Let $F$ be a relatively closed subset of the unit disc $D$. If $A$ is any of the Hardy spaces $H^p(D)$, $0 &lt; p &lt; \infty$, $\overline{A|_F}$ denotes the functions on $F$ being uniform limits of elements from $H^p(D)$. Let $\tilde F$ consist of all $z\in D$ such that $|f(z)|\le\sup \{|f(z)| z\in F\}$ for any bounded analytic function in $D$. It is proved that $\overline{A|_F}$ consist of all functions $f$ that can be decomposed as $f=u+v$, where $u$ belongs to $H^p(D)$ and $v$ is a uniformly continuous function on the set $\tilde F$, analytic at interior points of $\tilde F$

Stray, Arne

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Publicacions Matema`tiques, Vol 39 (1995), 61–69.MERGELYAN TYPE THEOREMS FOR SOMEFUNCTION SPACESArne StrayAbstractLet F be a relatively closed subset of the unit disc D. If A isany of the Hardy spaces Hp(D), 0 < p < ∞, A|F denotes thefunctions on F being uniform limits of elements from Hp(D). LetF˜ consist of all z ∈ D such that |f(z)| ≤ sup{|f(z)|z ∈ F} forany bounded analytic function in D. It is proved that A|F consistof all functions f that can be decomposed as f = u + v, where ubelongs to Hp(D) and v is a uniformly continuous function on theset F˜ , analytic at interior points of F˜ .Let A be a linear space of analytic functions and F a subset of thecomplex plane C such that each f ∈ A is defined on F . We denote byA|F the functions being uniformly approximable on F by sequences fromA. The aim with this paper is to give a partial solution to problem 8.5no. 2 in [7]. If A is any of the classical Hardy spaces Hp(D), 0 < p <∞,our main result is that A|F coincides (modulo the approximating space)with a well defined algebra of uniformly continuous analytic functionson F .Before giving a precise formulation of the main result, we need somedefinitions.Let Cua(F ) denote the functions on F being analytic in its interiorF 0 and admitting continuous extension to the extended complex planeC∪{∞}. If F is a compact subset of C and P consists of the polynomials,a famous theorem of S. N. Mergelyan [7] can be formulated asP |F = Cua(F˜ )where F˜ is the union of F and the bounded components of C/F .62 A. StraySuppose now we replace P by the set H(C) consisting of all entirefunctions. Also allow F to be a closed but possibly unbounded subset ofC. Then it can be proved that(I): H(C)|F = H(C) + Cua(F˜ )where F˜ is the union of F and certain components of C/F . A componentV is to be included in F˜ if and only if V ∪{∞} is not arcwise connectedin C ∪ {∞}. For details see [8], [9] and [10].In general A may contain unbounded functions. For this reason itis natural to look for an identity like (I) if we seek to describe A|F interms of uniformly continuous analytic functions. Let us use the notation‖g‖B = sup{|g(x)| : z ∈ B} if g is a function defined on the set B. Wealso define the hull of F with respect to A:FˆA = {z : |f(z)| ≤ ‖f‖F , f ∈ A}.We look for spaces A satisfying the following:(∗): A|F = A+ Cua(FˆA).Our main result is the (∗) is valid for the classical Hardy sapces HP (D)in the unit disc D, 0 < p < ∞, when F is any relatively closed subsetof D. Also note that the two introductory examples are special cases of(∗).We refer to [3] or [5] for the basic theory of Hp(D), 0 < p ≤ ∞.In particular H∞(D) denotes the bounded analytic functions in D. IfF ⊂ D is relatively closed, letFˆ = {z ∈ D : |f(z)| ≤ ‖f‖F , f ∈ H∞(D)}.Our main result isTheorem 1. If 0 < p <∞, then Hp(D)|F = Hp(D) + Cua(Fˆ ).Proof of Theorem 1: If f ∈ Hp(D) it is easy to find {fn} ⊂ H∞(D)such that |fn(z)| ≤ |f(z)| and fn(z) → f(z) for z ∈ D. This shows thatFˆA = Fˆ if A = Hp(D), 0 < p <∞.Let us first prove that Hp(D)|F ⊂ Hp(D) + Cua(Fˆ ). If g ∈ Hp(D)|Fis bounded, we may assumeg =∑nfn, fn ∈ Hp(D)Mergelyan type theorems for some function spaces 63in the sense that∑n ‖fn‖F <∞.There are two special classes of sets F where a short proof of thedecomposition of g can be found. It may be instructive to consider thesecases prior to the general proof.Let us firs assume that F is a Farrell set for Hp(D). (See [8] for defini-tion and various properties of these sets). Then we can find polynomialspn, n = 1, 2 such that‖pn‖F ≤ ‖fn‖F + 2−nand‖pn − fn‖Hp ≤ 2−nfor n = 1, 2, . . . . This gives a decompositiong =∑n(fn − pn) +∑npn|Fˆas claimed.In our second example, we assume that F can be written as a Blaschkesequence S = {ζν}, meaning that(1)∑ν1− |ζν | <∞.Then it is well known that the Blashcke productB(z) = Π|ζν |ζνζν − z1− ζ¯νzconverge in D. Using cofinite subproducts Bn of B, we can obtain‖(1−Bn)fn‖Hp(D) < 2−n, n = 1, 2, . . .and again we have a decompositiong =∑n(1−Bn)fn +∑nBnfn|F .The geometric properties of the set F are quite different in the twocases just discussed. To clarify this, let Fnt denote the non tangentialclosure of F on the unit circle T . So z ∈ Fnt if z ∈ T and z is a limitof a sequence {zn} ⊂ F satisfying |z − zn| ≤ C(1 − |zn|), n = 1, 2, . . . ,where C may depend only on z. We also define F t = F ∩ T\Fnt.64 A. StrayIt is well known that F is a Farrell set forHp(D) if and only if the linearmeasure |F t| of F t is zero ([8]). On the other hand, the condition (1) iseasily seen to imply that |Fnt| = 0.We have thus obtained the decomposition of Hp(D)|F in two ratherdifferent situations. The general proof will be divided into parts reflectingthe “geometry” of the cases considered above. We shall argue as in theproof where F was a Farrell set, but the polynomials pn will be replacedby functions from Hp(D) having a uniformly continuous restriction toF .The key part of the proof is an approximation argument related to theset Fnt:Lemma 1. Given f ∈ Hp(D), 0 < p < ∞, and  > 0, there is anopen set V and f1 ∈ Hp(D) with the following properties:(i) ‖f − f1‖Hp(D) < (ii) ‖f1‖F < ‖f‖F + (iii) f1 extends continuously to F ∩ V(iv) |Fnt\V | = 0.For the moment we take Lemma 1 for granted. Consider the “tangen-tial” part F t of F ∩ T . Let K be a compact subset of F t. We assumethere is a number δ = δ(K) such that Iz ∩K = φ if z ∈ F , |z| > 1 − δ,and Iz denotes the arc Iz = {eiθ : |z − eiθ| ≤ 2(| − |z|)}.By a construction due to J. Detraz ([2, Prop. 3.1]), we can find anouter function GK ∈ H∞(D) such that|GK | ≤ 1 and GK(z) → 0 if z → K and z ∈ F . Moreover, GK extendsto be continuous and non zero at any eiθ ∈ T\K.We can find an increasing sequence of such sets Kn ⊂ F t\V withcorresponding outer functions Gn, such that |F t\V \Kn| → 0 and suchthat G=∏nGn has the following properties(i) 0 < |G(z)| ≤ 1, z ∈ D(ii) G extends to be continuous at any eiθ ∈ V ∩ T .It also follows from the construction of {Gn} thatG(z) → 0 if z ∈ F and z → z0 ∈ ∪nKn.Consider finally the setL = (F⋂T )\V \⋃nKn.Mergelyan type theorems for some function spaces 65It is evident that the linear meausre |L| of L is zero. By a generalversion of the Rudin-Carleson theorem ([2]) there is H ∈ H∞(D) withcontinuous extension to L ∪ (T\L) such that H = 0 in D and H = 0 onL.For n = 1, 2, . . . we consider the functions Un in Hp(D) given byUn = G1nH1n f1where f1 satisfies the conclusions of Lemma 1.It follows from the construction of f1, G and H, that Un|F is uni-formly continuous. This implies that Un|Fˆ ∈ Cua(Fˆ ), by the maximumprinciple. To be a little bit more specific, suppose z0 ∈ F ∩ T and thatUn(z) → 0 as z → z0 and z ∈ F . Then |Un| <  in F ∩ ∆(z0), forsome disc centered at z0. Choose a polynomial p peaking at z0 such that|Unp| <  on F . Then if p(z0) = 1, we havelim supz→z0z∈Fˆ|Un(z)| = lim supz→z0z∈Fˆ|Un(z)p(z)| ≤ since ‖Unp‖Fˆ ≤ ‖Unp‖F ≤ .We turn to the proof of Theorem 1. If f ∈ Hp(D) and  > 0 isgiven, we have shown (modulo proving Lemma 1) that there is a functionU = Un ∈ Hp(D) ∩ Cua(Fˆ ) with n so large that‖f − U‖Hp(D) < ‖U‖ ≤ ‖f‖F + .The proof of Theorem 1 now follows the introductory argument wegave in the special case where F is a Farrell set.Let us finally prove Lemma 1. We may assume that f is bounded inD.So given f ∈ H∞(D), and  > 0, we consider a compact set K ⊂ Fnt.We shall require several properties of K related to f . If 0 < α < π,T (θ, α) denotes the cone in D with opening angle α, terminating at eiθ,and being symmetric with respect to the radius {reiθ, 0 ≤ r < 1}. Weassume thatf(eiθ) = lim f(z)holds uniformly in eiθ ∈ K as z → eiθ inside T (θ, α). Now fix p ∈ (0,∞).Since f ∈ Hp(D), the radial limits f(eiθ), 0 ≤ θ < 2π, belong to Lp(dθ).We assume that K is included in the Lebesgue set for f and that(3)12r∫ θ+rθ−r|f(eiϕ)− f(eiθ)|p dϕ→ 066 A. Strayuniformly in eiθ ∈ K as r → 0. Such a set K can be found with |Fnt\K|as small we please.Fix δ > 0 so small that |f(eiθ) − f(z)| <  if eiθ ∈ K, |z| > 1 − δand z ∈ T (θ, α). We are now in a convenient position for applyingVitushkin’s scheme for approximation (see [13] or [4]). Let {∆j}Nj=1 bea finite collection of open discs with centers zj ∈ K and a common radiusr < δ. Following Vitushkin’s scheme, let ϕj ∈ C10 (∆j) be chosen suchthat 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 in ∆1j ={z : |z − zj | < r2}. As a preliminaryapproximation to f we definefK = f −GKwhere GK =∑jTϕj(f−f(zj))−rj is a finite sum which we shall explainin some detail.We assume f is defined outside of D by f(z−1) = f(z). For generalproperties of the Tϕ-operator we refer to [11] or [4, page 30]. Here weonly note thatTϕj(f − f(zj))(ς) = ϕj(ς)(f(ς)− f(zj))− 1π∫∆j∫f(z)− f(zj)z − ς∂ϕj∂zdx dy(z)= Uj + Vj say.We assume∣∣∣∂ϕj∂z∣∣∣ ≤ Ar , where A is a numerical constant. Since f ∈Hp(D), we have in particular that f ∈ Lp(dx dy) locally. Therefore theconvolution term Vj is continuous as a function of ς. If α is close to π,Ho¨lders inequality gives that|Vj(ς)| ≤ , ς ∈ C j = 1, 2 . . . N.Note also that Vj is analytic outside ∆j . According to Vitushkin’sscheme, the functions rj should be analytic outside a compact subsetof ∆j\D and with the property that (Vj − rj)(ς) has a zero of order 3 at∞. In addition we should require(4): ‖rj‖∞ ≤ A1‖Vj‖∞ ≤ A1, j = 1, 2, . . . , Nwhere A1 is a numerical constant. In our simple situation, the existenceof {rj} is rather evident ([4, page 210–214]). From the individual bounds(3), it is part of Vitushkin’s scheme that(5):∥∥∥∥∥∥N∑j=1(Vj − rj)∥∥∥∥∥∥∞≤ A2Mergelyan type theorems for some function spaces 67for some numerical constant A2. We have not claimed that {∆1j}Nj=1cover all of K. In fact we shall assume that ∆j ∩ ∆k = φ if j = k. Inaddition we assume that∣∣∣∣∣K ∩N⋃1∆1j∣∣∣∣∣ ≥ A3|K|for some numerical constant A3, where ∆1j ={z : |z − zj | ≤ r2}. Weremark that fK = f −GK is analytic near ∆1j ∩ T for 1 ≤ j ≤ N . Thisis seen by writingfk = (f − Uj)− Vj −∑i =j(Ui + Vi) +n∑i=1riand inspecting these four terms separately.Note that the (Fatou) boundary values f(zj) satisfy |f(zj)| ≤ ‖f‖F .SincefK = f(1−∑ϕj)+∑jϕjf(zj) +∑j(Vj − rj)we have(6) ‖fK‖F ≤ ‖f‖F +A2.From (3) and (5) we also get‖f − fK‖Hp(D) = ‖GK‖Hp(D) ≤ (1 +A2)if r is sufficiently small.The function fK satisfies the conditions for f1 in Lemma 1 exceptthat fK is only analytic (and hence continuous) near a subset PK of K.But since |PK | ≥ A3|K|, Lemma 1 follows by repeating our constructioncountably many times. The main reason why repetition works, is thatthe Tϕ-operator preserves continuity and analyticity ([4, page 30]).It remains to show that Cua(Fˆ ) ⊂ Hp(D)|Fˆ . Let B denote the Banachalgebra H∞(D)|Fˆ . Also put X = (Fˆ ). If V is a component of C\X, theremus exist h ∈ H∞(D) such that1 = h(z0) > ‖h‖Ffor some z0 ∈ V . But then 1− h is invertible in B and since1− h = (z − z0)g, g ∈ H∞(D)68 A. Straywe conclude that (z−z0)−1|Fˆ ∈ B. This means that R(X)|Fˆ ⊂ B, whereR(X) is the uniform closure on X by the rational functions with polesoff X.But if {Vj} are the components os C\X, the maximum principle gives∂Vj∩T = φ, j = 1, 2, . . . and hence ∂X = ∪∞1 ∂Vj . For such sets X (withempty “inner boundary”) Vitushkin has proved that R(X) = Cua(X)([4, page 219]), and hence Theorem 1 is proved.This solves completely problem 8.5 no. 2 in [7] for the space Hp(D),0 < p <∞. For p = ∞ the problem is still open.For p = ∞, some information about H∞|F can be obtained from thework by Carl Sundberg in [12]. If f ∈ BMOA and f |F is bounded,Sundberg shows that f ∈ H∞|F . On the other hand, our proof aboveshows that any f ∈ H∞|F can be written as f = u + v with u ∈ H∞and v ∈ ∩p>0Hp(D). Several questions arises from this. Here we onlymention the following: Let f ∈ BMOA be bounded on a relatively closedset F ⊂ D.Is there g ∈ H∞ such that the restriction (f − g)|F is uniformly con-tinuous on F?References1. A. M. Davie and A. Stray, Interpolation sets for analytic func-tions, Pacific J. Math. 42(1) (1972).2. J. Detraz, Algebres de fonctions analytiques dans le disque, Ann.Sci. Ecole Norm. Sup. 4e serie (1970), 313–352.3. P. L. Duren, “Theory of Hp spaces,” Academic Press, 1970.4. T. W. Gamelin, “Uniform Algebras,” Prentice Hall, EnglewoodCliffs, N. J., 1969.5. J. Garnett, “Bounded Analytic Functions,” Academic Press, 1980.6. S. N. Mergelyan, Uniform approximation to functions of a com-plex variable, Urephi Mat. Nauk. 7(2) (1952), 31–122.7. ?, “Linear and Complex Analysis Problem Book,” Lecture Notes inMathematics 1043, Springer Verlag, 1984.8. F. Perez-Gonzalez and A. Stray, Farrell and Mergelyan setsfor Hp-spaces, 0 < p < 1, Michigan Math. J. 36 (1989), 379–386.9. A. Stray, Decomposition of approximable functions, Ann. of Math.120 (1984), 225–235.10. A. Stray, Approximation by analytic functions which are uni-formly continuous on a subset of their domain of definition, Ameri-can J. Math. 99 (1977), 787–800.Mergelyan type theorems for some function spaces 6911. A. Stray, Characterization of Mergelyan sets, Proc. Amer. Math.Soc. 44 (1974), 347–352.12. C. Sundberg, Truncations of BMO functions, Indiana UniversityMath. I. 33(5) (1984), 749–779.13. A. G. Vitushkin, The analytic capacity of sets and problems inapproximation theory, Russian Math. Surveys 22 (1967), 139–200.Department of MathematicsUniversity of BergenAlle´gt 555007 BergenNORWAYPrimera versio´ rebuda el 2 de Setembre de 1993,darrera versio´ rebuda el 9 de Febrer de 1995

Mergelyan type theorems for some function spaces

A. Stray

Crossref

Publicacions Matemàtiques

Stray, A.

Revistes Catalanes amb Accés Obert

Publicacions Matema`tiques, Vol 39 (1995), 61–69.
MERGELYAN TYPE THEOREMS FOR SOME
FUNCTION SPACES
Arne Stray
Abstract
Let F be a relatively closed subset of the unit disc D. If A is
any of the Hardy spaces Hp(D), 0 < p < ∞, A|F denotes the
functions on F being uniform limits of elements from Hp(D). Let
F˜ consist of all z ∈ D such that |f(z)| ≤ sup{|f(z)|z ∈ F} for
any bounded analytic function in D. It is proved that A|F consist
of all functions f that can be decomposed as f = u + v, where u
belongs to Hp(D) and v is a uniformly continuous function on the
set F˜ , analytic at interior points of F˜ .
Let A be a linear space of analytic functions and F a subset of the
complex plane C such that each f ∈ A is deﬁned on F . We denote by
A|F the functions being uniformly approximable on F by sequences from
A. The aim with this paper is to give a partial solution to problem 8.5
no. 2 in [7]. If A is any of the classical Hardy spaces Hp(D), 0 < p <∞,
our main result is that A|F coincides (modulo the approximating space)
with a well deﬁned algebra of uniformly continuous analytic functions
on F .
Before giving a precise formulation of the main result, we need some
deﬁnitions.
Let Cua(F ) denote the functions on F being analytic in its interior
F 0 and admitting continuous extension to the extended complex plane
C∪{∞}. If F is a compact subset of C and P consists of the polynomials,
a famous theorem of S. N. Mergelyan [7] can be formulated as
P |F = Cua(F˜ )
where F˜ is the union of F and the bounded components of C/F .
62 A. Stray
Suppose now we replace P by the set H(C) consisting of all entire
functions. Also allow F to be a closed but possibly unbounded subset of
C. Then it can be proved that
(I): H(C)|F = H(C) + Cua(F˜ )
where F˜ is the union of F and certain components of C/F . A component
V is to be included in F˜ if and only if V ∪{∞} is not arcwise connected
in C ∪ {∞}. For details see [8], [9] and [10].
In general A may contain unbounded functions. For this reason it
is natural to look for an identity like (I) if we seek to describe A|F in
terms of uniformly continuous analytic functions. Let us use the notation
‖g‖B = sup{|g(x)| : z ∈ B} if g is a function deﬁned on the set B. We
also deﬁne the hull of F with respect to A:
FˆA = {z : |f(z)| ≤ ‖f‖F , f ∈ A}.
We look for spaces A satisfying the following:
(∗): A|F = A+ Cua(FˆA).
Our main result is the (∗) is valid for the classical Hardy sapces HP (D)
in the unit disc D, 0 < p < ∞, when F is any relatively closed subset
of D. Also note that the two introductory examples are special cases of
(∗).
We refer to [3] or [5] for the basic theory of Hp(D), 0 < p ≤ ∞.
In particular H∞(D) denotes the bounded analytic functions in D. If
F ⊂ D is relatively closed, let
Fˆ = {z ∈ D : |f(z)| ≤ ‖f‖F , f ∈ H∞(D)}.
Our main result is
Theorem 1. If 0 < p <∞, then Hp(D)|F = Hp(D) + Cua(Fˆ ).
Proof of Theorem 1: If f ∈ Hp(D) it is easy to ﬁnd {fn} ⊂ H∞(D)
such that |fn(z)| ≤ |f(z)| and fn(z) → f(z) for z ∈ D. This shows that
FˆA = Fˆ if A = Hp(D), 0 < p <∞.
Let us ﬁrst prove that Hp(D)|F ⊂ Hp(D) + Cua(Fˆ ). If g ∈ Hp(D)|F
is bounded, we may assume
g =
∑
n
fn, fn ∈ Hp(D)
Mergelyan type theorems for some function spaces 63
in the sense that
∑
n ‖fn‖F <∞.
There are two special classes of sets F where a short proof of the
decomposition of g can be found. It may be instructive to consider these
cases prior to the general proof.
Let us ﬁrs assume that F is a Farrell set for Hp(D). (See [8] for deﬁni-
tion and various properties of these sets). Then we can ﬁnd polynomials
pn, n = 1, 2 such that
‖pn‖F ≤ ‖fn‖F + 2−n
and
‖pn − fn‖Hp ≤ 2−n
for n = 1, 2, . . . . This gives a decomposition
g =
∑
n
(fn − pn) +
∑
n
pn|Fˆ
as claimed.
In our second example, we assume that F can be written as a Blaschke
sequence S = {ζν}, meaning that
(1)
∑
ν
1− |ζν | <∞.
Then it is well known that the Blashcke product
B(z) = Π
|ζν |
ζν
ζν − z
1− ζ¯νz
converge in D. Using coﬁnite subproducts Bn of B, we can obtain
‖(1−Bn)fn‖Hp(D) < 2−n, n = 1, 2, . . .
and again we have a decomposition
g =
∑
n
(1−Bn)fn +
∑
n
Bnfn|F .
The geometric properties of the set F are quite diﬀerent in the two
cases just discussed. To clarify this, let Fnt denote the non tangential
closure of F on the unit circle T . So z ∈ Fnt if z ∈ T and z is a limit
of a sequence {zn} ⊂ F satisfying |z − zn| ≤ C(1 − |zn|), n = 1, 2, . . . ,
where C may depend only on z. We also deﬁne F t = F ∩ T\Fnt.
64 A. Stray
It is well known that F is a Farrell set forHp(D) if and only if the linear
measure |F t| of F t is zero ([8]). On the other hand, the condition (1) is
easily seen to imply that |Fnt| = 0.
We have thus obtained the decomposition of Hp(D)|F in two rather
diﬀerent situations. The general proof will be divided into parts reﬂecting
the “geometry” of the cases considered above. We shall argue as in the
proof where F was a Farrell set, but the polynomials pn will be replaced
by functions from Hp(D) having a uniformly continuous restriction to
F .
The key part of the proof is an approximation argument related to the
set Fnt:
Lemma 1. Given f ∈ Hp(D), 0 < p < ∞, and  > 0, there is an
open set V and f1 ∈ Hp(D) with the following properties:
(i) ‖f − f1‖Hp(D) < 
(ii) ‖f1‖F < ‖f‖F + 
(iii) f1 extends continuously to F ∩ V
(iv) |Fnt\V | = 0.
For the moment we take Lemma 1 for granted. Consider the “tangen-
tial” part F t of F ∩ T . Let K be a compact subset of F t. We assume
there is a number δ = δ(K) such that Iz ∩K = φ if z ∈ F , |z| > 1 − δ,
and Iz denotes the arc Iz = {eiθ : |z − eiθ| ≤ 2(| − |z|)}.
By a construction due to J. Detraz ([2, Prop. 3.1]), we can ﬁnd an
outer function GK ∈ H∞(D) such that
|GK | ≤ 1 and GK(z) → 0 if z → K and z ∈ F . Moreover, GK extends
to be continuous and non zero at any eiθ ∈ T\K.
We can ﬁnd an increasing sequence of such sets Kn ⊂ F t\V with
corresponding outer functions Gn, such that |F t\V \Kn| → 0 and such
that G=
∏
n
Gn has the following properties
(i) 0 < |G(z)| ≤ 1, z ∈ D
(ii) G extends to be continuous at any eiθ ∈ V ∩ T .
It also follows from the construction of {Gn} that
G(z) → 0 if z ∈ F and z → z0 ∈ ∪nKn.
Consider ﬁnally the set
L = (F
⋂
T )\V \
⋃
n
Kn.
Mergelyan type theorems for some function spaces 65
It is evident that the linear meausre |L| of L is zero. By a general
version of the Rudin-Carleson theorem ([2]) there is H ∈ H∞(D) with
continuous extension to L ∪ (T\L) such that H = 0 in D and H = 0 on
L.
For n = 1, 2, . . . we consider the functions Un in Hp(D) given by
Un = G
1
nH
1
n f1
where f1 satisﬁes the conclusions of Lemma 1.
It follows from the construction of f1, G and H, that Un|F is uni-
formly continuous. This implies that Un|Fˆ ∈ Cua(Fˆ ), by the maximum
principle. To be a little bit more speciﬁc, suppose z0 ∈ F ∩ T and that
Un(z) → 0 as z → z0 and z ∈ F . Then |Un| <  in F ∩ ∆(z0), for
some disc centered at z0. Choose a polynomial p peaking at z0 such that
|Unp| <  on F . Then if p(z0) = 1, we have
lim sup
z→z0
z∈Fˆ
|Un(z)| = lim sup
z→z0
z∈Fˆ
|Un(z)p(z)| ≤ 
since ‖Unp‖Fˆ ≤ ‖Unp‖F ≤ .
We turn to the proof of Theorem 1. If f ∈ Hp(D) and  > 0 is
given, we have shown (modulo proving Lemma 1) that there is a function
U = Un ∈ Hp(D) ∩ Cua(Fˆ ) with n so large that
‖f − U‖Hp(D) < 
‖U‖ ≤ ‖f‖F + .
The proof of Theorem 1 now follows the introductory argument we
gave in the special case where F is a Farrell set.
Let us ﬁnally prove Lemma 1. We may assume that f is bounded in
D.
So given f ∈ H∞(D), and  > 0, we consider a compact set K ⊂ Fnt.
We shall require several properties of K related to f . If 0 < α < π,
T (θ, α) denotes the cone in D with opening angle α, terminating at eiθ,
and being symmetric with respect to the radius {reiθ, 0 ≤ r < 1}. We
assume that
f(eiθ) = lim f(z)
holds uniformly in eiθ ∈ K as z → eiθ inside T (θ, α). Now ﬁx p ∈ (0,∞).
Since f ∈ Hp(D), the radial limits f(eiθ), 0 ≤ θ < 2π, belong to Lp(dθ).
We assume that K is included in the Lebesgue set for f and that
(3)
1
2r
∫ θ+r
θ−r
|f(eiϕ)− f(eiθ)|p dϕ→ 0
66 A. Stray
uniformly in eiθ ∈ K as r → 0. Such a set K can be found with |Fnt\K|
as small we please.
Fix δ > 0 so small that |f(eiθ) − f(z)| <  if eiθ ∈ K, |z| > 1 − δ
and z ∈ T (θ, α). We are now in a convenient position for applying
Vitushkin’s scheme for approximation (see [13] or [4]). Let {∆j}Nj=1 be
a ﬁnite collection of open discs with centers zj ∈ K and a common radius
r < δ. Following Vitushkin’s scheme, let ϕj ∈ C10 (∆j) be chosen such
that 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 in ∆1j =
{
z : |z − zj | < r2
}
. As a preliminary
approximation to f we deﬁne
fK = f −GK
where GK =
∑
j
Tϕj(f−f(zj))−rj is a ﬁnite sum which we shall explain
in some detail.
We assume f is deﬁned outside of D by f(z−1) = f(z). For general
properties of the Tϕ-operator we refer to [11] or [4, page 30]. Here we
only note that
Tϕj(f − f(zj))(ς) = ϕj(ς)(f(ς)− f(zj))
− 1
π
∫
∆j
∫
f(z)− f(zj)
z − ς
∂ϕj
∂z
dx dy(z)
= Uj + Vj say.
We assume
∣∣∣∂ϕj∂z
∣∣∣ ≤ Ar , where A is a numerical constant. Since f ∈
Hp(D), we have in particular that f ∈ Lp(dx dy) locally. Therefore the
convolution term Vj is continuous as a function of ς. If α is close to π,
Ho¨lders inequality gives that
|Vj(ς)| ≤ , ς ∈ C j = 1, 2 . . . N.
Note also that Vj is analytic outside ∆j . According to Vitushkin’s
scheme, the functions rj should be analytic outside a compact subset
of ∆j\D and with the property that (Vj − rj)(ς) has a zero of order 3 at
∞. In addition we should require
(4): ‖rj‖∞ ≤ A1‖Vj‖∞ ≤ A1, j = 1, 2, . . . , N
where A1 is a numerical constant. In our simple situation, the existence
of {rj} is rather evident ([4, page 210–214]). From the individual bounds
(3), it is part of Vitushkin’s scheme that
(5):
∥∥∥∥∥∥
N∑
j=1
(Vj − rj)
∥∥∥∥∥∥
∞
≤ A2
Mergelyan type theorems for some function spaces 67
for some numerical constant A2. We have not claimed that {∆1j}Nj=1
cover all of K. In fact we shall assume that ∆j ∩ ∆k = φ if j = k. In
addition we assume that
∣∣∣∣∣K ∩
N⋃
1
∆1j
∣∣∣∣∣ ≥ A3|K|
for some numerical constant A3, where ∆1j =
{
z : |z − zj | ≤ r2
}
. We
remark that fK = f −GK is analytic near ∆1j ∩ T for 1 ≤ j ≤ N . This
is seen by writing
fk = (f − Uj)− Vj −
∑
i =j
(Ui + Vi) +
n∑
i=1
ri
and inspecting these four terms separately.
Note that the (Fatou) boundary values f(zj) satisfy |f(zj)| ≤ ‖f‖F .
Since
fK = f
(
1−
∑
ϕj
)
+
∑
j
ϕjf(zj) +
∑
j
(Vj − rj)
we have
(6) ‖fK‖F ≤ ‖f‖F +A2.
From (3) and (5) we also get
‖f − fK‖Hp(D) = ‖GK‖Hp(D) ≤ (1 +A2)
if r is suﬃciently small.
The function fK satisﬁes the conditions for f1 in Lemma 1 except
that fK is only analytic (and hence continuous) near a subset PK of K.
But since |PK | ≥ A3|K|, Lemma 1 follows by repeating our construction
countably many times. The main reason why repetition works, is that
the Tϕ-operator preserves continuity and analyticity ([4, page 30]).
It remains to show that Cua(Fˆ ) ⊂ Hp(D)|Fˆ . Let B denote the Banach
algebra H∞(D)|Fˆ . Also put X = (Fˆ ). If V is a component of C\X, there
mus exist h ∈ H∞(D) such that
1 = h(z0) > ‖h‖F
for some z0 ∈ V . But then 1− h is invertible in B and since
1− h = (z − z0)g, g ∈ H∞(D)
68 A. Stray
we conclude that (z−z0)−1|Fˆ ∈ B. This means that R(X)|Fˆ ⊂ B, where
R(X) is the uniform closure on X by the rational functions with poles
oﬀ X.
But if {Vj} are the components os C\X, the maximum principle gives
∂Vj∩T = φ, j = 1, 2, . . . and hence ∂X = ∪∞1 ∂Vj . For such sets X (with
empty “inner boundary”) Vitushkin has proved that R(X) = Cua(X)
([4, page 219]), and hence Theorem 1 is proved.
This solves completely problem 8.5 no. 2 in [7] for the space Hp(D),
0 < p <∞. For p = ∞ the problem is still open.
For p = ∞, some information about H∞|F can be obtained from the
work by Carl Sundberg in [12]. If f ∈ BMOA and f |F is bounded,
Sundberg shows that f ∈ H∞|F . On the other hand, our proof above
shows that any f ∈ H∞|F can be written as f = u + v with u ∈ H∞
and v ∈ ∩p>0Hp(D). Several questions arises from this. Here we only
mention the following: Let f ∈ BMOA be bounded on a relatively closed
set F ⊂ D.
Is there g ∈ H∞ such that the restriction (f − g)|F is uniformly con-
tinuous on F?
References
1. A. M. Davie and A. Stray, Interpolation sets for analytic func-
tions, Paciﬁc J. Math. 42(1) (1972).
2. J. Detraz, Algebres de fonctions analytiques dans le disque, Ann.
Sci. Ecole Norm. Sup. 4e serie (1970), 313–352.
3. P. L. Duren, “Theory of Hp spaces,” Academic Press, 1970.
4. T. W. Gamelin, “Uniform Algebras,” Prentice Hall, Englewood
Cliﬀs, N. J., 1969.
5. J. Garnett, “Bounded Analytic Functions,” Academic Press, 1980.
6. S. N. Mergelyan, Uniform approximation to functions of a com-
plex variable, Urephi Mat. Nauk. 7(2) (1952), 31–122.
7. ?, “Linear and Complex Analysis Problem Book,” Lecture Notes in
Mathematics 1043, Springer Verlag, 1984.
8. F. Perez-Gonzalez and A. Stray, Farrell and Mergelyan sets
for Hp-spaces, 0 < p < 1, Michigan Math. J. 36 (1989), 379–386.
9. A. Stray, Decomposition of approximable functions, Ann. of Math.
120 (1984), 225–235.
10. A. Stray, Approximation by analytic functions which are uni-
formly continuous on a subset of their domain of deﬁnition, Ameri-
can J. Math. 99 (1977), 787–800.
Mergelyan type theorems for some function spaces 69
11. A. Stray, Characterization of Mergelyan sets, Proc. Amer. Math.
Soc. 44 (1974), 347–352.
12. C. Sundberg, Truncations of BMO functions, Indiana University
Math. I. 33(5) (1984), 749–779.
13. A. G. Vitushkin, The analytic capacity of sets and problems in
approximation theory, Russian Math. Surveys 22 (1967), 139–200.
Department of Mathematics
University of Bergen
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Primera versio´ rebuda el 2 de Setembre de 1993,
darrera versio´ rebuda el 9 de Febrer de 1995


Mergelyan type theorems for some function spaces

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