By LARs INGE HEDBERG Weighted mean square approximation in plane regions, and generators of an algebra of analytic functions ARKIV FOR MATEMATIK BAND 5 nr 36

A R K I V FOR MATEMATIK B A N D 5 nr 36

1.64 16 1 C o m m u n i c a t e d 2 D e c e m b e r 1964 b y O. FROSTMAN a n d L. CARLESO~r

Weighted mean square approximation in plane regions, and generators of an algebra of analytic functions

By LARs INGE HEDBERG

1. Introduction

If D is a region in the complex plane, and

a(z)

is a continuous, positive function in D, we denote b y

H~(a; D)

the set of all analytic functions,

h(z),

in

D,

which have the p r o p e r t y t h a t

II h I1~ = j DI h(z) I~a(z) dA <

where

dA

denotes plane Lebesgue measure.

D is called a Carathdodory region if it is simply connected, bounded, and its boundary, ~D, coincides with the boundary of the infinite component, D~r of the complement of the closure of D.

I n 1934 Markuw and Farrell proved independently t h a t for a n y Carathdodory region, D, the polynomials are complete in HZ(1; D). I t is well known t h a t this pro- p e r t y need not hold for non-Carath$odory regions. (See e.g. [3]). The result has been generalized to spaces with weight functions other t h a n the identity b y various, n o t a b l y Soviet, mathematicians. A survey of this theory is given in Mergeljan's p a p e r [3]. Most of the results, however, deal with non-Carathdodory regions, and because of this the weight function

a(z)

is required to tend to zero, when z ap- proaches the boundary.

For Carathdodory regions much more can be said, and the first p a r t of this p a p e r is devoted to this problem. The result is stated in Theorem 1.

I n the second p a r t we shall s t u d y the related problem of finding generators of

xT"oo w n

the algebra, A, of all analytic functions,

g(w) = .Lo gn

, in the unit disc, such t h a t the norm, Jig II = ~ l g ~ I, is finite. B y a generator of A we m e a n a function, % in the algebra A, such t h a t the polynomials (with constant term) P(~) are dense in A.

F o r a function to be a generator of A it is obviously necessary t h a t it is univalent in the closed unit disc, b u t whether this condition is also sufficient is an open prob- lem. D . J . N e w m a n proved [5] t h a t a univalent function which m a p s the unit disc onto a region with rectifiable b o u n d a r y is a generator of A, a n d a simpler proof of this was given b y H . S . Shapiro [7]. See also [6]. As a corollary to Theorem 1 we get another sufficient condition which we state as Theorem 2, and t h e n we show b y means of examples (Theorems 3 and 4) t h a t our result neither includes, nor is included in N e w m a n ' s .

I wish to acknowledge m y great indebtedness to Professor L e n n a r t Carleson, who has contributed i m p o r t a n t ideas to this work.

37:5 541

L. I. HEDBERG, Weighted mean square approximation in plane regions 2. Polynomial approximation

I f D is a simply c o n n e c t e d region, we d e n o t e b y / ( z ) a function which m a p s D conformally o n t o t h e u n i t disc, a n d we d e n o t e the inverse f u n c t i o n t o / ( z ) b y ~(w).

W e d e n o t e b y 6(z) t h e distance f r o m z to ~D. T h e n we h a v e t h e following theorem.

T h e o r e m 1: Let D be a Carathgodory region and a(z) a continuous, positive/unction in D. Then the polynomials are complete in H~(a; D) i/ the weight, a(z), satis/ies the /ollowing two conditions:

(a) f a(z)4(logo~z---))SdA<oo

(b) the polynomials are complete in H2(a(q~(w)); ]w] < 1).

Remark: Little seems to be k n o w n a b o u t when condition (b) holds, except the easily p r o v e d fact t h a t it holds w h e n a(q~(w)) depends only on I w [, i.e. w h e n a(z) is c o n s t a n t on e v e r y level curve ]/(z) I = K. (See [3]).

As for condition (a) it would be a n interesting task to t r y to replace it b y the clearly necessary condition ~ D a(z) dA < oo.

W e need the following l e m m a , which it of course well known, b u t since there seems to be no c o n v e n i e n t reference, we include its proof.

L e m m a : For every bounded, simply connected region, D, there is a constant K such that the mapping/unction/(z) satis/ies

/or all z in D.

1 - ]/(z) [ ~< K {5(z)} 89

Proo/: T h e proof is a simple application of t h e B e u r l i n g - N e v a n l i n n a estimates of h a r m o n i c measures.

L e t the d i a m e t e r of D be d, a n d choose a positive n u m b e r ~ < d / 6 . L e t z 0 be a p o i n t on a level curve I/(z) I = 1 - ~ / a n d let a(z0) be t h e disc with radius ~ a n d centre

%. Then, if to(z) is the h a r m o n i c measure of ~D n a(z 0) with respect to D at the p o i n t z,

~(%) to(z~ >~ 2 aresin ~ + r '

b y the B e u r l i n g - N e v a n l i n n a t h e o r e m ([4] p. 104 ft.). I t follows t h a t

1 - to(zo) < ~ {~(%)}~.

W h e n D is m a p p e d o n t o I w [ < 1, OD fl a(zo) corresponds to a set, Sz., on [w [ = 1, a n d because of the invariance of t h e h a r m o n i c measure co(z) = wl(/(z); Sz.), where toa is t h e h a r m o n i c measure with respect to t h e u n i t circle.

N o w OD c a n be covered b y a finite n u m b e r of discs, al, a 2 .. . . . aN, with r a d i u s ~.

OD always contains a point, Zl, with [ z 1 - z o [ >~ d/2. z a is c o n t a i n e d in a disc a, with centre a,, a n d t h e n [a, - z o [ >~ d/2 - 0 > 2Q. Thus, for e v e r y z o in D there is a a, such t h a t a(Zo) a n d a, are disjoint. B u t all t h e sets ~D N a, correspond to sets of positive

ARKIV FOR MATEMATIK, B d 5 nr 3 6

measure on ] w ] = 1, otherwise their harmonic measures with respect to D would be identically zero. I t follows from the Poisson representation t h a t there is a constant K, independent of %, such t h a t

1 ^-- (DI(/(Z0) , Sz, ) ~ KT].

Hence there is a constant K such t h a t ~ ~< K ((~(z0)}~.

Proo/ o/ Theorem 1:

The proof depends mainly on methods of Bers and Carleson.

We assume D to be Carathdodory, a n d s t a r t b y observing, with Mergeljan, [3J p. 136, t h a t it is enough to show t h a t for every n > 0 and every ~ > 0 there is a polynomial,

P(z),

such t h a t

fDi/n(Z)/'(Z) P(z) I ~a(z) dA < e.

For if

h(z)

is a r b i t r a r y in

H2(a;

D),

h(cf(w)) ~f'(w)

is clearly i n

He(a(q~(w));

I wl < 1), and thus, b y condition (b), there is a polynomial, Q(w), such t h a t

f DI h(z) - O(/(z) ) /' (z) l'a(z) dA = /,w,<, I h(~(w) ) ~' (w) - o(w) [~a(~(w) ) dA < ~"

B u t

Q(/(z))/'(z)

is a linear combination of functions

In(z)/'(z),

and thus there is a polynomial,

P(z),

such t h a t

fj l Q(l(z))l'(z) P(z) I sa(z) dA < s.

9

I t follows t h a t ~or every s > 0 there is a

P(z)

such t h a t

fD I h(z) - P(z)I~

a(z)

dA

< s, which proves this first assertion.

A n y bounded linear functional,

L,

on

H2(a; D)

can be expressed in the form

L ( h ) =

fob(z) ^dA,

where tt(z) is a function satisfying

f j [~(Z)[2a(z)-idA < ~ . (1)

9

We are thus required to prove t h a t if the function

m(z)- ~ F(r dA=O - J ~ - z

37* 543:

L. I. HEDBERG,

Weighted mean square approximation in plane regions

for all z in

Doo,

t h e n (1) implies t h a t

f /n(z) ['(z) #(z) dA = O,

i)

n>~O.

Now, for q > 0 we define a C ~ function,

oJq(z),

in D w i t h t h e following properties:

0

<~ ooq(z) <<.

1,

eo~(z)=O for 6(z)~<q, coq(z)=l for ~(z)>~2q,

I g r a d ~oq(z) [ ~<

K/q

for some c o n s t a n t K .

Such a function obviously exists, for t h e function 5(z) itself satisfies [(~(zl) - c~(z2) I ~<

[zx - z2 I- W e denote b y Dq t h e set {z; q ~< 5(z) ~< 2q}.

W e a s s u m e for the m o m e n t t h a t / ~ ( z ) E Coo a n d is zero outside a c o m p a c t subset of D. T h e n the f u n c t i o n

m(z)

is continuous in t h e whole p l a n e a n d

Ore(z)

~ z~ . tt(z)

in D (see e.g. [9], p. 29). Now, following L. Bers [1], we a p p l y Green's f o r m u l a to a region D ' c D, such t h a t 0]9' is s m o o t h a n d contained in t h e set where

6(z) < q. By

the a n a l y t i e i t y o f / " ( z )

]'(z)

^{we find}

--~t f D,~%(Z ) f'(Z) /'(Z) I~(Z) dA = f D, Wq(Z) 3~ (/~(Z) /'(z)m(z))dA

=-f~,O~(O/"(z)/'(z)m(z)dA-foDo,o(z)/"(z)/'(z)m(z)dz.

F r o m the definition of

~oq(z)

it follows t h a t the b o u n d a r y integral is zero, a n d t h a t we can replace D ' b y D in the o t h e r integrals. T h u s

~ o ~ ( z ) .

~Lo~q(z)/"(z)/'(Z)l.t(z)dA=~q~-z/ (z)/'(z)m(z)dA.

(2) L e t t i n g q - + 0 we find t h a t t h e integral on t h e left tends to

~D/n(z)/'(Z) #(Z)dA,

a n d hence we h a v e to p r o v e t h a t t h e integral on the right tends to zero.

B y the definition of

wq(z)

a n d t h e Sehwarz i n e q u a l i t y

LqO~q~(z) /n(z) /'(z)m(z)dA '

q Joq

H e r e

f Dq I/'(z) l'(z) [2 ~(z)- 89 dA <~ Kq- 89 fa(z)<_2q I l"(z) l'(z) 12 dA

<Kq-~( I/'(z) 12dA=Kq-~ ( dA<K,

Jl-Is189 J O<l-lwl<Kq 89

ARKIV FOR MATEMATIK. B d 5 n r 3 6

where t h e second i n e q u a l i t y follows f r o m t h e l e m m a . ( T h r o u g h o u t t h e p a p e r we use K to d e n o t e different constants.) Because of this a n d t h e definition of

Dq

the left- h a n d side in (3) is m a j o r i z e d b y

q-~-~DqJm(z)l~dA,

a n d we h a v e to p r o v e t h a t

S Dq ] m(z) ]2dA = o(qi).

B y a t h e o r e m of Sobolev [8] a f u n c t i o n

( #(~) dA

belongs to

L2(D)

if/~(z) E

L~(D)

for s o m e p > 1 such t h a t 89 >~ 1 / p + ~t/2 - 1, a n d t h e n

{f Jm~(z)l~dAl'<K{~l~(z)l'dA} "p,

⁽⁴⁾

where K d e p e n d s on

p, ~

a n d D.

B y tt61der's inequality, for p < 2

Ljla(z)jPdA<{fDj/~(z)j~a(z)_idA},12{fDa(z),~(2 ^{p)}l ,/2 (5)}

T h e second integral on t h e right is finite as soon as

p / ( 2 - p )

< 4 , i.e. p < 8/5, b y a s s u m p t i o n (a), a n d so/~(z)

ELY(D)

for p < 8 / 5 , b y (1).

N o w we can r e m o v e the r e g u l a r i t y h y p o t h e s i s on/~(z) a n d p r o v e t h a t (2) holds for all /~(z) satisfying (1). F o r if ~ = 1, (4) holds for all p > 1, a n d thus b y (3), (4), a n d (5)

-~ ! (z)i (z)m(z)dA <gq-~jDJ,U(Z)J~a(z)-adA. (

I f we a p p l y t h e Schwarz i n e q u a l i t y t o t h e l e f t - h a n d side in (2) we find

L oq(z) /~(z) /, (z) ,u(z) d A e < L j f~(z) /, (z) j~ a(z) dA L j t~(z) l~ a(z)_l dA"

Because for any/~(z) satisfying (1) t h e r e is a sequence {#~(z))~ of functions in C ~ such t h a t i'D J/~(Z) --/~(Z) J2

a(z) -~ dA--~ O,

these two inequalities show t h a t (2) holds for all such #(z).

A s s u m i n g t h a t

re(z)=0

for z E D~r we shall n o w e s t i m a t e

re(z)in

D b y m e a n s of a device due to Carleson, [2], a n d show t h a t

.~Dq re(z) 2dA = o(q~).

F i x a

zED

a n d let

zoE~D

be such t h a t

z - z o =~(z).

L e t

Cz

be the disc w i t h centre z 0 a n d radius O(z). E v e r y circle J s - z0J = Q intersects the o p e n set D : c b e c a u s e of the C a r a t h 6 o d o r y p r o p e r t y , a n d hence t h e r e is a m e a s u r e

da(s)

which is s u p p o r t e d b y linear s e g m e n t s in D ~ N C~, such t h a t

f ls_ zol<_qda(s) = e

for all e < 5(z). As

re(z)= 0

in D ~ we find 1

O[Z)JD~--Z

^J~--s

545

L; I. IIEDBEllG, Weighted mean square approximation in plane regions

where the change in the order of integration is permissible, because it follows from the estimates below t h a t

f l~(~)ldA (da(s)

_{j l r < oo.}

I n (6) we always have

I~- ~ I

< 2a(~). F o r (~(~) >~ }5(z) (and I r z

I

< 4(~(z)) we have

I ~ - s]/> ~(~)/> }(~(z), and hence, in this case

~) f ~z~d~(s) <4. ⁽⁷⁾

If I r ~,a,~(,) we have

1r162189162

a n d thus

a~) ~:~do(~)

< 1 ~ - ~ - - - ~ (s)

o - - - - - - 9 ~ P - - o =

J ~o - ~o I = to, t h e n I ~o - ~ 1/> Ir - ro I, a n d it f o l l o w s t h a t I ~ - ~ I >1 ~ I ~ - ro I if I r - ~o I/>

2(~(~). Hence, b y t h e definition of da,

f _ _ f__+dr 2 ( d r ,

JIc-~l .to(c) ^JIr-rol

where t h e first integral is t a k e n over all r with ] r - r 0 [ ~ 2~(~), and the second over all r with I r - r 0 [ ~> 2(~($) a n d 0 ~< r ~< O(z). T h e second integral is clearly greatest when r 0 = iS(z), and it follows t h a t

c ~ 1 , f ~ - - ~ da(s) <~ ~1 I "~'' ,og ~ ~(z) + K 2 ~ K 1 log 5 ~ + K2, 1

(9)

since (~(z) is b o u n d e d b y a constant.

N o w re(z) can be w r i t t e n as t h e sum of t h r e e integrals m~(z), i = 1, 2, 3, corre- sponding to the domains A~ where (7), (8), a n d (9) hold respectively. I t is thus suffi- cient to show t h a t for i = 1, 2, 3

B y (7) we find

f ~ [ m d z ) p d A = o(q+ ).

.-f'olm~(~)pdA<K( ~ (

J-oU-~.I $ - z ~ ' ~ : ] IZ'(~)I . . "F "

dA~

I/-*( )[

d A ] d

j . o L J . l ~ - i l ~ ~f

~.

ARKIV FOR MATEMATIK. B d 5 n r 3 6

B u t b y (4) a n d (5)

JDIJDI~--Zl J

a n d it follows t h a t if we p u t q = 2 -~, v = 1, 2 . . .

which p r o v e s our assertion for

ml(z ).

F o r

ms(z )

we find b y (8) t h a t

[#(~)

i~dAc} dA~

I~Wl ~ ~I ~=o(q~),

as above.

Similary, in t h e t h i r d case it suffices to p r o v e t h a t

fDq {fD'l~

[ ~ - ~ r"

[/x(~)[ dA:}~dAz =

o(q+).

I f we replace the i n e q u a l i t y (5) b y

f I~(~)l" I log ~(~)pdA

<<. { f D, ~(z)12 a(z)-i dA}'/2 { f Da(z) P~(2-')] log ~(z)]Uv'(~-') dA} ~-~/2,

this case follows as before, b y a s s u m p t i o n (a), a n d t h e proof of T h e o r e m 1 is complete.

Remark:

I t is easily seen f r o m t h e proof t h a t if ~D is so r e g u l a r t h a t 1 -

I/(z) [

K {(~(z)} ~ for s o m e a in 89 < a ~< 1, a s s u m p t i o n (a) c a n be r e p l a c e d b y

f D(a(z) [ log 6(Z)12)21~ dA < ~ .

3. A p p l i c a t i o n t o a g e n e r a t o r p r o b l e m , a n d e x a m p l e s

Applied to the g e n e r a t o r p r o b l e m s t a t e d in the introduction, T h e o r e m 1 gives the following result. F o r n o t a t i o n see the introduction.

T h e o r e m 2:

A /unction qD(w) = ~ q~nw n is a generator/or Ai/ it is univalent in Iwl <<. 1, and i/, /or some

cr 12,

547

L. I. HEDBERG,

Weighted mean square approximation in plane regions

o o

~. n(log n)~l ~ . I ~ < oo.

2

Remark 1:

Note t h a t the condition t h a t q is in A and is univalent implies t h a t

Remark 2:

I n the case when ~

n~+~l~l~<

~ for some a > 0, It. S. Shapiro has recently obtained a simple direct proof of the above theorem (private communication).

Proo/: ep

m a p s the unit disc onto a region D which is bounded b y a J o r d a n curve.

L e t the inverse of ~ b e / .

I t is easy to prove b y means of Parseval's relation a n d elementary estimates, t h a t for a n y

g(w) = ~'~ g,~ w n

the condition ~ n(log n) ~ I gn I ~ < o~ is equivalent to

I g'(w)

12 log

dA

< ~ . (10)

wl<l

Thus, if we a p p l y this to ~, and pass to D, we find

L( ¹ i; d A < oo,

log 1 - I I(~)

for some a > 12. I t follows, by the lemma and b y the r e m a r k following Theorem 1, t h a t the weight function

1 )1+~

as(z )

⁼ 1 § log 1 -

I1(~)1

^{, z e D ,}

satisfies all the conditions in Theorem 1, whenever fl ~< a / 4 - 3.

Furthermore, b y Cauchy's inequality,

I I g l l = Z l g . l v l g 0 1 + 0 n + n(log n) 1+~ 1 89 ~r (n + n(log n) a+~)

[gn

I ~ 89 ( l l ) I t is enough to show t h a t for every e > 0 there is a polynomial P such t h a t

Hw-P(q)(w))ll<e.

B u t b y (10) and (11), for f l > 0 ,

f f , ] 89

H w -

P(~(w)) II

< I p(~(0))I

+ K __.iJ,~l< 11- P'(~(w))q~ (w)12at3(cf(w))dA I

, 89

= 'P(cf(O)), + K { L '/'(z)- P (z)'2a~(z)dA} .

Here the last integral can be made less t h a n s (if fl ~< a / 4 - 3) b y Theorem 1, for

f]/,(z)i2aa(z)dA=flwl<l(l+/)

flog 1 ~ _ i [ ) 1 w

~l+fl~) dA,

which is certainly finite. Then we can choose P(~(0)) = 0, which proves the theorem.

ARKIV FOR MATEMATIK. B d 5 n r 3 6 Our result is neither included in, n o r does it include, N e w m a n ' s theorem. This is a consequence of the following two constructions.

Theorem 3: There is a region D, bounded by a non recti/iable Jordan curve, such that the Riemann mapping ]unction ](z) satis/ies

fD(1 - I ] ( z ) <

I)-~dA

/or every ~ < 1.

Proo/: W e shall c o n s t r u c t i n d u c t i v e l y a sequence of regions, {Dn)~, such t h a t Dn c Dn+l, a n d t h e n define D = [J n~_0 Dn.

11 D2

Fig. 1

See Fig. 1. W e let D o be a square with sides of u n i t length. W e divide one of the sides in three parts so t h a t the length of t h e middle p a r t is 1/~/2, a n d the lengths of the o t h e r p a r t s are 89 - 1/2 ~/2. W e choose a n u m b e r , N1, a n d divide the middle p a r t in N 1 equal parts, a n d t h e n we let each of these p a r t s be t h e base of a n isosceles r i g h t triangle, which lies outside D 0. The u n i o n of D o a n d these N 1 triangles is D 1.

To c o n s t r u c t D 2 we first divide each of the 2 N 1 legs of the isosceles triangles of D 1 - D o in three p a r t in the same p r o p o r t i o n s as above. T h e n we choose a n u m b e r , N2, (to be d e t e r m i n e d later) which is a multiple ( >~ 2N1) of N 1, a n d divide each of t h e middle p a r t s in N2/N 1 equal parts, a n d a d d isosceles r i g h t triangles lying outside D 1 as above. The length of t h e legs of one of these triangles is clearly 1/4N 2. T h e u n i o n of D 1 a n d these 2 N 2 triangles is D 2.

N o w assume t h a t Dn is c o n s t r u c t e d a n d t h a t D ~ - D ~ - I consists of 2 n - l N ~ isos- celes right triangles with legs 1/2nNn. To c o n s t r u c t D~+I we choose a multiple, N~+I ( ~> 2N~), of/V~ a n d divide the 2~N~ legs of the triangles c o n s t i t u t i n g D~ - D ~ - I in three p a r t s as before. T h e n we divide t h e middle p a r t s in Nn+l/Nn equal p a r t s a n d a d d isosceles r i g h t triangles lying outside D~ as above. W e e v i d e n t l y h a v e

2n2u such triangles whose legs are 1/2~+1Nn+1.

I t is easy to see t h a t t h e ~D so c o n s t r u c t e d is a J o r d a n curve. T h e difference in 549

L. I. HEDBERG,

Weighted mean square approximation in plane regions

length between ~Dn+ 1 and ODn is 1 - 1/]/2 for all n, and thus ~D is not rectifiable.

We shall show that the numbers N= can be chosen so t h a t D satisfies the require- ment in the theorem.

I t is enough to show t h a t the Green's function,

g(z),

of D, with the centre of D O as pole, satisfies

.~Dg(z)-~dA

< c~ for all a < 1.

We shall determine the sequence {N=}~ r inductively. We assume t h a t the numbers N~, 1 < i < n, are already chosen. L e t the Green's function of D~ with pole at the centre of D O be

gn(z).

Then, if In is the subset of

ODn

to which the triangles of D=+I are to be joined, the inner normal derivative,

Og~(z)/On,

^of

g,~(z)

is continuous on and near In, a n d

eg.(z)

Min = 2~/~ > 0.

ZEIn

Thus there exists an

Sn > O,

such t h a t if

z o E

In, if z E Dn lies on the normal to ~D n through %, and if [ z - z0[ <

en,

^then

g.(z)>~.lZ-~ol.

We choose N . + I ~> Max

(~/2n+len,

exp (1/~]n)), and complete the construction of D b y choosing N1 arbitrarily.

For every n, 0 <

gn(Z) < g(z)

in Dn. Thus, for ~ > 0,

1 2 n - D n - I

If one of the triangles in

Dn-D~-I

is extended into D~-I b y a square (which then has the side

]/2/2~Nn),

we have on the side, l~, of the resulting pentagon, P~, which faces the triangle,

gn(Z) > g n - l ( Z ) > ~ n 1

~/2"N.,

for b y the choice of

Nn, ]/2/2nNn <~s=-l.

Hence in P~, gn(Z) > ( ~ n - 1

~ / 2 " N . )

tOn(Z),

where con(z) is the harmonic measure of In with respect to Pn. B u t

f2. co.(z)-'dA=K(1/2"N.)2fplo~l(z)-'dA,

b y the invarianee of the harmonic measure, and the last integral is finite for all

< 1, because the angles in P1 are all greater or equal to ~ / 2 . I t follows t h a t

fm~_Dn_lgn(Z)-~dA ~'~

< . . . o n ( ~ - l ) IT~r . ~ . _ ~ < K 2 - ~ n(~r N . ~--1 ( l o g N . ) ,

ARKIV FOR MATEMATIK. B d 5 n r 3 6 b y t h e choice of N~. B u t N 1-~ (log N~) ~ is b o u n d e d as n--> oo. H e n c e ~D g(z)-~ d A < c~

for all ~ < 1.

Theorem 4: Let k(t) be a d e c r e a s i n g / u n c t i o n / o r 0 < t ~ 1, such that limt-~0/~(t) = oo.

T h e n there is a region D, bounded by a recti/iable Jordan curve, which is such that the m a p p i n g / u n c t i o n / ( z ) satis/ies

fl/~(1 - [/(z)[) d A = ^cx).

)

Proo/: L e t {a~}~ be a sequence of discs w i t h centres a t t h e points as on t h e x-axis, a n d radii r~ w i t h

for all i. L e t D~ = (J ~ o~ a n d D = U ~ o~. See Fig. 2. W e shall p r o v e t h a t t h e se- quences {a,} a n d

{r~}

c a n be chosen so t h a t D fulfils t h e requirements.

Fig. 2

W e first choose {r~} so t h a t ~ r~ < ~o. T h e n D is clearly b o u n d e d b y a rectifiable J o r d a n curve.

F o r given (as} we let to(z) be t h e h a r m o n i c measure w i t h respect to D - % of

~o 1 (] 00. I f g(z) is t h e Green's f u n c t i o n of D with pole a t a0, g ( z ) i s b o u n d e d on

~o 1 N 00 b y a c o n s t a n t C, which is i n d e p e n d e n t of t h e choice of a~, i > 1. This follows f r o m t h e fact t h a t t h e r e exists a region which has ~ol N o 0 as a p a r t of its b o u n d a r y , a n d which contains D - o 0 for e v e r y choice of a~, i > 1. T h e n g(z) <~ Cre(z) in D - 00.

Also, if we assume t h a t / ( a o ) = 0, 1 - I/(z) I ~< log 1/I/(z) I = g(z). I t is therefore e n o u g h t o show t h a t we c a n m a k e ~D-aa k(C6o(z))dA = oo.

I n a disc o~, i >~ 1, re(z) is always less t h a n t h e h a r m o n i c measure with respect to o~ of t h e p a r t of ~a~ which is c o n t a i n e d in o1-1 tJ o~+1. T h a t is, co(as) is m a j o r i z e d b y t h e s u m of t h e central angles corresponding to t h e arcs ~a~ n o~-1 a n d ~o~ N o~+1. Thus,

t oo

for a n y given positive sequence, { ~}2, we c a n clearly choose t h e as inductively in such a w a y t h a t 2Cre(a 0 ~t~, i > 1.

B y H a r n a c k ' s inequality re(z) ~< 2re(a~) in t h e disc, o~, with centre as a n d radius r~/3, a n d it follows t h a t

_o/r d A >1 -- d A >1 Y. It(t,) r~ ~.

2 a s

t oo

B u t t h e sequence { ~}~ c a n be chosen so t h a t ~ k(t~)r~ ~ 0% a n d this p r o v e s t h e theorem.

551

L. I. HEDBERG, Weighted mean square approximation in plane regions

R E F E R E N C E S

1. BEgs, L., An approximation theorem, New York Univ., Courant Inst. Math. Sci., R e p o r t IMM-NYU 317, 1964.

2. CARLESO~r L., Mergelyan's theorem on uniform polynomial approximation, Math. Scand., to appear.

3. MERGEL;rAN, S. I~T., On the completeness of systems of analytic functions, Amer. Math. Soc.

Translations (2) 19, 109-166 (1962). (Usp. Mat. Nauk (N.S.) 8 no. 4 (56), 3-63, (1953)).

4. I~TEVA~LIZqNA, R., Eindeutige Analytisehe Funktionen, 2nd ed., Springer, Berlin, 1953.

5. I~TEWMA)Z, D. 5., Generators in ll, Trans. Amer. Math. Soc. 113, 393-396 (1964).

6. N~WMAN, D. J . , SC~WA]~Wz, J . T., a n d SKA~'IRO, H. S., On generators of t h e Banaeh algebras 11 and LI(0 , co), Trans. Amer. Math. Soc. 107, 466-484 (1963).

7. S~API~O, H . S . , Weakly invertible elements in certain function spaces, a n d generators in ll, Michigan Math. J. 11, 161-165 (1964).

8. SOBOLEV, S. L.. Sur u n th~or@me de l'analyse fonetionnelle, Dokl. Akad. ]Nauk SSSI:t 20, 5-9 (1938).

9. VEKVA, I. N., Generalized analytic functions, P e r g a m o n Press, 1962. (Obob~Sennye analitiSeskie funkcii, Fizmatgiz, Moscow, 1959.)

Tryckt den 14 april 1965

Uppsala 1965. Ahnqvist & "Wiksells Boktryckeri AB