In the present note we examine possible extensions of Runge, Mergelyan and Arakelian Theorems, when the uniform approximation is meant with respect to the metric ϱ of a metrizable compactification (S, ϱ) of the complex plane C.peerReviewe

Fragoulopoulou, M.

Nestoridis, V.

Extracta Mathematicae

English

Institutional Repository University of Extremadura

E extracta mathematicae Vol. 31, Num. 1, 109 { 117 (2016)
A Note on Rational Approximation with Respect to
Metrizable Compactications of the Plane
M. Fragoulopoulou, V. Nestoridis
Department of Mathematics, University of Athens
Panepistimiopolis, Athens 157 84, Greece
fragoulop@math.uoa.gr, vnestor@math.uoa.gr
Presented by Manuel Maestre Received February 10, 2015
Abstract : In the present note we examine possible extensions of Runge, Mergelyan and
Arakelian Theorems, when the uniform approximation is meant with respect to the metric
% of a metrizable compactication (S; %) of the complex plane C.
Key words: compactication, Arakelian's theorem, Mergelyan's theorem, Runge's theorem,
uniform approximation in the complex domain.
AMS Subject Class. (2010): 30E10.
1. Introduction
It is well known that the class of uniform limits of polynomials in D =
fz 2 C : jzj  1g coincides with the disc algebra A(D). A function f : D ! C
belongs to A(D) if and only if it is continuous on D and holomorphic in the
open unit disc D. It is less known (see [3, 7]) what is the corresponding
class when the uniform convergence is not meant with respect to the usual
Euclidean metric on C, but it is meant with respect to the chordal metric 
on C[f1g. The class of {uniform limits of polynomials on D is denoted byeA(D) and contains A(D). A function f : D ! C[f1g belongs to eA(D) if and
only if f  1, or it is continuous on D; f(D)  C and fjD is holomorphic.
The function f(z) = 11 z ; z 2 D, belongs to eA(D), but not to A(D); thus,
it cannot be uniformly approximated on D, by polynomials with respect to
the usual Euclidean metric on C, but it can be uniformly approximated by
polynomials with respect to the chordal metric .
More generally, if K  C is a compact set with connected complement,
then according to Mergelyan's theorem [10] polynomials are dense in A(K)
with respect to the usual Euclidean metric on C. We recall that a function
f : K ! C belongs to A(K) if and only if it is continuous on K and holomor-
phic in the interior K of K.
109
110 m. fragoulopoulou, v. nestoridis
An open problem is to characterize the class eA(K) of {uniform limits of
polynomials on K.
Conjecture. ([1, 6]) Let K  C be a compact set with connected com-
plement Kc. A function f : K ! C[f1g belongs to eA(K) if and only if it is
continuous on K and for each component V of K, either f(V )  C and fjV
is holomorphic, or fjV  1.
Extensions of this result have been obtained in [5] when Kc has a nite
number of components and K is bounded by a nite set of disjoint Jordan
curves. In this case, the {uniform approximation is achieved using rational
functions with poles out of K instead of polynomials. Furthermore, exten-
sions of Runge's theorem are also proved in [5]. Finally a rst result has
been obtained in [5] concerning an extension of the approximation theorem of
Arakelian ([2]).
Instead of considering the one point compactication C[f1g of the com-
plex plane C, we can consider an arbitrary metrizable compactication (S; %)
of C and investigate the analogues of all previous results. This is the content
of the present paper.
2. Preliminaries
We say that (S; %) is a metrizable compactication of the plane C, if %
is a metric on S, S is compact, S  C and C is an open dense subset of S.
Obviously, SnC is a closed subset of S. We say that the points in SnC are
the points at innity.
Let (S; %) be a metrizable compactication of C with metric %. Many such
compactications can be found in [1]. The one point compactication C[f1g
with the chordal metric  is a distinct one of them. We note that in this case,
the continuous function  : S ! C[ f1g, such that (c) = c, for every c 2 C
and (x) =1, for every x 2 SnC, is useful.
Another metrizable compactication is the one dened in [8] and con-
structed as follows: consider the map
 : C  ! D = f 2 C : j < 1g
z 7 ! z
1 + jzj
;
which is a homeomorphism. A compactication of the image D of  is D, the
closure of D, with the usual metric. This leads to the following compactica-
rational approximation of the plane 111
tion of C
(2.1) S1 := C [
1ei# : 0  #  2	;
with metric d given by
d(z; w) =
 z1 + jzj   w1 + jwj
 if z; w 2 C ;
d
 
z;1ei# =  z1 + jzj   ei#
 if z 2 C; # 2 R ;(2.2)
d
 1ei#;1ei' = ei#   ei' if #; ' 2 R :
In what follows, with a compactication (S; %) of C, we shall always mean
a metrizable compactication.
An important question for a given compactication of C is, whether for
c 2 C and x 2 SnC, the addition c+ x is well dened. In other words, having
two convergent sequences fzng; fwng in C, such that zn ! c and wn ! x does
the sequence fzn + wng have a limit in S?
If the answer is positive for any such sequences fzng; fwng in C, then the
limit y 2 S of the sequence fzn + wng is uniquely determined and we write
c+ x = y = x+ c. We are interested in compactications (S; %), where c+ x
is well dened for any c 2 C and x 2 S (it suces to take x 2 SnC). In this
case, the map C S ! S, (c; x) 7! c+ x, is automatically continuous.
Indeed, let x 2 SnC, y 2 C and w = x+ y 2 SnC. Let fzng in S and fyng
in C, such that zn ! x and yn ! y. If all but nitely many zn belong to C,
then by our assumption zn + yn ! x + y. Suppose that innitely many zn
belong to SnC. Without loss of generality we may assume that all zn belong
to SnC and by compactness we can assume that zn + yn ! l 6= w = x+ y.
Let d = %(l; w) > 0. Then there exists n0 2 N, such that
%(zn + yn; l) <
d
2
for all n  n0 :
Fix n  n0. Since, zn + yn is well dened, there exists z0n 2 C, such that
%(zn; z
0
n) <
1
n
and %(zn + yn; z
0
n + yn) <
1
n
:
It follows that
%(z0n; x)  %(z0n; zn) + %(zn; x) <
1
n
+ %(zn; x) ! 0 :
112 m. fragoulopoulou, v. nestoridis
Hence, z0n ! x; yn ! y and z0n; yn 2 C. By our assumption, it follows that
z0n + yn ! x+ y = w. But
%(z0n + yn; l)  %(z0n + yn; zn + yn) + %(zn + yn; l)
 1
n
+ %(zn + yn; l) <
1
n
+
d
2
! d
2
:
Thus, for all n large enough we have
%(z0n + yn; l) 
3d
4
< d = %(l; w) :
It follows that %(z0n+ yn; w)  d4 , for all n large enough. Therefore, we cannot
have z0n + yn ! w.
Consequently, one concludes that the addition map is continuous at every
(x; y) with x 2 SnC and y 2 C. Obviously, it is also continuous at every (x; y)
with x and y in C. Thus, addition is continuous on S  C. Furthermore, the
following holds:
Let K  C be compact. Obviously, the map K  S ! S, (c; x) 7! c + x,
is uniformly continuous.
Remark 1. The preceding certainly holds for the compactication (S1; d)
(see (2.1)), since
c+1ei# =1ei# for all c 2 C and # 2 R ;
and we have continuity.
Remark 2. If we identify R with the interval ( 1; 1), up to a homeomor-
phism, then C = R2 is identied with the square ( 1; 1)( 1; 1). An obvious
compactication of C is then the closed square with the usual metric. The
points at innity are those on the boundary of the square, for instance, those
points on the side f1g  [ 1; 1]. If x 2 f1g  ( 1; 1) and c 2 C, then c + x
is a point in the same side; if Im c 6= 0, then c + x 6= x. If x = (1; 1) and
c 2 C, then x + c = x. If Im c > 0, then c + x lies higher than x in the side
f1g  ( 1; 1).
In this example, the addition is well dened and continuous, but the points
at innity are not stabilized as in Remark 1.
Question. Is there a metrizable compactication of C such that the
addition c+ x is not well dened for some c 2 C and x 2 SnC ?
rational approximation of the plane 113
The answer is \yes". An example comes from the previous square in
Remark 2, if we identify all the points of f1g  [ 12 ; 12 ] and make them just
one point.
3. Runge and Mergelyan type theorems
In this section using a compactication of C satisfying all properties dis-
cussed in the Preliminaries, we obtain the following theorem, that extends
[5, Theorem 3.3].
Theorem 3.1. Let 
  C be a bounded domain, whose boundary consists
of a nite set of pairwise disjoint Jordan curves. Let K = 
 and A a set
containing one point from each component of (C [ f1g)nK. Let (S; %) be a
compactication of C, such that the addition + : C S ! S is well dened.
Let f : K ! S be a continuous function, such that f(
)  C and f 
 is
holomorphic. Let " > 0. Then, there exists a rational function R with poles
only in A and such that %(f(z); R(z)) < ", for all z 2 K.
Proof. If 
 is a disk, the proof has been given in [1]. If 
 is the interior of
a Jordan curve, the proof is given again in [1], but also in [6]. In the general
case, we imitate the proof of [5, Theorem 3.3]. Namely, we consider the Lau-
rent decomposition of f , given by f = f0+f1+   +fN (see [4]). The function
f0 is dened on a simply connected domain, bounded by a Jordan curve, and
it can be uniformly approximated by a polynomial or a rational function R0
with pole in the unbounded component. Similarly, f1 is approximated by a
rational function R1 with pole in A and so on. Thus, the function R0 +R1 +
  +RN approximates, with respect to %, the function f = f0+ f1+   + fN .
This is due to the fact that at every point z all the fi's, i = 1; 2;    ; N ,
except maybe one, take values in C and the one, maybe has as a value,
an innity point in SnC. In this way, the addition map C  S ! S,
(c; x) 7! c+ x, is well dened and uniformly continuous on compact sets and
so we are done.
Another Runge{type theorem is the following, where we do not need
any assumption for the compactication S, or the addition map
+ : C S ! S.
Theorem 3.2. Let 
  C be open, f : 
 ! C be holomorphic and
(S; %) a compactication of C. Let A be a set containing one point from
each component of (C [1)n
. Let " > 0 and L  
 compact. Then, there
114 m. fragoulopoulou, v. nestoridis
exists a rational function R with poles in A, such that %(f(z); R(z)) < "
for all z 2 L.
Proof. Clearly the subset f(L) of C is compact. Then, from the classical
theorem of Runge, there exist rational functions fRng, with poles only in A,
converging uniformly to f on L, with respect to the Euclidean metric j  j.
Hence, there is a positive integer n0 and a compact K, such that
f(L)  K  C and Rn(L)  K for all n  n0 :
But on K the metrics j  j and % are uniformly equivalent. Therefore, Rn ! f
uniformly on L, with respect to %. To conclude the proof, it suces to put
R = Rn, for n large enough.
Theorem 3.2 easily yields the following
Corollary 3.3. Under the assumptions of Theorem 3.2 there exists a
sequence fRng of rational functions with poles in A, such that Rn ! f ,
%{uniformly, on each compact subset of 
.
Remark. According to Corollary 3.3, some of the %{uniform limits, on
compacta, of rational functions with poles in A, are the holomorphic functions
f : 
! C. Those are limits of the nite type. The other limits of sequences
fRng as above may be functions f : 
 ! SnC of innite type, continuous
(but maybe not all of them, as the Example (S1; d) shows; cf. [8]).
Question. Is a characterization possible for such limits f : 
! S1nC ?
An imitation of the arguments in [8, p. 1007] gives that f must be of the
form f(z) =1ei#(z), z 2 
, where # is a multivalued harmonic function.
The following extends [5, Section 5].
Theorem 3.4. Let 
  C be open and f a meromorphic function on 
.
Let B denote the set of poles of f . Let (S; %) be a compactication of C, such
that the addition + : C  S ! S is well dened. Let " > 0 and K  
 be a
compact set. Then, there is a rational function g, such that %(f(z); g(z)) < ",
for every z 2 KnB.
Proof. Since B\K is a nite set, the function f decomposes to f = h+w,
where h is a rational function with poles in B\K and w is holomorphic on an
open set containing K. By Runge's theorem there exists a rational function R
rational approximation of the plane 115
with poles o K, such that jw(z) R(z)j < "0 on K. Since w(K) is a compact
subset of C and the addition + : C S ! S is well dened, a suitable choice
of "0 gives
%
 
[h(z) + w(z)] ; [h(z) +R(z)]

< " on KnB :
We set g = h+R and the result follows.
4. Arakelian sets
A closed set F  C is said a set of approximation if every function
f : F ! C continuous on F and holomorphic in F  can be approximated
by entire functions, uniformly on the whole F . This is equivalent to the fact
that F is an Arakelian set (see [2]), that is (C [ f1g)nF is connected and
locally connected (at 1).
We can now ask about an extension of the Arakelian theorem in the context
of metrizable compactifcations. A result in this direction is the following
Proposition 4.1. Let F  C be a closed Arakelian set with empty in-
terior, i.e., F  = ;. We consider the compactication (S1; d) of C (see (2:1)
and (2:2)) and let f : F ! S1 be a continuous function. Let " > 0. Then,
there is an entire function g such that d(f(z); g(z)) < ", for every z 2 F .
Proof. According to (1.1), the compactication S1 is homeomorphic to
D = fz 2 C : jzj  1g. For each 0 < R < 1 let us dene
R : D  ! fz 2 C : jzj  Rg  D
z 7 !
8<: z ; if jzj  R ;Rzjzj ; if R  jzj  1 :
In other words, the whole line segment

Rei#; ei#

is mapped at the end point
Rei#. The function R is continuous and induces a continuous function eR :
S1 ! S1. It suces to take eR := T 1  R  T , where T : S1 ! fw 2 C :
jwj  1g is dened as follows
T (z) :=
z
1 + jzj for z 2 C  S1 ;
T
 1ei# := ei# for # 2 R :
116 m. fragoulopoulou, v. nestoridis
If " > 0 is given, then there exists R" < 1, such that for R"  R < 1 and
z 2 S1, we have d
 
z; eR(z) < "2 .
Let now f be as in the statement of the Proposition 4.1. Then,
d

f(z);
 eR  f(z) < "
2
for all z 2 F :
Moreover, the function eR  f : F ! C is continuous. Since F is a closed
Arakelian set, with empty interior, and
 eR  f(F )  K, is included in a
compact subset K of C, there exists g entire, such that eR  f(z)  g(z) < "0 for all z 2 F :
Since
 eR  f(F ) is contained in a compact subset K of C, for a suitable
choice of "0, it follows that
d
 eR  f(z); g(z) < "
2
for all z 2 F :
The triangle inequality completes the proof.
An analogue of Proposition 4.1 for the one point compactication C[f1g of
C has been established in [5].
References
[1] I. Androulidakis, V. Nestoridis, Extension of the disc algebra and
of Mergelyan's theorem, C.R. Math. Acad. Sci. Paris 349 (13{14) (2011),
745 { 748.
[2] N.U. Arakelian, Uniform approximation on closed sets by entire functions,
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[3] L. Brown, P.M. Gauthier, W. Hengartner, Continuous boundary
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rational approximation of the plane 117
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A note on rational approximation with respect to metrizable compactifications of the plane

Dehesa. Repositorio Institucional de la Universidad de Extremadura

E extracta mathematicae Vol. 31, Núm. 1, 109 – 117 (2016)A Note on Rational Approximation with Respect toMetrizable Compactifications of the PlaneM. Fragoulopoulou, V. NestoridisDepartment of Mathematics, University of AthensPanepistimiopolis, Athens 157 84, Greecefragoulop@math.uoa.gr, vnestor@math.uoa.grPresented by Manuel Maestre Received February 10, 2015Abstract : In the present note we examine possible extensions of Runge, Mergelyan andArakelian Theorems, when the uniform approximation is meant with respect to the metricϱ of a metrizable compactification (S, ϱ) of the complex plane C.Key words: compactification, Arakelian’s theorem, Mergelyan’s theorem, Runge’s theorem,uniform approximation in the complex domain.AMS Subject Class. (2010): 30E10.1. IntroductionIt is well known that the class of uniform limits of polynomials in D ={z ∈ C : |z| ≤ 1} coincides with the disc algebra A(D). A function f : D → Cbelongs to A(D) if and only if it is continuous on D and holomorphic in theopen unit disc D. It is less known (see [3, 7]) what is the correspondingclass when the uniform convergence is not meant with respect to the usualEuclidean metric on C, but it is meant with respect to the chordal metric χon C∪{∞}. The class of χ–uniform limits of polynomials on D is denoted byÃ(D) and contains A(D). A function f : D → C∪{∞} belongs to Ã(D) if andonly if f ≡ ∞, or it is continuous on D, f(D) ⊂ C and f|D is holomorphic.The function f(z) = 11−z , z ∈ D, belongs to Ã(D), but not to A(D); thus,it cannot be uniformly approximated on D, by polynomials with respect tothe usual Euclidean metric on C, but it can be uniformly approximated bypolynomials with respect to the chordal metric χ.More generally, if K ⊂ C is a compact set with connected complement,then according to Mergelyan’s theorem [10] polynomials are dense in A(K)with respect to the usual Euclidean metric on C. We recall that a functionf : K → C belongs to A(K) if and only if it is continuous on K and holomor-phic in the interior K◦ of K.109110 m. fragoulopoulou, v. nestoridisAn open problem is to characterize the class Ã(K) of χ–uniform limits ofpolynomials on K.Conjecture. ([1, 6]) Let K ⊂ C be a compact set with connected com-plement Kc. A function f : K → C∪{∞} belongs to Ã(K) if and only if it iscontinuous on K and for each component V of K◦, either f(V ) ⊂ C and f|Vis holomorphic, or f|V ≡ ∞.Extensions of this result have been obtained in [5] when Kc has a finitenumber of components and K is bounded by a finite set of disjoint Jordancurves. In this case, the χ–uniform approximation is achieved using rationalfunctions with poles out of K instead of polynomials. Furthermore, exten-sions of Runge’s theorem are also proved in [5]. Finally a first result hasbeen obtained in [5] concerning an extension of the approximation theorem ofArakelian ([2]).Instead of considering the one point compactification C∪{∞} of the com-plex plane C, we can consider an arbitrary metrizable compactification (S, ϱ)of C and investigate the analogues of all previous results. This is the contentof the present paper.2. PreliminariesWe say that (S, ϱ) is a metrizable compactification of the plane C, if ϱis a metric on S, S is compact, S ⊃ C and C is an open dense subset of S.Obviously, S\C is a closed subset of S. We say that the points in S\C arethe points at infinity.Let (S, ϱ) be a metrizable compactification of C with metric ϱ. Many suchcompactifications can be found in [1]. The one point compactification C∪{∞}with the chordal metric χ is a distinct one of them. We note that in this case,the continuous function π : S → C∪ {∞}, such that π(c) = c, for every c ∈ Cand π(x) = ∞, for every x ∈ S\C, is useful.Another metrizable compactification is the one defined in [8] and con-structed as follows: consider the mapϕ : C −→ D = {λ ∈ C : |λ < 1}z 7−→ z1 + |z|,which is a homeomorphism. A compactification of the image D of ϕ is D, theclosure of D, with the usual metric. This leads to the following compactifica-rational approximation of the plane 111tion of C(2.1) S1 := C ∪{∞eiϑ : 0 ≤ ϑ ≤ 2π},with metric d given byd(z, w) =∣∣∣∣ z1 + |z| − w1 + |w|∣∣∣∣ if z, w ∈ C ,d(z,∞eiϑ)=∣∣∣∣ z1 + |z| − eiϑ∣∣∣∣ if z ∈ C, ϑ ∈ R ,(2.2)d(∞eiϑ,∞eiφ)=∣∣∣eiϑ − eiφ∣∣∣ if ϑ, φ ∈ R .In what follows, with a compactification (S, ϱ) of C, we shall always meana metrizable compactification.An important question for a given compactification of C is, whether forc ∈ C and x ∈ S\C, the addition c+ x is well defined. In other words, havingtwo convergent sequences {zn}, {wn} in C, such that zn → c and wn → x doesthe sequence {zn + wn} have a limit in S?If the answer is positive for any such sequences {zn}, {wn} in C, then thelimit y ∈ S of the sequence {zn + wn} is uniquely determined and we writec+ x = y = x+ c. We are interested in compactifications (S, ϱ), where c+ xis well defined for any c ∈ C and x ∈ S (it suffices to take x ∈ S\C). In thiscase, the map C× S → S, (c, x) 7→ c+ x, is automatically continuous.Indeed, let x ∈ S\C, y ∈ C and w = x+ y ∈ S\C. Let {zn} in S and {yn}in C, such that zn → x and yn → y. If all but finitely many zn belong to C,then by our assumption zn + yn → x + y. Suppose that infinitely many znbelong to S\C. Without loss of generality we may assume that all zn belongto S\C and by compactness we can assume that zn + yn → l ̸= w = x+ y.Let d = ϱ(l, w) > 0. Then there exists n0 ∈ N, such thatϱ(zn + yn, l) <d2for all n ≥ n0 .Fix n ≥ n0. Since, zn + yn is well defined, there exists z′n ∈ C, such thatϱ(zn, z′n) <1nand ϱ(zn + yn, z′n + yn) <1n.It follows thatϱ(z′n, x) ≤ ϱ(z′n, zn) + ϱ(zn, x) <1n+ ϱ(zn, x) → 0 .112 m. fragoulopoulou, v. nestoridisHence, z′n → x, yn → y and z′n, yn ∈ C. By our assumption, it follows thatz′n + yn → x+ y = w. Butϱ(z′n + yn, l) ≤ ϱ(z′n + yn, zn + yn) + ϱ(zn + yn, l)≤ 1n+ ϱ(zn + yn, l) <1n+d2→ d2.Thus, for all n large enough we haveϱ(z′n + yn, l) ≤3d4< d = ϱ(l, w) .It follows that ϱ(z′n+ yn, w) ≥ d4 , for all n large enough. Therefore, we cannothave z′n + yn → w.Consequently, one concludes that the addition map is continuous at every(x, y) with x ∈ S\C and y ∈ C. Obviously, it is also continuous at every (x, y)with x and y in C. Thus, addition is continuous on S × C. Furthermore, thefollowing holds:Let K ⊂ C be compact. Obviously, the map K × S → S, (c, x) 7→ c + x,is uniformly continuous.Remark 1. The preceding certainly holds for the compactification (S1, d)(see (2.1)), sincec+∞eiϑ = ∞eiϑ for all c ∈ C and ϑ ∈ R ,and we have continuity.Remark 2. If we identify R with the interval (−1, 1), up to a homeomor-phism, then C ∼= R2 is identified with the square (−1, 1)×(−1, 1). An obviouscompactification of C is then the closed square with the usual metric. Thepoints at infinity are those on the boundary of the square, for instance, thosepoints on the side {1} × [−1, 1]. If x ∈ {1} × (−1, 1) and c ∈ C, then c + xis a point in the same side; if Im c ̸= 0, then c + x ̸= x. If x = (1, 1) andc ∈ C, then x + c = x. If Im c > 0, then c + x lies higher than x in the side{1} × (−1, 1).In this example, the addition is well defined and continuous, but the pointsat infinity are not stabilized as in Remark 1.Question. Is there a metrizable compactification of C such that theaddition c+ x is not well defined for some c ∈ C and x ∈ S\C ?rational approximation of the plane 113The answer is “yes”. An example comes from the previous square inRemark 2, if we identify all the points of {1} × [−12 ,12 ] and make them justone point.3. Runge and Mergelyan type theoremsIn this section using a compactification of C satisfying all properties dis-cussed in the Preliminaries, we obtain the following theorem, that extends[5, Theorem 3.3].Theorem 3.1. Let Ω ⊂ C be a bounded domain, whose boundary consistsof a finite set of pairwise disjoint Jordan curves. Let K = Ω and A a setcontaining one point from each component of (C ∪ {∞})\K. Let (S, ϱ) be acompactification of C, such that the addition + : C× S → S is well defined.Let f : K → S be a continuous function, such that f(Ω) ⊂ C and f Ω isholomorphic. Let ε > 0. Then, there exists a rational function R with polesonly in A and such that ϱ(f(z), R(z)) < ε, for all z ∈ K.Proof. If Ω is a disk, the proof has been given in [1]. If Ω is the interior ofa Jordan curve, the proof is given again in [1], but also in [6]. In the generalcase, we imitate the proof of [5, Theorem 3.3]. Namely, we consider the Lau-rent decomposition of f , given by f = f0+f1+ · · ·+fN (see [4]). The functionf0 is defined on a simply connected domain, bounded by a Jordan curve, andit can be uniformly approximated by a polynomial or a rational function R0with pole in the unbounded component. Similarly, f1 is approximated by arational function R1 with pole in A and so on. Thus, the function R0 +R1 +· · ·+RN approximates, with respect to ϱ, the function f = f0+ f1+ · · ·+ fN .This is due to the fact that at every point z all the fi’s, i = 1, 2, · · · , N ,except maybe one, take values in C and the one, maybe has as a value,an infinity point in S\C. In this way, the addition map C × S → S,(c, x) 7→ c+ x, is well defined and uniformly continuous on compact sets andso we are done.Another Runge–type theorem is the following, where we do not needany assumption for the compactification S, or the addition map+ : C× S → S.Theorem 3.2. Let Ω ⊂ C be open, f : Ω → C be holomorphic and(S, ϱ) a compactification of C. Let A be a set containing one point fromeach component of (C ∪∞)\Ω. Let ε > 0 and L ⊂ Ω compact. Then, there114 m. fragoulopoulou, v. nestoridisexists a rational function R with poles in A, such that ϱ(f(z), R(z)) < εfor all z ∈ L.Proof. Clearly the subset f(L) of C is compact. Then, from the classicaltheorem of Runge, there exist rational functions {Rn}, with poles only in A,converging uniformly to f on L, with respect to the Euclidean metric | · |.Hence, there is a positive integer n0 and a compact K, such thatf(L) ⊂ K ⊂ C and Rn(L) ⊂ K for all n ≥ n0 .But on K the metrics | · | and ϱ are uniformly equivalent. Therefore, Rn → funiformly on L, with respect to ϱ. To conclude the proof, it suffices to putR = Rn, for n large enough.Theorem 3.2 easily yields the followingCorollary 3.3. Under the assumptions of Theorem 3.2 there exists asequence {Rn} of rational functions with poles in A, such that Rn → f ,ϱ–uniformly, on each compact subset of Ω.Remark. According to Corollary 3.3, some of the ϱ–uniform limits, oncompacta, of rational functions with poles in A, are the holomorphic functionsf : Ω → C. Those are limits of the finite type. The other limits of sequences{Rn} as above may be functions f : Ω → S\C of infinite type, continuous(but maybe not all of them, as the Example (S1, d) shows; cf. [8]).Question. Is a characterization possible for such limits f : Ω → S1\C ?An imitation of the arguments in [8, p. 1007] gives that f must be of theform f(z) = ∞eiϑ(z), z ∈ Ω, where ϑ is a multivalued harmonic function.The following extends [5, Section 5].Theorem 3.4. Let Ω ⊂ C be open and f a meromorphic function on Ω.Let B denote the set of poles of f . Let (S, ϱ) be a compactification of C, suchthat the addition + : C × S → S is well defined. Let ε > 0 and K ⊂ Ω be acompact set. Then, there is a rational function g, such that ϱ(f(z), g(z)) < ε,for every z ∈ K\B.Proof. Since B∩K is a finite set, the function f decomposes to f = h+w,where h is a rational function with poles in B∩K and w is holomorphic on anopen set containing K. By Runge’s theorem there exists a rational function Rrational approximation of the plane 115with poles off K, such that |w(z)−R(z)| < ε′ on K. Since w(K) is a compactsubset of C and the addition + : C× S → S is well defined, a suitable choiceof ε′ givesϱ([h(z) + w(z)] , [h(z) +R(z)])< ε on K\B .We set g = h+R and the result follows.4. Arakelian setsA closed set F ⊂ C is said a set of approximation if every functionf : F → C continuous on F and holomorphic in F ◦ can be approximatedby entire functions, uniformly on the whole F . This is equivalent to the factthat F is an Arakelian set (see [2]), that is (C ∪ {∞})\F is connected andlocally connected (at ∞).We can now ask about an extension of the Arakelian theorem in the contextof metrizable compactifcations. A result in this direction is the followingProposition 4.1. Let F ⊂ C be a closed Arakelian set with empty in-terior, i.e., F ◦ = ∅. We consider the compactification (S1, d) of C (see (2.1)and (2.2)) and let f : F → S1 be a continuous function. Let ε > 0. Then,there is an entire function g such that d(f(z), g(z)) < ε, for every z ∈ F .Proof. According to (1.1), the compactification S1 is homeomorphic toD = {z ∈ C : |z| ≤ 1}. For each 0 < R < 1 let us defineϕR : D −→ {z ∈ C : |z| ≤ R} ⊂ Dz 7−→ z , if |z| ≤ R ,Rz|z| , if R ≤ |z| ≤ 1 .In other words, the whole line segment[Reiϑ, eiϑ]is mapped at the end pointReiϑ. The function ϕR is continuous and induces a continuous function ϕ̃R :S1 → S1. It suffices to take ϕ̃R := T−1 ◦ ϕR ◦ T , where T : S1 → {w ∈ C :|w| ≤ 1} is defined as followsT (z) :=z1 + |z|for z ∈ C ⊂ S1 ,T(∞eiϑ):= eiϑ for ϑ ∈ R .116 m. fragoulopoulou, v. nestoridisIf ε > 0 is given, then there exists Rε < 1, such that for Rε ≤ R < 1 andz ∈ S1, we have d(z, ϕ̃R(z))< ε2 .Let now f be as in the statement of the Proposition 4.1. Then,d(f(z),(ϕ̃R ◦ f)(z))<ε2for all z ∈ F .Moreover, the function ϕ̃R ◦ f : F → C is continuous. Since F is a closedArakelian set, with empty interior, and(ϕ̃R ◦ f)(F ) ⊂ K, is included in acompact subset K of C, there exists g entire, such that∣∣∣(ϕ̃R ◦ f)(z)− g(z)∣∣∣ < ε′ for all z ∈ F .Since(ϕ̃R ◦ f)(F ) is contained in a compact subset K of C, for a suitablechoice of ε′, it follows thatd((ϕ̃R ◦ f)(z), g(z))<ε2for all z ∈ F .The triangle inequality completes the proof.An analogue of Proposition 4.1 for the one point compactification C∪{∞} ofC has been established in [5].References[1] I. Androulidakis, V. Nestoridis, Extension of the disc algebra andof Mergelyan’s theorem, C.R. Math. Acad. Sci. Paris 349 (13–14) (2011),745 – 748.[2] N.U. Arakelian, Uniform approximation on closed sets by entire functions,Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1187 – 1206 (Russian).[3] L. Brown, P.M. Gauthier, W. Hengartner, Continuous boundarybehaviour for functions defined in the open unit disc, Nagoya Math. J. 57(1975), 49 – 58.[4] G. Costakis, V. Nestoridis, I. Papadoperakis, Universal Laurentseries, Proc. Edinb. Math. Soc. (2) 48 (3) (2005), 571 – 583.[5] M. Fragoulopoulou, V. Nestoridis, I. Papadoperakis, Some re-sults on spherical approximation, Bull. Lond. Math. Soc. 45 (6) (2013),1171 – 1180.[6] V. Nestoridis, Compactifications of the plane and extensions of the discalgebra, in “Complex Analysis and Potential Theory ” , CRM Proc. LectureNotes, 55, Amer. Math. Soc., Providence, RI, 2012, 61 – 75.[7] V. Nestoridis, An extension of the disc algebra, I, Bull. Lond. Math. Soc.44 (4) (2012), 775 – 788.rational approximation of the plane 117[8] V. Nestoridis, N. Papadatos, An extension of the disc algebra, II,Complex Var. Elliptic Equ. 59 (7) (2014), 1003 – 1015.[9] V. Nestoridis, I. Papadoperakis, A remark on two extensions of the discalgebra and Mergelian’s theorem, preprint 2011, arxiv: 1104.0833.[10] W. Rudin, “Real and Complex Analysis ”, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966.

A note on rational approximation with respect to metrizable compactifications of the plane

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