ) , s u c h t h a t B ( f ( x ) f(y)) < E,
whenever
pB(x
x E K
-
and
-
y) < 6. Since
( P r o p o s i t i o n 1.15), t h e n f o r e a c h
f l U n ~ B is
x E K,
continuous
is
there
6x
>
0,
A x 5 d i S t E ( K , C E (U n E B ) ) , s u c h t h a t B ( f ( X I - f (y)) < E/ 2 , for B B n pB(x y ) < 6 x . S i n c e K C U 17 EB i s compact i s EB , K C . L J B(xi,GXi), 1=1 f o r some s e t {x, xn} C K . ( B ( a , r ) = { x E EB; p B ( x a ) C r , when
-
,...,
and
a E EB
-
Define
r > 0)).
y(x) = sup { 6
Then
y :K
+
Now f o r any B(x,6)
C
B(xi,6
‘i
Since E has the
for a l l
x E K.
for all
x
Let
Uo =
-
pB(x
-
...,n }
xi);
i=l,
i s c o n t i n u o u s a n d y > 0. L e t
R
x E K
E
xi
),
and
y E B(x, 6 )
I
for
x E K.
6 = i n f { y ( x ) ; x E K).
there
is
some
i
with
thus
S.a.p.,
there is
T E E* Q E
such that ~ f , ( T ( x- )x ) c 6 ,
By the a b o v e , w e g e t t h a t
K. L e t
{gl
, .. . , gn}
be a b a s i s i n
T ( E ) and l e t
U n EB n T ( E ) . S i n c e f i s S i l v a - h o l o m o r p h i c ,
f
can
be
c o n s i d e r e d as a h o l o m o r p h i c mapping from t h e f i n i t e d i m e n s i o n a l balanced s e t
Uo
into
F,
PAQUES
n f(z) = f ( B
i=l
where (z,,
..., z n )
f
ECn,
subsets of Uo. S i n c e is
E
P
F
5
zigi)
=
z
IPl= 0
ZPf
P '
and c o n v e r g e n c e i s uniform on compact
T(K) C U
(1
EB
and i s compact i n
there
Uo,
M E IN, such t h a t
Thus, i f
x E K,
Since
t h e proof i s complete. NOW, w e g i v e a n e x t e n s i o n of t h e p r e v i o u s theorem
class of s u b s e t s of
2 . 2 DEFINITION:
t o be
Let
s a i d t o be d i n i t e d y S
2 . 3 REMARK:
E ,
If
U be a non-void open s u b s e t of
Pb(E;C)
- Runge
(Paques [ 111)
2 . 4 THEOREM:
E,
said
T ~ ) .U
is
i n E i f for e a c h f i n i t e dimensional sub-
Eo
*
i s a Banach s p a c e , t h i s d e f i n i t i o n c o i n c i d e s w i t h
t h e D e f i n i t i o n 2 . 1 of A r o n - S c h o t t e n l o h e r [ 2 open s u b s e t o f
is
E. U
is d e n s e i n ( J C s ( U ; ( c ) ,
i s S-Runge i n
U n Eo
E
another
E.
S-Runge in E i f
s p a c e Eo of
to
then
U
1
.
If
U
is
a
i s f i n i t e l y S-Runge and S-Runge
balanced in
E.
. L e t E have t h e
S.a.p.
and L e t
U
b e an o p e n
nubbet
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
06
Jfs(U;C)
E which i h 4 i n i t e d y S-Runge. Then
JCs(U; F)
604
Q
F
i b
383
T
S
-denhe i n
F.
e v e h y l k ~ c a e e y convex hpace
F o r t h e p r o o f o f Theorem 2 . 4 i t w i l l b e n e e d e d
the
following
p r o p o s i t i o n , which h a s i m p o r t a n t c o r o l l a r i e s .
2 . 5 PROPOSITION:
L e t F b e a dpace s a t i d d y i n g t h e doLeowing
tion: I d K i n a compact bubded
v e x huLL
06
F , t h e n t h e ceohed
06
(Pb(nE;F),
( a na lo g o u s 6ohmuLaA hold a b o h a t
f o r all
to
T : 3CS(U;F)
Let
+
f E XS(U;F),
I$
f o r each
JCs(U;C).
E
JCs(U;C) F ' and
E
U
Lo a nun-void
x
f E Jcs(U;F)
ous. Indeed, l e t
U.
and
Clearly,
n E IN).
doh
+
Xs(U;C)
seminorm on
K
x E K}, where
(Tf) ($)
belongs
$ E F'.
Tf : FA
p be a rS-continuous
p ( g ) = sup { Ig(x) 1 ;
-tS),
b e d e f i n e d by (Tf) ($1 ( x ) = ( $ of) (x),
F
W e now show t h a t t h e l i n e a r map
by
F. 7 6
abno.ecl*eLy con-
E , then
o p e n 6 u b d e t 06
PROOF:
06
r ( K ) , i n a compact n u b s e t
K,
C
U
is continu-
JCs(U;C)
defined
is a strict
compact
s e t . By h y p o t h e s i s , t h e closed a b s o l u t e l y convex h u l l o f compact s u b s e t o f f i n e d by
for all
fine
F.
C a l l it
$ E F'.
Hence
Now
E
f(K)
q b e t h e seminorm on
Let
F'
is a de-
L). I t f o l l o w s t h a t
Tf E Z ( F & ; 3 C s ( U ; C ) ) .
A E Xs(U;C)
g ( x ) E (FA)' = F
$ E F'.
L.
q ( $ ) = s u p { II$(t) I; t
L e t now
condi-
E F = L(F;,
JC,(U;C)).
by t h e formula
g is weakly S-holomorphic,
For each
x E U, de-
g ( x ) ( @ ) = ( A @ )( x ) ,
hence
S
for
- holomorphic
all by
PAOUES
364
C l e a r l y , Tq = A, a n d t h e r e f o r e T is onto Xs(U;C) E F .
P r o p o s i t i o n 1.20.
On t h e o t h e r hand, T
i s i n j e c t i v e by t h e Hahn-Banach Theorem.
r e m a i n s t o show t h a t
T i s a homeomorphism.
Let T(g) = s u p
6
I
E c s ( F ) and
Ig(x) I;
x
E
K C U
KI,
b e a s t r i c t compact
g E Jcs(U;Cl.
It
subset.
Let
t h e n , f o r every f EJCS(U;F),
we have by t h e Hahn-Banach Theorem, t h a t
This completes t h e p r o o f .
2.6 COROLLARIES OF THE PROPOSITION 2 . 5 :
nubnet a 6
a)
16 U i d
U
- void
nun
Open
E , we h a v e :
16
F i b a c o m p k t e bpace a n d
F oh (X,(U;C),
T
~
)hub
the
a p p t o ximatio n p ~ ~ o p e t t yt ,h e n
I n p a t t i c u l a t id E had d i n i t e d i m e n n i o n a n d F
i d
a com-
pLete d p a c e , t h e n
b)
16 F had t h e a p p t o x i m a t i o n p t o p e t t y a n d
condition
06
Pmpodition 2.5, then
JCs(U;C)
dadiddied
B F
the
d Ts-deue
i n XS(U;F). c)
(X,(U;C),
ill
T,)
Jc,(U;C)
npuced
F.
had 8 F
t h e a p p t o x i m a t i o n p k o p e t b y id a n d i d
Ts-denAe i n
JcS(U;F),
doh
only
a l e am&
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
366
The proof of a) f o l l o w s from P r o p o s i t i o n 2.5 and
( a ) . The
1.33
Proposition
proof of b ) f o l l o w s from P r o p o s i t i o n 2 . 5 a n d
Proposition
1 . 3 3 (b); and c ) f o l l o w s from P r o p o s i t i o n 2.5 and P r o p o s i t i o n 1 . 3 3 ( c ) and P r o p o s i t i o n 1 . 2 2 .
PROOF OF THEOREM 2 . 4 :
Let
be a s t r i c t compact s e t , B E c s ( F )
K C U
f E JCs(U;F). By h y p o t h e s i s , t h e r e i s
and
EB
,
so t h a t g i v e n
pB(T(x)
-
x) <
and is compact i n satisfying
then
6
5
d i s t E (K,C
B
-
B(f(x)
f(y)) <
f i n i t e dimension a) )
,
@
where
f l E X(Uo;(c)
Let
(JC(Uo;@),T ~ )
fl =
m
2:
'P. 8 2 .
where
There a r e
zj
...,m,
since
i s S-Runge i n
T(E)
It5
-
Uo
E
I'
-
with
JCS(T(E) ;C) 0 F r it f o l l o w s t h a t
2 . 1 THEOREM:
Let E 06
E.
16
S.a.p.
then E
ha6 t h e
PROOF:
W e show t h a t
.
=
-
,
E,
Gj
8 F,
-
2.
3
E
- zj) m
c, j = 1...,m. < ~ / 2 m , and with
-
'Pj
@
j
f o r a l l yET(N.
E,
we g e t f o r a l l
hab t h e
is
there
E JCS(T(E) ;C)
f2(y)) <
has
y E T(K).
for
f 2 = j-1 B
Let
6,
<
T(E)
1 uo
3C(Uo;6) and
are
x)
is
(by C o r o l l a r y 2.6 f
q u a b i - c o m p t e t e Apace and
(JCS(U;C),-rs)
E$
?'
9.11 T ( K ) - B ( z j
11
B(fl(y)
h = f 2 o T I U E JCs(U;C)
open b u b b e t
3
...,m.
Thus l e t t i n g
(I
~ p .E
, there
pjllT(K) B ( z j ) < ~ / 2 m , j =1,
be
pg(T(x)
Thus f o r
F.
f 3 ( f ( U o ( y )- f l ( y ) ) <
with
Q F r
j=l I F, j = 1 ,
-
(JC(Uo;C), T ~ GE )
is t h e completion of
U n EB
there
Uo = U n UB n T ( E ) . S i n c e
Let
E.
C
T E E* Q E
is
x E K,
( U n E B ) ) , such t h a t i f
EB
K
x E K.
As i n t h e proof of Theorem 2 . 1 , whenever 6 > 0,
that
SL&I
> 0, there
E
for a l l
E,
B E BE
x
E
K,
U be a n o n
- void
apphoximatiofl phopehty,
( E * , T ~ ) i s a complemented
subspace
of
386
PAQUES
(Xs(U;C),T
:) hence E$ h a s t h e a p p r o x i m a t i o n p r o p e r t y . From
~
w e have t h a t , i f a
for
f E XS(U;C),
t h e mapping
U,
clear t h a t
Da : ( J c s ( U ; C ) ,
To show c o n t i n u i t y , l e t
a
Then t h e r e i s
E.
B E BE
6 > 0 , be s u c h t h a t
Let
-1
defined by Da(f) = 6 f ( a ) ,
T ~ +E;, )
i s a c o n t i n u o u s p r o j e c t i o n onto
D2 = Da.
s u b s e t of i n EB.
i s a q u a s i - c o m p l e t e s p a c e , t h e n E h a s t h e S.a.p..
E
For
this,
+
compact
K C EB and i s c o m p a c t
From Cauchy
C U n EB.
SK
Indeed, it is
K be a s t r i c t
such t h a t
a
E;.
in-
e q u a l i t i e s , ( C o r o l l a r y 1.18) i t f o l l o w s t h a t
for all
f
E Jcs(U;C).
Then
i s continuous.
Da
w e show t h a t E h a s t h e S . a . p .
NOW,
.
Since
Ei
has the
p r o x i m a t i o n p r o p e r t y , t h e n f o r e v e r y b a l a n c e d convex compact of
1 E
,
EZ
f o r e v e r y s t r i c t compact subset K o f
> 0, t h e r e i s
p E 1. S i n c e
g
E
g E (E;)
m g =
Since, f o r each
for of
Bi
B E
m
E
BE
U
Fo r
so t h a t
1=1
D E
BE
vi
.., m ,
vi
(EZ)', xi
E
E, a n d f o r
-
pIIK <
E
i s c o n t i n u o u s Ipi(lp) I (cipi(p),
,f o r
some s t r i c t compact s u b s e t Li
.+
Li
C
EB
and are compact i n
i
Ipi(lp)
B C D
is
and
K C EB
and
I
fore, f o r
..
so t h a t
i =1,. , m
Bi
,
compact
in
,
pi
.
EB
Let
5 c I I I ~ I I,~ where c i s a c o n s t a n t . Bi
Banach Theorem, t h e r e are, f o r e a c h
,
E
S i n c e K i s a s t r i c t compact s u b s e t o f E , t h e r e
K). Hence
pi
for every
E,
E*, i = l , . . . , m .
C D,
IGi(lp)
E ((ED):)
i = 1,.
I
2 I .
IIpIIL
..,m ,
,
i =l,...,m
for
b a l a n c e d convex compact s e t i n t h e Banach s p a c e E D .
e x t e n s i o n s of
every
Q:
: E;
be such t h a t
L - r(.U Li
Ilg(p)
subset
8 E,
pi(vP) = 11 911 L~
such t h a t
BE
I
such t h a t
8 E*,
pi Q x i ,
i =1,.
i = l , .. . , m .
f o r each
is
i =1
9 E E*, where E. L e t
(EZ)
ap-
,
for
By
pi : (ED) p E
L
is a
the
Hahn-
-+
Q: linear
(ED) I . There-
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
Let
: Ei
(ED);
-+
I
be d e f i n e d by
i s l i n e a r and c o n t i n u o u s . Hence
J,
compact s u b s e t of
write
D
=
E ED
we get
K
and then
equicontinuous.
Hence, w e
can
I
pD(v)
5 6 1 , f o r some
6 > 0). Hence
PE
= Voo=V.
Sine
ED,
C
g =
I
f o r T E E*. ED J , ( I ) = I D i s a b a l a n c e d convex
$(T) = TI
V", where V i s a c l o s e d a b s o l u t e l y 0-neighborhood in ED.
( V = {v
where
(ED);
367
Z pi i=l
@ xi.
Hence,
Therefore
and t h e n
that is,
-
S i n c e , g E E* 8 E the
6
i s i n d e p e n d e n t of
i t f o l l o w s t h a t E has
S.a.p..
2 . 8 DEFINITION:
L e t E be a l o c a l l y convex complex Hausdorff s p a c e .
i s s a i d t o have t h e S - h o l a m o h p h i c a p p h o x i m a t i o n p h o p e h t y (S.H.a.p.1
E
K C E, a s t r i c t compact set, t h e r e i s
i f given K
and
C
EB
and i s compact i n EB and g i v e n
such t h a t
pB(g(x)
-
x) <
E,
for a l l
E
B E
BE
> 0, there is g
x E K.
such t h a t E
JCs(E;&) B E
PAQUES
368
I t is clear t h a t i f
E has t h e
S.a.p.,
t h e n E h a s t h e S.H.a.p..
For t h e converse i t i s needed t h a t E be a quasi-complete space, t h a t
i s , w e have t h e f o l l o w i n g theorem, which c o n t a i n s t h e p r e v i o u s t h e o -
E, which i s f i n i t e l y S-Runge.
rem f o r an open s u b s e t U o f
2 . 9 THEOREM:
U b e an open
which i d h i n i t e l y S-Runge. Then t h e 6 o & l o w i n g conditionh
E,
d u b b e d 06
b e a q U a d i - C O m p l E t e d p a c e and l e t
Let E
ahe e q u i v a l e n t : a)
E
S.H.a.p..
b)
Foh eweny l o c a l l y convex d p a c e
had t h e
in c)
(xs(u;C),T
d)
E
)had
~
only i n
c)
+
+
E t o be a quasi-complete space i s
needed
d).
c) i s p a r t (c) o f C o r o l l a r y 2.6, which i s t r u e f o r
open s u b s e t of
E.
c)
+
d) i s Theorem 2.7.
remains o n l y t o show t h a t proof o f Theorem 2 . 1 , ( c f . D e f i n i t i o n 2.8)
2.10 COROLLARY:
S.a.p.
t h e a p p k o x i m a t i o n ptopekty.
S.a.p..
had t h e
The assumption of
b)
63 F i d -rs-dende
3ES(U;F).
REMARK:
PROOF:
F, JCs(U;C)
a)
+
Let
E
i 6 and o n l y id,
+
a ) i s obvious.
b ) . T h i s proof i s analogous
substituting
.
d)
g
E
for
HS(E;C) C3 E
be a q u a d i - c o m p l e t e d p a c e . Then 60k
each
n E IN,
(Pb(%;C),
-rs)
any It
t o the
T E E* Q E
E
had
had t h e
the ap-
pho ximation phopehty.
PROOF:
If
E has t h e
any open s u b s e t U of
S.a.p.,
i t f o l l o w s by Theorem 2 . 9 ,
E , which i s f i n i t e l y S-Runge,
h a s t h e approximation p r o p e r t y . S i n c e f o r each
n
E
that
(X,(U;C),
for T ~ )
1N, ( P b ( n E ; C ) ,rS
THE APPROXIMATION PROPERTY FOR SPACES OF HOLOMORPHIC MAPPINGS
i s a complemented subspace of
( X S (U;(c), ' I ~ )
, we
have t h a t
369
(Pb(%;O),
'I~)
h a s t h e approximation p r o p e r t y . Conversely, i n p a r t i c u l a r , E* h a v i n g t h e a p p r o x i m a t i o n property, has the S.a.p.
E
i n t h e proof of Theorem 2 . 7 )
.
By t h e p r e v i o u s C o r o l l a r y , w e have t h a t
2.11 REMARK: quasi-complete S-Runge,
(as
s p a c e and U i s an open s u b s e t o f
El
if
E
a
is
which is f i n i t e l y
h a s t h e approximation p r o p e r t y , i f and only t h e n (Ws(U;C), T ~ ) n E IN,
i f , f o r each
(Pb("E;C), ' I ~ ) h a s t h e approximation p r o p e r t y .
REFERENCES
I1 1
R. ARON,
Tensor p r o d u c t s o f holomorphic f u n c t i o n s , Indag. Math.
35, (1973) I 1 9 2 [ 21
- 202.
R. ARON and M. SCHOTTENLOHER, Compact holomorphic mappings Banach s p a c e s and t h e Approximation p r o p e r t y , J. t i o n a l Analysis 21,
[ 31
[ 4
I
1
51
(1976) , 7
- 30.
P . ENFLO, A counterexample t o t h e approximation p r o p e r t y Banach s p a c e , A c t a Math. 130 (1973) , 309 317.
-
A.
on
Func-
in
Phoduitd ten6o&ie& t a p o e o g i q u e d e t eApace6 n u c . t e a i h e 6 , Memoirs Amer. Math. SOC., 1 6 ( 1 9 5 5 ) .
GROTHENDIECK,
C. P. GUPTA, Malgrange theorem f o r n u c l e a r l y e n t i r e f u n c t i o n s o f bounded t y p e on Banach s p a c e . D o c t o r a l D i s s e r t a t i o n , U n i v e r s i t y of R o c h e s t e r , 1 9 6 6 . Reproduced by I n s t i t u t o de Matemgtica Pura e A p l i c a d a , Rio de J a n e i r o , B r a s i l , Notas de Matemgtica, N Q 37 ( 1 9 6 8 ) .
[ 61
M. C. MATOS, Holomorphically b o r n o l o g i c a l s p a c e s and
infinite d i m e n s i o n a l v e r s i o n s o f H a r t o g s theorem, J . London Ma*. SOC. ( 2 ) 17 (19781, t o a p p e a r .
370
I 71
PAQUES
L. NACHBIN, Recent developments i n i n f i n i t e dimensional
holo-
morphy, B u l l . Amer. Math. SOC. 79 ( 1 9 7 3 1 , 6 2 5 - 6 4 0 . [ 81
In:
L. NACHBIN, A glimpse a t i n f i n i t e d i m e n s i o n a l h o l o m o r p h y ,
P h a c c e d i n g h o n ' I n , 3 i n i t e D i m e n d i o n a L Holomokphy, U n L v m i t y
0 6 Kentucky
1 9 7 3 , ( E d i t e d by T. L. Hayden and
T.
J.
S u f f r i d g e ) . L e c t u r e Notes i n Mathematics 3 6 4 , S p r i n g e r Verlag B e r l i n - H e i d e l b e r g - N e w York 1 9 7 4 , p p . 69 - 79.
I91
L . NACHBIN, TopoLogy o n S p a c e d 0 6 Holomo/rpkic M a p p i n g h , . E r g e b ~ s s e der M a t h e m a t i k und ihrer Grenzgebtete, B a n d 47, Springer
-Verlag New York I n c . 1 9 6 9 .
[lo ]
Ph. NOVERRAZ, P d e u d a - c v n v e x i t e , c a n v e x i t i i p o l y n o m i d e eA d o m d n u d ' h o L o m o h p h i e en d i m e n h i o n indinie, ca 4 8 , North-Holland,
[111
0. T. W.
Notas de M a t e m s t i -
Amsterdam, 1 9 7 3 .
PAQUES, P h o d u t o h t e n d o h i a i d d e dunqoe.4 Silva-hvlomok-
6ah
e a
p h o p h i e d a d e d e a p h o x i m a ~ i i a , Doctoral Dissertation,
Universidade E s t a d u a l de C a m p i n a s , C a m p i n a s ,
Brasil,
1977. [12 1
In: Analyhe , 3 v n c t i a n e l l e e t a p p l i c a t b n h (L. N a c h b i n , e d i t o r ) . Hermann, Paris,
D. PISANELLI, S u r l a L F - a n a l i t y c i t g . 1 9 7 5 , pp. 2 1 5 - 2 2 4 .
I131
J . B. PROLLA, A p p k o x i m a t i o n
06
Vectak Valued F u n c t i o n h ,
d e Maternztica 6 1 , N o r t h - H o l l a n d , [14]
L . SCHWARTZ, T h d o r i e des d i s t r i b u t i o n s
Notas
Amsterdam, 1 9 7 7 . valeurs
vectorielles
I , Ann. I n s t . F o u r i e r 7 ( 1 9 5 7 1 , 1 - 1 4 1 .
[151
M.
SCHOTTENLOHER, €-product a n d c o n t i n u a t i o n o f a n a l y t i c
map-
pings, I n : Anaeybe F o n c t i o n e l l e e t AppRicationn, (L. N a c h b i n , e d i t o r ) Hermann, P a r i s , 1 9 7 5 , p p . 2 6 1 - 2 7 0 . [161
J. S. SILVA, C o n c e i t o h
calmente
d e dunciio diddenenci&~eL em
COnULXVh,
L i s b o a , 1957.
C e n t ro de E s t u d o s
ebpacob
Matemsticos
lade
Approximation T h e o q and Functional AnaZyaie J . B . ProZZa (ed.) QNor th-Hc Z land Pub t i s h i n g Company, 19 79
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
JOAO B . PROLLA Depar tamento d e M a temstica U n i v e r s i d a d e E s t a d u a l de Campinas Campinas, S P , B r a z i l
1. INTRODUCTION Throughout t h i s p a p e r X i s a Hausdorff s p a c e s u c h t h a t C&(X;X)
(IK = I R o r
C)
s e p a r a t e s t h e p o i n t s of
X,
and
E
i s a non-zero locally
convex s p a c e . Our aim i s t o p r o v e t h a t c e r t a i n function spaces L C C(X;E) have t h e approximation p r o p e r t y as soon as E h a s t h e
approximation
p r o p e r t y . W e show t h i s f o r t h e c l a s s of a l l Nachbin s p a c e s C V m ( X ; E ) . Such s p a c e s i n c l u d e
C ( X ; E ) w i t h t h e compact-open t o p o l o g y ;
w i t h t h e s t r i c t topology:
, Bierstedt
that
CVm(X;IK)
that
X i s a completely r e g u l a r
[ 11
,
w i t h t h e uniform t o p o l o g y .
Co(X;E)
E = IK
v E V
When
u s i n g t h e t e c h n i q u e of E-products, had proved
h a s t h e approximation p r o p e r t y , under t h e h y p o t h e s i s k m - s p a c e , and t h a t t h e f a m i l y V o f
w e i g h t s i s such t h a t g i v e n a compact subset weight
Cb (X;E)
such t h a t
v(x)
1
for a l l
K C X, one c a n f i n d
x
a
E K.
The t e c h n i q u e w e u s e h e r e was s u g g e s t e d by t h e p a p e r
151
G i e r z , who proved t h e analogue o f Theorem 1 below f o r t h e c a s e of
of X
compact and b u n d l e s o f Banach s p a c e s . T h i s t e c h n i q u e of " l o c a l i z a t i o n " of t h e approximation p r o p e r t y was used by B i e r s t e d t , i n t h e c a s e t h e p a r t i t i o n by a n t i s y m m e t r i c s e t s ( B i e r s t e d t [ 2 1 1 , b u t
the
of
main
i d e a of r e p r e s e n t i n g t h e s p a c e o f o p e r a t o r s of L as a n o t h e r Nachbin s p a c e o f cross s e c t i o n s i s due t o G i e r z . However o u r p r e s e n t a t i o n is 371
372
PROLLA
much s i m p l e r , i n p a r t i c u l a r w e do n o t u s e t h e concept of a C (X)-convex
C ( X ) -module.
locally
I n t h e I n t r o d u c t i o n t o h i s paper, Gierz said
t h a t h i s method could be a p p l i e d t o t h e v e c t o r f i b r a t i o n s i n t h e sense of [ 8]
,
and t h i s l e d t o o u r e f f o r t a t s i m p l i f y i n g
his
proof
and
adapting it t o our context.
2. THE APPROXIMATION PROPERTY FOR NACHBIN SPACES A v e c t o h d i b h a t i o n o v e r a Hausdorff t o p o l o g i c a l space
p a i r ( X , ( F x ) x E X ) ,where each F,
i s a v e c t o r space over
X
the
is a field
IK (where K = IR or a ) . A c k o d b - ~ e c t i o nis then any element f o f t h e C a r t e s i a n product o f t h e s p a c e s A w e i g h t an
Fxl i . e .
X i s a f u n c t i o n v on
norm o v e r Fx f o r each L of c r o s s - s e c t i o n s
f
.
f = ( f (x)I x
I
X such t h a t
v ( x ) is a semi-
LVm i s a v e c t o r space
x E X. A Nachbin b p a c e
such t h a t t h e mapping
is upper semicontinuous and n u l l a t i n f i n i t y on X f o r each weight v
be onging t o a d i h e c t e d b e t V of weights ( d i r e c t e d means t h a t , given v1
, vz
E
v
V , t h e r e is some
( i = 1,2) f o r a l l
x
f
E
V
and
X > 0 such t h a t v i ( x ) 5 Av(x)
X); t h e space L is then equipped
with
the
topology d e f i n e d by t h e d i r e c t e d s e t of seminorms
and i t i s denoted by
LVa
.
S i n c e only t h e subspace w e may assume t h a t
L(x) = F,
L(x) = { f ( x ) ; f f o r each
x
C(X;IK)
L} C Fx i s relevant,
E X.
The C a r t e s i a n p r o d u c t of t h e s p a c e s F, C ( X ; M ) -module, where
E
h a s t h e s t r u c t u r e of a
denotes t h e r i n g of
all
continuous
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
i f we d e f i n e t h e p r o d u c t
IK-valued f u n c t i o n s on X I
Q E C(X;IK)
for a l l
and e a c h c r o s s - s e c t i o n
x E X. I f
W
C
373
f
Of
each
by
B C C(X;IK) is a
is a v e c t o r subspace and
L
for
s u b a l g e b r a , w e s a y t h a t W i s a B-module,
i f BW = { $ f ; $ EB, f
W ) CW.
E
W e recall t h a t a l o c a l l y convex s p a c e E h a s t h e a p p h o x i m a t i o n
p h o p e h t y i f t h e i d e n t i t y map e on E can be approximated,
uniformly
on e v e r y t o t a l l y bounded s e t i n E, by c o n t i n u o u s l i n e a r maps of f i n i t e rank. T h i s i s e q u i v a l e n t t o s a y t h a t t h e space
E ' @ E i s dense i n
L(E) w i t h t h e topology o f uniform convergence on
bounded s e t s of
E.
Let
Ec(E),
totally
c s ( E ) b e t h e s e t of a l l c o n t i n u o u s seminom
,
d e n o t e t h e spacz E s e m i P normed by p. I f , f o r e a c h p E c s ( E ) , t h e s p a c e E h a s t h e a p p r o x i P mation p r o p e r t y , t h e n E h a s t h e a p p r o x i m a t i o n p r o p e r t y . on E .
For each seminorm
THEOREM 1:
p
E cs(E)
Suppabe t h a t , d o h each
Fx equipped w&h
x E X, t h e bpace
{v(x); v E V l
hab
B c C b ( X ; I K ) be a b e l d - a d j o i n t
and
t h e t o p o l o g y dedined by t h e damily
t h e apphoximation p h o p e h t y . L e t
let E
06
beminohnb
b e p a h a t i n g b u b a l g e b h a . Then any Nachbin d p a c e
which
LVm
id
a
B-modute hab t h e apphoximation p h o p e h t y . The i d e a o f t h e p r o o f i s t o r e p r e s e n t t h e s p a c e W = LV,
being
,
a s a Nachbin s p a c e of c r o s s - s e c t , i o n s o v e r
XI
e(W),
where
each
fiber
L(W;Fx), and t h e n a p p l y t h e s o l u t i o n o f t h e Bernstein-Nachbin
a p p r o x i m a t i o n problem i n t h e s e p a r a t i n g and s e l f - a d j o i n t bounded case. B e f o r e p r o v i n g theorem 1 l e t us s t a t e some c o r o l l a r i e s .
COROLLARY 1: Fx
L e t X be a Hauddohdd b p a c e , and
604
each
be a nohmed b p a c e w i t h t h e apphoximation p h o p e h t y .
Cb(X;IK)
be a b e t i - a d j o . i n t and b e p a h a t i n g b u b a l g e b h a .
let
x E X
Let
B
C
374
PROLLA
L e t L be a v e c t o t s p a c e
x)
(X; (F,)
chodb
06
-Aectiand
pehtaining
to
nuch t h a t
x
f E L , t h e map
(1) doe evetry
+
Ilf(x)II 0 u p p a demicontinuoirn
and nuLL a t i n d i n i t y ;
i n a B-rnoduLe;
(2)
L
(3)
L(X) = F,
60%
x E
each
x.
Then L equipped w i t h nohm IIf 1 I = sup fIlf(x)lI; x E X I
had t h e
apphoximation p t o p e h t y . PROOF:
Consider t h e w e i g h t v on X d e f i n e d by
f o r each
II
f
x E X.
II = sup
REMARK:
Then
{ IIf ( x ) II ; x E
LVm
is
just
L
v ( x ) = norm of
equipped
with
FX’
norm
x).
From C o r o l l a r y 1 i t f o l l o w s t h a t a l l “ c o n t i n u o u s sums”,
t h e s e n s e of Godement [ 6
1 or
[7
in
1 , of Banach s p a c e s w i t h the approxi-
mation p r o p e r t y have t h e approximation p r o p e r t y , i f t h e X
the
i s compact and i f such a “ c o n t i n u o u s sum” i s a
“ b a s e space“
Cb(X;IK)
-module.
I n p a r t i c u l a r , a l l “ c o n t i n u o u s sums“ o f H i l b e r t s p a c e s and of C*-alg e b r a s , i n t h e sense of D i x m i e r and Douady [ 3 tion property, i f
1
have t h e approxima
X i s compact. Indeed, a ” c o n t i n u o u s sum“
sense of [ 3 1
is a
COROLLARY 2 :
Let X b e a Hauddohdd dpace buch t h a t
-
i n the
C ( X ; I I o -module.
k a t i n g ; L e t V b e a dikected b e t demicontknuoub dunctiand o n
X;
04
C b ( X ; x ) 0 bepa-
&eat-vaLued, n o n - n e g a t i v e , uppek
and l e t E be a lacuLLy convex pace
w i t h t h e apphoximation p h o p e h t y . Then C V m ( X ; E ) had t h e apphaxha.tLun pto pehty
.
PROOF:
By d e f i n i t i o n , CVm(X;E) = { f E C ( X ; E ) ;
finity, for a l l
v
€
vf
vanishes
at
in-
V), equipped w i t h t h e topology d e f i n e d
by
the
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
376
f a m i l y o f seminorms
where
v E V Let
and
p
E
denote
Lv
cs(E).
C V m ( X ; E ) equipped w i t h t h e topology d e f i n e d by
t h e above seminorms when either
or
Lv(x) = 0
by t h e seminorms
v
E
V
Lv(x) = E
{v(x)p; p
i s k e p t f i x e d . Then, for e a c h x E X , equipped with t h e topology defined
E CS(E)
1 . Hence i n b o t h c a s e s , L v ( x ) h a s
t h e approximation p r o p e r t y . I t remains t o n o t i c e spaces a r e
Cb(X;JK)-modules. T h e r e f o r e
property. Since
v E V
Lv
that
has
was a r b i t r a r y , C V m ( X ; E )
the has
all
Nachbin
approximation t h e approxima-
t i o n property.
COROLLARY 3:
(a)
Let X and E b e an i n CoaoLLaay 2 . T h e n
C(X;E)
w i t h t h e compact-open t o p o L o g y h a d t h e a p p h o x i m a -
t i o n phopehty. (b) C o ( X ; E )
N i t h t h e uni6oam t o p o L o g g had
the
appkoximation
pkopehtg.
REMARK:
I n ( a ) above, i t i s s u f f i c i e n t t o assume t h a t
C(X;IK)
is
separating.
COROLLARY 4 :
(Fontenot [ 4 1 )
A p a c e , and Let E
Let
X
b e a LocaLLy compact
be a L o c a L L y convex Apace w i t h
p a o p e h t y . T h e n c ~ ( x ; E )w i t h t h e n t a i c t t o p o e o g y
the
Haundoa66
appaoximation
B had t h e a p p k o x i -
m a tio n p h o p e h t y .
PROOF:
Apply C o r o l l a r y 2 , w i t h
COROLLARY 5:
Ale Nachbin spaced
V = {v E Co(X;JR);
06
v
0).
continuoun ncaLak-vaLu&d duncfiond
376
PROLLA
h a v e t h e apphoximation p h o p e k t y .
I n Corollary 2, take
PROOF:
E = IK.
3 . PROOF OF THEOREM 1
Let
W = LV,
Let
vo
E
For e a c h
and l e t and
V T
w
be a t o t a l l y bounded s e t .
be g i v e n .
> 0
E
J(W)
E
A C
c o n s i d e r t h e map
E ~ O T : W + F ,
for
x
E
X I where
for all
f
STEP 1:
sX o T
E
E~
:W
+
W.
E L(W;Fx).
Just notice t h a t
PROOF:
is t h e e v a l u a t i o n map, i.e., ~ ~ (= ff( x) ) ,
F,
E,
E C ( W ; F ~ ) #s i n c e
v ( x ) [ ~ ( x5 ) ~1 I f
f o r every
f o r any
v
U(x)
T E
f o r any
v E V.
T E C (W), c o n s i d e r t h e c r o s s - s e c t i o n
F o r each
and f o r each
IIv ,
E V
E
E o ~T)
c o n s i d e r t h e weight ? on X d e f i n e d by
C(W;Fx).
e(W).
?=(
Then
377
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
STEP 2:
x * ~ ( x ) I ? ( x ) ]i b uppetc . b e m i c o n t i n u o u b and vanishes
T h e map
at i n d i n i . t g o n X , d o & e a c h
PROOF:
Let
Choose
h"
xo E X
and
there exist
such t h a t
{ 1 , 2 , ...,rn}
x
x E Vi
Let
X. L e t
h')
. Then
6 > 0. Since
A
such t h a t , given
E
u
=
T(A) i s totally bounded, f E A,
there
is
such t h a t
+
v ( x )[
( T f i ) ( x )]
neighborhoods of
V2,...,Vm
for all
-
fl,f 2,...,fm
Since
in
h'
6 = 2(h"
Let
V1,
a n d assume
~ ( x o ) [ ? ( x o )
(1)
i
T E E(W).
is upper semicontinuous, t h e r e are xo
such t h a t
(i = 1 , 2 , . . . , m ) .
V1 n V2
x E U
( 2 ) i s t r u e . Then
... n V, .
and l e t
Then
f E A . Choose
U i s a neighborhood of i
E
{1,2,
...,rn}
xo
such t h a t
378
PROLLA
On t h e o t h e r hand, by ( 1 1 , w e h a v e
Hence
v ( x ) ( ( T f ) (x)] < h"
Therefore
for a l l
f
E A,
f(x)[?(x)] 5 h " < h , f o r a l l
L e t us now p r o v e t h a t t h e mapping
and
x E U.
x 6 E U. x
+
C(x)[ $(x) ] v a n i s h e s
at
infinity. Let
be g i v e n and d e f i n e
6 > 0
Since
K6 = @, i f
sup { I I T f l l v ; f
E A]
sup { I I T f l l v :
f E A] < 6 ,
f E A, there is
t h a t , given
may
assume
2 6.
T ( A ) i s t o t a l l y bounded, t h e r e are
Since
we
i
E
{l,...,m]
fll...lf,
E A
such
such t h a t
m
U { t E X; v ( t ) [ ( T f i ) ( t ) ]2 6 / 2 1 . i=l Then K i s compact, s i n c e e a c h o f t h e f u n c t i o n s x + v ( x ) [ ( T f i ) (x)]
Let
K =
v a n i s h e s a t i n f i n i t y . L e t now
Choose
fi E A
Therefore Since
K6
6/2
x
E
K6
f E A
and choose
such
that
s a t i s f y i n g ( 4 ) . Then
< v ( x ) [ (T f i ) ( x ) ] a n d so
i s c l o s e d , t h i s e n d s t h e proof.
x
E
K , i.e.,
K6
C K.
THE APPROXIMATION PROPERTY FOR NACHBIN SPACES
The above two s t e p s show t h a t t h e image L(W)
under t h e map T
over
X,
1
+
i s a IJachbin s p a c e
=
eVm
p e r t a i n i n g t o t h e v e c t o r f i b r a t i o n (X;
w e take a s family V of weights t h e family
STEP 3:
F o h ewehy
PROOF:
f
Let
L e t now
E
e(W),
T E
v
V =
379
{'?; T
,
if
v E V)
A. Then
F =
I?;
T E W'
8 W).
sup IIT f f EA
- f IIv
< 0
T E W' 8 W
E
such t h a t
.
Hence, by S t e p 3, it i s enough t o p r o v e t h a t
P
of
of c r o s s sections
1 (W;Fx) )
c;
1
E V,
Our aim i s t o p r o v e t h a t w e can f i n d
where
g(W)
11 ? - ^I llc
< c, 0
=
By t h e bounded c a s e of t h e Bernstein-Nachbin approximation prab-
l e m (Theorem 11, I 8 1 ,
STEP 4 :
F
STEP 5 :
Fot each
i b
pg. 314) i t i s enough t o p r o v e t h a t
a B-modde.
x E
x,
F(x) i n denbe i n
t h e t a p o e o g y dedined b y t h e
bemiI'I0hmb
L(W;Fx), e q u i p p e d
{ C ( x ) ; v E V).
with
PROLLA
380
To prove t h a t
PROOF:
for a l l
M
Q
: W
!i! +
W
E
F i s a B-module,
F, i.e., for a l l i s d e f i n e d by
i t i s enough t o prove t h a t
T E W' 8 W
and f o r a l l
N ( f ) = I$ f , f o r a l l
4
f
E
Q E B;
and
W.
Now t o prove (71, one h a s t o prove t h a t
for all
x E X. However,
And, f o r a l l
f E W
one has
T h i s ends t h e proof o f s t e p 4 . T o prove s t e p 5 , w e f i r s t n o t i c e t h a t , s i n c e each
w i t h t h e topology d e f i n e d by property, then
W'
8
Fx
{v(x); v E V)
is d e n s e i n
w i t h t h e topology o f t h e seminorms
has
the
F x equipped approximation
Lc(W;Fx), a f o r t i o r i i n L(W;Fx) { + ( x ) ; v E V}.
r n E APPROXIMATIONPROPERTY FORNACHBIN SPACES
Hence, a l l w e h a v e t o p r o v e i s t h a t f o r each
x
E
381
F(x) contains
W'
8 Fx,
X.
Let then
be a c o n t i n u o u s l i n e a r o p e r a t o r o f
T E W' Q Fx
fi-
n i t e rank, say T =
ei
where
and
E W'
n
2
i =1
vi E F ,.
+i8vi
Since
W(x) = Fx , c h o o s e
fi E W
such
that
for
i = 1,2,.
n
u = z
Define Then
. ., n . i =1
U E W'
W;
8
$
i
SO
8 f i '
fi
E
F.
NOW
and t h e r e f o r e 6 ( x ) (f) =
for a l l
f
E
n
C
i=1
Qi(f)fi(x)
=
n 2: + i ( f ) v i = T ( f ) i=l
W.
REFERENCES
[ 1]
K-D.
BIERSTEDT, G e w i c h t e t e
Raume
stetiger
vektorwertiger
F u n k t i o n e n und d a s i n j e k t i v e T e n s o r p r o d u k t . I , J . r e i n e angew. Math. 259 (1973), 186
[ 2
I
K-D.
- 210.
BIERSTEDT, The a p p r o x i m a t i o n p r o p e r t y f o r w e i g h t e d f u n c t i o n s p a c e s , BonnerMath. S c h r i f t e n 8 1 ( 1 9 7 5 ) , 3 - 25.
382
PROLLA
[ 3 ]
J. DIXMIER a n d A. DOUADY, Champs c o n t i n u s d ' e s p a c e s hilbertiens e t d e C * - a l g s b r e s , Bull. SOC. Math. F r a n c e 9 1 (1963), 227 - 284.
[41
R. A. FONTENOT, S t r i c t t o p o l o g i e s for v e c t o r - v a l u e d f u n c t i o n s , Can. J . Math.
15
I
G.
26 (1974), 841 -853.
G I E R Z , R e p r e s e n t a t i o n of s p a c e s of compact
operators
a p p l i c a t i o n s t o t h e approximation p r o p e r t y , Nr.
[6
1
R.
GODEMENT,
335, Techn. Hochschule D a r m s t a d t , Feb. 1 9 7 7 . T h g o r i e g g n i r a l e d e s sommes c o n t i n u e s d ' e s p a c e s d e
Banach, C . R . [ 7 ]
R.
Acad. S c i . P a r i s 228 (19491, 1321-1323.
GODEMENT, Sur l a t h i o r i e d e s & p r e s e n t a t i o n s
of Math.
[81
and
Preprint
u n i t a i r e s , Ann.
53 (19511, 68 -124.
L. NACHBIN, S . MACHADO a n d J . B. PROLLA, Weighted approximation, v e c t o r f i b r a t i o n s and a l g e b r a s of o p e r a t o r s , J . p u r e s e t a p p l . 50 ( 1 9 7 1 )
,
299
- 323.
Math.
Approximation Theory and Functional Analysis J.B. Prolla (Ed.)
@North-Holland Publishing Company, 1979
ON CARDINAL SPLINE SMOOTHING
I. J. SCHOENBERG Mathematics Research Center University of Wisconsin Madison, Wisconsin 53706, USA.
INTRODUCTION The present paper describes the methods, with some changes be mentioned below, whereby I solved the numerical problem
to
assigned
to me at the Ballistics Research Laboratories in Aberdeen, Maryland, during the second World War (see [ 4
]
1.
The problem was to smooth very
extended equidistant tables of drag functions (or drag coefficients) by approximating them by very smooth functions that were easily computable with their first and second derivatives. The first question to be answered was this: When may a " m o w h g awefiage" o p e f i a t i o n L e g i t i m a t e L y b e c a L l e d a n m o o t h i n g 6ofimuLa ?
An
answer is given in Section 1 of Part I. That it is a reasonable
one
is shown in Section 2 of Part I. These ideas were later greatly generalized by Fritz John in his important work on parabolic differential equations (see [ 3 1 ) . The connection is briefly
mentioned
in
Section 2 of Part I. The second essential ingredient of our results of Part
I1
is
the process of c a f i d i n a e h p t i n e intehpolation (see [ 9 1 ). The required results are described in Section 3 of Part I. The changes from my war-time approach are developed completely in Part 11. This is the new contribution of the paper, and it arises 383
384
SCHOENBERG
i f w e a d a p t t h e i d e a o f E. T. Whittaker (see [lo 1 and [ 7
1 ) to
the
problem of smoothing a b i - i n f i n i t e sequence of e q u i d i s t a n t d a t a . The r e s u l t i s t h e cahd.inaL 4 p L i n e s m o o t h i n g phOces4. L e t m e thank P r o f e s s o r I l i o G a l l i g a n i f o r a s k i n g m e
to
visit
Rome i n May of 1 9 7 7 , where I gave a s h o r t 10-hours c o u r s e o n C a r d i n a 1 Spline Interpolation a t h i s I s t i t u t o per l e applicazioni d e l Calcolo "Mauro Picone".
PART I .
PRELIMINARIES
1. WHAT IS A SMOOTHING FORMULA?
The s e v e r a l smoothing ( o r gradua
-
t i o n ) methods d i s c u s s e d i n [lo] are a l l o f t h e "moving a v e r a g e " t y p . By t h i s w e mean t h a t w e are g i v e n a sequence o f r e a l and
symmetric
weights
which w e a p p l y t o t h e d a t a (x,)
t o produce t h e i r smoothedversicln (y,)
,
by t h e formula
yn
--
m
C
v=-co
an-Vxy
, (n=O,
f 1,
...) .
I n o t h e r words, w e c o n v o l u t e t h e sequences (a,)
and (x,)
,
an o p e r a
-
t i o n s y m b o l i c a l l y d e s c r i b e d by
(3)
By d e r i v i n g a formula ( 2 ) a c c o r d i n g t o a d e f i n i t e i d e a , s u c h a s a l e a s t s q u a r e s formula, o r W h i t t a k e r ' s formulae (see [ 1 0 , XI
1
),
Chapter
we may r e a s o n a b l y e x p e c t t h a t it w i l l t r a n s f o r m a g i v e n
se-
quence ( x ) i n t o a smoother sequence (y,). T h i s p o i n t becomes doubtn f u l l however, i n c a s e t h e formula ( 2 ) was o t h e r w i s e o b t a i n e d , e . g . ,
385
ON CARDINAL SPLINE SMOOTHING
by some a p p r o x i m a t i n g p r o c e d u r e t h a t i s n o t s t r i c t l y
interpolatory.
An example of such a p r o c e d u r e w i l l be o u r c a r d i n a l s p l i n e smoothing o f P a r t 11. A c r i t e r i o n f o r ( 2 ) t o be r e g a r d e d a s a b m o o t h i n g ~0hmlLea w a s proposed by t h e a u t h o r i n [ 4 , pp. 50
- 54 1
and p r o c e e d s a s f o l l w s .
To b e g i n w i t h , w e assume t h e L a u r a n t series
(4)
I
t o converge i n some r i n g c o n t a i n i n g t h e u n i t c i r c l e
z j =1. S e t t i n g
i u , we c a l l t h e even f u n c t i o n z = e
d(u)
(5)
iu
= F(e
)
=
a
0
+
2al cos u
+
2a2 c o s 2u
+
...
t h e chahactehintic ( u n c t i o n of t h e formula ( 2 ) . The r e g u l a r i t y of F(z) on
I
IIm
UI
z
I
= 1, i m p l i e s t h a t
$(u)
is regular
in
a
certain
strip
< a , whence t h e v a l i d T a y l o r e x p a n s i o n m
T h i s e x p a n s i o n a l l o w s us t o e x p r e s s e a s i l y an i m p o r t a n t p r o p e r t y ( 2 ) : I t s hephoductive p o w e f ~ o , r deghee
06
exactnebb.
We s a y t h a t ( 2 ) h a s t h e d e g r e e o f e x a c t n e s s i s a n a t u r a l number, p r o v i d e d t h a t ( 2 ) r e p r o d u c e s
yn
- xn
values
f o r a l l n , a sequence (x,) P(n) of
,
of
where
2m-1,
exactly,
m
i. e.
,
i f t h i s sequence r e p r e s e n t s t h e
a polynomial of d e g r e e
2m-1,
b u t d o e s n o t have
t h i s p r o p e r t y f o r any h i g h e r d e g r e e . T h a t t h e d e g r e e of e x a c t n e s s i s always odd, f o l l o w s from t h e symmetry c o n d i t i o n ( 1 ) . I n t e r m s of
(6)
w e have t h e f o l l o w i n g e a s i l y e s t a b l i s h e d p r o p o s i t i o n : T h e doamuta ( 2 ) hub t h e degkee
id t h e e x p a n n i o n ( 6 ) i n
06
the 6otm
06
eXaCtflebb
2m-1,
i6 and onLy
SCHOENBERG
386
$(u) = 1
(7)
-x
.,.
+
U2m
(A
#
0).
An example. L e t (1) be t h e sequence a
0
= 5/8,
al = 1 / 4 ,
a2 =
-
1/16,
an = 0
if
n > 2,
when t h e c o n v o l u t i o n ( 2 ) assumes t h e form
From (5) w e o b t a i n
(9)
$ ( u ) = ( 5 + 4 cox u
-
and ( 7 ) i s s e e n t o h o l d w i t h mula ( 8 ) has t h e d e g k e e
06
cos 2u)/8 = 1
rn = 2 , X
=
1/16.
- 7
u4
+
...
I t follows that the f o r -
exactness 3 .
May ( 8 ) be r e g a r d e d as a smoothing formula ? The g e n e r a l
cri-
t e r i o n i s d e s c r i b e d by t h e f o l l o w i n g .
DEFINITION:
kJe 4ay t h a t ( 2 ) Lo a s m o o t h i n g botmuLa, p h o v i d e d t h a t La2
chahactchihtic dunction
d(u) 4atiAdieb
This condition evidently implies t h a t t h e coed6icien-t
peaking i n ( 7 ) ,
A ,
ap-
io p o s i t i v e . The c h a r a c t e r i s t i c f u n c t i o n ( 9 ) i s e a s -
i l y s e e n t o s a t i s f y (lo), and so ( 8 ) i s a smoothing f o r m u l a a c c o r d i n g t o o u r d e f i n i t i o n of t h e t e r m . The c r i t e r i o n (10) raises t h e f o l l o w i n g k i n d o f e q u a t i o n :
Do
t h e numerous smoothing formulae a s g i v e n i n 110 1 s a t i s f y our criterion
301
ON CARDINAL SPLINE SMOOTHING
(lo)? S e e G r e v i l l e ' s p a p e r [ l 1 where s o m e of t h e s e q u e s t i o n s a r e answered a f f i r m a t i v e l y . I n [ 4 , p p . 50
-
54 ] good a r g u m e n t s i n s u p p o r t of
( 1 0 ) are p r e s e n t e d . Even m o r e c o n v i n c i n g r e a s o n s
the criterion
w e r e given i n
[5,
P a r t I ] . I n t h e n e x t s e c t i o n we r e p r o d u c e t h e s e a r g u m e n t s a s p r e s e n t e d i n [ 6 , pp. 2 0 0 - 2 0 4 1 .
2 . THE BEHAVIOR OF THE ITERATES O F A MOVING AVERAGE FORMULA:
THE
PROBLEM OF DE FOREST. A c o n c l u s i v e argument i n s u p p o r t o f t h e n e c e s s i t y of o u r c o n d i t i o n ( 1 . 1 0 )
i s f u r n i s h e d by t h e s o l u t i o n o f t h e f o l -
l o w i n g problem f i r s t s t a t e d a n d a t t a c k e d
by
Erasmus
L. D e
(1834-1888). I f we s u b j e c t t h e g i v e n s e q u e n c e ( x u ) n times
Forest
i n suc
-
c e s s i o n t o t h e same t r a n f o r m a t i o n ( 1 . 2 ) , we o b t a i n a l i n e a r transformation
which i s t h e n - f o l d 06
the
i t e r a t e of ( 1 . 2 ) . W h a t i n R h e a o y m p t o t i c
c a e d 6 i c i e n t n 0 6 (1) an
n
+
m ?
T h i s q u e s t i o n was
behaviah
answered
by D e F o r e s t a n d by G. B. D a n t z i g ( f o r r e f e r e n c e s see151 ) f o r t h e c a s e when a l l c o e f f i c i e n t s of that
m=l
( 1 . 2 ) a r e non - n e g a t i v e , h e n c e n e c e s s a r i l y
i n ( 1 . 7 ) . A g e n e r a l s o l u t i o n i s as follows.
L e t ( 1 . 2 ) be s u c h t h a t ( 1 . 1 7 )
hence t h a t
,
( l . l O ) , and (1.71, are s a t i s f i e d ,
X > 0. L e t
-V
which i s t h e normal f r e q u e n c y f u n c t i o n
2m
cos vx d v ,
SCHOENBERG
388
if
m =1, o t h e r w i s e (m = 2,3,.
.
.)
Gm(x) i s an e n t i r e f u n c t i o n
having
i n f i n i t e l y many zeros, a l l r e a l . T h e caeddicients a d (1) satis6q t h e a s y m p t o t i c h e L a t i o n h
--
--
1 1 --1 a ( n ) = ( i n ) 2m Gm(v(hn) 2m ) + , o ( n 2m)
(4)
a4
V
where t h e " l i t t e e
n +
m ,
v.
or' dymbok? hoed4 unidaamly d o h n l L i n t e g e h d
For a proof see ( 5 , P a r t I ] , where i t i s a l s o shown byexamples (1.10),
t h a t ( 4 ) no l o n g e r h o l d s i f t h e e q u a l i t y s i g n i s a l l o w e d i n and t h a t t h e c o e f f i c i e n t n = 2k
aAn) d i v e r g e s e x p o n e n t i a l l y t o
t e n d s t o i n f i n i t y t h r o u g h even v a l u e s , i f
( 1 . 1 0 ) are r e v e r s e d anywhere i n t h e i n t e r v a l
the
0 < u < 2.rr
+
m
,
as
inequalities
.
The f o l l o w i n g d i s c u s s i o n , w h i l e n o t d i r e c t l y r e l a t e d
to
our
s u b j e c t of smoothing, w i l l show t h e c o n n e c t i o n of t h e a s y m p t o t i c rel a t i o n ( 4 ) w i t h t h e w i d e r f i e l d of p a r a b o l i c d i f f e r e n t i a l e q u a t i o n s . Observe t h a t ( 2 ) i m p l i e s t h a t
(5)
--1
--1
U(x,t) = t 2m G m ( x t 2m) =
-
-tv
2m
+ ixv
d v , ( t > 0).
The f u n c t i o n under t h e i n t e g r a l s i g n i s immediately s e e n t o s a t i s f y for all v
, the
d i f f e r e n t i a l equation
which r e d u c e s t o t h e f a m i l i a r h e a t e q u a t i o n i f
also -plane
m = l . I t follows t h a t
U ( x , t ) , d e f i n e d by ( 5 ) , is a s o l u t i o n of (6) i n t h e upper h a l f t > 0 . On t h e o t h e r hand, a p p l y i n g t o ( 2 ) F o u r i e r ' s inversion
formula and s e t t i n g
v = O r we f i n d that
ON CARDINALSPLINE SMOOTHING
These r e m a r k s imply t h e f o l l o w i n g : 'I 6
1x 1
say, a s
+
LA a b o t u , t i o n
a,
06
f (x)
389
cantinuow and a ( I X I - * )
,
then
t h e d i , j d e ~ e n . t i a l e q u a t i o n ( 6 ) Aattin6qLng t h e boundmy
condition
This p a r t i c u l a r s o l u t i o n
u ( x , t ) may now a l s o be
approximated
by t h e f o l l o w i n g n u m e r i c a l p r o c e d u r e : Draw i n t h e ( x , t ) - p l a n e
the
rectangular lattice of p o i n t s
(WAX,
n At)
(w
= 0, k 1
, ...
;
D e f i n e on it a l a t t i c e f u n c t i o n
n = 0,1,2
u
v
, ...) .
by s t a r t i n g w i t h
uw ,o = f ( v Ax) ,
and computing t h e v a l u e s a l o n g e a c h h o r i z o n t a l l i n e from those on the l i n e below i t , by means o f t h e t r a n s f o r m a t i o n (1.2). T h i s
evidently
amounts t o i t e r a t i n g (1.21, a n d a f t e r n s t e p s w e o b t a i n
(10)
For any g i v e n x a n d
t > 0,
( 1 0 ) w i l l go o v e r i n t o ( 8 ) i f w e
f o l l o w i n g : We 6 h A t c o n n e c t the. m e o h - n i z e b
Ax and
A t
do the
by ,the h d a t i a n
SCHOENBERG
380
A t = X (Ax) 2m.
(11)
Id t h e i n t e g e k b
n ahe buch t h a t
v and
VAX
+
x,
and
n A t
.+
t
an
Ax
0.
+
then
U
v,n
T h i s follows r e a d i l y from r e l . a t i o n (4): ( 1 0 ) d i f f e r s
U(X,t).
+
1 0 ) and ( 8 1 , i n view o f t h e
asymptotic
from a Cauchy-Riemann sum f o r tk integral
( 8 1 , by a q u a n t i t y t h a t t e n d s t o z e r o due t o t h e u n i f o r m i t y i n
v of
t h e error t e r m o f ( 4 ) . I t i s i n t e r e s t i n g t o n o t e t h a t it d o e s n o t matter which
for-
mula ( 1 . 2 ) w e u s e i n t h i s c o n s t r u c t i o n , as l o n g as it i s o f the degree of exactness
2m-1,
i.e.,
i t s a t i s f i e s (1.71, and above a l l t h a t i t
s a t i s f i e s t h e s t a b i l i t y c o n d i t i o n (1.10)
,
t h e t e r m "stabi1ity"meaning
h e r e s t a b i l i t y on i t e r a t i o n . F o r t h e g e n e r a l t h e o r y of F. J o h n , which t h e e q u a t i o n ( 6 ) i s a s p e c i a l example, see [ 3 1
.
of
I n t h i s s e c t i o n w e d e a l t e x c l u s i v e l y w i t h f o r m u l a e ( 1 . 2 ) which s a t i s f y t h e symmetry r e l a t i o n . I n [ 2 ]
T.
N.
E. G r e v i l l e d e a l t
with
t h e more d i f f i c u l t c a s e o f unsymmetric f o r m u l a e .
3 . CARDINAL SPLINE INTERPOLATION (see [ 9 , L e c t u r e s 1
l e m o f caadinal intehpolation i s t o f i n d s o l u t i o n s
-
4 1 ) . T h e prob-
f ( x ) of t h e i n -
t e r p o l a t i o n problem
(1)
f ( v ) = Y"
,
for all i n t e g e r s
v
,
where ( y v ) are t h e d a t a . A f o r m a l s o l u t i o n i s f u r n i s h e d b y t h e series
391
ON CARDINAL SPLINE SMOOTHING
i n v e s t i g a t e d i n 1 9 0 8 by de l a V a l l G e P o u s s i n , also l a t e r
by
E. T.
W h i t t a k e r , who c a l l e d i t t h e cahdinad b e h i e b . The d i f f i c u l t y w i t h ( 2 )
“i: :y
i s t h e s l o w decay o f t h e f u n c t i o n
as
x
-. A
+
s o l u t i o n of (1) i s t h e p i e c e w i b e l i n e a h i n t e h p o t h z t
much s i m p l e r g i v e n by
S1(x)
m
(3)
where
M2(x) i s t h e roof f u n c t i o n d e f i n e d by
in
M2(x) = x + l
,
[-1,01
M (x) = 1
2
-x
i n [ O , l l and%(x)
=o
The p u r p o s e o f cahdinad b p l i n e i n t e h p o & z t i o n i s t o b r i d g e between t h e p i e c e w i s e l i n e a r
if 1x1 ’1.
the
gap
S1(x) d e f i n e d by (31, a n d t h e c a r d i n a l
series ( 2 ) . I t a i m s a t r e t a i n i n g s o m e of t h e s t u r d i n e s s a n d s i n p l i c i t y of ( 3 ) , a t t h e same t i m e c a p t u r i n g some of t h e s m o o t h n e s s a n d s o p h i s t i c a t i o n of
(2).
Le-t m be a natuhad numbeh, and d e b
(4)
S2rn-l
b e t h e cLadb
06
= {S(X)3
cahdinad b p d i n e d
S(x)
0 6 deghee
2m-1
dedined
by
the two conditionb:
(5)
The h e s t h i c t i o n whete
v
i d
04
S ( x ) -to e u e h y u n i t i n t e n v a l
a n i n X e g e h , i b apolynomia!.
(v ,v
0 6 deghee 2
2m
+11,
-
1.
392
SCHOENBERG
For
m =1
we f i n d
S1
t o be i d e n t i c a l w i t h t h e c l a s s ( 3 )
c o n t i n u o u s p i e c e w i s e l i n e a r f u n c t i o n s . Observe t h a t t h e c l a s s o f p o l y n o m i a l s of d e g r e e s n o t e x c e e d i n g The r o l e o f t h e r o o f - f u n c t i o n t h e s o - c a l l e d centha.! B-npLine
M 2 m ( ~ ) : Waiting
SZmml c o n t a i n s 2m-1.
of (3)
M2(x)
of
x+
i s t a k e n o v e r by = max ( x , O ) ,
it
may be d e d i n e d b y
Clearly port
M2m(~)
€
S2m-l; w e also f i n d t h a t
M2m(~) > 0
i n its
sup-
- m < x < m. The B - s p l i n e s h o u l d be f a m i l i a r i n view of the fun-
damental i d e n t i t y
which a l s o shows t h a t
IM2,(x)dx
= 1 if
w e choose
f (x) = x
The r e p r e s e n t a t i o n ( 3 ) a l s o g e n e r a l i z e s , and eueAny S ( x )
2m E
. SZm-l
admitb a unique hepheoentation m
S(x) = c c
~
M*m(X
--m
whehe t h e
-
v)
I
c v ahe c o n n t a n t n . T h i s i s t h e s o - c a l l e d ntandahd heptebefl-
t a t i o n . The c o n v e r s e i s clear: Every series (8) f u r n i s h e s an e l e m e n t of
SZm-1
ments o f
.
W e now t r y t o s o l v e t h e i n t e r p o l a t i o n problem (1) b y e l e -
S2m-l.
I n t h i s d i r e c t i o n t h e r e are t w o d i f f e r e n t k i n d s o f
results.
A. T h e d a t a (y,) s e q u e n c e (y,)
ahe
06 poweh
g h o w t h (See [ 8
i s of p o w e h g n o w t h , and w r i t e
1). W e s a y t h a t t h e
ON CARDINAL SPLINE SMOOTHING
(y,)
(9)
393
E PG,
provided t h a t
y,
(10)
v
+
E
PG,
= ~ ( l v l y ) as
f
m,
f o r some
y
2
0.
y1
2
Similarly, we w r i t e
f(x)
(11)
provided t h a t
f ( x ) = O ( l x l y l ) as
x
+
f o r some
f m,
Below w e e x c l u d e t h e t r i v i a l c a s e when
m=l,
l e m i s s o l v e d by ( 3 ) w i t h o u t any r e s t r i c t i o n on t h e
THEOREM 1:
16 t h e heqUenCt
(y,)
i h
0.
s i n c e o u r prob
-
(y,,).
a d pawet g t u w t h , t h e n t h e i n t e h -
palation p t a b l e m
huh a u n i q u e h o l u t i o n
S(x)
huch t h a t
The a s s u m p t i o n ( 9 ) o f Theorem 1 i s a rough one; i t admits,e.g., a l l bounded s e q u e n c e s ( y V ) , w i t h
y = 0 i n ( 1 0 ) . The s e c o n d assump-
t i o n t o which w e now p a s s , i s much more s e l e c t i v e , and
takes
a c c o u n t t h e f i n e r s t r u c t u r e of t h e sequence: i n f a c t i t a d m i t s a narrow subclass of t h e s e q u e n c e s of
PG.
into only
As u s u a l , w i t h strongeras-
sumptions, s t r o n g e r c o n c l u s i o n s are p o s s i b l e : The i n t e r p o l a n t w i l l e x h i b i t a n i m p o r t a n t extremum p r o p e r t y .
S(x)
SCHOENBERG
394
m
B. T h e c a n e when
IAmyv12 <
C
-m
(See [ 9 , L e c t u r e
m
6] )
.
We
i n t r o d u c e t h e classes o f s e q u e n c e s and f u n c t i o n s a s follows:
(14)
(15)
L:={f(x);
Li
Of c o u r s e
f,
...,f (m-l)
and
W e may also d e s c r i b e
ments of L :
a r e a b s o l u t e l y continuous, f(")(x) ELZ@)
are t h e f a m i l i a r
L;
l;
L2 and
1.
Lzl respectively.
as t h e c l a s s o f s e q u e n c e s o b t a i n e d f r a n e l e -
L2 by n s u c c e s s i v e summation. S i m i l a r l y t h e e l e m e n t s o f
are o b t a i n e d from t h o s e o f
THEOREM 2.
L2
by
n
successive integrations.
76
t h e n t h e i n t e h p o l a t i o n phobLem
has a u n i q u e n o L u t i o n n u c h t h a t
Jhio dolution f(x)
(19)
and
i b
S(x)
han t h e 6oLLowing exthemum p h o p e h t y :
an a t b i t a a h y 6unction ouch t h a t
76
396
ON CARDINAL SPLINE SMOOTHING
f ( v ) = y,
v ,
hat all
then
(21)
1-U7
f (x) =
UnLebh
J-m
x.
d o t aLL keaL
S(X)
I n words: If (Y,)
€
R2
t h e n t h e s p l i n e i n t e r p o l a n t S ( x ) mini-
r
mizes t h e i n t e g r a l
(22)
among a l l s u f f i c i e n t l y smooth i n t e r p o l a n t s o f If
y, = P ( v ) f o r a l l
v , where P ( x ) E
(y,).
7
,~t h e -n
~
m P(x) ~ ES2nrlnL2,
and t h e r e f o r e
S ( x ) = P ( x ) by t h e u n i c i t y of t h e s o l u t i o n i n Theorem m 2 . However, h e r e I(S) = 0 . I n t h e g e n e r a l c a s e of (y,) E L 2 wemay
therefore say t h a t
S ( x ) i s among a l l i n t e r p o l a n t s o f
t h a t " i s most n e a r l y " a p o l y n o m i a l o f d e g r e e If P(X)
y,
E SZm-1
= P ( u ) , where
n PG, and so
P
(XI
E
IT^^-^ ,
-
1.
P(x)
9
2 m
but
~
~
the
-
~
f ( x ) such t h a t
I (f) <
m
How d o we a c t u a L L y c o n n t h u c t t h e
9 :l
.
S(x)
04
E
ll ,
AOLUtiOnb
hence a f o r t i o r i
(y,)
E
L2
.
T h i s i n s u r e s t h e c o n t i n u i t y of t h e p e r i o d i c f u n c t i o n m
T(u) =
C -m
y,e
ivu
is
.
assume t h a t
(y,)
r again t h a
There
these
t e h p o l a t i a n p k o b l e m n ? To answer t h i s q u e s t i o n l e t u s f o r t h e
(23)
one
S ( x ) = P ( x ) i s t h e unique s o l u t i o n o f The-
orem 1. Theorem 2 does n o t a p p l y h e r e b e c a u s e (y,) no i n t e r p o l a n t
(y,),
in-
moment
SCHOENBERG
366
which w e c a l l t h e g e n e h a t i n g
6uncLLon of t h e sequence ( y v ) . Here and
below w e d e n o t e t h e r e l a t i o n s h i p between a sequenceand its g e n e r a t i n g f u n c t i o n s y m b o l i c a l l y by w r i t i n g
We a l s o r e q u i r e t h e g e n e r a t i n g f u n c t i o n o f t h e sequence ( M 2 m ( ~ ) ) r which
is
Z2,(u)
m- 1
=
C
v=- (m-1)
ivu MZm(v)e
T h i s i s a c o s i n e polynomial o f o r d e r
I
x
I 2
m-1,
m . I t i s r e a d i l y e v a l u a t e d by ( 7 )
@,(u) = 1 , p14(u) = -1 ( ~ + c o s u), 3
Z,(U) =
I
.
because
MZm(x) = 0
if
and w e f i n d t h a t
1 ~ ( 3 3 + 2 6cos
... .
U + C O S ~ U ) ,
It a l s o has the property t h a t
(27)
0 < d 2 m ( ~5) d2,.,,(u)
5
Z2m(0) = 1
for all
u.
I t f o l l o w s t h a t i t s r e c i p r o c a l h a s an expansion
w i t h real c o e f f i c i e n t s
w
,W-v
= W v # t h a t decay e x p o n e n t i a l l y .
Let
us f i n d t h e s t a n d a r d r e p r e s e n t a t i o n
o f t h e s o l u t i o n o f t h e i n t e r p o l a t i o n problem ( 1 7 ) , which r e q u i r e s that
ON CARDINAL SPLINE SMOOTHING
-
C c . M2m(v j J
(30)
j) = y,
397
v.
for all
Furthermore l e t
b e t h e a s y e t unknown g e n e r a t i n g f u n c t i o n o f t h e ( c . ) . S i n c e t h e con-
I
v o l u t i o n o f two s e q u e n c e s h a s a g e n e r a t i n g f u n c t i o n t h a t i s t h e produ c t o f t h e g e n e r a t i n g f u n c t i o n s o f t h e two s e q u e n c e s , w e see by (241, (26) , and (31) , t h a t t h e r e l a t i o n s
( 3 0 ) are e q u i v a l e n t t o t h e rela
-
tion
* (wv)
Now ( 2 8 ) shows t h a t ( c v ) = (y,)
c
(33)
V
c y j wv-j
=
and t h e r e f o r e
v.
for all
j
T h e b e ake t h e c o e 6 d i c i e n A b a d t h e intekpaLating s p l i n e ( 2 9 ) .
EXAMPLES:
1. 16 m = l ,
we o b t a i n
c v = y,
Section
v
f o r all
.
0
= 1 , wv = O ( v # 0 )
,
and
16 m = 2 , w e f i n d ( S e e [ 9 , L e c t u r e 4 ,
51 ) t h a t
W
V
XIv1,
=
2. I f w e choose
shows t h a t
(34)
t h e n $,(u) = 1 , hence w
cv = w
V
.
y,
-
where
6" , w h e r e
X
= -2
+
47
=
-.26795.
6 o = 1, 6 v = O ( V
Therefore t h e s p l i n e
# O),then (33)
388
SCHOENBERG
i s t h e s o l u t i o n o f t h e i n t e r p o l a t i o n problem
L*m-l(4
(35)
=
6v
I
for a l l
v.
The f u n c t i o n ( 3 4 ) i s t h e dundamental & u n c t i o n o f t h e p r o c e s s , and t h e S(X)
A O h t i O M
o f t h e g e n e h U l p k o b l e m (17) LO g i v e n b y m
T h i s c a r d i n a l i n t e r p o l a t i o n f o r m u l a b r i d g e s t h e gap between the linear i n t e r p o l a n t ( 3 ) a n d t h e c a r d i n a l series ( 2 ) . I n f a c t , n o t i c e t h a t i f
m = 1 t h e n ( 3 6 ) r e d u c e s t o ( 3 ) , w h i l e w e have
l i m S2m-1(~)= m+m
(37)
Also every d e r i v a t i v e
sin
TIX
TIx
(k) (x) c o n v e r g e s t o t h e corresponding derivaS2m-l
t i v e of t h e r i g h t s i d e of
(37) I uniformly f o r a l l real
x
I n o u r d i s c u s s i o n w e have assumed t h a t ( 2 3 ) h o l d s .
the
tULatbJMd
. However,
( 3 3 ) , ( 2 9 ) , and ( 3 6 ) a k e v a l i d d o % b o t h T h e o t e m d 1 and
2, undeh t h e i t t e n p e c t i w e a d d u m p t i a n d .
PART 11.
THE CARDINAL SMOOTHING SPLINE
1. STATEMENT OF THE PROBLEM:
We assume now t h a t
(1)
and r e s t r i c t o u r s e l v e s t o r e a l - v a l u e d d a t a and f u n c t i o n s .
We
also
r e c a l l t h e d e f i n i t i o n s ( 3 . 1 4 ) and ( 3 . 1 5 ) o f P a r t I , o f t h e c l a s s e s ly and
L:
.
I n view o f t h e i n c l u s i o n r e l a t i o n s
ON CARDINAL SPLINE SMOOTHING
( S e e [ 9 , p. 1 0 4 1 )
,
399
w e o b s e r v e t h a t (1) i m p l i e s t h a t (y,)
satisfiesthe
a s s u m p t i o n s o f Theorem 2 f o r a l l m .
We a t e g i v e n m and a n m o o t h i n g patrameteh
THE PROBLEM:
E < 0.
Among
aLL 6unctionn
we w i b h t o d i n d t h e . b o l u t i o n
I
m
J(f) =
(4)
E
*
06
t h e phobLem
2
m
+
( f ( m ) (x)) d x
C
-m
-03
-
y V l 2 = minimum.
I n bOlVing t h e minimum p t o b l e m ( 4 ) we may tenLkiot t h e choice
LEMMA 1: 06
(f (v)
adminnible dunctianb
f ( x ) t o t h e eeementb
06
(5)
PROOF:
If
f ( x ) is such t h a t
J(f) <
m
,
then ( f ( v )
a p p l y Theorem 2 t o t h e s e q u e n c e ( f ( v ) ), and l e t
be such t h a t
s(v)
= f
(v) for a l l
v
.
But t h e n
-
yv) E L2
.
We
4w
SCHOENBERG
and so
in view of the extremum property of
S ( x )
as expressed by (3.21)
Theorem 2. Therefore, for any
f ( x ) , the spline
f (x), produces a value
J(f).
Let
U6
J(s)
of
s(x) that interpolates
thehedote d i n d t h e nolutian
o d t h e m i n i m u m ptobLem
I, m
J(S) =
(8)
E
m
(S(m))2dx+ C
-w
(S(v)
-
Y,)~ = minimum.
Here we need another
LEMMA 2:
7 6 ( 7 ) o a t i d d i e n S(x)
E
L2 (R),
(S(m)(x))2dx = -00
PROOF: From (7) we find that
hence aLno (c.) E L2 , t h e n
C yj-” c . c j, v
3
,
ON CARDINAL SPLINE SMOOTHING
401
i s t h e e v e n s e q u e n c e d e f i n e d by
where (y,)
where, t o s i m p l i f y n o t a t i o n s w e dropped t h e s u b s c r i p t 2m o f M 2 m ( ~ ) . I n t e g r a t i o n s by p a r t s show t h a t
(-ilm-' M(2m-1) x)
Observe t h a t
Jm
-m
M i ( X ) M ( ~ ~ -( x~ )- r ) d x
.
i s a s t e p f u n c t i o n assuming i n c o n s e c u t i v e
u n i t i n t e r v a l s t h e v a ue s
... 1 0 ,
(14)
011,
-
(2m-1 1
)
I
(2m-1 2 )
...
I
1 - 1 1
0, 0,
... .
T h i s sequence h a s t h e g e n e r a t i n g f u n c t i o n
except f o r a s h i f t f a c t o r
eiuk which w e d i s r e g a r d . N o w ( 1 3 ) indicates
t h a t ( y ) i s t h e c o n v o l u t i o n o f t h e sequence ( 1 4 ) w i t h t h e sequence
r
However, i n (13) t h e s e q u e n c e Cu avbv-r
.
If
(yr)
appears
as a sum o f
w e pass from ( a v ) t o t h e r e v e r s e d s e q u e n c e
o b t a i n a genuine c o n v o l u t i o n
Cva-vbv-r
.
L e t us t h e r e f o r e
the
form
(a_"),
we
reverse
t h e f i r s t s e q u e n c e (14). As w e o b t a i n t h e g e n e r a i n g f u n c t i o n o f t h e r e v e r s e d sequence by c h a n g i n g
u into
- u i n its o r i g i n a l generating
function, we f i n d the generating function of factor
eiuk) t h e p r o d u c t
(yr
t o b e ( u p t o a shift
402
-ium
= e
S i n c e (y,)
(2 sin
u 2m 7)
Z2,(u).
i s an e v e n s e q u e n c e , i t s g e n e r a t i n g f u n c t i o n m u s t b e e v e n ,
and t h e r e f o r e
e s t a b l i s h i n g (10).
2 . SOLUTION OF THE PROBLEM:
From ( a ) ,
( 9 1 , and ( 7 ) w e f i n d t h a t
L e t us minimize t h i s f u n c t i o n of t h e (c,).
tions, we differentiate
-a 2 ack
J(S) =
E
To o b t a i n t h e normalequa-
J(S) obtaining
Z yj-kcj+
C { Z c . M ( w - j ) - y w ) M ( w - k ) = O (kEZ).
j
v
j
7
I f w e sum w i t h i n t h e double-sum o n l y w i t h r e s p e c t t o
where
v , we obtain
ON CARDINAL SPLINE SMOOTHING
(3)
(au)
403
2
.
(d2m(u))
+
The normal e q u a t i o n s t h u s become
or
(4)
C j
+
(clj-k
E
Y,
yj-k)~j=
M2m(~
-
k)
(k
E
However, by ( 3 ) a n d ( 1 . 1 0 ) w e f i n d
and w r i t i n g
(6)
(c,,)
+
C(u),
(y,)
+
T(u)
,
w e f i n d t h e normal e q u a t i o n s (4) t o b e e q u i v a l e n t t o t h e r e l a t i o n
i ( p ~ , ~ ( u +) )E~( 2 s i n + ) 2 m
whence
This e s t a b l i s h e s
~ , , ( u ) ) c ( u ) = ~ ( u 4) 2 m ( u ) t
if).
SCHOENBERG
404
THEOREM 3:
I n tehmh
06
whehe t h e c a e d d i c i e n t n dicientn (c.) 7
04
t h e expannion
w"(E)
=
w-"(E)
d e c a y e x p o n e n t i a L L y , t h e coed-
t h e naLution
0 6 t h e minimum p h o b l e m ,
ahe
W e c a l l t h e s o l u t i o n ( 9 ) t h e cahdinad smoothing n p l i n e .
3.
A dew p h o p e h t i e b A.
06
t h e cahdinad nmoothing n p d i n e
W e have assumed above t h a t
E
S(X) =s(x;E).
> 0 . However, i f w e s e t
E
=O
i n (2.81, i t becomes
and a comparison w i t h t h e e x p a n s i o n ( 3 . 2 8 ) o f P a r t I , w v ( 0 ) = wv
for all
v : T h i n nhawn t h a t S (x)
inteapolafing caadinal npline B.
What
n t h e eddect
a n t h e ahiginai? dada sequence
(S( v )
,
(y,) ?
06
06
shows
that
S ( x ; O ) = S ( x ) &educed A0 t h e
Theohem
2.
t h e nmoathing n p l i n e
S(x) = S(x;
E)
T h i s w e answer by determining the "smoothed"
t o compare i t w i t h ( y , ) .
By ( 2 . 9 ) and (2.10)we find
406
ON CARDINAL SPLINE SMOOTHING
a n d t h e r e f o r e , by ( 2 . 7 )
I
I n terms o f t h e e x p a n s i o n
1 (2 sin
(3)
=
u 2m
c
eivu
uv(E)
V
Z2,(U)
+
( 2 ) shows t h a t t h e s e q u e n c e ( S ( V ; E ) ) a h i n e n d h o m t h e d a t a (y,)
by t h e
n m o o t h i n g dohmuea
Observe t h a t by ( 2 . 8 ) a n d ( 3 ) t h e c o e f f i c i e n t s i n terms of
W v ( ~ )
by
u"(E)
= C MZm (v
j
-
j)
U ~ ( E )
are e x p r e s s e d
W.(E).
3
Is (4) a s m o o t h i n g f o r m u l a a c c o r d i n g t o o u r d e f i n i t i o n o f P a r t I , S e c t i o n l ? T h a t it i s o n e w e see i f w e i n s p e c t i t s c h a r a c t e r i s t i c
function
K(u;E)
(5)
1 u 2m (2 sin T )
=
!d2m(u)
+
f o r it is evident t h a t
0 < K(u;E)
(6)
C. cheabing
< K(O;E)
T h e b m o o t h i n g poweh E
.
06
=
1
for
0 < u < 2r
.
t h e 6ohmuLa ( 4 ) i n c h e a b e n w i t h
I n [ 4 , D e f i n i t i o n 2 , p . 5 3 1 w e g a v e good r e a s o n s
in-
for
406
SCHOENBERG
t h e f o l l o w i n g d e f i n i t i o n : Of two d i f f e r e n t smoothing formulae h a v i n g the characteristic functions
d ( u ) and
$(u)
, we
s a y t h a t t h e second
h a s g r e a t e r smoothing power, p r o v i d e d t h a t
(7)
/J(U)
However, i f
I 5
0 <
E
Id(u)I
<
El
for a l l
u, excluding e q u a l i t y f o r a l l
u.
i t i s c l e a r by ( 5 ) t h a t
and t h e c r i t e r i o n ( 7 ) i s s a t i s f i e d .
D.
The deghee
06
eXUCtneAA
0 6 ,the nmootking 6omonuRa
(4) A = h - l .
T h i s f o l l o w s from ( 1 . 7 ) o f P a r t I , b e c a u s e ( 5 ) shows t h a t w e h a v e t h e e x p a n s i o n i n powers o f
E.
06
u
I f w e d r o p o u r a s s u m p t i o n (1.1) , and assume o n l y t h a t ( y v )
poweh ghowth, Rhen
b y t h e dohmutae(2.8)
,
#in conb&ucfi#n
06 t h e bmvotking
(2.10) , and (2.9) , hemaind appticabLe.0f murse, J ( S ) , of
i t s e a r l i e r connection with t h e funtional holds. I n f a c t we w i l l f i n d t h a t sumably, it i s s t i l l t r u e t h a t o u r t h a t (y,)
npfine S ( x ) = S ( x ; E )
J(S) =
m
( 1 . 8 ) , no l o n g e r
f o r a l l s p l i n e s S . Pre-
S ( X ; E ) minimizes
J ( S ) , provided
s a t i s f i e s the condition
o f Theorem 2 . However, t h i s I was n o t a b l e t o e s t a b l i s h . I n any c a s e I recommend t h e c a r d i n a l smoothing s p l i n e ( S ( X ; E ) ) , which r e p r e s e n t s t h e m o d i f i c a t i o n , found more t h a n 30 y e a r s l a t e r , o f may war-time approach t o t h e problem o f c a r d i n a l smoothing.
407
ON CARDINALSPLINE SMOOTHING
REFERENCES
[ 11
T. N.
E.
GREVILLE, On s t a b i l i t y o f l i n e a r s m o o t h i n g
[ 21
T. N.
E.
GREVILLE, On a p r o b l e m of E .
SIAM J. N u m . A n a l y s i s , 3 ( 1 9 6 6 ) , p p . 1 5 7 - 1 7 0 .
s m o o t h i n g , SIAM J . Math. A n a l . [ 31
,
L.
De Forest i n iterated
5(1974) , pp.
376
FRITZ J O H N , On i n t e g r a t i o n o f p a r a b o l i c e q u a t i o n s by m e t h o d s , Corn. on P u r e a n d Appl. Math.
[ 41
formulas,
I.
J.
,
- 398. difference
5 ( 1 9 5 2 ) ,pp.155-211.
SCHOENBERG, C o n t r i b u t i o n s t o t h e p r o b l e m o f approximation
o f e q u i d i s t a n t d a t a by a n a l y t i c f u n c t i o n s , Q u a r t . o f Appl. Math., [ 51
4 ( 1 9 4 6 ) , P a r t A, p p . 45 - 9 9 ,
P a r t B , pp. 1 1 2 - 1 4 1 .
I. J. SCHOENBERG, Some a n a l y t i c a l a s p e c t s o f t h e p r o b l e m s
of
s m o o t h i n g , C o u r a n t A n n i v e r s a r y volume ".Sfidh% and en nay^", New York, 1 9 4 8 , p p .
351 - 3 7 0 .
[ 61
I . J. SCHOENBERG, On s m o o t h i n g o p e r a t i o n s a n d t h e i r g e n e r a t i n g
[ 71
I . J . SCHOENBERG, S p l i n e f u n c t i o n s a n d t h e p r o b l e m o f g r a d u a -
f u n c t i o n s , B u l l . Amer. Math. SOC., 59 ( 1 9 5 3 ) , p p . 1 9 9
t i o n , Proc. N a t . [ 81
Acad. S c i . 5 2 ( 1 9 6 4 ) , pp.
- 230.
947 - 9 5 0 .
I . J . SCHOENBERG, C a r d i n a l i n t e r p o l a t i o n and s p l i n e
functions
11. I n t e r p o l a t i o n o f d a t a o f power g r o w t h , J . Approx. The-
o r y , 6 ( 1 9 7 2 ) , pp. 4 0 4 [ 9
1
- 420.
I . J . SCHOENBERG, C a k d i n a b h p l i n e . i n t e t p o e a t i o n ,
Reg.
Conf.
Monogr. NQ 1 2 , 1 2 5 p a g e s , SIAM, P h i l a d e l p h i a , 1 9 7 3 .
[lo]
E . T . WHITTAKER a n d G.
ROBINSON, T h e caecueun o d o b n e n v a t i a n n ,
B l a c k i e a n d Son, London, 1924.
D e p a r t m e n t of M a t h e m a t i c s U n i t e d S t a t e s M i l i t a r y Academy
West P o i n t , N e w York 1 0 9 9 6
This Page Intentionally Left Blank
Approximation Theory and FunctionaZ AnaZysis J.B. Prolla led.) @North-HolZand Publishing Company, 1979
A CHARACTERIZATION OF ECHELON KOTHE-SCHWARTZ SPACES
M. V A L D I V I A
Facultad Paseo
de C i e n c i a s a 1 Mar, 1 3
Valencia
I n 11
1 ,
A.
(Spain)
G r o t h e n d i e c k a s k s i f e a c h q u a s i - b a r r e l l e d (DF)-space
i s b o r n o l o g i c a l . W e gave a n answer t o t h i s q u e s t i o n i n [ 5 ] s t r u c t i n g a c l a s s of quasi-barrelled
(DF)-spaces which
b o r n o l o g i c a l nor b a r r e l l e d . I n t h i s p a p e r , i n
by
neither
are
the context
of
K o t h e ‘ s e c h e l o n s p a c e s which are M o n t e l , w e c h a r a c t e r i z e t h e of Schwartz u s i n g c e r t a i n non-bornological
con-
spaces
b a r r e l l e d spaces.
consequence, w e p r o v e t h e e x i s t e n c e of non - b o r n o L o g i c a l
the
As
a
barrelled
(DF) - s p a c e s . The v e c t o r s p a c e s w e u s e h e r e a r e d e f i n e d on t h e f i e l d t h e r e a l o r complex numbers. I f
(E,F) is a dual p a i r , we
p ( E , F ) t h e Mackey t o p o l o g y on E
.
If
E
of
denote
by
i s a t o p o l o g i c a l vector space,
E’ is its topological dual. I n the sequel
and
K
X w i l l b e a n e c h e l o n space
A X i t s a - d u a l . L e t us s u p p o s e t h a t t h e s t e p s d e f i n i n g h
a r e all p o s i t i v e , t h e y f o r m a n i n c r e a s i n g s e q u e n c e
and,
for
each
a ( q ) # 0 . L e t E~ be P the s e q u e n c e s u c h t h a t all i t s t e r m s v a n i s h e x c e p t n - t h whose v a l u e
index p,
t h e r e e x i s t s and i n d e x q s u c h t h a t
i s one. G e n e r a l l y , w e f o l l o w t h e terminology o f [ 2 1 f o r t h i s of spaces. I n p a r t i c u l a r ,
9
i s t h e s p a c e g e n e r a t e d by
the
kind vectors
VALDlVlA
410
E~
,
AX[
LI
n = 1, 2 , (AX, X)
I
.
... .
w e always c o n s i d e r
Here
a subspace of
P = 11, : n = l , 2 , . . . } be a p a r t i t i o n of t h e s e t N o f
Let
In i s i n f i n i t e , n = 1 , 2 , .
t u r a l numbers, such t h a t
f i l t e r o f a l l t h e subsets o f tary i n
as
I n of
F n In
N
such t h a t , i f
is finite, n =1,2,... N f i n e r than
f o r some
# @, n = 1 , 2 , . . .
E
J, then
M n In
.
E
F
{Fj : j
Let
.
F be the
t h e complemen-
F so t h a t , i f
t h e s e t o f a l l t h e f i l t e r s on j
F
.. . L e t
na-
E
be
J}
M E F
I t f o l l o w s immedi
j
-
a t e l y t h a t , with t h e r e l a t i o n of inclusion, t h i s set is inductiveord e r e d . Using Z o r n ' s lemma, l e t
PROOF: A1
U
Let
A1
and A2 b e t w o non-empty subsets of
A2 = I n , and
A1
A2 =
0. T h e r e f o r e ,
i n t e r s e c t s a l l t h e e l e m e n t s of
A u~ [ u { I
A =
belongs t o
U and
For each Xx[~(Xx,X)] If
U1
U b e a maximal e l e m e n t .
P
one o f t h e s e s e t s , s a y A1
1111
Xx(U)
t h e s e c t i o n a l subspace o f
X
X (U)={a=(al,a2 c . . . r a n l . . . ) : a E
and U 2 b e l o n g t o
that
This completes t h e p r o o f .
U C N , w e d e n o t e by
d e f i n e d by
such
U and t h e n
: p E N, p #
A 17 I n = Al.
In
U i t follows t h a t
and
and, t h e r e f o r e , L = u {XX(U) :
u
E U}
X
U1 n U 2
X
, an=O, WnE belongs to
U}. U
A CHARACTERIZATION OF ECHELON KOTHE-SCHWARTZ SPACES
i s a s u b s p a c e of gy of
c o n t a i n i n g p . L e t us s u p p o s e t h a t t h e t o p o l o -
AX
L i s t h e one i n d u c e d by
PROPOSITION 2:
v(AX,X).
X i n a M a n t e L Apace a n d T i n a batrtree -in L , it
16
abnotrbn .the b o u n d e d n u b n e t n
PROOF:
41 1
p n AX(N
06
-
n
I n ) , doh each
L e t us s u p p o s e t h a t t h e r e e x i s t s i n
p
n XX(N
E
- In)
N.
a bounded
T . W e now i n d u c t i v e l y mn-
normal subset B which i s n o t a b s o r b e d by
s t r u c t a s e q u e n c e ( y ) i n B i n t h e f o l l o w i n g way: L e t 9 t h a t w e have a l r e a d y o b t a i n e d t h e e l e m e n t s y1,y2,...,yq
us
suppose
in
B such
that
yp
where
N(1)
,
9 pT, yp =
N(2),
j o i n t s , such t h a t
.. . ,N(q) N(1)
c
r EN(p)
U
N(2)
The s p a c e
El
Let
B
2
p
I
... U N ( r ) .
n
XX(N
In , m u t u a l l y d i s -
n which d o e s n o t l i e i n M(p
- In) is
- M(q))
B1 be t h e p r o j e c t i o n o f b e the p r o j e c t i o n of
,
K , p = 1 , 2 , ...,q
c o n t a i n s t h e f i r s t e l a n t of In, and N ( I p ) r P’1r
U
= p n XX(N
E
are f i n i t e s u b s e t s of
c o n t a i n s t h e f i r s t element of M(r) = N(1)
ar Er ’ a r
normal s e t i t f o l l o w s t h a t
t h e t o p o l o g i c a l d i r e c t sum o f
B1
according t o
E2 a c c o r d i n g t o U
B2
-
E2 = p n A X “
and
B o n t o El
B onto
- l),being
C
El.
B. Moreover, B1
(In
E2
- M(q))).
,
and
B is
Since
+ B2
2
B.
let a
B1
i s a bounded s u b s e t o f t h e f i n i t e - d i m e n s i o n a l s p a c e E l , h e n c e T abs o r b s B1. S i n c e B i s n o t a b s o r b e d by c a n f i n d an e l e m e n t
yq+l E B2
Yq+l
p
C
B
T,
neither
such t h a t
(q + 1 ) T .
B2. T h e r e f o r e , w e
VALDlVlA
412
The e l e m e n t
where
yq+l
c a n b e w r i t t e n i n t h e form
N(q +1) i s a f i n i t e s u b s e t o f
---
I n , d i s j o i n t from
set
each
q ) and t h a t it c o n t a i n s t h e f i r s t e l e m e n t of I which n i s n o t c o n t a i n e d i n M ( q ) . The sets of t h e sequence ( N ( q ) ) d e f i n e a
"1)
i
N(2)
p a r t i t i o n of
In
Let
S i n c e t h e r e s t r i c t i o n of an
U E U
such t h a t
U on
U n In
In i s an u l t r a f i l t e r , t h e r e
coincides with
P1
or
.
P2
,
exists
U n I n =P1,
E Xx(U) , q = 1 , 2 , . . . The s p a c e Xx(U) is bary2q r e l l e d , b e c a u s e i s a s e c t i o n a l s u b s p a c e o f Ax[ 1~ (Xx,X) 1; hence T ab-
say. Therefore,
sorbs the set
{ y 2 , y 4 1 . . . l y 2 q . . . ) and i t c o n t r a d i c t s
S i n c e t h e normal h u l l o f e v e r y bounded s u b s e t of bounded, i t f o l l o w s t h a t
PROPOSITION 3 :
9
n XX(N
-
is
In)
T a b s o r b s every bounc?.t?d subset of 9 n X X ( N - I n ) .
1 6 X i b a MonteL b p a c e and T i n a b a m e L i n L , it
abboabn eweky bounded n u b n e t o d 9 .
PROOF:
L e t us suppose t h a t t h e r e i s i n
s e t B n o t a b s o r b e d by is n o t i n T
.
9
T . L e t us choose i n
a bounded and normal subB an element
y1
By a r e c u r r e n t p r o c e s s , l e t u s d e f i n e a sequence
which (y,)
413
A CHARACTERIZATION OF ECHELON KOTHESCHWARTZ SPACES
i n B.
yp
where
are a l r e a d y d e f i n e d , such t h a t
y1,y2,.-.,yq
If
N(1)
,
p pT, yp
N(2),
.. .
=
z
r EN(p)
a
a
E
r r'
r
E K,
p =1,2
N ( q ) are f i n i t e subsets o f
,... , q
,
N so t h a t
...................
Let
K
4
= U {Hp
: p =l12,...,q}.
Then
~p
i s t h e t o p o l o g i c a l d i r e c t sum
of
Let
B1
b e t h e p r o j e c t i o n of
p r o j e c t i o n of
Bl
B onto
El a c c o r d i n g t o E 2 and B 2 the
E2 a c c o r d i n g t o
Moreover, B1
U B2 C B.
B onto
+
B
2
El.
Since
B
is
normal,
3 B.
From t h e p r e v i o u s p r o p o s i t i o n , i t f o l l o w s e a s i l y t h a t T absorbs B1.
S i n c e B is n o t a b s o r b e d by
an e l e m e n t
Then
y
q+l
E
B2
C
B
T I neither
such t h a t
B2
,
hence w e
can
find
414
VALDlVlA
being
N(q
+
1) a f i n i t e s u b s e t o f
N which n o t i n t e r s e c t s K
P = {In : n
is a p a r t i t i o n of number
nq+l
,
l a r g e r than
n
q'
.Since
...I
= 1,2,
N(q + 1 ) i s f i n i t e , w e can f i n d
N and
q
a
natural
such t h a t
N ( q + l ) C I n +1 9
Let
M = U {N(q) : q =1,2,.
. . . Then
s e t w i t h f n i t e complement i Since
-
Ax(N
i t follows t h a t
y
9
-
N
y
E AX(N
$2 qT, q = 1 , 2 ,
C
n in a
U.
and i t
con-
T absorbs each
9.
X[p(X,Xx)
I
- M) , q = 1 , 2 , . . . ,
... . C l e a r l y ,
q.e.d.
1 a Monte1 s p a c e which i s n o t Schwartz. Therefo&,
t h e r e e x i s t s a p o s i t i v e i n t e g e r k such t h a t , i f s u b s e t o f a l l t h e n a t u r a l numbers
(-)
n so t h a t
a;)
M is the
ordered
# 0 , t h e sequence
nE M
does n o t converges towards z e r o , p = k d e f i n e a n i n c r e a s i n g sequence i n M
such t h a t
F
M E
q {yl,y2,...,yq,...}
T absorbs the set
bounded s u b s e t o f Let
I n . Hence
M ) i s b a r r e l l e d and
t r a d i c t s t h e f a c t of
i n t e r s e c t s each
M
N
+ 1,
k
+ 2,.
. . , [2,p.422] . L e t
us
416
A CHARACTERIZATION OF ECHELON KOTHE-SCHWARTZ SPACES
Since
i s a Monte1 s p a c e w e c a n s e l e c t a s u b s e q u e n c e (mi)
h[u(X,Xx)]
k l > k + l
( q i ) so t h a t , f o r a p a r t i c u l a r number
of
1 (kl) -
lim
i+m
[ 2, p. 421
I.
a
mi
Let
. . .} .
{ml,m2,
I1 b e t h e s e t
Obviously, M
f i n i t e s e t . L e t us s u p p o s e t h a t w e h a v e c o n s t r u c t e d N,
Il,12,...,I
9 ,
so t h a t
I
n Ir
=
P
fi
8,
-
is an
I1
subsets
inof
i s a n i n f i n i t e s e t and
M
p # r,
.
p, r = 1 , 2 , . . . , q
I = ~ r l , r 2 , . . . , r i , . . . ~s u, p p o s e a l s o t h a t t h e r e a r e two n a t u r a l P numbers k > k + p, i so t h a t P P '
If
1i m i +m
Let
H
= c
a (k+p) r
P
# O ,
i
= U {Ip : p =1,2,...,q}.
q a sequence
n l < n 2 <
we obtain, for
u > k
P '
1i m i +m
= 0,
i
i
P
.
'i
I f w e a r r a n g e t h e t e r m s of
...
< n . <
p =1,2, ...,q
From (1) and t h e c o n d i t i o n of space, it follows t h a t
a
(kp)
I
H
9
nM as
...
that
X [ p ( X I Ax) 1 n o t b e i n g a Schwartz
416
VALDlVlA
M
- Hq
= {sl,s2,..
., s i t . . . I
i s a n i n f i n i t e s e t and t h e s e q u e n c e
does n o t c o n v e r g e s t o z e r o . T h e r e f o r e , w e c a n s e l e c t
( t i ) of ( s i ) and a p o s i t i v e i n t e g e r
kq+l
> k
+
q
+
a 1
subsequence so t h a t
b e t h e s e t { t l l t 2 , . . . , t i , . .. } t o g e t h e r w i t h the f i r s t q+l I n t h i s way w e o b t a i n a p e e l e m e n t o f N which does n o t l i e i n H q t i t i o n P = { I n : n =1,2,...} o f N such t h a t I n i s i n f i n i t e , whose Let
I
.
p r o p e r t i e s w i l l b e used i n t h e s e q u e l .
THEOREM 1:
in i n
16 t h e MonteL A p a c e
Xx[~(Xx,X)l
i n n o t Schwaatz,
X[p(X,Xx)]
a dense nubnpace
G
theae
w h i c h i n batr4eLLed and non boa-
naLogicaL.
PROOF:
Using t h e number
k and t h e p a r t i t i o n
t h e space L as w e d i d a t t h e b e g i n n i n g of
construct
and t h e s u b s p a c e G o f and t h e v e c t o r
Ax[
a ( k ) . W e w i l l prove t h a t
bounded subset o f
,
hence
G
.
paper L
G i s b a r r e l l e d and non br-
B y P r o p o s i t i o n 3 , T a b s o r b s every
9 . On t h e o t h e r hand, T n 9
this
we
which i s t h e l i n e a r h u l l of
p(Xx,X)]
n o l o g i c a l . L e t T be a b a r r e l i n
[ 3, p. 324 ]
P o b t a i n e d above,
9
is a bornological
i s a neighbourhood o f t h e o r i g i n
space in
9
.
A CHARACTERIZATION OF ECHELON KOTHE-SCHWARTZ SPACES
S i n c e 9 i s dense i n
G
,
the closure of
T
and i t i s a neighbourhood o f t h e o r i g i n i n r e l l e d . S i n c e L i s a subspace of l a r g e r t h a n one, t o see t h a t r e s u l t o f Mackey [ 4
1 ,
I)
41 7
p i n G is c o n t a i n e d i n T
Therefore, G
G .
G whose codimension i n
it i s s u f f i c e s t o prove t h a t
L e t us suppose t h a t ( B ( " ) )
ber
G
is not
G i s n o t bornological, according
i s a s e q u e n c e of
to a
a(k) is not the
l i m i t i n t h e s e n s e o f Mackey, o f a s e q u e n c e l y i n g i n
to
is bar-
L
.
L which c o n g e r g e s
a ( k ) i n t h e s e n s e of Mackey. C o n s e q u e n t l y , t h e r e i s a n a t u r a l nmp such t h a t ( 6 ' " ) )
pology of t h e norm
11
11
B(")
= (bin) ,bin)
W e can f i n d
U E U
p) f o r the to-
r so t h a t
,.. . ,bq( n ) ,. . . )
such t h a t
,
w e have t h a t
B(n)
E Xx(U). S i n c e
n o t f i n i t e , w e can o b t a i n a p o s i t i v e i n t e g e r
r , such t h a t
+
XE(k
deduced from t h e u n i t b a l l
W e can f i n d a p o s i t i v e i n t e g e r
Given
a (k) i n
converges t o
s
in
I
PI
U n I
larger
is
P
than
bLn) = 0 . Then
and w e o b t a i n a c o n t r a d i c t i o n . Therefore,
G i s not bornological.
VALDlVlA
418
L e t E be a Fhechet-Schwahtz bpace.16 F 0 a bahn&ed
PROPOSITION 4 :
bubbpace
PROOF:
06
E' [ p ( E ' , E )
1 , then F
bohnolagical.
i b
L e t ( A ) b e an i n c r e a s i n g fundamental sequence of compact suh-
n
. Let
sets of
E'
Let
be t h e c l o s u r e o f
Bn
u ( E ' ,E)
1
us suppose f i r s t t h a t F is dense in E ' [ p ( E ' , E ) ] .
h e normed s p a c e
c l o s u r e of
in
An r~ F
FAn
E'[ v ( E ' , E l ]
. Let
Fn
i n t h e Banach s p a c e
t h e t o p o l o g y i n d u c e d by t h e one a s s o c i a t e d t o
be the
E'
.
An
with
Since F is a An (DF)-space, ( B n ) i s a fundamental sequence of canpact sets i n E ' [ p ( E ' , E ) I ,
[ 2, p.
402
tegers Then A
I.
Given a p o s i t i v e i n t e g e r
q and P
r such t h a t
A
E'[u(E',E)]
spaces. L e t
u n of
E'
proved o b t a i n i n g t h e c l o s u r e
with
PROOF:
Ar'
u to
F
F n Fn
. Since
Banach
of
is d e n s e n can b e e x t e n d e d t o a F n F
Fn. Evidently, there e x i s t s
a linear
.. .
i n Fn , n = 1 , 2 , . Then n E' [ p ( E ' , E ) 1 and, consequently, i t s r e s t r i c t i o n
F i s c o n t i n u o u s . Hence,
that
and B i s a compact s e t i n E ' q q and t h e r e f o r e w e c a n a f i r m that
which c o i n c i d e s w i t h
i s c o n t i n u o u s On
in-
B
u be a bounded l i n e a r form on
c o n t i n u o u s l i n e a r form v n i n form v on
p t h e r e a r e two p o s i t i v e
i s the i n d u c t i v e l i m i t o f t h e sequence (F,)
i n Fn, the r e s t r i c t i o n
to
C
P i s a compact s e t i n Fr
El
v
F i s b o r n o l o g i c a l . The g e n e r a l case
of
F in
E'[p(E',E)]
and
It
t h e orthogonal subspace of
E
to
F.
f o l l o w s from Theorem 1 and P r o p o s i t i o n 4 .
u
is
proving
i s t h e d u a l t h e Mackey o f t h e Fr6chet-Schwartz s p a c e E / F F I
v
I,
A CHARACTERIZATION OF ECHELON KOTHESCHWARTZ SPACES
419
REFERENCES
[
1]
A. GROTHENDIECK, Sur les spaces (F) et (DF), Summa Brasil. (19541, 57 - 123.
-
3,
[ 2]
G. KOTHE, Topological vector spaces I. Berlin-Heidelberg York. Springer: 1969.
[ 3]
T. KOMURA and Y. KOMURA, Sur les espaces parfaits de suites et Japan 15(1963) , 319-338. leurs g6nGralisations. J. Math.*.
[4]
G. MACKEY, On infinite dimensional linear spaces, Proc. Acad. Sci. USA 29 (1943), 216 -221.
[5]
M. VALDIVIA, A class of quasi-barrelled (DF)-spaces which not bornological, Math. 2. 136 (1974), 249 - 251.
New
Nat.
are
This Page Intentionally Left Blank
Approximation Theory and Functional Analysis J.B. Prolla ( e d . ) 0 North-Holland Publishing Company, 1979
THE RATIONAL APPROXIMATION OF REAL. FUNCTIONS
DANIEL WULBERT M a t h e m a t i c s Department University of California L a J o l l a , C a l i f o r n i a 92093, USA
I.
INTRODUCTION
This paper i s c l o s e l y r e l a t e d t o t h e classical theory
of
u n i f o r m a p p r o x i m a t i o n o f c o n t i n u o u s f u n c t i o n s by q u o t i e n t s of nomials. That i s , l e t
f
best poly-
b e a c o n t i n u o u s r e a l f u n c t i o n on [ 0 , l l
and
let
(1.1)
p where
/q
irreducible1
Pn d e n o t e s t h e r e a l p o l y n o m i a l s of d e g r e e less t h a n o r
equal
n. I t i s c l a s s i c a l l y known t h a t t h e r e i s a n b e s t approximation t o
f
IIf
(1.2)
r
E
RZ
[ s e e f o r example Walsh, 1935
- rI1
= dist (f
I
is
which
. That
a
is,
,RE).
F u r t h e r m o r e t h e a p p r o x i m a t i o n i s c h a r a c t e r i z e d by f
-
-9
hav-
i n g t h e z e r o f u n c t i o n a s a b e s t a p p r o x i m a t i o n from t h e l i n e a r s p a c e
qPm where
N = max { a q
+
m,
+
pPn = PN
ap
+
.
n}. Hence 421
r i s a b e s t approximation t o
422
WULBERT
f i f and o n l y i f
f - r
has an extremal a l t e r n a t i o n of length
(Achieser [ 1 9 3 0 ] ) . I t follows t h a t b e s t approximations
are
N+2. always
is
unique. I n t h i s s e t t i n g however t h e b e s t a p p r o x i m a t i o n o p e r a t o r n o t g e n e r a l l y continuous. I n f a c t , it i s continuous a t if
f
h a s a normal p o i n t
e i t h e r ap = m o r t i o n s , R:(C),
p/q
f
i f andonly
a s a b e s t approximation, t h a t i s ,
aq = n (Werner
[ 1965
if
1 1 . The complex rational fun*
a r e defined s i m i l a r l y with
Pm and
Pn
with
replaced
P m ( C ) a n d Pn (C), t h e p o l y n o m i a l s w i t h complex c o e f f i c i e n t s . A complex f u n c t i o n d e f i n e d on [ 0 , 1 1 s t i l l h a s e x a c t l y one b e s t
a p p r o x i m a t i o n from
w e l l understood.
, but
i n R:(C)
?,(a).
However a p p r o x i m a t i o n from R m ( C ) i s n o t as n
It i s s t i l l t r u e t h a t a b e s t approximations e x i s t s
Walsh [ 19311 h a s c o n s t r u c t e d a n example t o show t h a t i f
t h e domain of t h e f u n c t i o n s i s a p a r t i c u l a r “ c r e s c e n t moon”
shaped
r e g i o n o f t h e complex p l a n e ( i n s t e a d o f t h e i n t e r v a l [ O , l ] as i n o u r s e t t i n g ) , t h e n t h e r e i s a complex f u n c t i o n w i t h more t h a n one
m
approximation i n R n ( C ) .
best
More r e c e n t l y E . S a f f and R . Varga made
s u r p r i s i n g observation t h a t i n f a c t (x 1 a p p r o x i m a t i o n s from R1 (C) [ 1 9 761
.
-
the
1/212 h a s nonunique
best
I f t h e f u n c t i o n b e i n g approximated i s t h e r e a l f u n c t i o n f , t h e n
i t s b e s t approximation i n
P n ( C ) i s a l s o r e a l , and t h e
r e d u c e d t o t h e t h e o r y of a p p r o x i m a t i o n from
Pn.
problem
But f o r t h e r a t i o
n a l f u n c t i o n s t h e Saff-Varga example shows t h a t t h e a n a l o g o u s
R e R:(C)
-
-
reduc-
t i o n i s n o t v a l i d . It appears n a t u r a l t o consider approximations f from
is
to
t h e real p a r t s of R t ( C ) functions.
T h i s p a p e r i s a n e x p o s i t i o n o f p a r t o f such a s t u d y .
The
de-
t a i l e d p a p e r w i l l a p p e a r e l s e w h e r e . A s i t t u r n s o u t t h e t h e o r y of app r o x i m a t i o n from
R e R t i s , a n i n t r i g u i n g mix of t h e r e g u l a r i t y
a p p r o x i m a t i o n from R E w i t h t h e p a t h o l o g y o f t h a t from a r e a l s o some a p p l i c a t i o n s t o a p p r o x i m a t i o n from
R;(~I.
R:(C).
of There
THE RATIONAL APPROXIMATION OF REAL FUNCTIONS
423
11. EXISTENCE OF BEST APPROXIMATIONS
I n t h e c l a s s i c a l s e t t i n g s , t h e e x i s t e n c e of a b e s t approximat i o n i s e a s y t o e s t a b l i s h . The i d e a is that a minimizing sequence r
i
for f
d i s t (f , R E ) ) h a s a s u b s e q u e n c e w i t h c o n v e r g i n g
nu-
m e r a t o r s and d e n o m i n a t o r s . C a n c e l l i n g common z e r o s o f t h e l i m i t
nu-
( i . e . I l l i -fll
+
merator a n d d e n o m i n a t o r p r o d u c e s a b e s t a p p r o x i m a t i o n t o f . Here t h e p r o b l e m i s t h a t t h e l i m i t f u n c t i o n may n o t b e i n F o r example, f o r e a c h
R e R:((T.).
> 0
E
(2.1)
so
I
(2.2
xm+n
But , o n e e a s i l y shows t h a t
i s n o t o f t h e form
(2.3)
Hence t h e t h e o r y i s a c t u a l l y a b o u t a p p r o x i m a t i o n from t h e c l o s u r e o f R e R t ( C ) ( d e n o t e d h e r e f t e r by R:).
m c h a r a c t e r i z e Rn Clearly,
.
5
:R
Q+:,
The f i r s t p r o b l e m , t h e n ,
,
where
+
1, t h e n
is
to
In fact:
2
m
1.
PROPOSITION: 16 n
2.
PROPOSITION: F o h age m ,
age
f
E
C [0,11
.
and n
Rt ,Rt
=
c+n.
admito bent apphoximationb t o
WULBERT
424
111. CHARACTERIZATIOX AND U N I C I T Y OF APPROXIMATIONS FROM
:2
As i n t h e c h a r a c t e r i z a t i o n o f a p p r o x i m a t i o n from R:
the idea
i s t o change t h e problem t o t h a t o f a p p r o x i m a t i o n f r o m a m o r e computa b l e s e t . W e w i l l f i r s t s t a t e a s p e c i a l case so t h a t t h e g e n e r a l c a s e
a b
w i l l a p p e a r less a b s u r d . Suppose t h a t
a
no common f a c t o r s and t h a t t h e d e g r e e s o f 2n
+
aa 5 ab
c
E
t h a t a and
I
and
b
b
have
a r e such
that
+ m.
Let
H ( a , b ) = {h E PM : sgn h(x) =
(3.1) where
3.
M = ab
+
m
i d
Now i n g e n e r a l suppose
b 2 0).
a - 6
f
f
a
c.
E
id and o n l y
f .
id
H(a,b).
dkam
From t h e d e f i n i t i o n o f
a and b have no common q u a d r a t i c f a c t o r s
a and b have some
However it may be p o s s i b l e t h a t
r e a l zeros. L e t
x E z(b)}
for
a b e n t appkoximatian t o
z e h o i6 a b e n t a p p h o x i m a t i o n t o
w e may assume t h a t
sgn a ( x )
Z ( f ) d e n o t e s t h e z e r o set o f a f u n c t i o n
and
$
PROPOSITION:
-
F be t h e g r e a t e s t monic common d i v i s o r o f
(i.e. common
a a n d b.
Put (3.2)
a.
= a / F
and
bo = b / F .
Now p u t M = max { ab
(3.3)
0
= dim { b P
+
o m
For
a 6
E
:2
+
aoP2n)
+
-
2nl
1.
w e now d e f i n e Z(bo) nlR
(3.4)
m, aao
if
2n
+ aa 5
ab
+m
Z(a,b) = [z(bo) n n ] u I-lu{--)
if
2n
+
aa > a b + m
THE RATIONAL APPROXIMATION OF REAL FUNCTIONS
426
For convience w e w i l l w r i t e f(-)
(3.5)
l i m f(x) x+m
for
and
when t h e s e l i m i t s e x i s t s . Now d e f i n e :
for
4 . COMMENT:
x E z(a, b)}.
is
With t h e above n o t a t i o n p r o p o s i t i o n 3 above
still
valid. Our i n t e r e s t i n p r o p o s i t i o n 3 i s t h a t o n e c a n compute t h e numb e r o f p o s s i b l e s i g n c h a n g e s of members of H ( a , b )
and
use t h i s
to
d e r i v e an e x t r e m a l a l t e r n a t i o n t y p e o f c h a r a c t e r i z a t i o n f o r a p p r o x i -
c.
m a t i o n s from
p e n d i n g o n t h e numbzr and p a r i t y of t h e p o i n t s i n and i n
-
However t h e r e s u l t s e p a r a t e s i n t o many c a s e s d e
Z(a,b) n [ l
,
a ) .
Rather than p r e s e n t i n g
Z(a,b) n (the
-
,01
complicated
s t a t e m e n t o f t h e a l t e r n a t i o n t h e o r e m , w e w i l l g i v e some of t h e con
-
sequences.
5.
COROLLARY:
g e n t apphoximationn 6hom
6.
COROLLARY:
S u p p o b e a , a n d b have no corninion dactuhb, m
and Z ( b ) n R =
.id
{horn
2
7.
+
max{m
+
@I
.
Then
f-
and o n l y -id ab, 2n
COROLLARY:
+ aal.
f
:Q
ahe unique.
+
ab 2 2n
+ aa
i n a b e n t apphoximatian t o f E C [ 0
, 11
-
2 b
A conntant dunction
han an e x t h e m d d t e m a t i o n 0 6 l e n g t h
i d
a b e n t apphoximation,
t o
a
426
WULEERT
ContinUOUb d u n c t i o n , d h o m
id and o n l y id t h e ehhoh d u n c t i o n han
2;
an exthemal a l t e h n a t i o n 0 6 l e n g t h
8.
COROLLARY:
r E
n 2 1 thehe
16
2 + m a x { m , 2111.
i n a continuoun bunction
f
and
an
nuch t h a t
IV.
(i)
r i n a b e n t apphoximation 0 6
(ii)
-r
f
but
i n n o t a bebt apphoximation t o
APPROXIMATION FROM
I n some special
R:(C)
f
-
2r.
:
cases
= dist
d i s t (f,R:)
Although
(f,R:(C)).
t h i s p h e n o m e n o n o n l y occurs i n a r e s t r i c t e d s e t t i n g , t h e r e a r e
applications to t h e theory for a p p r o x i m a t i o n f r o m
For -1e
R:(C).
w e can e a s i l y produce r e a l f u n c t i o n s w h i c h have nonunique b e s t proximations from
9.
p h o x i m a t i o n dhom
ap-
R'"(c).
Let
THEOREM:
some
f E C[O
R;(C)
,11
and
2E
. 16
R :
a
i n a bent
up -
then
II f - a II 2 d i s t ( f , Rmn )
whehe
10.
N = min { n
COROLLARY:
-
ab, m
Let
m
- aa].
2 n, p
E
Pm-n , and
f
E
C [0
,l1;the
dollow-
i n g ahe e q u i v a l e n t :
11. .a
i d
(i)
p
i n a b e n t apphoximatian t o
f
Ahom
R:(C)
(ii)
p i n a b e n t apphoximation t o
f
dhom
R.:
COROLLARY:
Let
m
2 n
and
a b e n t apphoximation @om
f E C[ 0
R:(C)
, 11.
,
A condtant
id and o n l y id
f
-
dunc-tion a
han a n
THE RATIONAL APPROXIMATION OF REAL FUNCTIONS
+
2.
For e v e r y
m
exthemaL a l t e h n a t i o n 0 5 L e n g t h
12.
EXAMPLE:
(Saff
- Varga)
m
+
n
n
427
1. 1
there
.are
con-
t i n u o u s r e a l f u n c t i o n s which have nonunique b e s t a p p r o x i m a t i o n s from R:(C).
13.
COROLLARY:
(Saff -Varga)
be t h e b e n t a p p h o x i m a t i a n .to
a n exthemaL a l t e a n a t i o n 2
+
f
6hom
E
Then
R:(Cl.
$(Z(a) f
n Z(b) = @)
-
munt have
L e n g t h at Lean2
06
m
Lef n l m + l . L e t ?
+
min I n
-
ab, m
- aal.
REFERENCES
[
11
[ 21
N.
I . ACHIESER, On extremal p r o p e r t i e s o f c e r t a i n r a t i o n a l f u n c -
t i o n s . Doklady Akad Nauk SSSR ( 1 9 3 0 ) , 4 9 5 - 4 9 9
E . W.
CHENEY,
(Russian).
McGraw
l n t h o d u c t i o n t o Apphoximation Theohy.
H i l l , N e w York 1 9 6 6 . [ 31
E . W.
CHENEY, A p p r o x i m a t i o n by g e n e r a l i z e d r a t i o n a l f u n c t i o n s ,
P h o c e e d i n g b Symponium o n t h e A p p h o x i m a t i o n a 5 G e n e r a l Motors, E l s e v i e r P u b l i s h i n g C o . ,
101 [ 41
E . W.
- 110.
CHENEY
C.
Amsterdam 1964,
a n d H . L . LOEB, G e n e r a l i z e d r a t i o n a l f u n c t i o n s
SIAM J o u r n a l [ 51
Funcfionn,
Numerical Anal.
1 ( 1 9 6 4 ) , 11 - 2 5 .
,
S u r l e s polynomes d ' a p p r o x i m a t i o n e t l a r e p r d s e n t a t i o n ' a p p r o c h 6 e d ' u n a n g l e , Acad. Royale
J. DE LA VALLEE POUSSIN,
d e B e l g i q u e , B u l l d e l a Classe d e s s c i e n c e s 1 2 ( 1 9 1 0 ) .
61
A. A.
GOLDSTEIN, R a t i o n a l a p p r o x i m a t i o n s on f i n i t e p o i n t sets, Sympanium o n t h e A p p h o x i m a t i o n 0 6 F u n c t i o n n , Szneral Motors, E l s e v i e r P u b l i s h i n g C o . ,
Amsterdam 1 9 6 4 .
420
WULBERT
[ 7)
A. N.
KOLMOGOROFF, A remark c o n c e r n i n g t h e p o l y n o m i a l s o f P.L. T s c h b y c h e f f which d e v i a t e t h e l e a s t from a g i v e n f u n c -
t i o n ( R u s s i a n ) Uspekhi Math. Nauk 3 ( 1 9 4 8 ) , 2 1 6 - 2 2 1 . 81
G. MAINARDUS a n d R. S . VARGA, Chebyshev r a t i o n a l a p p r o x h m t b n t o c e r t a i n e n t i r e f u n c t i o n s i n [ 0 , +m1 ,J.Approx. Theory 3 ( 1 9 7 0 ) , 300
- 309.
[ 91
J. A. ROULIER a n d G. D. TAYLOR, R a t i o n a l Chebyshev approximat i o n of [ O , + m 1,J.Approx. Theory l l ( 1 9 7 4 ) , 208-215.
[lo]
E.
B.
SAFF and R. S. VARGA, Nonuniqueness o f b e s t complex rat i o n a l a p p r o x i m a t i o n s t o r e a l f u n c t i o n s on r e a l i n t e r -
vals (1976) , p r e p r i n t .
[Ill
J. L. WALSH, I n t e r p o l a t i o n a n d a p p r o x i m a t i o n b y r a t i o n a l f u n c -
t i o n s i n t h e complex domain, Amer. Math. SOC. Cbllcquim P u b l i c a t i o n s 20, P r o v i d e n c e R . I . , [121
J . L. WALSH,
1935.
On t h e o v e r c o n v e r g e n c e o f s e q u e n c e s of
f u n c t i o n s , Amer. J . Math. 54 (19321, 559
rational
- 570.
[131
J. L. WALSH, The e x i s t e n c e of r a t i o n a l f u n c t i o n s of best
[141
H . WERNER, On t h e l o c a l b e h a v i o r of t h e r a t i o n a l T s c h b y s c h e f f o p e r a t o r , B u l l . Amer. Mat. SOC. 70(1964) , 554 555.
[15]
D. E . WULBERT, The r a t i o n a l a p p r o x i m a t i o n of
app r o x i m a t i o n , T r a n s . Amer. Math. SOC. 3 3 ( 1 9 3 1 ) , 477-502.
-
Amer. J . Math.
,
t o appear.
real
functions,
Approximation Theory and Functional AnaZysis J.B. ProlZa (Ed.) 0 Nor&-Holland Publishing Company, 1979
FUNDAMENTAL SEMINORMS
G U I D O ZAPATA"
I n s t i t u t o de Matemstica Universidade F e d e r a l
do Rio de J a n e i r o
R i o de J a n e i r o ,
Brazil
1. INTRODUCTION
Here w e w i l l c o n s i d e r a g e n e r a l p r o b l e m o f p o l y n o m i a l a p p r o x i -
of
m a t i o n i n e u c l i d e a n n - d i m e n s i o n a l s p a c e . The subject
polynomial
a p p r o x i m a t i o n w a s i n i c i a t e d i n 1885 w i t h t h e f i r s t v e r s i o n
of
the
Weierstrass t h e o r e m f o r u n i f o r m a p p r o x i m a t i o n o n compact sets o f euc l i d e a n s p a c e . The non-uniform a p p r o x i m a t i o n p r o b l e m
on
the
whole
s p a c e was i n i c i a t e d w i t h t h e B e r n s t e i n p a p e r of 1 9 2 4 [ 2 ]
and
con-
t i n u e d t o b e d e v e l o p e d i n t h e so c a l l e d B e r n s t e i n problem. C l a s s i c a l l y t h i s problem h a s b e e n s t u d i e d f r o m t h e p o i n t of view
of
continuous
and i n t e g r a b l e f u n c t i o n s i n t h e n a t u r a l c o n t e x t o f weighted S e e , f o r i n s t a n c e [ 1 I, [ 4
I,
[7
I,
[9
spaces.
I and (12I f o r related developnents
a n d a d d i t i o n a l r e f e r e n c e s . More r e c e n t l y , B e r n s t e i n ' s
problem
has
a l s o b e e n c o n s i d e r e d i n t h e c o n t e x t of w e i g h t e d s p a c e s of d i f f e r e n t i a b l e f u n c t i o n s a n d d i s t r i b u t i o n s . S e e , f o r i n s t a n c e [ l o ] a n d [131. I n t h i s approach w e u s e t h e u n i f y i n g
notion
of
fundamental
seminorm i n c o n s i d e r i n g a p o l y n o m i a l a p p r o x i m a t i o n problem which amt a i n s a l l t h e above m e n t i o n e d cases of B e r n s t e i n ' s problem. F u r t h e r , t h i s a p p r o a c h p u t s i n f o c u s t h e seminorm p o i n t o f view i n approximation
*
The author w a s p a r t i a l l y s u p p o r t e d by FINEP, B r a z i l . 429
ZAPATA
430
t h e o r y which h a s been u n d e r t a k e n f o r i n s t a n c e i n 1 3 1 .
semi-
The main r e s u l t s a r e a c h a r a c t e r i z a t i o n o f fundamental norms o n t h e r e a l l i n e (Theorem l), a q u a s i - a n a l y t i c c r i t e r i o n
for
s u c h seminorms (Theorem 2 ) and a t e n s o r p r o d u c t c r i t e r i o n f o r fundam e n t a l seminorms i n t h e g e n e r a l c a s e (Theorem 3 ) .
We f i n i s h by l i s t i n g some i n t e r e s t i n g open problems,
some
of
them u n s o l v e d even i n t h e c l a s s i c a l case.
2.
PRELIMINARIES and k w i l l d e n o t e e l e m e n t s
I n t h e following, n, m
IN
U
I
and
1
INn
tk = t kl l '
t E IR",
respectively. W e put
...
kn tn
*
.
o u s l y d i f f e r e n t i a b l e f u n c t i o n s on Jklf kl
. ..
kn
(ax,) (axn) a l g e b r a s o f Cm(lRn) : P(IR")
.
= {p E C ~ ( I R " ) I p
mn)
II 11,
on
I1 f ( x ) 1 I x
C:(lRn)
1
E
lRn. I f
+
. . , + kn
f E Clkl(lRn),
lRn
s u p p o r t of
and i f
T.
then
f
-
akf sub-
on
f
i s compact]
vanishes a t i n f i n i t y , f o r all ke I?) IR"
1. A l s o f o r
we l e t
m
I1 f 11
denote
E IN w e d e f i n e
the
the
norm
by
The t o p o l o g y d e f i n e d by t h e f a m i l y of norms d e n o t e d by
= kl
,
i s a polynomial}
I akf
= ( f E Cm(lRn
For a bounded complex f u n c t i o n sup
1
We w i l l c o n s i d e r a l s o t h e f o l l o w i n g
= if E C~(IR")
number
k
lN*
is t h e a l g e b r a of a l l complex v a l u e d m - t i m e s c o n t i n u
Cm(lRn)
means
1
in
1 I Ilm , m
E
IN, w i l l
Unless e x p l i c i t l y s t a t e d , t h i s i s t h e t o p o l o g y
be
t o be
43 1
FUNDAMENTAL SEMINORMS
c o n s i d e r e d on
REMARK 1: to
.
Ci(lRn)
t
The f u n c t i o n s
C i ( I R ) . Hence f o r a l l
~ 1 2 ... ) ~ (1
+
p/(l
+
1/(1
+
p E P(lRn)
-+
t/(l
there exists
m 2 m0
a n g E Co(lR 1 ,
gp,
t h a t t h e s e t of a l l p r o d u c t s
t
f o r any
E C:(IRn)
t
t 2 ) and
mo
+
t2) belong
E
IN such t h a t
. From this f o l l w s
p E P(IRn)
contains
t h e sum o f a n y t w o s u c h p r o d u c t s a n d a l s o a n y p o l y n o m i a l .
The B e k n s R e i n s p a c e o n
DEFINITION 1: B
d e n o t e d by
n = 1 , i s t h e complex v e c t o r s p a c e
when
g E
IR",
cz(mnj, Let
all
Then
xa
p E p(mn).
a b e a seminorm o n
c l o s u r e of
of
A
Bn,
A
C
Bn.
i n t h e seminormed s p a c e ( B n , a )
DEFINITION 2:
A seminorm
a on
Bn
E Bn . a
Bn
or simply
gp,
products
w i l l denote the
.
is p o e y n o m i a l l y c o m p a t i b l e i f t h e
module o p e r a t i o n s
are c o n t i n u o u s . SPC(lRn) w i l l d e n o t e t h e ( d i r e c t e d ) s e t o f a l l p o l y n o m i a l l y c o m p a t i b l e seminorms o n B n .
EXAMPLE 1:
when Since
Ifl 5 lgfl
a be a n i n c h e a o i n g A e m i n o h m on Bn t h a t is a ( f ) c a ( g )
Let
lg
5
I
. Then
IlgII
1
f
I
a E SPC(IRR). I n f a c t , l e t g E C i m n ) , then
a(gf)
5 Ilglla(f).
Also
f
Bn-
a ( z ) = a ( f ) since
= i f l . I t i s c l e a r t h a t f i n i t e p o s i t i v e l i n e a r c o m b i n a t i o n s of increasi n g seminorms are a l s o i n c r e a s i n g seminorms. EXAMPLE 2:
Let
m
E
IN*, ak
,i
k
1 5
m,
be
a
family of i n c r e a s i n g
432
ZAPATA
seminorms on B ki
5 k; ,
Then
n such t h a t i =l,...,n. Let
akl
5 c o n s t a n t . ak
a E SPC(IRn). I n f a c t , l e t
formula, t h e f a c t t h a t
when
k 5 k', t h a t i s
g E C mo ( I R n ) , f E Bn. Using L e i b n i t z ' s
ak i s i n c r e a s i n g , a n d t h e c o n d i t i o n
on
the
family, it follows t h a t
Hence
a(gf)
REMARK 2:
5
T h e r e e x i s t seminorms
types described
a(?) = a ( f ) .
114 I l m a ( f ) . A l s o , it is clear that
constant
i n Examples 1 o r
a E SPC(IRn) which a r e n o t of 2. For instance,
this
the
t h e case
is
f o r t h e seminorm d e f i n e d by
Then f o r a l l
g E Cz(IR)
PROPOSITION 1:
PROOF:
+gPo E B
be such t h a t Then
a E SPC(IR").
Let
w e have
f E B
Then
go E Cz(lRn) , po E P (IR")
Let
g E C,"(IRn)
and
0
2
Bm(l,
in
emgo + go
n,a
Ci(lRn) i a d e n b e i n
be g i v e n .
the
let emeC:(lF?)
fJm(x)=l when l l x l l ~ m and IIak BmII
L i 1i ,
Hence
9,gop0
E
l(\kILm.
Cz(IRn) f o r a l l
emgoPo + gopo
in
DEFINITION 3 :
a E SPC(IRn) is d u n d a m e n t a L when
.
mapping
is continuous. For m = 1 , 2 , . . .
C:(IRn).
Bn,a
Then
Bn, a
m and
*
P(IRn)
i s dense i n
FUNDAMENTAL SEM INORMS
B
nIa
. we
433
a i s a Be4nAtein beminoam o n
s a y a l s o that
mn.
Beanbtein'd
n-dimenbional p f i o b t e m c o n s i s t s i n d e s c r i b i n g B e r n s t e i n seminorms
on
.
lRn
A l l t h e cases o f B e r n s t e i n a p p r o x i m a t i o n problem mentioned
REMARK 3:
a t t h e I n s t r o d u c t i o n c o n s i s t i n a s k i n g for n e c e s s a r y and c o n d i t i o n s i n order t h a t some c o n v e n i e n t seminorm
sufficient be
a E SPC(IR")
fundamental. H e r e is a u s e f u l r e s u l t .
PROPOSITION 2 : OR
Let
b e a cvmpLex 4eminahmed Apace
(E,B)
IR" b u c h t h a t
didZRibuZiond on
Adbume R h a t t h e h e e x i b t n
m E IN
U
i n E and t h e induced t a p o b a g y o n
c E
Bn m
1
a = 6 ] Bn
and
bUCh
dUnC,t~OnA
06
E
SPC(Rn).
C2(IRn) i d d e n b e
that
Cz(IRn) i~ ueakek than .the inductive
L i m i t topabogy.
Then P(lRn) i n denbe i n E i 6 and o n l y .id Necessity i s obvious. Conversely, s i n c e
PROOF: in
dundamentat.
From t h i s i t f o l l o w s t h a t
dense
is
Ci(lR")
also
is
it
P(IRn) i s d e n s e i n E l
since
is fundamental.
BIBn
u be
Let
EXAMPLE 3:
o n IRn s u c h t h a t say t h a t
m
u(t)tk
upper-semicontinuous
u i s a w e i g h t on l R n ) . L e t
a t i n f i n i t y , seminormed b y
m = 1,2,.
i n g seminorm. For
5 Om 5
-
1,
emf)
+
em(x) = 1 i f 0
when
nonnegative
vanishes a t i n f i n i t y f o r a l l
m+m.
B ( f ) = IIu f 11.
..
let
IIxlI Also
k E
Then
BIBn
I
we say t h a t
(We
Then f o r
is an increas-
be any
such
f E
B ( f ) 5 IIuII IIf II f o r a l l
Thus t h e c o n d i t i o n s of P r o p o s i t i o n 2 are s a t i s f i e d . When d e n s e i n Cu,(IRn)
I"".
s u c h t h a t uf v a n i s h e s
E Cc ( IRn
5 m.
function
E = Cum(IRn) b e t h e vector space
o f a l l complex c o n t i n u o u s f u n c t i o n s f o n Rn
f3(f
i b
Cz(lRn) i n the i n d u c t i v e l i m i t t o p o l o g y [ll], t h e n
d e n s e i n E.
0
CI
that
Cuw(lRn)
fECc(Rn). P(lRn)
u i s a dundamentab w e i g h t .
is
434
ZAPATA
EXAMPLE 4 :
Let
b e a p o s i t i v e Bore1 measure on
p
is p-integrable f o r a l l 6 = LP-seminorm.
B
E and
5
(f)
Then
15
k E INn,
f
+
m
.
11
for all
E = Lp( p ) ,
Let
is increasing. A l s o
BIB,
p (lRn) l/plI
p <
tk
IRn s u c h t h a t
Cc(IRn)
f E C,(IRn).
is dense i n
Thus t h e c o n d i -
t i o n s of P r o p o s i t i o n 2 a r e s a t i s f i e d .
EXAMPLE 5:
1k] 5
IN*, uk
E
uk 5 c o n s t a n t
such t h a t of a l l
m
Let
f E Cm(IRn)
, I kl 5
ukI
-
O(f
emf)
0 , when
-+
B(f) 5 for all
5
k. L e t
E
be t h e vector space
uka f vanishes a t i n f i n i t y f o r a l l
. Then .. i s
k
11 uka f l l lkl9 i n Example 2. I f ,,B m=1,2, 1, t h e n
k'
k
such t h a t
m, B ( f ) =
if
m, b e a f a m i l y o f w e i g h t s o n Rn
m+-,
B IBn
i s of t h e t y p e
a s i n t h e p r o o f of for all
k,
described Proposition
f E E . Also
(
f E C t ( I R n ) . Thus t h e seminormed s p a c e ( E , B ) s a t i s f i e s
the
h y p o t h e s i s of P r o p o s i t i o n 2 .
EXAMPLE 6:
Let
m and %
n o t e s Lebesgue measure on Given
p,
15 p <
tributions
f
+
m,
on lRn
, ik I 5 lRn,
let
m, b e as i n Example 5 . I f dpk = uk d h
dX
de-
1k1 5
for all
m.
l e t E b e t h e v e c t o r s p a c e of a l l complex dissuch t h a t
akf E zp(pk)
for all k, , k
5 m,
( f / a k f l P d p k ) l / p . Then B i B n i s a l s o o f t h e t y p e deIklcm s c r i b e d i n Example 2. F u r t h e r m o r e Cz(IRn) i s d e n s e i n E . The p r o o f
B(f)
=
of t h i s f a c t i s s i m i l a r t o t h a t used i n p r o v i n g d e n s i t y s p a c e s 1111. A l s o
B(f) 5
(
in
Sobolev
a x pk(lRn) l") 11 fit, f o r a l l f E Cz(lRn)
1 kl5m
.
Once a g a i n t h e h y p o t h e s i s of P r o p o s i t i o n 2 are s a t i s f i e d .
PROPOSITION 3:
1)
rb
Let
CI
be a dundamentat heminotm on IR".
B E SPC(IR~)i b buck t h a t
hundamentae.
B 5 constant
a, t h e n B i b
FUNDAMENTALSEMINORMS
436
1) i s a n immediate c o n s e g u e n c e of P r o p o s i t i o n 1. I n t h e case
PROOF:
P ( I R n ) o q = P ( I R n ) , cZ(IRn)o 9 = C:(ntn)
of 21, o b s e r v e t h a t
.
3 . M A I N RESULTS I n t h e c h a r a c t e r i z a t i o n of d e n s e s u b a l g e b r a s i n s p a c e s o f d i f f e r e n t i a b l e f u n c t i o n s t h e following is a c r u c i a l r e s u l t .
LEMMA 1 ( N a c h b i n ' s Lemma):
Let
m 2 1, b e a n e t 06 k e d
A C Cm(IRn),
~ u n c . t i a n b baLib6ying t h e doelawing c o n d i t i a n d : 1)
Fah any
x, y
duch t h a t
2)
F o h any
3)
Foh
x
IR"
E
and
E A
h
f E Cm(IRn) E
e Cm(lR ) ,
f = h(gl,
See [ 8
DEFINITION 4:
g E A
thehe i b
o u c h . t hat
a n y x, v E IR", v # 0 , t h e h e i d g
Then g i v e n any
PROOF:
x # y , thehe i d ' g
buch
E A
g(x) # g(y).
that
gl, . . . , g t
E IR"
I
and K C IR"
h(0) = 0,
...,g L )
on
E A
g(x) # 0.
buch t h a t $(x)
compact
thehe
#O.
exidt
duck $that K.
.
A set
A
C
Cm(lRn), m
2 1, s a t i s f i e s c o n d i t i o n s
(N)
a held-adjoint
dub-
i f 11, 2 ) a n d 3 ) above are true.
LEMMA 2 :
Let
~1 E
SPC(IR").
Id
A
C
C:(IRn)
i b
a l g e b x a t h a t batiddied c o n d i t i a n b (N), t h e n A
PROOF:
i d
dende i n
From P r o p o s i t i o n 1 i t i s enough t o show t h a t
Bn,a
CE(IRn) C
. ia.
436
ZAPATA
F u r t h e r , s i n c e t h e t o p o l o g y d e f i n e d by
we need o n l y show t h a t t h e c l o s u r e of
T,
contains
C i s a subset o f
If
i n t h e topology if
gl,...,gl
then
.
C: ( IR")
i s weaker t h a n
Ci(IRn)
latter
A i n the
w e w i l l d e n o t e by
C:(IRn)
its closure
a r e r e a l and
h ( g l , ...,g
c.
E
L
h
I n f a c t , i t i s clear t h a t h(gl,...,gL)
G = {gl(x)
,..., g l ( x ) , x E
i = 1,.
INn,
E
e
i n t h e topology Let
. . ,ge)
p(gl,.
be g i v e n . A s s u m e
s a t i s f i e s conditions (N)
Also
such t h a t
hl
E
.
f
# 0
0.
g ( x ) # 0 . Choose
91
If
E
A1
and
r >0
x E H
then
.., g t )
on K .
x.
DEFINITION 5: d e f i n e d by
A. For
thereexists
Hence
h(g(x)) > O
Hence by
E
El.
compact-
1-
on a n e i g h -
[ r , + m ) , hl = O
fl = 1
Since
h o g
on
H
and
fl E
i1
h a s compact s u p p o r t , s a y E.
f = fl
%,
L
h e C"(W 1, h(O) = O
h(gl,..
and from t h e remakk on s u b a l g e b r a s i t f o l l o w s t h a t E
be its
g > r on H, gl(0) = O .
such t h a t
hl = 1 on
f l = hl o g l ,
f = h(gl,.
., g L )
. .,gL)
A andalso
h E C"(lR) such that h,O,
Then from Nachbin's Lenuna,there exist gl,...,gt
h(gl,..
all
for
H
i s a subalgebra of
s i n c e t h i s i s a c l o s e d s u b a l g e b r a . A l s o , fl
such t h a t
con-
h(gl,.
let
and
I n p a r t i c u l a r , f o r any
C m ( I R ) b e such t h a t
bourhood o f
IRn
i s bounded
approximates
i s p o s i t i v e on a neighbourhood of
hog
n e s s , there e x i s t
K.
without
h ( 0 ) = 0 . From the above remark, it follows t h a t
and
Let
gi
functions,
be t h e s e t o f r e a l p a r t s o f f u n c t i o n s i n A.
i s a s e l f - a d j o i n t a l g e b r a , t h e n A1
g E A1
k
1.
e R ,using
T.
f E CE(IRn)
s u p p o r t . L e t A1 A
Hence
p E P(lR )
a
h ( 0 ) = 0. Furthermore,
. . ,e.
E Corn
IRn } i s bounded i n
w e c a n approximate h on G by p o l y n o m i a l s s t a n t t e r m , since
h(0) = 0, m n
i s such t h a t
Cm(lR )
t h e W e i e r s t r a s s a p p r o x i m a t i o n theorem f o r d i f f e r e n t i a b l e
k
topology
A s s u m e a l s o t h a t C i s a s u b a l g e b r a . I n t h i s case,
7.
E C
S i n c e the set
on
c1
f
., g L )
on
A,
since
E
N o w t h e p r o o f i s complete.
z E C\lR
, let
gz
b e t h e complex f u n c t i o n on
W
FUNDAMENTAL SEMINORMS
g,(x) I t i s clear t h a t
PROOF:
g,
E
E = P(DUa. W e
In fact, for
m=O
claim t h a t
gp(IR) C E
for all
m E IN.
t h i s i s e v i d e n t . Assume t h a t t h e p r o p o s i t i o n
m E IN. L e t
t r u e f o r some
x E IR.
ci(IR).
-
Let
1 x - 2
=
437
p E P(IR). Since
-
q = gz(p
is
p ( z ) ) EP(IR),
t h e n from t h e a s s u m p t i o n i t f o l l o w s t h a t
Now t h e mapping
f E Ba
+
g f:
i s continuous hence
E Ba
S i n c e E i s a complex v e c t o r s p a c e w e h a v e
So t h e claim i s p r o v e d . F u r t h e r , t h e mapping
continuous, hence E is s e l f - a d j o i n t s i n c e for all
g:(;,)'
1
E
f
E
P(IR)
Ba
+
is.
'f
SO
E
(4,)
is
Ba
-.7E E -g,
I N : whence
E E:g
c g:
P(B)
a C
E
for all
m , n E IN.
From t h i s i t f o l l o w s t h a t t h e complex a l g e b r a A g e n e r a t e d by g, a n d
-g2
i s c o n t a i n e d i n E.
t i o n s (N) s i n c e
{gz}
Also
A
i s s e l f - a d j o i n t and s a t i s f i e s condi-
s a t i s f i e s c o n d i t i o n s (N).
From
Lemma
2
it
438
ZAPATA
P(IR) is d e n s e i n B a r t h a t i s ,
follows t h a t
LEMMA 4 :
1e.t
con6.tan.t
CzIZl
PROOF:
Let
p
,
a E SPC(IR)
z, z'
c
\ IR.
is fundamental.
T h e n t h e 4 e e x i s t 6 a pa4,itive
such ,tha.t
E
~ ( m ) .Since
gzp
i t follows
From t h e d e f i n i t i o n of
If
E
CI
r = ~ y r n z . t~h,e n
=
42 42
a, there exists
11 gzIlm =
Z
k-0
k! k+l
g Z I p = (1 + ( z
C > 0
=
cz
and m E IN s u c h t h a t
and
T o f i n i s h , i t i s enough t o o b s e r v e t h a t t h e number Cz
d o e s n o t depend on
PROOF:
Assume t h a t
- z')g2)gzlp
12'
= l + Iz-z'I CC,
p.
P a ( z ) i s unbounded. L e t
p E P(IR) be such t h a t
FUNDAMENTAL SEMINORMS
then
q
E P
(IR) and
q
-
gz =
.
g ZP
p(z)
By c h o o s i n g a c o n s t a n t C Z r i > 0
a s i n Lemma 4 i t follows t h a t
Since
P ( z ) i s unbounded, t h e n a
gz
and from Lemma 3
E P")-
is
c1
f undamen t a 1.
C o n v e r s e l y assume t h a t n
E
IN* be g i v e n . S i n c e
that
a(gZ
-
p)
a(gzq) = n a ( g Z
5
-
3.
pn
E
E
P(IR)",
q = n(l
Let
a(giq)
~ ( n, cl(gipn) ) 5
5
CirZ
1 and
-
there exists (x
-
.
pn(z) =
THEOREM 2 ( q u a s i - a n a l y t i c c r i t e r i o n ) :
a
PROOF:
on
Q: \
and
IR
p E P(IR)
Then
To f i n i s h w e l e t
unbounded.
then
ZIP).
z E
q
E
such
P ( I R ) and
i s a p o s i t i v e c o n s t a n t as is Lsrr
P) 5 1. If Ci,z
m a 4 it follows t h a t Then
gz
i s fundamental. L e t
c1
n . Hence 'i,z
Let
a
pn =
'i,z
P,(z)
is
E SPC(lR), Id
in 6undamenZaL.
Let
P(sX).
T b e a c o n t i n u o u s l i n e a r form on Let
D d e n o t e t h e s e t o f complex
Ba
s u c h t h a t T vanishes
numbers
such
that
on D. I n f a c t assuming t h i s ,
from
z
Imz < 1. D e f i n e h ( z ) = T ( g Z ) , z E D.
I t i s enough t o p r o v e t h a t
h =O
440
ZAPATA
Hahn-Banach a
-
g2 E P ( l R J a for a l l
theorem i t f o l l o w s t h a t
i s fundamental from Lemma 3 . Let
z E D, n E IN.
S i n c e T v a n i s h e s on is also t r u e f o r
If
n =O.
z, zo
E D, z
#
zo.
h(z) =
Hence
a, t h e r e e x i s t
S i n c e II g 2 I l m 5 ( m + l ) !
Let
n 2 1 then
P(lRR) i t follows t h a t
From t h e d e f i n i t i o n of
for a l l
Then
gz
=
4,
From t h i s i t follows t h a t
h
i s holomorphic on D and
n=l
m E IN such t h a t
(2
-
zO)4,g2
0
.
i s holomorphic on D . Since
m
z
~ ( g ~ x " ?his ). zn
we have t h a t
z E D
-
and
C > 0
0
h
z E D. Then
n-
4
1
=
+
(*)
is t r u e ,
m,
a(x")
t h e n Denjoy c o n d i t i o n s i n Watson's problem are s a t i s f i e d , v a n i s h e s on
Hence
D ( [ 6 1 ) . N o w t h e proof i s complete.
hence
h
441
FUNDAMENTAL SEMINORMS
COROLLARY 1: t h e h e ahe
Let A be t h e
aLl neminohmb
d e t 06
p o n i t i v e conntantd
C
I
N , m E IN
and
c
a E SPC(IR) doh
which
(ddepending o n a)
A U C ~t h a t
... - log,
a ( x n ) 5 c(c n log n whehe log,
doh
log, n = n and
dedined b y
i d
n) n
aLL log,"
n 2 N = l o g ( 1 o gm - l n)
.id
m 2 1. Then A
PROOF:
s e t 06
id a dihected
6undamentaL neminohmd.
T h i s i s a d i r e c t consequence o f Theorem 2 o b s e r v i n g t h a t t h e
"moments" o f any t w o such seminorms have a common e s t i m a t e of the sirme type
a
Let
THEOREM 3:
SPC(IRn). 1 6 t h e k c e x i d t 6undamentaL
E
JemiMohmd
~ 1 ~ , . . .E~ SPC(IR) a ~ duch t h a t
a(fl then
for
... B
f n ) 5 a 1( f 1
...
*
an(fn)
aee
doh
flI . . . I f n E B t
6undamenXaL.
a i d
PROOF:
(9
Let
n :B x . . . x Ban a1 f l l . . . l f n E B. Then
-+
Bnta
be defined by m(fl
,...,f n) =f,@ ...@ f n ,
i i s fundamental and .rr i s c o n t i n u o u s . Hence i f t h e complex s u b a l g e b r a g e n e r a t e d by
s i n c e each
a
T(C;(IR)
then
x
-a
A C P(IRn)
...
x
C;(IR))
. Since
A
viz
A = C ~ ( I R Io R)
A
... o C ~ ( I R )
is a s e l f - a d j o i n t s u b a l g e b r a o f
I
C:(IRn)
is
442
ZAPATA
and a l s o s a t i s f i e s c o n d i t i o n s (N) , from Lemma 2 i t f o l l o w s is dense i n
Hence
Bn,cl.
a
that
A
i s fundamental.
4 . OPEN PROBLEMS
1.
2.
Give i n t e g r a l c r i t e r i a l i k e t h o s e i n [ 7 ]
for characteriz-
i n g fundamental seminorms on
IR.
Under what c o n d i t i o n s on
SPC(IR) i s i t t r u e t h a t a i s
fundamental i f and o n l y i f
a
E m
L:
i=ly a ( x n )
= + a ?
3.
If
CY
E SPC(lR) i s n o t f u n d a m e n t a l , d e s c r i b e
4.
If
a E SPC(IR) i s n o t f u n d a m e n t a l , a r e t h e r e p o s i t i v e con-
s t a n t s c , C such t h a t f o r a l l
z 5.
E C
we have
p E P(IR) , a ( p )
a
SPC(IR) i s i t t r u e t h a t
fundamental i f and o n l y i f t h e s e t { p
6.
the
E
a is
P ( I R ) , a ( p ) (1) i s
s p a c e o f e n t i r e f u n c t i o n s on
Q:?
Give a c h a r a c t e r i z a t i o n o f fundamental seminorms o n
n 7.
in
and
Ip(z) 1 5 C e C I Z I ?
Under what c o n d i t i o n s on
unbounded
5 1
IRn
,
2 2.
I s t h e set of a l l fundamental seminorms on
Same on
R
directed?
R”?
REFERENCES
[ 11
N.
AKIEZER,
On t h e w e i g h t e d a p p r o x i m a t i o n o f c o n t i n u o u s
func-
Amer. t i o n s by p o l y n o m i a l s on t h e e n t i r e number a x i s , Math. SOC. T r a n s l a t i o n s , S e r i e s 2 , v o l . 22 (1962) , 95 - 138. [ 21
S . BERNSTEIN, Le problgme d e l ‘ a p p r o x i m a t i o n d e s f o n c t i o n s con-
t i n u e s s u r t o u t l ’ a x e r 6 e l e t l ‘ u n e de ses a p p l i c a t i o n s , B u l l . SOC. Math. F r a n c e 52 ( 1 9 2 4 ) , 399 -410.
443
FUNDAMENTAL SEMINORMS
[ 31
J. P . FERRIER, Suk k ? ' a p p k o x i m a t i o n pond&ce, moderne, Univ. de Sherbrooke, 1972.
[ 4]
P. GEETHA, On Bernstein approximation problem, J. Math. and Appl. 25 (1969), 450 - 469.
[ 51
P . MALLIAVIN, L'approximation polynomiale pondGr6e sur un
[ 6]
S. MANDELBROJT, S & i e n a d h h e n t n , k z g u l a h i z a t i o n den nuiteA,app t i c a t i o n d , Gauthier-Villars, 1952.
[ 71
S. MERGELYAN, Weighted approximation by polynomials, Arwr. Math. SOC. Translations, Series 2, vol. 10 (19581, 59 -106.
[ 81
L. NACHBIN, Sur les algzbres denses de fonctions diffgrentia-
Sem.
Analysis
espace localement compact,Amx.Journal Math. 81(1959), 605-612.
bles sur une variatg, Comptes Rendus Acad. t. 228 (1949), 1549 -1551.
1 91
d'Analyse
Sc.
Paris,
05 a p p h o x i m a t i o n t h e o h y , D. Van Nostrand, 1967. Reprinted by R. Krieger Co., 1976.
L. NACHBIN, Elementd
[lo] N. SIBONY, Problsme de Bernstein pour les fonctions contintment diffgrentiables, Comptes Rendus Acad. Sc. Paris, t. 270 (19701, 1683-1685. [ll] F. TReVES, T o p o l o g i c a l v e c t o l r n p a c e d , d i n t k i b u t i o n n and KehneRs, Academic Press, 1967. [121
K. UNNI, Lectuked o n B e k n b t e i n a p p k o x i m a t i o n phob.tem, in Analysis, Madras, 1967.
[131
G. ZAPATA, Bernstein approximation problem for differentiable functions and quasi-analytic weights.Transactions Amer. Math. SOC. 182 (19731, 503-509.
[141
G. ZAPATA, Weighted approximation, Mergelyan theorem and quasianalytic weights, Arkiv for Matematik 13 (1975), 255-262.
Seminar
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INDEX
A
a l g e b r a i c convolution i n t e g r a l s
71
almost simple
214
a p p r o x i m a t i o n , non-archimedean
121
a p p r o x i m a t i o n on p r o d u c t
46
sets
37,
approximation p r o p e r t y approximation, r a t i o n a l
4 21
approximation, r e s t r i c t e d range
226
approximation, simultaneous
227
B
- differentiable
161
B e r n s t e i n problem
433
B e r n s t e i n seminorm
4 33
Be rnstein space
4 31
Birkhoff c on d i t i o n
192
B i r k h o f f i n t e r p o l a t i o n problem
189
Birkhoff' s kernel
222
b
C
c a r d i n a l series
391
cardinal s p l i n e i n t e r p o l a t i o n
39 0
c o a l e s c e n c e of matrices
1 98
c o e f f i c i e n t of c o l l i s i o n
200
compactly
-
291
regular
446
280,
373
446
INDEX
c o n d i t i o n (L)
167
cross -section
372
D Dedekind c o m p l e t i o n
64
degree o f e x a c t n e s s
385
differentiability type
164
d i f f e r e n t i a h i l i t y t y p e , compact
165
E e c h e l o n Kothe-Schwartz s p a c e s E
409
- product
Fe j&
- Korovkin
37, 269
F
kernel
78,
f o r m a l power series
354
fundamental seminorm
4 32
f undamen t a 1 w e i g h t
4 33
f u s i o n lemma
143
G Gaussian m a t r i x
2 31
G e 1f and t h e o r y
3 36
generating function
396
I i n c r e a s i n g seminorm
4 31
i n t e r c h a n g e number
202
i n t e r p o l a t i o n matrix
189
i n t e r p o l a t i o n matrix, p o i s e d
189
interpolation matrix, regular
189
79,
88
INDEX
447
K
Korovkin a p p r o x i m a t i o n
19
Korovkin c l o s u r e
20
Korovkin s p a c e
20
Korovkin' s theorem
63
L level functions
199
M meromorphic uniform a p p r o x i m a t i o n
139
N Nachbin s p a c e non-archimedean
3 72 spaces
121
0
order regularity
189
P
plurisubharmonic f u n c t i o n
34 3
p o i d s de B e r n s t e i n
237
point r6gulier
238
Pdlya c o n d i t i o n
192
P6lya f u n c t i o n s
191
p o l y n o m i a l l y c o m p a t i b l e seminorm
4 31
power growth
392
p r o p e r t y (B)
168
pseudodifferential operator
13
INDEX
446
q
Q
- regular
quasi
- analytic
229
4 39
criterion
R
r a t i o n a l approximation
421
regular interpolation matrix
189
r e l a t i v e Korovkin a p p r o x i m a t i o n
28
r e l a t i v e Korovkin c l o s u r e
28
r e s t r i c t e d range approximation
226
Rogosinski summation method
103
Rolle set
209
S S-approximation p r o p e r t y ( S . a . P . 1
359
seminorm, B e r n s t e i n
433
seminorm , fundamental
4 32
seminorm, i n c r e a s i n g
4 31
seminorm, p o l y n o m i a l l y c o m p a t i b l e
431
s h e a f o f F-morphic f u n c t i o n s
40
shift
203
S-holomorphic a p p r o x i m a t i o n p r o p e r t y (S.H.a.p.1
367
Silva-bounded n-homogeneous polynomial
353
Silva-bounded n - l i n e a r map
352
S i 1va- bounded po 1ynomi a 1
35 4
Silva-holomorphic
35 5
S i l v a - h o l o m o r p h i c , weakly
356
simple
21 3
s i n g u l a r i n t e g r a l of de l a v a l l d e P o u s s i n
99
singular Integral of Fej6r
98
s i n g u l a r i n t e g r a l of Landau-Stieltjes
93
449
INDEX
s i n g u l a r i n t e g r a l of Weierstrass
96
smoothing f o r m u l a S
386
- Runge
36 2
s t r i c t compact
3s 7
supported sequence
194
V V*- a l g e b r a
339
vector fibration
372
v e r y compact
275
w weakly S i l v a - h o l o m o r p h i c
355
weight
372,
w e i g h t , fundamental
4 33
433
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