Polynomial Approximation on Compact Sets

Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977

POLY NOE I 1PL AP P ROS I ElRT I ON 3 N COE 1PACT S CTS

By I I A R T I N S C H O T T E N L O H E R "

I n T h i s n o t e a s i m p l e p r o o f of t h e f o l l o w i n g r e s u l t g i v e n : A pseudoconvex, f i n i t e l y Runge open s e t U i n a

is

locally

convex Hausdorf f s p a c e w i t h t h e a p p r o x i m a t i o n p r o p e r t y i s p o l y n o m i a l l y convex. T h i s g e n e r a l i z e s r e s u l t s of

Dineen

141

and

of

the

Noverraz [13]. W e a l s o p r o v e a n a p p r o x i m a t i o n theorem Runge t y p e f o r s u c h domains.

F i n a l l y , i n t h e l a s t s e c t i o n of t h i s n o t e , we d i s c u s s

a

s t r o n g form o f t h e Oka-Weil a p p r o x i m a t i o n theorem, which i s u s e f u l i n t h e s t u d y o f t h e Nachbin t o p o l o g y T~ on t h e s p a c e

@ (U)

o f holomorphic f u n c t i o n s on U (see blujica [ll]).

1. NOTATIONS AND PRELIE~IINARIES L e t E b e a l o c a l l y convex Hausdorff s p a c e o v e r 6

s h o r t : l c s ) , and l e t c s ( E ) d e n o t e t h e set of c o n t i n u o u s norms on E . F o r a

€

cs ( E ) , x

€

E and r > o t h e " a - b a l l "

( for

semiabout

............................................................... *

Research s u p p o r t e d by t h e Brazilian-German C o o p e r a t i o n

Aqre-

- Gesellschaft fur

Mathe

ement (Conselho N a c i o n a l d e P e s q u i s a s m a t i k und D a t e n v e r a r b e i t u n g )

. 379

M.

380

SCHOTTENLOHER

x with the radius r is (x,r) = {y

B~

E EI

The "a-boundary distance" dz

cr(x-y) < rI :

U-

. for an open set U t E

[o,m]

is defined by :d

For B

(x) = sup {r >

c U we

" put dU

tance function

:

Ba(x,r)

( B ) = inf {d:

U

x E+

~ ~ ( x , a= )sup {r >

1x1

01

01

C

U}

(x)

I

,x x

U.

E

. Another

E B}

dis-

[o,m] is given by

x + ha

u

E

c r}, (x,a) E U

x

x

for all

E C,

E.

d" is continuous, while 6 u is in qeneral only lower semicontinu U

ous

.

An open set U C E is called pseudoconvex if -1oq plurisubharmonic, i.e. if the restrictions o f -1oq 6u plex lines in U

E are subharmonic. Let @ ( U )

x

is to

(resp.

denote the set of plurisubharmonic (resp. continuous

com-

@c(U)) plurisuh-

harmonic) functions on U, and let @4U) clenote the vector space of holomorphic functions on U. For

n

C

@(U)

and

I< C

U

the

"Q-convex hull" of K is defined by

KO

-

=

{x

UI

E

and for A c @ ( U ) KA

whereby

=

{x

I I fI I

v(x) 5 sup v(y) for all v YEK the "A-convex hull" of K is

UI

E

=

If(x)I 5 IlfllK for all f

sup

[ f (y)I Iy

E

E

01,

E A),

K l . Prom the characterization

of pseudoconvex sets in finite dimensional spaces one can

de-

duce (cf. [17]): PROPOSITION 1 a4e

Fo4

an o p e n s e t U

C

I.: t h e . ~ o ~ l o w i n gp t o p e a t i e n

equivalent 1? 20

U i b pbeudocanwex. a -log du i 6 p l u t i h u b h a a m o n i c

o n {x

E U I d$

(x)>o}

404

POLYNOMIAL APPROXIMATION a E cs(E)

euehy

30

d:

381

.

F o h e v e h y compuct

K C U

With

t h e h e i6

a E cs(E)

604 CVV.hY

compact K~u.Hehe,

( ? Q ( ~ ) ) > 0.

4Q

ii

phecompucf i n U

Q ( ~ )

L C U iA c u t l e d p4ecompuct i n U i.6 L iA phecampuci uvld i d Rhehe a E cs(E) with

d;(L)

exinto

> 0.

I t i s an open q u e s t i o n whether t h e above e q u i v a l e n c e s h o l d

if

@(U)

.

i s r e p l a c e d by G>c(U) A partial answer is given in section 2.

U i s c a l l e d h o l o n o r p h i c a l l y convex i f

k

(iT) is p r e m p a c t

i n U f o r e v e r y compact K C U . A h o l o m o r p h i c a l l y c o n v e x open s e t

U cE

i s p s e u d o c o n v e x . The c o n v e r s e i s t r u e f o r A a n d a!

p r o b l e m ) , f o r a! (IN) [ 7 ]

spaces E with a basis (cf. L O ] ,

[2]

,

E =

an

(Levi

and f o r c e r t a i n s e p a r a b l e

[S!, 1 1 7 1 ) . I t i s a n o p e n q u e s -

t i o n whether t h e c o n v e r s e h o l d s i n g e n e r a l . A p a r t i a l answer i s given i n t h e n e x t s e c t i o n .

u

c E i s polynomially convex i f

compact K C U, whereby

TI

C. B ( U )

Gn

u

i s precmpact i n

for all

denotes t h e space of a l l

con-

t i n u o u s p o l y n o m i a l s f r o m 1: t o a!. S i n c e n i s d e n s e i n Z ( E ) w i t h r e s p e c t t o t h e canpact o p n topology on

KT

kTr

( E ) , w e have

k@(E)

=

*

i s contained i n

2

=

{x

/ p ( x ) I 5 1Ip

F EI

lK

for all

p

n).

E

-

-

W e d o n ’ t know, w h e t h e r f o r a p o l y n o m i a l l y c o n v e x U, K = K

TI

is

t r u e i n g e n e r a l . I n t h e case of a F r 6 c h e t s p a c e E w i t h t h e a p p r o x i m a t i o n p r o p e r t y t h i s w i l l b e p r o v e d i n s e c t i o n 3. Closely r e l a t e d w i t h p o l y n o m i a l c o n v e x i t y o f a Runge o p e n s e t

U

E.

U i s c a l l e d Runge i f

s ( U ) w i t h r e s p e c t t o t h e compact open t o p o l o g y .

same a s t o s a y t h a t

8 (E)

is

the TI

notion

i s dense i n

This is

the

i s d e n s e i n z ( U ) . F i n a l l y , rJ i s c a l -

l e d f i n i t e l y Runye ( r e s p . f i n i t e l y p o l y n o m i a l l y c o n v e x )

i f for

382

M.

SCHOTTENLOHER

a l l f i n i t e d i m e n s i o n a l v e c t o r s u b s p a c e s F of E l U f l P i s

Runqe

( r e s p . p o l y n o m i a l l y convex) i n p .

2. POLYNOFIIAL CONVEXITY Throughout t h i s s e c t i o n l e t E b e a l c s w i t h t h e approxima t i o n property, i.e. every

E

cz. i f

and a ( x

m

Let

PROPOSITION 2 €3.

Then

doh

- $(x))<

K63(U) l e t

e(E)

for a l l x E K.

open

U I Xo&

XoE

K6qU)

-

with

@(U)

E

with

e u c h y compact R C U :

.

Then t h e r e i s v

E

U be a pbe.udoconifcx, d i n i t e l y Runge

I n o r d e r t o show K

v(xo) > sup v ( y ) . Y EK Due t o t h e s e m i c o n t i n u i t y of v t h e r e a r e a s >

e v e r y a E C S ( E ) and

> o t h e r e e x i s t s a c o n t i n u o u s l i n e a r map 4 : E+E

dimc $(El <

A e t in

f o r e v e r y compact F

cs(E), q > o and

o such t h a t f o r a l l x E K a B a ( x , 2 s ) ~ i land v ( y ) < v ( x0 1- rl f o r y E R ( ~ ~ 2 s ) .

Because 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 t h e r e e x i s t s a c o n t i n u

ous l i n e a r

+:

x E K . Now

IJJ

E+E =

4

+

w i t h dimc $ ( C ) <

xo

-

and u ( @ ( x ) - x) < s f i r

m

@(xo) s a t i s f i e s n ( @ ( x )- x ) < 2 s

x E K , hence j (I;) C

I.'

f ? I'

where f = s p (@(I?)

v o @(x) < v(xo) I t follows f o r

VI

-

= vlr>

rl

1

f o r a l l x E I<.

e,:.f)1 ' )

:

and

for

POLYNOMIAL APPROXIMATION

Consequently, xo

4

.

A

~ ( K ) Q ( ~ ~ Since ~ ) U n F is pseudoconvex and

Runve, there exists a polynomial 9 19(xo) I

'1

383

:

F+@

with

141 I$(K) (cf. [9, p. 5 3 1 ) . Now 9 o JI E

19

(xo)I =

$

9(xo)l

-

hence x

'

11"1$(K)

= 114

and

T

$11,

I

0 !$ %CE) From the propositions 1 and 2 we deduce:

A p b e u d o c a n v e x , I j i n i t e L y Runge open b e t

COROLLARY 1

UC E

i b

hoLomotphicaLly convex. REPIARK

Since for domains R C C : " the main step in solving

Levi problem is to show that pseudoconvexity implies

the

holomor-

phic convexity, corollary 1 is a certain contribution to the s o lution of the Levi problem. However, in infinite

dimensional

spaces E a holomorphically convex domain need not be a

domain

of holomorphy. In fact, Josefson [lo] gives an example of

a

pseudoconvex domain Q in I? = co(A), A uncountable, which is not

a domain of holomorphy. Ploreover, since this domain can be

de-

it

is

fined by a global plurisubharmonic function v E @ ( E ) , finitely Runge and hence holomorphically convex. But

in

certain

infinite

dimensional

spaces,

for example in Silva spaces [15] , every holomorphically convex domain is a domain of holomorphy or even a domain of existence.

Fo4

TIIEOWH 1

a17 o p e n

b e t U C I?, t h e doli!owing p h o p e t t i e s a 4 e

eq u i va i! e n t : 10

U LO p b e u d o c o n v e x a n d d i n i . t e i ! y Runge.

2Q

U

A &

3?

U

i a pvlqnomiaf!Py convex.

h o l c m o 4 p h i c a l l q conL'ex and 17unge.

M.

384

PROOF

According t o c o r o l l a r y 1, U i s holomorphi-

20".

"19

SCHOTTENLOHER

c a l l y convex. T h e r e f o r e , i t s u f f i c e s t o show t h a t a

finitely

Runqe s e t i s a Runge s e t . T h i s was proved i n [I] i n a more g e n e r a l c o n t e x t . I n o u r s i t u a t i o n t h e p r o o f o f [l] i s a s

,

Let f E g ( U )

I<

c U compact,

follows:

> 0. Because of t h e

E

continuity

o f f t h e r e are a E c s ( E ) and s > o s u c h t h a t f o r a l l x

1 f(x) -

Ba ( x , s ) C U and

E

f (y) ] < 2 f o r y E B

There e x i s t s a c o n t i n u o u s l i n e a r 4 : E + E and a ( + ( x )

-

a

(x,s).

w i t h dimc @ ( E l < m

x ) < s f o r a l l x E K . IIence

$(K) c

U

n r,

where F = 4 ( E l

Ilf -

f

E

411,(?

0

,

and

*

Since U f l I' i s f i n i t e l y Runqe t h e r e i s a polynomial

IIf

with

u/II.'

I If -

E K

9

-

ql

0

$IIK 5 I I f -

<

E

5

. Now g f

0

+I!<

g

k I L jt h)e,r e

is f

E

+

IIf

0

4- g

0

+'I1;

%(U)

with ] f ( x o ) l

<

-

Now I<

- PI/K U{xo) =

-

kiu, is

<

1

2 (lf(xo)1-1\f\lK)8

>IlfllK. .

xo $

precompact i n U s i n c e U i s

E.

Fc86(L:).

S i n c e U i s Runge, t h e r e e x i s t s a polynomial p E n w i t h

/If

6:

o cp E n and

L e t Y C t' be compact. E v i d e n t l y T'N!.

"2?+3?".

I f xo E U , xo

I+(K)

g: ??*

'%[C)'

bloanorphically

convex. '13'.'*40"

i s t r i v i a l and " 4 ? 4 1 ? ' " f o l l o w s from

the

f i n i t e d i m e n s i o n a l r e s u l t s [9, p . 531. Applying a r e s u l t o f Noverraz [14, t h . 31 w c see

that

f o r a F r 6 c h e t s p a c e E w i t h t h e appro::imation p r o p e r t y t h e

fol-

REPmRI<

lowing i s t r u e : For any d e n s e v e c t o r s u b s p a c c s e c t i o n of a l l pseudoconvex domains '' C

r c I:

with

r

the

Cii,

intei i s equal

385

POLYNOMIAL A P P R O X I M A T I O N

t o t h e i n t e r s e c t i o n o f a l l domains o f e x i s t e n c e R C E ,

with

I n o t h e r w o r d s , t h e p s e u d o c o n v e x c o m p l e t i o n FQ of P aare

F C R.

es w i t h t h e h o l o m o r p h i c c o m p l e t i o n

of P.

3 . POLYNOMIAL PPPROXIMATION

I n t h i s s e c t i o n l e t C be a m e t r i z a h l e l c s w i t h

the

ap-

p r o x i m a t i o n p r o p e r t y . Moreover, l e t E b e h o l o m o r p h i c a l l y p l e t e , i . e . E = E o ( c f . f o r example [ 1 4 ] )

. We

need t h e

corn

follow-

i n n c h a r a c t e r i z a t i o n [14, P r o p . 101 : F m e t r i z a b l e l c s E i s ho-

se

l o m o r p h i c a l l y c o m p l e t e i f f o r e v e r y n o n - c o n v e r v e n t Cauchy E there exists f E %(E)

v i t h sup

I f (xn) 1

q u e n c e (x,)

in

THEOREM 2

L e t U b e n p e u d o c o n v e x , 4 i n i R e t r r Runge o p e n n e t i n

-

C, a n d l e t I: = I<

B(U)

=

m.

be n c o m p a c t b u h b e t o S 11. T h e n e v e h r r 4iinc-

t i o n f w h i c h i n holornoflphic i n a neiohbofihood 0 4 K can be

p t o x i m a t e d uni{ohmtr/ o n

1 :

brr c o n t i n u o u d po4trnominPd

I?g m

N o t e t h a t f o r any compact K C U ,

on R.

s p a c e s . Hence

BiU)

i s compact s i n c e

K

-

BW) -

accog

i s compact

d i n 9 t o t h e above C h a r a c t e r i z a t i o n of h o l o m o r p h i c a l l y

I<

np-

KB$iEf

complete

r

( ropos i ti on

2).

PROOF Or TIIEOREM 2 L e t E > 0 . T h e r e a r e a E cs ( C ) and

s >

0

s u c h t h a t f i s h o l o m o r p h i c o n T J = IC + n a ( O , s ) IT, and f o r E a x E: I ( , If ( x ) f ( y ) I < 3 i f 7 7 E R ( x , s ) . T*Te f i r s t show

all

T h e r e e x i s t s a f i n i t e r a n k l i n e a r o p e r a t o r rt, : E

d E

-

(*) fi

with .9

& q J

(1:) )

c l !

and n(Q\fx) - x) < s f o r all x E K .

L e t ( a n ) khe a n i n c r e a s i n n s e q u e n c e of c o n t i n u o i i s norms on E , a

5 (L n ' whjch n e n e r a t c s

t h e t o p o l o n v of R .

semiSince

E

386

SCHOTTENLOHER

M.

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 t h e r e a r e c o n t i n u o u s l i n e a r maps

@n

x

dimc a n ( E ) <

: E-E,

A

Assume t h a t $ J n ( K )

E K.

fi

xn

E 4 , n ( ~ )\

v.

m,

s u c h t h a t an(4,,(x)-

@ i7 f o r

P u t E~ = s p (I;

u

a

x) <

for all

are

a l l n E I N . Then t h e r e [ r t l n ( ~ )

denote t h e c o m p l e t i o n of Eo. For a l l q

F

I n

F

IN)) and l e t E~

I

@(El),

(xn) 1

K), s i n c e $ n ( E l i s a f i n i t e d i m e n s i o n a l , h e n c e p l e m e n t e d s u b s p a c c of E l .

Pccordinn t o c u r choice of

(an)

( x n ) i s bounflinn i n E l a n J , s i n c e El i s a

Hence,

Frgchet space,

(x,)

xnj

rectly)

. Now

Wnj Thus xg E E ,

is

that

relatively

t h e proof i n [16] can h e t r a n s f c r r e c l

di-

s a hounrlinq Cauchy s e n u c n c e i n C : ,(I<)+ ! l n \ l I c f o r a11 n E % ( c ) . n1 I: i s h o l o m o r p h i c a l l y c o m p l e t e . P i n a l l v ,

< lim 1 1 0 o + n j I 1 1 < = I J n l Icr(xo) I = l i m I n ( x n j ) I j +m j +m T h i s i s a c o n t r a d i c t i o n t o xn $ V f o r a l l n E: I?!. TIC t h u s proved ( * )

anc?

xo.

(This is a straiqhtforward oeneralization of t h e r e s u l t

compact: f o r

cnm-

se p ar a b l e

h a s a converrrent suhseciuence

e v e r y bounilinq s e t i n a s e p a r a b l e Banach s p a c e

5

for

have

.

i s h o l o m o r p h i c i n T ' f-l I', whcrehv n hence i n a neinhhorhood o f L = 4 (I<) C L e t q:

Now f

%.)

=

(c),

r+

be

r

387

POLYNOMAIL APPROXIMATION

p 4 o p e h t i e b l? - 40 i n t h e o h e m 1 atie

The

COROLLARY 3

eqiiiva-

PetiR R c 5?

K

c I1

404

a!P c o m p a c t

=>

moor

Cvidentlv " 5 ~

K C [J.

k% ( E l

3911, s i n c e

.

I

C

ic.

To s h o v

a r c compact s i n c e E i s l e t I: C [J b e c o m p a c t . Then K and 1:. &YE) holomorphically complete and s i n c e U is p o lv n o mially convex. A L

k n (E\U)

so, KO =

i s compact. Now a h o l o m o r p h i c f u n c t i o n f can

-

-

u KO

be d e f i n e d i n a n e i n h h o r h o o d of K = Kg(r,

equal t o

0

i n a n e i n h b o r h o o d o f KBe(c)

and

f

by p u t t i n s

eoual t o

1

f

a

in

n e i a h b o r h o o d o f KO. C o r o l l a r v 2 i m p l i e s t h e e x i s t e n c e of a con t i n u o u s p o l y n o m i a l p: E + C with I - pi 1~ < 71 T h u s , I IpI 1 i.e. KO = d , < 2 < Ip (x)1 f o r a l l x F KO. I t follows

.

If

- . .

K = I<

%(C)

IK<

= "-

4 . Theor rn 2 c a n be f o r m u l a t e d i n t h e f o l l o w i n n way.For a

compact

T C I1

r

4

d e n o t e t h e s p a c e o f T e r m s of

%rL)

holomor-

p h i c f u n c t i o n s on L . Then, i f U i s a p s e u d o c o n v e x ,

finitely

PunFe open s e t i n a h o l o m o r p h i c a l l y c o m p l e t e , m e t r i z a b l e

Ics

w i t h t h e a p p r o x i m a t j on p r o p e r t y , t h e f o l l o w i n a h o l d s : ( 1 ) T o eue4lj c o m p a c t I , C J,

map"

@ (11) LA

-

c o n t a i n ~ n gK ,

denbe

412

%(L)

(Take L =

1'

c1

A U C ~t h a t

thehe

c o ~ ~ e n p o nad ~

compact

t h e imnqe undeh t h e " t e n t t i c t i o n

g ( L )

4enpec.t

1 ~ 4 t h

k,

,UI

).

t o ,the & u p n n h m

t o p o ~ o n r ro n

The same a p p r o x i m a t i o n r e s u l t

shown f o r a pseudoconvex domain U s p r e a d over a r r 6 c h e t

%a).

can 01

be

Cilva

3 88

M.

SCHOTTENLOHER

s p a c e w i t h a f i n i t e dimensional Schauder decomposition (aaain with L =

A

1.

i n t h e case of C = Q:

n

F\

,a

[IT]

s t r a i n h t f o r w a r d r e a s o n i n n shows

that

s h a r p e r v e r s i o n (1) holds f o r a p s e u d o

convex domain U (acrain w i t h L = K): ( 2 ) T o evehq

L

C

compact I.: cc

t h e f i e c o t 4 e ~ p o ~ dan

U, c o n t a i n i n g I<, ~ u c ht h a t t h e i m a g e u n d e t t h e

conip.c.t

“fiesttiction

map”

%$(u)id

%(Lf

A e y u e n t i a L L y d p n b e w i t h 4ebpec-t t o t h e n a t u f i a t i n d u c t i v e t i n-1

it R o p o L o g y o n % ( L ) , The i n d u c t i v e l i m i t t o p o l o n v on g ( L ) i s d e f i n e d by = l i m inr1 V Ff(L1

@(L)

where &(L)

8%

(v) ,

i s a base of open n e i q h b o r h o o d s of L and i%”(V)

is

t h e Banach s p a c e of bounded h o l o m o r p h i c f u n c t i o n s on V w i t h t h e s u p norm. ( 2 ) FOR A PSEUDOCONVEX DOFIRIN

PROOF OF

compact and L = I<

Let f E %(L).

t h a t f i s h o l o m o r p h i c on L ( s ) = L

hence I q ( x n ) l 5 xnj

Ilnl IL(+)

+

for a l l

L e t I<

CU

he

There c x i s t s s > o

such

Assume

that

B(o,s)

E

Cr”:

1’

c I I .

%(u).

€ 0 11o v r s

It

L f o r a subsequence (x

) o f (x,). Contradiction. nj Now l e t 1 8 ( t \C L ( s ) f o r o < t < s . A c c o r d i n q t o t h e approximaj xo E A

t i o n t h e o r e m o f 01:a-Vcil

\Ifm-

flIL(t)+o,

striction

mdr)

[ 9 , p. 911, t h e r e e x i s t fn

i.e. fm j

8%”(Iz( t ))-+

?%(I,)

t h e qerms o f fm c o n v e r n e t o f in

f in %‘(I(t)).

6

Rfll)v i t h

Since t h e

rc

i s c o n t i n u o u s , it follows that Bil,,.

W e d o n ’ t know w h e t h e r o r n o t ( 2 ) h o l d s for

all

open

POLYNOMAIL APPROXIMATION

s u b s e t s of Cn.

389

T h e r e e x i s t domains i n Cn s a t i s f y i n 9 ( 2 ) h u t n o t

b e i n ? pseudoconvex. ror e x a m p l e , i f 0 i s pseudoconvex and K C 2 i s compact s u c h t h a t U = Q \ K

connected, then U s a t i s f i e s ( 2 ) .

P r o p e r t y (2) i s p a r t i c u l a r l y i n t e r e s t i n n i n t h e

infinite

d i m e n s i o n a l case: T f an open s e t U i n a norme?. s p a c e E satisfies

(2), t h e n t h e Nachhin topo1oc.y j e c t i v e l i m i t of a11

,

(%(u)

T

w

T~

s(~ , K )c u

pro-

[12] i s o b t a i n e d a s t h e

compact:

) = lir, r ? v J T’C cpt.

(K).

T h i s i s d i s c u s s e d i n [ll] and [ 3 ! . T h e r e a r e o n l y few e x a m p l e s of o p e n s e t s i n i n f i n i t e m e n s i o n a l s p a c e s €or which i t i s known t h a t ( 2 ) i s e.n.

r

f o r a b a l a n c e d open s e t i n a n o r h i t r a y l c s

di-

satisfied,

cer-

and for

t a i n R e i n h a r d t open s e t s i n a n a n a c h s p a c e w i t h a n u n c o n d i t i o n d b a s i s [ll]. U n f o r t u n a t e l y , t h e methods p r e s e n t e d above as as t h a t of [17!

proof of

we1 1

do n o t p r o v i d e more e x a m p l e s . A l s o , t h e

( 2 ) f o r a pseudoconvex U C C

n

above

cannot be t r a n s f e r r e d t o

it can

domains i n i n f i n i t e d i m e n s i o n a l n o r n e d s p a c e s . IIowever, b e t r a n s f e r r e d t o t h e case of a pseudoconvex c7olnain U

in

an

a r b i t r a r y p r o d u c t CpA of l i n e s , b e c a u s e s u c h a domain i s t h e p r o duct R

c”,

x

CA’ o f a pseudoconvex

R in a certain C

n

an? t h e

space

A ’ = ~ \ { l , . . . n ~~ 2 1 .

ACKNOWLEDGEMENT:

I want t o t h a n k M.C.

Matos for h e l p f u l c a m n e n t s .

REFERENCES

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PI.

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.

POLYNOMIAL APPROXIMATION

[12]

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39 1

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nappincrs"

.

S p r i n n e r - V e r l a n , New York 1 9 6 3 . [13] P!!.

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23

convex

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~ € 7 . M ~ ~ V E R P . A Z ,? s e u d o - c o n v e x i t g

e.1.s.

I n : S6m. L e l o n n 13/74.

4 7 4 (19751, 6 3

[16] br.

e t h a s e de Schaucler clans les

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S p r i n n e r L e c t u r e Notcs

82.

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(Symposium

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[ 171 M. SCHOTTENLOHER, P a s

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in

Ann.

Inst.

Fourier.

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D

8

39

E'liinchen 2