Publicacions Ma em`a iques, Vol 39 (1995), 61–69.
MERGELYAN TYPE THEOREMS FOR SOME
FUNCTION SPACES
A ne S ay
Abs ac
Le Fbe a ela i ely closed subse o he uni disc D.I Ais
any o he Ha dy spaces Hp(D), 0 <p<∞,A|Fdeno es he
unc ions on Fbeing uni o m limi s o elemen s om Hp(D). Le
˜
Fconsis o all z∈Dsuch ha | (z)|≤sup{| (z)|z∈F} o
any bounded analy ic unc ion in D. I is p o ed ha A|Fconsis
o all unc ions ha can be decomposed as =u+ , whe e u
belongs o Hp(D) and is a uni o mly con inuous unc ion on he
se ˜
F, analy ic a in e io poin s o ˜
F.
Le Abe a linea space o analy ic unc ions and Fa subse o he
complex plane Csuch ha each ∈Ais defined on F. We deno e by
A|F he unc ions being uni o mly app oximable on Fby sequences om
A. The aim wi h his pape is o gi e a pa ial solu ion o p oblem 8.5
no. 2 in [7]. I Ais any o he classical Ha dy spaces Hp(D), 0 <p<∞,
ou main esul is ha A|Fcoincides (modulo he app oxima ing space)
wi h a well defined algeb a o uni o mly con inuous analy ic unc ions
on F.
Be o e gi ing a p ecise o mula ion o he main esul , we need some
defini ions.
Le Cua(F) deno e he unc ions on Fbeing analy ic in i s in e io
F0and admi ing con inuous ex ension o he ex ended complex plane
C∪{∞}.I Fis a compac subse o Cand Pconsis s o he polynomials,
a amous heo em o S. N. Me gelyan [7] can be o mula ed as
P|F=Cua(˜
F)
whe e ˜
Fis he union o Fand he bounded componen s o C/F.
62 A. S ay
Suppose now we eplace Pby he se H(C) consis ing o all en i e
unc ions. Also allow F o be a closed bu possibly unbounded subse o
C. Then i can be p o ed ha
(I): H(C)|F=H(C)+Cua(˜
F)
whe e ˜
Fis he union o Fand ce ain componen s o C/F. A componen
Vis o be included in ˜
Fi and only i V∪ {∞} is no a cwise connec ed
in C∪ {∞}. Fo de ails see [8], [9] and [10].
In gene al Amay con ain unbounded unc ions. Fo his eason i
is na u al o look o an iden i y like (I) i we seek o desc ibe A|Fin
e ms o uni o mly con inuous analy ic unc ions. Le us use he no a ion
gB= sup{|g(x)|:z∈B}i gis a unc ion defined on he se B.We
also define he hull o Fwi h espec o A:
ˆ
FA={z:| (z)|≤ F, ∈A}.
We look o spaces Asa is ying he ollowing:
(∗): A|F=A+Cua(ˆ
FA).
Ou main esul is he (∗) is alid o he classical Ha dy sapces HP(D)
in he uni disc D,0<p<∞, when Fis any ela i ely closed subse
o D. Also no e ha he wo in oduc o y examples a e special cases o
(∗).
We e e o [3]o [5] o he basic heo y o Hp(D), 0 <p≤∞.
In pa icula H∞(D) deno es he bounded analy ic unc ions in D.I
F⊂Dis ela i ely closed, le
ˆ
F={z∈D:| (z)|≤ F, ∈H∞(D)}.
Ou main esul is
Theo em 1. I 0<p<∞, hen Hp(D)|F=Hp(D)+Cua(ˆ
F).
P oo o Theo em 1: I ∈Hp(D) i is easy o find { n}⊂H∞(D)
such ha | n(z)|≤| (z)|and n(z)→ (z) o z∈D. This shows ha
ˆ
FA=ˆ
Fi A=Hp(D), 0 <p<∞.
Le us fi s p o e ha Hp(D)|F⊂Hp(D)+Cua(ˆ
F). I g∈Hp(D)|F
is bounded, we may assume
g=
n
n,
n∈Hp(D)
Me gelyan ype heo ems o some unc ion spaces 63
in he sense ha n nF<∞.
The e a e wo special classes o se s Fwhe e a sho p oo o he
decomposi ion o gcan be ound. I may be ins uc i e o conside hese
cases p io o he gene al p oo .
Le us fi s assume ha Fis a Fa ell se o Hp(D). (See [8] o defini-
ion and a ious p ope ies o hese se s). Then we can find polynomials
pn,n=1,2 such ha
pnF≤ nF+2
−n
and
pn− nHp≤2−n
o n=1,2,... . This gi es a decomposi ion
g=
n
( n−pn)+
n
pn|ˆ
F
as claimed.
In ou second example, we assume ha Fcan be w i en as a Blaschke
sequence S={ζν}, meaning ha
(1) 
ν
1−|ζν|<∞.
Then i is well known ha he Blashcke p oduc
B(z)=Π
|ζν|
ζν
ζν−z
1−¯
ζνz
con e ge in D. Using cofini e subp oduc s Bno B, we can ob ain
(1 −Bn) nHp(D)<2−n,n=1,2,...
and again we ha e a decomposi ion
g=
n
(1 −Bn) n+
n
Bn n|F.
The geome ic p ope ies o he se Fa e qui e diffe en in he wo
cases jus discussed. To cla i y his, le Fn deno e he non angen ial
closu e o Fon he uni ci cle T.Soz∈Fn i z∈Tand zis a limi
o a sequence {zn}⊂Fsa is ying |z−zn|≤C(1 −|zn|), n=1,2,...,
whe e Cmay depend only on z. We also define F =F∩T Fn .
64 A. S ay
I is well known ha Fis a Fa ell se o Hp(D) i and only i he linea
measu e |F |o F is ze o ([8]). On he o he hand, he condi ion (1) is
easily seen o imply ha |Fn |=0.
We ha e hus ob ained he decomposi ion o Hp(D)|Fin wo a he
diffe en si ua ions. The gene al p oo will be di ided in o pa s eflec ing
he “geome y” o he cases conside ed abo e. We shall a gue as in he
p oo whe e Fwas a Fa ell se , bu he polynomials pnwill be eplaced
by unc ions om Hp(D) ha ing a uni o mly con inuous es ic ion o
F.
The key pa o he p oo is an app oxima ion a gumen ela ed o he
se Fn :
Lemma 1. Gi en ∈Hp(D),0<p<∞, and >0, he e is an
open se Vand 1∈Hp(D)wi h he ollowing p ope ies:
(i)  − 1Hp(D)<
(ii)  1F< F+
(iii) 1ex ends con inuously o F∩V
(i ) |Fn V|=0.
Fo he momen we ake Lemma 1 o g an ed. Conside he “ angen-
ial” pa F o F∩T. Le Kbe a compac subse o F . We assume
he e is a numbe δ=δ(K) such ha Iz∩K=φi z∈F,|z|>1−δ,
and Izdeno es he a c Iz={eiθ :|z−eiθ|≤2(|−|z|)}.
By a cons uc ion due o J. De az ([2, P op. 3.1]), we can find an
ou e unc ion GK∈H∞(D) such ha
|GK|≤1 and GK(z)→0i z→Kand z∈F. Mo eo e , GKex ends
o be con inuous and non ze o a any eiθ ∈T K.
We can find an inc easing sequence o such se s Kn⊂F Vwi h
co esponding ou e unc ions Gn, such ha |F V Kn|→0 and such
ha G=
n
Gnhas he ollowing p ope ies
(i) 0 <|G(z)|≤1, z∈D
(ii) Gex ends o be con inuous a any eiθ ∈V∩T.
I also ollows om he cons uc ion o {Gn} ha
G(z)→0i z∈Fand z→z0∈∪
nKn.
Conside finally he se
L=(
FT) V 
n
Kn.
Me gelyan ype heo ems o some unc ion spaces 65
I is e iden ha he linea meaus e |L|o Lis ze o. By a gene al
e sion o he Rudin-Ca leson heo em ([2]) he e is H∈H∞(D) wi h
con inuous ex ension o L∪(T L) such ha H=0inDand H=0on
L.
Fo n=1,2,... we conside he unc ions Unin Hp(D) gi en by
Un=G1
nH1
n 1
whe e 1sa isfies he conclusions o Lemma 1.
I ollows om he cons uc ion o 1,Gand H, ha Un|Fis uni-
o mly con inuous. This implies ha Un|ˆ
F∈Cua(ˆ
F), by he maximum
p inciple. To be a li le bi mo e specific, suppose z0∈F∩Tand ha
Un(z)→0asz→z0and z∈F. Then |Un|<in F∩∆(z0), o
some disc cen e ed a z0. Choose a polynomial ppeaking a z0such ha
|Unp|<on F. Then i p(z0) = 1, we ha e
lim sup
z→z0
z∈ˆ
F
|Un(z)|= lim sup
z→z0
z∈ˆ
F
|Un(z)p(z)|≤
since Unpˆ
F≤UnpF≤.
We u n o he p oo o Theo em 1. I ∈Hp(D) and >0is
gi en, we ha e shown (modulo p o ing Lemma 1) ha he e is a unc ion
U=Un∈Hp(D)∩Cua(ˆ
F) wi h nso la ge ha
 −UHp(D)<
U≤ F+.
The p oo o Theo em 1 now ollows he in oduc o y a gumen we
ga e in he special case whe e Fis a Fa ell se .
Le us finally p o e Lemma 1. We may assume ha is bounded in
D.
So gi en ∈H∞(D), and >0, we conside a compac se K⊂Fn .
We shall equi e se e al p ope ies o K ela ed o .I 0<α<π,
T(θ,α) deno es he cone in Dwi h opening angle α, e mina ing a eiθ,
and being symme ic wi h espec o he adius { eiθ,0≤ <1}.We
assume ha
(eiθ) = lim (z)
holds uni o mly in eiθ ∈Kas z→eiθ inside T(θ,α). Now fix p∈(0,∞).
Since ∈Hp(D), he adial limi s (eiθ), 0 ≤θ<2π, belong o Lp(dθ).
We assume ha Kis included in he Lebesgue se o and ha
(3) 1
2 θ+
θ−
| (eiϕ)− (eiθ)|pdϕ →0

66 A. S ay
uni o mly in eiθ ∈Kas →0. Such a se Kcan be ound wi h |Fn K|
as small we please.
Fix δ>0 so small ha | (eiθ)− (z)|<i eiθ ∈K,|z|>1−δ
and z∈T(θ,α). We a e now in a con enien posi ion o applying
Vi ushkin’s scheme o app oxima ion (see [13]o [4]). Le {∆j}N
j=1 be
a fini e collec ion o open discs wi h cen e s zj∈Kand a common adius
<δ. Following Vi ushkin’s scheme, le ϕj∈C1
0(∆j) be chosen such
ha 0 ≤ϕ≤1 and ϕ≡1in∆
1
j=z:|z−zj|<
2. As a p elimina y
app oxima ion o we define
K= −GK
whe e GK=
j
Tϕj( − (zj))− jis a fini e sum which we shall explain
in some de ail.
We assume is defined ou side o Dby (z−1)= (z). Fo gene al
p ope ies o he Tϕ-ope a o we e e o [11]o [4, page 30]. He e we
only no e ha
Tϕj( − (zj))(ς)=ϕj(ς)( (ς)− (zj))
−1
π∆j (z)− (zj)
z−ς
∂ϕj
∂z dx dy(z)
=Uj+Vjsay.
We assume 
∂ϕj
∂z ≤A
, whe e Ais a nume ical cons an . Since ∈
Hp(D), we ha e in pa icula ha ∈Lp(dx dy) locally. The e o e he
con olu ion e m Vjis con inuous as a unc ion o ς.I αis close o π,
H¨olde s inequali y gi es ha
|Vj(ς)|≤, ς ∈C j=1,2...N.
No e also ha Vjis analy ic ou side ∆j. Acco ding o Vi ushkin’s
scheme, he unc ions jshould be analy ic ou side a compac subse
o ∆j Dand wi h he p ope y ha (Vj− j)(ς) has a ze o o o de 3 a
∞. In addi ion we should equi e
(4):  j∞≤A1Vj∞≤A1, j =1,2,... ,N
whe e A1is a nume ical cons an . In ou simple si ua ion, he exis ence
o { j}is a he e iden ([4, page 210–214]). F om he indi idual bounds
(3), i is pa o Vi ushkin’s scheme ha
(5): 





N

j=1
(Vj− j)




∞
≤A2
Me gelyan ype heo ems o some unc ion spaces 67
o some nume ical cons an A2. We ha e no claimed ha {∆1
j}N
j=1
co e all o K. In ac we shall assume ha ∆j∩∆k=φi j=k.In
addi ion we assume ha

K∩
N

1
∆1
j
≥A3|K|
o some nume ical cons an A3, whe e ∆1
j=z:|z−zj|≤
2.We
ema k ha K= −GKis analy ic nea ∆1
j∩T o 1 ≤j≤N. This
is seen by w i ing
k=( −Uj)−Vj−
i=j
(Ui+Vi)+
n

i=1
i
and inspec ing hese ou e ms sepa a ely.
No e ha he (Fa ou) bounda y alues (zj) sa is y | (zj)|≤ F.
Since
K= 1−ϕj+
j
ϕj (zj)+
j
(Vj− j)
we ha e
(6)  KF≤ F+A2.
F om (3) and (5) we also ge
 − KHp(D)=GKHp(D)≤(1 + A2)
i is sufficien ly small.
The unc ion Ksa isfies he condi ions o 1in Lemma 1 excep
ha Kis only analy ic (and hence con inuous) nea a subse PKo K.
Bu since |PK|≥A3|K|, Lemma 1 ollows by epea ing ou cons uc ion
coun ably many imes. The main eason why epe i ion wo ks, is ha
he Tϕ-ope a o p ese es con inui y and analy ici y ([4, page 30]).
I emains o show ha Cua(ˆ
F)⊂Hp(D)|ˆ
F. Le Bdeno e he Banach
algeb a H∞(D)|ˆ
F. Also pu X=(ˆ
F). I Vis a componen o C X, he e
mus exis h∈H∞(D) such ha
1=h(z0)>hF
o some z0∈V. Bu hen 1 −his in e ible in Band since
1−h=(z−z0)g, g ∈H∞(D)
68 A. S ay
we conclude ha (z−z0)−1|ˆ
F∈B. This means ha R(X)|ˆ
F⊂B, whe e
R(X) is he uni o m closu e on Xby he a ional unc ions wi h poles
off X.
Bu i {Vj}a e he componen s os C X, he maximum p inciple gi es
∂Vj∩T=φ,j=1,2,... and hence ∂X =∪∞
1∂Vj. Fo such se s X(wi h
emp y “inne bounda y”) Vi ushkin has p o ed ha R(X)=Cua(X)
([4, page 219]), and hence Theo em 1 is p o ed.
This sol es comple ely p oblem 8.5 no. 2 in [7] o he space Hp(D),
0<p<∞.Fo p=∞ he p oblem is s ill open.
Fo p=∞, some in o ma ion abou H∞|Fcan be ob ained om he
wo k by Ca l Sundbe g in [12]. I ∈BMOA and |Fis bounded,
Sundbe g shows ha ∈H∞|F. On he o he hand, ou p oo abo e
shows ha any ∈H∞|Fcan be w i en as =u+ wi h u∈H∞
and ∈∩
p>0Hp(D). Se e al ques ions a ises om his. He e we only
men ion he ollowing: Le ∈BMOA be bounded on a ela i ely closed
se F⊂D.
Is he e g∈H∞such ha he es ic ion ( −g)|Fis uni o mly con-
inuous on F?
Re e ences
1. A. M. Da ie and A. S ay, In e pola ion se s o analy ic unc-
ions, Pacific J. Ma h. 42(1) (1972).
2. J. De az, Algeb es de onc ions analy iques dans le disque, Ann.
Sci. Ecole No m. Sup. 4e se ie (1970), 313–352.
3. P. L. Du en,“Theo y o Hpspaces,” Academic P ess, 1970.
4. T. W. Gamelin,“Uni o m Algeb as,” P en ice Hall, Englewood
Cliffs, N. J., 1969.
5. J. Ga ne ,“Bounded Analy ic Func ions,” Academic P ess, 1980.
6. S. N. Me gelyan, Uni o m app oxima ion o unc ions o a com-
plex a iable, U ephi Ma . Nauk. 7(2) (1952), 31–122.
7. ?,“Linea and Complex Analysis P oblem Book,” Lec u e No es in
Ma hema ics 1043, Sp inge Ve lag, 1984.
8. F. Pe ez-Gonzalez and A. S ay, Fa ell and Me gelyan se s
o Hp-spaces, 0 <p<1, Michigan Ma h. J. 36 (1989), 379–386.
9. A. S ay, Decomposi ion o app oximable unc ions, Ann. o Ma h.
120 (1984), 225–235.
10. A. S ay, App oxima ion by analy ic unc ions which a e uni-
o mly con inuous on a subse o hei domain o defini ion, Ame i-
can J. Ma h. 99 (1977), 787–800.
Me gelyan ype heo ems o some unc ion spaces 69
11. A. S ay, Cha ac e iza ion o Me gelyan se s, P oc. Ame . Ma h.
Soc. 44 (1974), 347–352.
12. C. Sundbe g, T unca ions o BMO unc ions, Indiana Uni e si y
Ma h. I. 33(5) (1984), 749–779.
13. A. G. Vi ushkin, The analy ic capaci y o se s and p oblems in
app oxima ion heo y, Russian Ma h. Su eys 22 (1967), 139–200.
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