Approximation in weighted Hardy spaces

A P P R O X I M A T I O N I N W E I G H T E D H A R D Y S P A C E S

By

A. BON1LLA, V. PI~REZ-GONZALEZ, A. STRAY AND R. TRUJILLO-GONZ,~LEZ

Dedicated to Professor Ndcere Hayek on the occasion of his 75th birthday

A b s t r a c t . This paper is concerned with several approximation problems in the weighted Hardy spaces HP(w) of analytic functions in the open unit disc D of the complex plane C. We prove that if X is a relatively closed subset of D, the class of uniform limits on X of functions in HV(w) coincides, modulo HV(w), with the space of uniformly continuous functions on a certain hull of X which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets for HV(w), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spaces Hv(D), 1 _< p < oo.

1 I n t r o d u c t i o n

In this paper we deal with some qualitative approximation questions in the

weighted Hardy spaces HV(w) of analytic functions in the open unit disk D of

the complex plane C. The definition of these spaces, and the structural properties

needed, are given in Section 2. We are interested in problems concerning the

decomposition of the space of uniform limits of functions in HP(w) on a relatively

closed subset of D and with simultaneous approximations in these spaces.

In 1984, A. Stray proved that i fX is any closed subset of C, the class of uniform

limits on X of entire functions can be written as

H(C)Ix = C~,a(X) + H(C),

where A" denotes the union of X and those components V of C \ X such that

V to {c~} is not arcwise connected in C to {oc}, and C~,a()() is the class of all

uniformly continuous functions on )~ which are analytic in its interior (see [St5,

Corollary 1]). Recently, Stray ([St6]) has proved that for the classical Hardy spaces

HP(D), 0 < p < c~, one has the similar decomposition

(1.1) Hv(D)Ix = C~a(2) + HP(D),

65 JOURNAL D'ANALYSE MATHEMAT1QUE, Vol. 73 (1997)

66 A. BONILLA ET AL.

where )f represents a hull of X. In Section 3 we generalize (1.1) to the weighted

Hardy spaces HP(w), 0 < p < oo.

In the early seventies, L. A. Rubel ([Ru]) posed problems of simultaneous

approximation by polynomials in linear spaces of analytic functions in D endowed

with a topology r such that analytic polynomials are r-dense. Nowadays, such problems are known as describing Farrell and Mergelyan sets for different function

spaces and have been studied by several authors. Among others, we mention that

L. A. Rubel, A. L. Shields and B. A. Taylor ([RST]) and A. Stray ([Stl, St3])

considered the space H(D) of all analytic functions in D equipped with the topology

of uniform convergence on compacta; A. Stray analysed them in the space H a (D)

with the bounded pointwise convergence topology, while a complete description of

such subsets for the classical Hardy spaces HP(D), 0 < p < oo, which even holds

for the Smirnov class N + with the usual metric, was given by F. Prrez-Gonz~ilez

and A. Stray ([PSI) (see also [RS], [Pe]). Recently, the Farrell and Mergelyan sets

for classes of harmonic functions in higher dimensions have been characterized in

[PT] and [BPT].

A well-known theorem of Beurling states that the outer functions are the cyclic

vectors of HP(]I)), p > 0. This means that any subspace generated by multiplying

an outer function by polynomials is dense in HP(]I)). Hence, it is quite natural to

consider simultaneous approximation problems involving cyclic vectors in classical

Hardy spaces. With this statement it is understood that the outer function will have

to satisfy certain compatibility conditions determined by the type of approximation considered (see Example 5.2). This topic is considered in Section 5, but its

development requires the study of simultaneous approximation by polynomials on

the weighted Hardy spaces HP(w), 1 < p < oo. We dedicate Section 4 to this matter,

showing that Farrell and Mergelyan sets for these spaces can be characterized by

a geometric condition independent o fp and ~o.

2 Pre l iminar ie s

We denote by ql" the unit circle. The Lebesgue measure of any subset E of qr

will be represented by [El, and, as usual, C will mean a constant which can change

in each occurrence.

The classical Hardy spaces HP(D), 0 < p < oo, of analytic functions in

ID can be defined as HP(D) = N + n LP(qF) where N + is the Smirnov's class

([Gar, page 72]).

Let w be a non_negative function o f LI(qF). For 0 < p < oo, LP(w) is the space

APPROXIMATION IN WEIGHTED HARDY SPACES 67

of all complex-valued Lebesgue measurable functions on "It such that

{ 1 / : }l/p (2.1) Ilfllp,~, = 27 I:(ei~176 < m ,

7r

and we define the weighted Hardy space HV(w) of analytic functions on D by

(2.2) n % o ) = N + n Z%o) = {f e N + : Ilfllp# < ~ } .

Weighted Hardy spaces have recently been considered in [McP, GLS, Zh]. We

recall some properties.

(1) Ifw - 1, then HP(w) agrees with the classical Hardy space HP(ID). (2) It follows from the integrability condition of ~o that H~(D) c HP(r for

allp > 0.

(3) For any 0 < a < 1, consider

I-a( w(e iO) = exp L \Tr + O/

defined on T with a > 0, so that w(e i~ is a positive function in LI(T) whose

logarithm is also integrable. Define

f(e i~ = exp [ \7-+-'0: J

with b > 0. Then, since log f E L 1 (T),

1 H(z, eiO) 1 logf(eiO)dO F ( z ) = e x p ~ ~ p

belongs to N +, where e i~ + z

H ( z , e i e ) = e i e - ;

is the Herglotz kernel. However, for any p > 0, f ~ LP(T), and consequently

F c N+ \ (Up>o Hp(D)).

On the other hand, for fixed p > 0, we can choose b > 0 such that F defines a

function in HP(w), Indeed, as

l:(ei~176 = exp (pb- a) \ 7-~ : J '

taking b < alp, we obtain that f ~ LP(w) and therefore F E HV(w). By a weight on T we mean a nonnegative function w defined in T such that

w E LI(T) and logw E LI(T). I r E is any measurable subset o f t we will use the

68 A. BON1LLA ET AL.

notation ~(E) := fE w(()d~. To each weight w we can associate the outer function

of H p(D):

[/; ] 1 g ( z , eiO) log~(eiO)dO (2.3) G(z) = exp ~ r

Such an outer function satisfies the identity IG(ei~ = w(ei~ 1/p at almost every

point of the unit circle and provides a close relationship between H p (~) and H p (D).

P ropos i t ion 2.1. Let p > O, w be a weight on T, and f be a function defined in D. The following assertions are equivalent:

(a) f �9 HP(w),

(b) f G �9 HP(D),

where G is defined by (2.3).

Proof. Let f �9 HP(~). Then log [f(ei~ �9 Ll(']r) and, by the Riesz-Herglotz

factorization theorem, f can be represented as f = B S F , B being a Blaschke

product, S a singular inner function, and F the outer function given by

[/; ] 1 H(z, e ~~ log If(e%l dO F(z)=exp ~ ~

Now,

1 f(z)G(z) = B(z)S(z)exp [-~ i_ H(z, ei~ logif(ei~176 dO ]

is analytic on D and belongs to HP(D) because f(ei~176 lip E LP(T). On the other hand, if fG �9 HV(D), f is defined almost everywhere in ~I" and

f �9 N +. Moreover, f �9 HP(w) since

}"" Ilfllp,• = 1 ~r If(ei~176 dO

7P

(2.4) }"" = I I ( . % G ( . % I p dO = I I fGI I . .

7r

[]

Proposition 2.1 provides an equivalent definition of weighted Hardy spaces as

H"(w) = { f �9 N + : f G �9 HV(D)}.

P ropos i t ion 2.2. Let w be a weight on .~. Then HP(w) is a complete metric

space with the metric H ' H~p,~, 0 < p < 1, and a Banach space with the norm N" Np,~, 1 < p < oo.


Proof. We have only to prove that HP(w) is complete with the metric of LV(w)

for any 0 < p < c~. Take f E HP(, 0, p > 0. By Proposition 2.1, Ilfl lp,~ = IIfGIIp where G is defined by (2.3); hence any Cauchy sequence {fn }~-1 in H p (w) produces

the Cauchy sequence {fnG}.~__l in Hp(D) and, therefore, there exists F E HP(D)

such that f~G , F in Hp(]I)). If f = F/G, then trivially f belongs to HP(w) and

[]fn - f[[~,~ = [[f~G - FIIpP ,0 as n ~ c~. []

The equivalence between weighted Hardy spaces and classical Hardy spaces

given in Proposition 2.1 will play a crucial role in our study of approximation

problems in H P ( w ) . In the next two propositions we give a necessary and sufficient

condition for convergence in HP(w) and prove that analytic polynomials are dense

in HP(w).

Proposition 2.3. A sequence {fn}n~ converges to f in Hv(w) i f and only i f

{f~}~=1 converges to f uniformly on compact subsets o f D and [If~ [Ip,~ ' I lf l lp,~ aS ~2 ---~ (YL

Proof. Necessity is clear from the convergence in HP(D) of {f,~G}~_ 1 to f G

since G(z) # 0 on D. The reverse implication follows directly from the hypothesis

and from [CMS, Thin. 2], keeping in mind that G is an outer function of liP(D). []

Proposition 2.4. Analytic polynomials are dense in HP(w).

Proof. By Beurling's theorem ([Gar, Chapter II, w we have that for any

function f E HP(a;) there exists a sequence {Pn}nC~=X of analytic polynomials such

thatpnG ~ f G in HP(D). But this just means that pn ~ f in Hp(~) as n ~ oz. []

We close this section by proving that H~176 is dense HP(w). Actually, it will

be shown that approximating functions can be chosen in such a way that they are

controlled by the function in HP(w) to be approximated.

Proposition 2.5. For any function f in HP(w) (0 < p < co) there exists a

sequence {fn}~=l in H~(D) such that

(1) [f~l <- Ill in D and at almost every point in "~, for all n E N,

(2) fn ~ f pointwise in D and almost everywhere on qF as n ~ oo, and

(3) fn ~ f in HP(w) as n ~ oo.

7 0 A. BONILLA ET AL.

Proof . Fix f c H"@). By the factorization theorem, f = B S F , t3 being a

Blaschke product, S a singular inner function and F the outer function

] F(z) = exp ~ H(z, u(e ~~ dO , 7r

where u( e iO) = log I f ( ei~ [. Put un ( e iO ) = min{ u( ei~ n }, n = 1,2 . . . . . The functions

f,~(z) = B(z)S(z)F,~(z), with

[12 ] Fn(z) = exp ~ H(z, e i~ un(e i~ dO , 7r

give a sequence o f bounded analytic functions satisfying (1) which converges

pointwise to f on ]I). Moreover, it is easy to see that f,~ ~ f in measure on "IF so

that, by passing to a subsequence i f necessary, (2) holds, while (3) follows from

the dominated convergence theorem. The proof is complete. []

3 D e c o m p o s i t i o n o f HP(w)

Let X be a relatively closed subset of II~ and A be a class of analytic functions

on D. I f f E A, as usual we put Ilfllx = sup{If(z)l : z E x } and define the hull

o f X with respect to A as

2 A = {z e D : If(z)l ___ Ilfllx for all f E A}.

In the case A = H~(D) we write )(H~(D) = Jr.

R e m a r k 3.1. From Proposition 2.5, we conclude that )~H,(,o) = ~-.

By Alx we will represent the class of all those functions which are uniform

limits on X of functions belonging to A. C,,~(X) denotes the class o f all analytic

functions in the interior o f X which admit a continuous extension to C u {oo}.

We can now state the main result of this section, which is a decomposition

theorem for weighted Hardy spaces.

T h e o r e m 3.2. For 0 < p < oo we have

(3.1) Hv(w)lx = C,~a(.~) + HP(w).

R e m a r k 3.3. It should be noted that HP@)Ix = HP(w)I2. Indeed, since to

any function f in Hv(w)lx there corresponds a sequence of functions in HP(w)

which is, of course, a Cauchy sequence in J(, it follows that f e ap(~o)l ~.


The proof o f Theorem 3.2 requires some lemmas. The next one can be easily

shown using standard techniques o f integration theory (see, e.g., [Fo, page 93]).

We omit its proof.

L e m m a 3.4. Consider w E L~(w) with w > 0 a.e.-[dO] on T and let f E

IT(w), 0 < p < co, I f

1 fO+r If(e ey) - f(ela)[P w(ei'~)d% Av(f, e i0, r) ---- ~ Ja - r

then Ap(f, e i~ r) is a measurable function ofeiO fo r any r > O, and Av(f, e i~ , r) , 0

as r ~ O fo r almost every point o f t with respect to Lebesgue measure.

In order to prove (3.1) we are required to approximate any function in HP(w)

by continuous functions on X. Denote by Xnr the nontangential boundary of X ,

i.e., the set o f all those points e ~~ E X Ca T that are nontangential limits o f points

of X. Its complement in X n T, the tangential boundary of X, will be denoted by

Xt. As a first step, we obtain approximation by continuous functions on a subset

of the X,u having full measure.

L e m m a 3.5. Let f E HP(a O, 0 < p < ~ , and let X be a relatively closed

subset o f D. For any e > 0 there exist an open set V and a function f E HP(w)

satisfying."

(1) Ilf - fllH,(~) < e,

(2) l l / ] lx -< Ilfllx + e,

(3) f can be extended continuously to -X N V, and

(4) IXnt \ V I = O.

P r o o f . In the case IX,ul = 0 we take V = 0 and f = f . Hence we can assume

that IX,~tl > 0 and, by Lemma 2.5, without loss o f generality that f is bounded

on D.

For a > 1 and 0 < s < 1, we denote by T(e i~ a, s) the truncated cone in I13 with

vertex at e ~~ amplitude a and height s, i.e.,

T(e i~ = {z �9 D : I z - ei~ I < c~(1 - Izl), Izl > s}.

For fixed a l > 1, we define the following measurable sets:

.Aj = {e i~ �9 Xnt : If(e iO) - f ( z ) l < e ,z �9 T(ei~ - l / j ) } ,

B j = {e i~ � 9 : Ap(f ,e i ~ g k > _ j } .

It is easy to check that these families o f sets satisfy


(a) Aj c Aj+I and Bj c Bj+I for any j e N, and

(b) IXnt \ (Uj~N .aj)l = IX,~ \ (U~eN B;)I = o.

Observe that {6j = Aj n Bj}]~ is an increasing sequence o f measurable sets

and that for j t large enough, we can choose a compact set K1 C C h satisfying 1 [X,~t \ Kxl < 7[X,~tl, and for every point e iO E K1

(i) ] f (d ~ - f(z)l < e for all z ~ T(ei~ l / j1) , and

(ii) Ap(f,e i~ 1/j~) < e.

Now take finitely many discs { A t }5 e J, with the same radii r~ < 1/ j l and centers

at {e i~ }jeJ~ C K1, such that A~ n Aj = 0 i f / r j and

I X Ix,,~ \ ( U aj ) l _< ~1 ,,,I. jEJ1

Let {~j }jeJ, C Co ~ (C) be a partition of unity subordinate to the family {2Aj}j~j,

such that, for each j E J1, supp ~y c 2Aj, ~oj -- 1 on Aj, and 110~3/0~1to~ _< C~/r~, C1 being an absolute constant. We extend f to C by f ( z ) = f(1/2) , for Izl > 1, so

that f is essentially bounded on C with respect to area measure. Define f~ = f - G1

with

jEJ1

where

1 f~ f(z) - f(e~ o~j dx dy

: v ~ + vj.

Here T~ is Vitushkin's localization operator, and Rj is a holomorphic function

off a compact subset of 2Aj \ D satisfying

(A1) Vj - Rj has a triple zero at infinity, and

(A2) IIR~tloo <_ C211vjlloo,

where C2 is a constant. In our situation, the existence o f such functions R~ is given

by Vitushkin's theorem applied to this case ([Ga, Corollary 7.3]).

I f we take a l sufficiently close to 7r, the boundedness of f and the uniform

approximation on T(e i~ , a l , 1 - rj) give IlVjll~ < e/2 for every j . By (A1) and


standard properties of the Vimshkin scheme for approximating analytic functions,

we obtain that

(3.2) II ~ (vj - R,.)lloo < c c jEJI

with C3 an absolute constant (see, e.g., [Ve, Lemma 2.2]).

Note that fl is analytic on D and near Uj~& Zxj, and it follows from our selection

C , of the centers e iej in X~t that l]f11[x <- llfl[x + 3~. With regard to G1, we have

Ilalll~,,~ ~ ~ . ,,.rot If('~) - f (e*OJ)F~a(5)d5 + (63) , . , (2Aj n"]r) .1

C

J 6 < a(4~ + c4)~ = c ~ .

Proceeding by induction, suppose we have chosen a finite collection of discs

{ A j } j e j ~ and constructed a function f,~ satisfying

( B I ) Xnt \( O( U Aj)) < ~nlXngl, t= l jEJ~

(B2) IIAIIx _< IlSllx + ca( s ~)~,1 /=1

(B3) Ilfn - A-*lip,., -< C5~, and

(B4) fn is analytic on D and continuous where f,~-i is continuous and on I.Jje j~ Aa.

To obtain fn+l, we exploit the arguments used to find fl. Indeed, having fixed n + l an angle c~,~+1, consider a compact subset K,~+I ofXn, \ (U~=I ([.J~eJ~ Aj)) in the

same way as we chose K1. Next, take a finite collection of discs {Aj}se&+ 1 with

radii rn+l and centers at e i~ E K~+t such that, for any j E J,~+l, I f ( d ~ - f(z)l < ~ for all z e T ( e i~ ~ + 1 , 1 - r j ) ,

and

,~p(I, ei~ , r ,+ l ) < -~ .

Moreover, Kn+l and r,~+l can be selected so that n + l

1 X x,,, \ ( U ( U A')) i <-- 2--~ 1 ,~,l. 1=1 jEJI


Then we define f,~+l from f,~ in the same way we obtained f l from f . Finally, from (B3) we have that {fn},~__a is a Cauchy sequence in HP(w) whose

limit f satisfies

Ill - f l l . ,~ -< ~ = ~, n = l

while the inequality

II lx IIfllx + A :

follows from the pointwise convergence in D and (B2).

I f V = Ut~l (Uj~j~ A j), it remains to show that f is continuous on V A ~. For

any z E V N X f3 'IF there exist no E N and a neighborhood E of z such that, for

all n > no, E does not meet any disc of the generation of discs used to define fn;

hence fn is continuous on E. Also, by (3.2), we have E

llf,, - fn+111E _< 2--;

So we can write f = f,~ + ~]~k~=n(fk+l -- fk) and conclude that {fn} converges

uniformly on E to f . Thus, f i s continuous on E and the proof is finished. []

We are now in a position to prove Theorem 3.2.

P r o o f o f T h e o r e m 3.2. We first prove that C~,a(~') + HV(w) c Hv(o.,)lx.

Since Hv(w)lx = HP(w)Iy " (see Remark 3.3), it will be enough to show that

C~,~(~') C Hv(~o)[~,. Let B be the Banach algebra H~(D)Ij? and let S = ~ . I f V is

a bounded component o f C \ S, then, for each point z0 E V, there exists a bounded

analytic function h on D such that 1 = h(zo) > IIhLIx- So 1 - h is invertible in B

and, since we can write 1 - h(z) = (z - zo)g(z) with g �9 H~(D) and g r 0 on X,

we conclude that (1) l:z �9

On the other hand, for any point z0 in the unbounded component of C \ S, it is

well-known that 1/(z - zo) can be uniformly approximated on X by polynomials.

These facts imply that R(S)I2 c B, where R(S) is the uniform closure on S of the

rational functions with poles off S.

I f {Vj}3~x are the components of C \ S, the maximum principle gives that

0Vj. M "Ii" r 0 for all j . In particular, for any z �9 OS and for 6 > 0 small enough, the

interior o f A(z, 6) \ S contains an arc with diameter bigger than 6/2. By [Ga, Cor.

8.4], it follows that R(S) = C,~a(S) and, since trivially (3f) ~ = S ~ we obtain that

R(S)I.~ = C,,,,(S)1~7 = C,,o(X) c B c HP(w)I.~.

We now prove the reverse inclusion Hp(w)lx c C,,,(.~) + HP(w). Given

f �9 Hp(w)lx, we can assume that f is bounded on X. Otherwise, i f f l �9 HP(w)


and Ilf - f i l l s < 1, writing f = ( f - f l ) + f l we note that i f the decomposit ion

holds for ( f - f l ) it also holds for f .

Since any element f E Hp(w)lx can be represented on X as f = ~n~__l f,~,

with {f,~}n~_l c HP(w) and ~--~,,~--1 IIf~llx < ~ , all we have to do is to construct a

h ~ C~,a(-~) such that sequence { n}n=l in HP(w) N

1 Ilfn- h,~llp,~ < ~-~, (3.3)

and

1 (3.4) Ilhnll~ ~ IIf~llx + 2,~

for each n. Indeed, i f this is the case, we can put f = 2nL1 hn + ~'~nC~=l(fn -- hn) ,

where En~x hr~ E Cua(2) by (3.4), and ~,~-_l(fn - hn) E HP(w) by (3.3).

Let us assume that f is a function o f HP(w) bounded on X and fix e > 0. Our

task is now to find a function h in HP(~) M C~a ()() such that

l l f - hllp,~ < ~ and IIh[l~ ~ Ilfllx + ~.

By Lemma 3.5, there exist an open subset V such that [Xnt \ V[ = 0 and a function f E HP(~), continuous on X fq V, close to f in HP(cz), and bounded

on X by IlflIx. Our goal is to find a function in HP(w), continuous on X O qF,

approximating ] ' in HP(~) and whose supremum over )( is less than II]'llx- Since ] ' is continuous on X N V, we may "repair" the function on Xt and on

Xnt \ V. Indeed, i f IXtl > o, we use a construction due to J. Detraz ([De, page

332]) which guarantees the existence o f a compact subset K o f Xt with IKI > 0

and an outer function GK E H ~ (]D) such that

(i) 0 < IGg(z)l ~ i for all z E D,

(ii) G K ( Z ) ----* 0 as z ~ ~ E K, z E X, and

(iii) GK extends continuously to "IF \ K and does not vanish there.

By induction, we can consider an increasing sequence o f compact subsets

{K,~}n~-i of Xt \ V such that [Xt \ (V U K,~)I ---* 0 as n ~ c~. Then, if Gn = GK,~ is

the outer function associated to Kn as above, G = I-I,~__l G,~ extends continuously

to V n ' r and 0 < Ia(z)l _< 1 onD.

If [Xtl = 0 we simply take G = 1. - - O O Let L = (X n 2") \ (V U (U,~=I K,~)). Then ILl -- 0 and, by a general Ru d in -

Carleson-type theorem ([De], [St6]; see also [Ho, page 80]), there exists a function

H E H~(D), with continuous extension to L u (qr \ L), and satisfying H r 0

76 A. B O N I L L A E T AL.

on D and H --- 0 on L. Consider now the sequence {U~},~__I in HP(w) given by

U,~ = G1/'~H1/'f ". It follows from Lemma 3.5 and the properties o f G, f and H

that (Un)ly is uniformly continuous. This means that (U,~)I 2 E C~a()~). Indeed,

let z0 be a point in Xf~ "s such that U~(z) ~ 0 as z ~ z0, z ~ X. Then IU,~(z)l <

on x • A for some disc A centered at zo. I f we choose a polynomial p(z) peaking

at z0, i.e., such that p(zo) = 1, Ip(z)l < x for any z # z0 and IU~pl < ~ on X, we

have that

l imsup If,~(z)l -- lim sup IU,~(z)p(z)l <_ ~, Z ~ Z 0 Z---~Z 0

ze2 ze2

since ]]U~p]] 2 _< HU~p]]y <_ E. Finally, since G~/'~H ~/n ---+ 1 at almost every point of'lr, it follows that {Un}n~176

converges to f i n HP(w). For n large enough, h = U,~ satisfies (3.3) and (3.4) and

the proof is finished. []

4 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 on H'(w)

In this section we characterize the Farrell and Mergelyan sets for HP(w) in

terms of a certain geometric condition that does not depend on w or p. From now

on, polynomial will mean analytic polynomial.

Def in i t ion 4.1. We say that a relatively closed subset X o f D is a Farrell set for HP(w) i f for any function f in HP(w) bounded on X there exists a sequence o f

polynomials {p~}n~=l such that

(F1): p,~ ~ f in HP(w),

(F2): IIP~llx ' IlYllx,

a s n ----~ ~O.

Analogously, we say that X is a Mergelyan set for H p (w) i f for any function f in

HP(w) uniformly continuous on X there exists a sequence {p,~}n~__x o f polynomials

such that (M1): pn , f inHP(w),

(M2): [IPn- fllx ,0, a s n - - - ~ cr

Def in i t ion 4.2. We say that a relatively closed subset X o f D satisfies the

nontangential condition (NTC) i f almost every point o f X n q1" with respect to the

Lebesgue measure on q" can be approached nontangentially from X.

We are now in a position to establish the main resut t in this section.


T h e o r e m 4.3. Let w be a weight in qF, 1 <_ p < co, and X a relatively closed subset olD. The following assertions are equivalent:

(i) X is a Farrell set for HP(w).

(ii) X is a Mergelyan set for HP(a;).

(iii) X satisfies the NTC.

(iv) For anyfimction f E HV(w) bounded on X and for any function 9 uniformly continuous on X, there exists a sequence {pn}n~z 1 of polynomials such that:

(1) Pn , y in HP(co),

<2) Ilpn - 9 [ I x ' I l l - 9 1 I x ,

asia- - - -* o c ,

Proof. First, we show that (i) :=> (iii). Suppose that (i) holds but (iii) fails.

Following an idea in [RS], we can choose a subset E of the tangential botmdary

X~ of X such that w(E) > 0 (because log w E LI(T)) and construct a function

f E Ha(D) satisfying [f(ei~ = 1 for all e ~~ E E and [If[Ix < 1. Let h(e i~ be defined in T by

h(eio)={~(ei~ ) i n E ,

in'ir \ E.

By (i), if {P,~}~=I is a sequence of polynomials satisfying (F1) and (F2) for

such a function f , it follows from (F1) that

// s (4.1) lim p,~(ei~ i~ w(e~~ = If(e~~ 2 w(ei~ = w(E). n - - - + o ~ 7r

On the other hand, (F2) yields

limsupn~ f~-~P~(ei~176176 <- limsuPn~e~ ~ ]P'~(ei~176176

_< lira sup []pn[[x f ]h(ei~176 n - - - ~ OO dE

< w(E),

which contradicts (4.1).

To see (ii) ~ (iii), we again suppose that (ii) holds but (iii) fails. By a construc-

tion due to J. Detraz ([De, page 332]), there exist a subset B o f Xt with [B[ > 0

and a function f in H ~ ( D ) , uniformly continuous on X, such that l i m z ~ f(z) = 0 z E X

for any ~ E B. Without loss of generality, we may assume that f (0) # 0.

78 A . B O N I L L A E T A L .

Let {p,~}n~_l be a sequence o f polynomials satisfying (M1) and (M2) for such

G ~ f G in HP(D) by a function f . Then, i fG(z) is as in (2.3), {pn },~=1 converges to

Propostion 2.1. Now, by Jensen's inequality we have

log Ip,~(0)G(0)l < ~ log Ip,~GIdO

= 2 ~ 1 [ i x . l~176176176

for any n. Next, we estimate each term in the right hand side. Since p,~G --~ f G in

HP(I~), we can conclude that

(4.2) sup [ loglp~CldO <_ sup [ Ip~Cl'dO <_ C < 0(3.

n JT\B n JT

On the other hand, since

G is integrable with IGI > 0 almost everywhere on "I" and pn , 0 uniformly on B

by (M2), we obtain

lim [ log IPnGIdO = - ~ . (4.3) n--* oo JB

Since fB l~ IGI dO < oc, (4.2) and (4.3) lead to

log If(0)G(0)l = lira log Ip,~(0)G(0)l = - ~ , ~--'-* OO

which would imply that f(0)G(0) = 0 and therefore f(0) = 0. This contradiction

shows that the NTC is necessary for X to be a Mergelyan set for HP(w). The implications (iv) ~ (i) and (iv) =, (ii) are immediate if in (iv) we take g _-__ 0

and g -- f i x , respectively. It remains to prove that (iii) =~ (iv).

Fix g E C(X) and fo E Hp(~o) bounded on X. For simplicity, we assume that

IIf011p,~ = I and Ill0 - gllx = ~ < oc. Consider the Banach space B = LP(a~) x C(X) endowed with the norm

II(fl, f2)ll = max{Ill1 IIp,,~, II f2 I1~-}.

Note that any continuous function f on D can be represented as an element o f

B by the pair f = (fhr, f i X ) . It will be convenient to denote by P the subspace o f

B consisting of all analytic polynomials.

For e > 0, let K , be the bounded convex subset of B defined as

K~ = {(fa, f2) E B : tlYltlp,~ < 1 + e , Ill -gllx < ~ + e } .


We make the following

Cla im. Fix any compact subset J o f D; fo can be uniformly approximated on

J by elements o f P N K~.

Take the claim for granted and let (Jn)n~_--I be an increasing sequence o f compact

subsets of D such that D = U,~__I Jn. Then, for each compact Jn and each real

~ = 1/n, n E N, the claim provides a sequence {q~}~-i C P satisfying

[[q~[[p,,o < 1 + l /n ,

[]q~ - 9]]x < 6 + 1/n, and

IIq~ - fo l lJ . < l / k ,

for k = 1, 2, . . . . By a diagonalization process, we obtain a sequence {p,~},~__l =

{q~}n~_~ c P such that, for any compact subset J o f D and any e > 0,

(4.4) 11;41p,~ < IlYollp,~ + ~,

(4.5) [[P,~ - g l lx < lifo - g112 + ~, and

(4.6) []Pn - follJ < e

for n large enough. Trivially, (4.6) leads to

(a) pn ' fo uniformly on compact subsets o f D .

Also, (4.6) with (4.4) gives

(~) Ilpnllv,,, ' ]lf01l,,~;

and again (4.6), but with (4.5), leads to

(7) I l p , ~ - g l l x ' t l f o - g l l x -

Now in (iv), (1) follows from (a), (/3) and Proposition 2.3, while (2) holds

by (7). []

P r o o f o f t h e C l a i m . Let J be a compact subset o f D; then we have to show

that f0[J can be uniformly approximated on J by elements from P n K , . By duality,

this is equivalent to proving that

Re A(fo) = Re f f o dA < 1, (4.7)

for any complex regular measure A supported on J such that Re A(p) < 1 whenever

pE P n K , .


Each of these measures A defines a continuous linear functional on the closure

T B of P on B which satisfies Re A(h) _< 1 for all f E pB n K,/2. Indeed, since G

does not vanish on ID, i f / = minz~j [G(z)[ > 0 and q E P, then

<- 7 [qG[ dia I <_ ][qG]lp _< Cllqllv:, _< CBIqllB,

where C is a constant that depends only on the compact subset J and G. Here we

have used the Poisson formula and the well-known pointwise estimate by the norm

for HP(ID)-functions ([Du, Chap. III, w On the other hand, i f h E pB n K,/2

there exists a sequence of polynomials {p,~}n~176 1 that converges to h in Hr'(a:) and

such that

[Ih-pni]p,~_<s/2 and ] [h-pnl [x <_e/2

for n large enough. Hence,

[IPnl[p,~ < l[ht[p,w "F s/2 ~ 1 -b s,

Ilpn - 91Ix -< Ilpn - h l ]x + I I h - 91Ix -< +

since h E K~/2. Thus, for n large enough, these polynomials belong to K, ,

Moreover, since convergence in H;(w) implies uniform convergence on compact

subsets of D, we obtain that A(pn) ---* A(h). Thus, Re A(h) _< 1.

We now need a Hahn-Banach-type theorem due to A. M. Davie, T. W. Gamelin

and J. B. Garnett.

L e m m a 4.4 (DGG, L e m m a 4.1). Let B be a real Banach space and M a

closed subspace o f B. I f K is a bounded convex subset o f B and r is a con-

tinuous linear functional on M satisfying r < a (a > O) on U~ n M with U, an

s-neighborhood o f K, then there exists an extension r o f r to B such that ~ < a on

K.

By Lemma 4.4, the functional A admits a continuous extension L to B satisfying

(4.8) Re L(h) < 1 for all h E K~,

with 0 < s' < s. The action of L on any element (fl , f2) o f B can be represented as

(4.9) L(fl'f2) : ~T flIx~dOJ- fX :2dl"~' where 11 E Lq(a:) and # is a finite Borel measure supported on X.

It is easy to see that the restriction o f # to X n qP is absolutely continuous

with respect to Lebesgue measure on the unit circle. Indeed, for any compact


a o o subset K o f t with zero Lebesgue measure, let { n}n=l be a sequence of functions

continuous on D and analytic on its interior such that an = 1 on K and an ---' 0 on D \ K ([Ho, page 80]). Since A(an) = L(a,~) for any n, (4.9) and the Lebesgue

dominated convergence theorem lead to

0 = lim A(an)= lim L(a~)= #(K), n -----r o o n - - ~

which proves the desired property for P]wX" It should be noted that this property of #, together with (iii) and (4.9), implies

that L(fo) is well defined, even though fo may be outside B.

To conclude the proof it will be sufficient to establish that

(4.10)

and

(4.11)

: L ( f o )

ReL(f0) _< 1.

Lemma 2.5 gives a sequence {f~}n~176 1 in H~(D) that converges to f0 in HP(w)

and such that, for any n E N, If,~(z)l <_ I/o(z)l for all z E D and for almost every

point in "ir. Thus, ,~(f,~) ~ ~(fo) and, since fo is bounded on X and almost

everywhere on X n ql" (here we again use the hypothesis (iii)), L(f,~) ---* L(fo). Now we want to show that ,~(f,~) = L(fn) (n = 1, 2, ...), which will give (4.10).

Since we are dealing with bounded functions, this is immediate i f we take its

dilations fn(rz), 0 < r < 1 (which trivially belong to T a ) , apply a dominated

convergence argument and recall that )~ - L on T a.

In order to prove (4.11) it will be enough to find a sequence {(f{~,/~')}n~__o c

K~, such that L(f~, f~) , L(fo). As the first component {f~},~--o, we take the

sequence of bounded analytic functions {fi~}~-o given by the Lemma 2.5 and used

in the proof o f (4.10). Since {f~}n~176 0 converges weakly to f0 in LP(a~), it follows

that

fT f~ ll wdO ) IT fo 11 ~dO (4.12)

a s n -----> 0 o .

To construct the sequence {f~ }~:o in C(X) we note again that, by assumption,

the estimate [fo - gl -< 5 holds on X and, by (iii), at almost every point o f X n "ii"

with respect to Lebesgue measure; therefore, it also holds at almost every point

of X with respect to #. By Lusin's theorem, we obtain a sequence {hn}~__0 in

C(X) uniformly bounded on X by [[hnlI-R <_ ~ and converging pointwise to f0 - g

at almost every point o f X with respect to #. This implies that

(4.13) , s

8 2 A. B O N I L L A E T A L .

a s n ~ .

Taking f~ = hn + g, we see that the sequence {(f~, f~)}~=0 belongs to Ke.

Hence, by (4.8),

(4.14) ReL(f~, f~) < 1

for any n E N. Moreover, by (4.12) and (4.13)

(4.15) L(f?, f~) ~ L(fo)

a s n -----~ o o .

Thus, (4.11) follows from (4.14) and (4.15).

A final comment should be made. The extension of the functional A which

Lemma 4.4 provides also exists in the case that the subspace does not meet the

convex set. In our situation, i f P B does not intersect K, , the claim above could also

be proved for any measure # with support on J, which would allow us to show that

Re f j f0 d# is less than any constant. Since this is clearly impossible, we conclude

that P n K~ cannot be empty.

5 Simultaneous approximation by cyclic vectors

A well-known theorem of Beurling states that any outer function G in HV(~))

generates a dense subspace {pG : p is an analytic polynomial } ([Gar, Chapter II,

w Hence it is quite natural to study simultaneous approximation by these

functions. The definitions of Farrell and Mergelyan sets in this case can be

reformulated as follows.

Definit ion 5.1. Let G be an outer function of H~(D), 1 < p < oo, and X a

relatively closed subset of D. We say that X is a Farrell set for HP(D) with respect

to G if for any function f E HP(D) bounded on X there exists a sequence {Pn}~=l

of polynomials such that

(FG1): p,~G , f inHP(D),

(FG2): IIP,~allx ' II/llx, a s h - - - - } o o .

Analogously, we say that X is a Mergelyan set for HP(D) with respect to G if

for any function f E HP(D) uniformly continuous on X there exists a sequence

{Pn },~----1 of polynomials such that

(MG1): p,~G ----+ f in HP(D),

(MG2): IlPna-/llx ,0, a s n - - - - > o o .


It should be noted that these kinds o f approximation require the outer function

to satisfy some rather evident compatibility conditions on X. We illustrate partially

the nature o f such conditions in the example below.

E x a m p l e 5.2. Consider a relatively closed subset X o f D and let E be a

Cantor-type subset o f X N "]r with positive measure. Let G be an outer function o f

HP(D) such that [G[ = �89 on E and Ial = 1 on qF \ E. I f {Pn}n~176 is a sequence o f

polynomials satisfying pnG " ' 1 in HP(D), then, by the pointwise convergence at

almost every point o f T , there exists a compact subset K c E where IP,~I > 2 - e

for n large enough. By continuity, IPn [ > 2 - 2e in a neighbourhood V o f K on

T. Since [G I = 1 on 7f \ E, [p,~G[ > 2 - 3e in a subset o f positive measure o f

V n (T \ E). Therefore, we can choose z E X close enough to V n ('IF \ E) such that

[pn(z)G(z)[ > 2 - 3e for n large. This makes it impossible that IIp,~Gllx - - ~ 1.

The characterizations of Farrell and Mergelyan sets with respect to outer

functions are stated in the following theorems.

T h e o r e m 5.3. Assume that i <_ p < oo and let G bean outer function of liP(D) with IGI uniformly continuous on X. Then a necessary and sufficient condition for X to be a Farrell set for HP(D) with respect to G is that X satisfies the NTC.

P roo f . Replacing polynomials by polynomial multipliers o f G, we see that

the same argument used in the p roof o f the implication (i)=~ (iii) o f Theorem 4.3

shows that the NTC is necessary.

Conversely, fix fo E Hp(]D) with Ilfollx < oo. To begin with, we assume that G

does not vanish on X. For any e > 0, define the Banach space B = Lv('IF) x C(X) endowed with the norm

{ lllfollp IIfollx1 } II(fl, f2)lIB -- max + ~llAIIp, + ~ IlY211x-

Note that the hypothesis o f continuity o f IGI over X guarantees that

Q = {(q[G[, q[G[) : q is an analytic polynomial}

is a subspace o f B. Denote by K , the unit ball of B, i.e.,

K~ = {(f l , f2) : IIflllp <~-- IISollp + ~, IIf211x- <- IIfollx + ~}-

Let us assume for the moment the following

Given any compact subset J of D, fol~_ I can be approximated Claim.

uniformly on J by elements from K, n Q .

84 A. B O N I L L A E T A L .

Let {Jn},~-i be an increasing sequence o f compact subsets o f D such that

D = Un~__t J,~. From the claim it follows that, for each compact J,~, there exists a

polynomial p~ such that

I@11 1 (5.1) I l p n l G I - fo J~ < - ,

n

1 (5.2) IIp,dalllp _< Ilfollp + - ,

n

1 (5.3) IIp,dGIIIR- -< Iifollx + - .

n

Since

IIP, IGI - f o L ~ l l J ~ = IIp~G - follJ~,

we deduce from (5.2), (5.3) and Proposition 2.3 (see also [CMS, Thm. 2]) that

{pn }n=l converges to fo in HP(D). On the other hand, from (5.2) and (5.3) it

follows that [IpnGIIR- ' I l f o l l x as n ---, oo. []

Hence it only remains to prove the claim above.

P r o o f o f t h e C l a i m . By duality, we must show that

}A(fo~-~)] < 1 ,

whenever )~ is any complex Borel measure supported on J satisfying I;~(h)l < i for

any h E K~ n Q.

Since K, is the unit ball of B, it is easy to see that ), defines a continuous linear

functional on P whose norm is bounded by one. Now, A can be extended to a

continuous linear functional L on B with IILII < 1, and L can be represented as

L(sl,s2) = f sloe0 (sl,s ) B,

with I 6 Lq(qr) ( l / p + 1/q = 1) and # a complex Borel measure compactly supported

on X. Moreover, since

II(b,0)il~=/ II(0,g)llB=l

= sup ~ bldO + sup f gdl~ Ilbllp=llYotl~ +~ IIgH~-=ll.follx+e

= sup [(llfollx +e)hldO + sup [(llfollx +~)kdu Ilhllp=l J~ Ilkll~-=l JR-

= (IILII + ~)lllllq + (llfollx + ~)11~11,


where [l#if denotes the total variation of/~, the estimate IIL[t <_ 1 means

(5.4) (llfollp + r + (llfollx + r - 1.

On the other hand, the same argument given in the proof of the implication (iii)

(iv) in Theorem 4.3 can be exploited again to show that the restriction ~l~nT o f

to "r is absolutely continuous with respect to Lebesgue measure on qF. Indeed, it

is enough to repeat the reasoning making use of the sequence { (an l G I, a~ IGI)}~-1, g ~o where { n },~=1 is the sequence of functions in the disk algebra given there.

Note that, even in the case that fol---~ - does not belong to B, L(fol---~ -] is well

defined by the NTC and the properties~of/~. Thus, the proof will ~ ~ / b e complete as

soon as we prove that

(c~) ~(folGG-~l) = L(fol--~-), and

To see that the identity in (c~) holds, we use our results on simultaneous ap-

proximation in the weighted Hardy spaces. In fact, since X satisfies the NTC and

~(e ~~ = IG(e~~ is a weight on "2, X is a Farrell set for HP(w) (Theorem 4.3).

Thus, since fo/G E HP(~o) and it is bounded on X, there exists a sequence {io,~ },~__1

of analytic polynomials satisfying

(1) Pn ~ fo/G in HP(~o), and

(2) IIP-llx ~ Ilfo/IGlllx. Now, from (1) and (2), it is easy to check that

(1') pnla[ , fol-~ - in LP(ql ") and uniformly on compact subsets o f D, and

(2') IlpnGtlx <_ K, K being a constant independent o fn .

By (1') and (2') it follows that

A(pnIGI)---* A(fo,G--~) and \ l(..x[ ,;

L(p,dGI) ~ L(fo~Gi) ash--* oe;

since A(PnlGI) = L(PnlG[) for all n, (a) holds. Finally, by (5.4) we obtain that

L(fo~--~Gi)l < Ilfollplltllq + llfollxll~l,

-< (IILIIp + ~)llltlq + (llfollx + ~)11~11 < 1 ,

8 6 A. B O N I L L A ET AL.

which proves (/7) as desired.

To prove the general case (in which G can vanish on X f~ "IF), we consider the

outer functions defined by

Gn(z) = exp [ ~----~ f ~ H(z' ei~ l~ ]Gn(ei~ '

where [Gn(eie)l = max{IG(eie)1,1/n}, n = 1, 2, . . . . Observe that, for all n,

IGn(z)l _> ta(z)l in D and at almost every point o fT . Note also that IGn(z)[ > 1/n for all z E D and Gn , G pointwise on D and at almost every point of T. Since

_ G c~ IIG~II~ < IICII~ + 2~/nP, { ~}~=x is a sequence of outer functions that converges to G in HP(•).

Furthermore, from the uniform continuity of IGI on X, it is not difficult to check

that each ]Gn I is also uniformly continuous on X.

Now let fo E HP(D) with Ilfollx < ~ and assume that X satisfies the NTC.

Arguing as in the previous case, we see that for each n there exists a polynomial

Pn such that

IlpnGn - flip < 1/n and IIp~a~llx ~ Ilfllx + 1/n.

Since Ip,~(ei~176 < Ipn(eio)G~(ei~ for almost every point in T, Ilp~Gnllp Ilfllp and pnG --, f almost everywhere on T, a dominated convergence argument

yields pnG ~ f in HP(D) (see, e.g., [Fo, page 57]). Finally, since IIp~allx <_ [IpnGnllx <- tlfllx + Un for all n, it follows that limsup IlpnGllx < Ilfllx. The proof is complete. []

T h e o r e m 5.4. Assume that 1 < p < ~ and let G be an outer function of HP(D) satisfying

(i) G is uniformly continuous on X,

(ii) G does not vanish on X.

Then a necessary and sufficient condition for X to be a Mergelyan set for HP(D) with respect to G is that X satisfies the NTC.

Proof . First suppose that X is a Mergelyan set but it does not satisfy the

nontangential condition. In this case, again using the construction due to J. Detraz

in [De, page 335], we can find a subset B of Xt with positive Lebesgue measure and

a bounded analytic function f uniformly continuous on X which satisfies f(0) ~ 0,

and f(z) ~ 0 as z ~ ~ for z E X and ~ E B. Then, i f {p,~}n~__x is a sequence o f


polynomials satisfying (MG1) and (MG2) for such f , by Jensen's inequality we have

log Ip,(0)a(0)[ < f l o g IPn(eiO)G(ei~ dO.

We split the integral on the right hand side into

f loglp,~G'dO=f~ l~ fBl~ \ B

Now

(5.5)

(5.6)

when n ~ ~ . As

L\Blog lPnGIdO <_ (L IPnGIPdO}I/P < oo,

for all n, because {pnG}~=l converges to f in HP(D). However, since log fpnG[ ~ 0 uniformly on B, it follows that fB log IpnG I ~ -oo. Hence f(0) = 0, and the

contradiction is evident.

Assume now that X satisfies the NTC. Consider the weight w(e i~ = [G(ei~ p and let f e HP(D) be uniformly continuous on X. Then f /G e nP(w) and (by

(i) and (ii)) it is also uniformly continuous on X. Since X is a Mergelyan set for

HV(w) (Theorem 4.3), there exists a sequence {Pn}n~176 of polynomials such that

f ) - - in H~'(w), and pn G

P n - - f X ~0

f lpn(eie)G(e i~ - f(eiO)lP de = fT p , ~ ( O )

= f~ Pn(eiO)

pnG that [IpnG - f[lx

f(e o) p v(e'o) IV(ei~ f(eiO) P G(eiO) w(ei~ dO,

, f in HP(D) by (5.5), while the boundedness of G on X and (5.6) show ,0. The proof is complete. []

Acknowledgments

The authors wish to express their sincere appreciation to the referee for some

valuable comments and suggestions leading to the revised version. The research of

the authors in La Laguna has been partially supported by a grant of the DGICYT

no. PB95-0749-A.


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dntonio Bonilla DEPARTAMENTO DE ANALISIS MATEM./~T1CO

UN1VERSIDAD DE LA LAGUNA 38271 LA LAGUNA, TENERIFE, SPAIN

email: abonilla~ull.es

Fernando P~rez-Gonzcilez DEPARTAMENTO DE ANALIS1S MATEM,~TICO

UNIVERSIDAD DE LA LAGUNA 38271 LA LAGUNA, TENERIFE, SPAIN

email: fpergon~ull.es Arne Stray DEPARTMENT OF MATHEMATICS

UNIVERSITY OF BERGEN N-5007 BERGEN, NORWAY

email: stray~mi.uib.no

Rodrigo Trujillo-Gonzfilez DEPARTAMENTO DE AN.~LISIS MATEMATICO

UNIVERSIDAD DE LA LAGUNA 38271 LA LAGUNA, TENERIFE, SPAIN

email: [email protected]

(Received October 7, 1996 and in revised form August 6, 1997)

Documents

Approximation in weighted Hardy spaces