E extracta mathematicae Vol. 31, Núm. 1, 109 – 117 (2016)

A Note on Rational Approximation with Respect to

Metrizable Compactifications 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
ϱ 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. Introduction

It 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 → 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 χ
on C∪{∞}. The class of χ–uniform limits of polynomials on D is denoted by
Ã(D) and contains A(D). A function f : D → C∪{∞} belongs to Ã(D) if and
only if f ≡ ∞, or it is continuous on D, f(D) ⊂ C and f|D is holomorphic.
The function f(z) = 1

1−z , z ∈ D, belongs to Ã(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 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 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 Ã(K) of χ–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 ∪ {∞} belongs to Ã(K) if and only if it is
continuous on K and for each component V of K◦, either f(V ) ⊂ C and f|V
is holomorphic, or f|V ≡ ∞.

Extensions of this result have been obtained in [5] when Kc has a finite
number of components and K is bounded by a finite set of disjoint Jordan
curves. In this case, the χ–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 first result has
been obtained in [5] concerning an extension of the approximation theorem of
Arakelian ([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 content
of the present paper.

2. Preliminaries

We say that (S, ϱ) is a metrizable compactification of the plane C, if ϱ
is 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 are
the points at infinity.

Let (S, ϱ) be a metrizable compactification of C with metric ϱ. Many such
compactifications 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 ∈ C
and 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 map

ϕ : C −→ D = {λ ∈ C : |λ < 1}
z 7−→

z

1 + |z|
,

which is a homeomorphism. A compactification of the image D of ϕ is D, the
closure of D, with the usual metric. This leads to the following compactifica-



rational approximation of the plane 111

tion of C

(2.1) S1 := C ∪
{
∞eiϑ : 0 ≤ ϑ ≤ 2π

}
,

with metric d given by

d(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 mean

a metrizable compactification.
An important question for a given compactification of C is, whether for

c ∈ C and x ∈ S\C, the addition c + x is well defined. In other words, having
two convergent sequences {zn}, {wn} in C, such that zn → c and wn → x does
the sequence {zn + wn} have a limit in S?

If the answer is positive for any such sequences {zn}, {wn} in C, then the
limit y ∈ S of the sequence {zn + wn} is uniquely determined and we write
c + x = y = x + c. We are interested in compactifications (S, ϱ), where c + x
is well defined for any c ∈ C and x ∈ S (it suffices to take x ∈ S\C). In this
case, 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 zn
belong to S\C. Without loss of generality we may assume that all zn belong
to 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 that

ϱ(zn + yn, l) <
d

2
for all n ≥ n0 .

Fix n ≥ n0. Since, zn + yn is well defined, there exists z′n ∈ C, such that

ϱ(zn, z
′
n) <

1

n
and ϱ(zn + yn, z

′
n + yn) <

1

n
.

It follows that

ϱ(z′n, x) ≤ ϱ(z
′
n, zn) + ϱ(zn, x) <

1

n
+ ϱ(zn, x) → 0 .



112 m. fragoulopoulou, v. nestoridis

Hence, z′n → x, yn → y and z′n, yn ∈ C. By our assumption, it follows that
z′n + yn → x + y = w. But

ϱ(z′n + yn, l) ≤ ϱ(z
′
n + yn, zn + yn) + ϱ(zn + yn, l)

≤
1

n
+ ϱ(zn + yn, l) <

1

n
+

d

2
→

d

2
.

Thus, for all n large enough we have

ϱ(z′n + yn, l) ≤
3d

4
< d = ϱ(l, w) .

It follows that ϱ(z′n + yn, w) ≥
d
4
, for all n large enough. Therefore, we cannot

have 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, 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 compactification (S1, d)
(see (2.1)), since

c + ∞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 obvious
compactification of C is then the closed square with the usual metric. The
points at infinity are those on the boundary of the square, for instance, those
points on the side {1} × [−1, 1]. If x ∈ {1} × (−1, 1) and c ∈ C, then c + x
is a point in the same side; if Im c ̸= 0, then c + x ̸= x. If x = (1, 1) and
c ∈ 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 points
at infinity are not stabilized as in Remark 1.

Question. Is there a metrizable compactification of C such that the
addition c + x is not well defined for some c ∈ C and x ∈ S\C ?



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 {1} × [−1

2
, 1
2
] and make them just

one point.

3. Runge and Mergelyan type theorems

In 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 consists
of a finite set of pairwise disjoint Jordan curves. Let K = Ω and A a set
containing one point from each component of (C ∪ {∞})\K. Let (S, ϱ) be a
compactification 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 is
holomorphic. Let ε > 0. Then, there exists a rational function R with poles
only 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 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 defined 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 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 and
so we are done.

Another Runge–type theorem is the following, where we do not need
any 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 from
each component of (C ∪ ∞)\Ω. Let ε > 0 and L ⊂ Ω compact. Then, there



114 m. fragoulopoulou, v. nestoridis

exists a rational function R with poles in A, such that ϱ(f(z), R(z)) < ε
for all z ∈ L.

Proof. Clearly the subset f(L) of C is compact. Then, from the classical
theorem 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 that

f(L) ⊂ K ⊂ C and Rn(L) ⊂ K for all n ≥ n0 .

But on K the metrics | · | and ϱ are uniformly equivalent. Therefore, Rn → f
uniformly on L, with respect to ϱ. To conclude the proof, it suffices 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 {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, on
compacta, of rational functions with poles in A, are the holomorphic functions
f : 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 the
form 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, such
that the addition + : C × S → S is well defined. Let ε > 0 and K ⊂ Ω be a
compact 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 an
open set containing K. By Runge’s theorem there exists a rational function R



rational approximation of the plane 115

with poles off K, such that |w(z)−R(z)| < ε′ on K. Since w(K) is a compact
subset of C and the addition + : C × S → S is well defined, a suitable choice
of ε′ gives

ϱ
(
[h(z) + w(z)] , [h(z) + R(z)]

)
< ε on K\B .

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 ∪ {∞})\F is connected and
locally connected (at ∞).

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 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 to
D = {z ∈ C : |z| ≤ 1}. For each 0 < R < 1 let us define

ϕR : D −→ {z ∈ C : |z| ≤ R} ⊂ D

z 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 point

Reiϑ. 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 follows

T(z) :=
z

1 + |z|
for z ∈ C ⊂ S1 ,

T
(
∞eiϑ

)
:= eiϑ for ϑ ∈ R .



116 m. fragoulopoulou, v. nestoridis

If ε > 0 is given, then there exists Rε < 1, such that for Rε ≤ R < 1 and
z ∈ S1, we have d

(
z, ϕ̃R(z)

)
< ε

2
.

Let now f be as in the statement of the Proposition 4.1. Then,

d
(
f(z),

(
ϕ̃R ◦ f

)
(z)

)
<

ε

2
for all z ∈ F .

Moreover, the function ϕ̃R ◦ f : F → C is continuous. Since F is a closed
Arakelian set, with empty interior, and

(
ϕ̃R ◦ f

)
(F) ⊂ K, is included in a

compact 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 suitable

choice of ε′, it follows that

d
((

ϕ̃R ◦ f
)
(z), g(z)

)
<

ε

2
for all z ∈ F .

The triangle inequality completes the proof.

An analogue of Proposition 4.1 for the one point compactification C ∪ {∞} of
C has been established in [5].

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