Mergelyan theorem

A theorem on the possibility of uniform approximation of functions of one complex variable by polynomials. Let $K$ be a compact subset of the complex $z$-plane $\mathbf C$ with a connected complement. Then every function $f$ continuous on $K$ and holomorphic at its interior points can be approximated uniformly on $K$ by polynomials in $z$.

This theorem was proved by S.N. Mergelyan (see [1], [2]); it is the culmination of a large number of studies on approximation theory in the complex plane and has many applications in various branches of complex analysis.

In the case where $K$ has no interior points this result was proved by M.A. Lavrent'ev [3]; the corresponding theorem in the case where $K$ is a compact domain with a connected complement is due to M.V. Keldysh [4] (cf. also Keldysh–Lavrent'ev theorem).

Mergelyan's theorem has the following consequence. Let $K$ be an arbitrary compact subset of $\mathbf C$. Let a function $f$ be continuous on $K$ and holomorphic in its interior. Then in order that $f$ be uniformly approximable by polynomials in $z$ it is necessary and sufficient that $f$ admits a holomorphic extension to all bounded connected components of the set $\mathbf C\setminus K$.

The problem of polynomial approximation is a particular case of the problem of approximation by rational functions with poles in the complement of $K$. Mergelyan found also several sufficient conditions for rational approximation (see [2]). A complete solution of this problem (for compacta $K\subset\mathbf C$) was obtained in terms of analytic capacities (cf. Analytic capacity), [5].

Mergelyan's theorem touches upon a large number of papers concerning polynomial, rational and holomorphic approximation in the space $\mathbf C^n$ of several complex variables. Here only partial results for special types of compact subsets have been obtained up till now.

References

[1]	S.N. Mergelyan, "On the representation of functions by series of polynomials on closed sets" Transl. Amer. Math. Soc. , 3 (1962) pp. 287–293 Dokl. Akad. Nauk SSSR , 78 : 3 (1951) pp. 405–408
[2]	S.N. Mergelyan, "Uniform approximation to functions of a complex variable" Transl. Amer. Math. Soc. , 3 (1962) pp. 294–391 Uspekhi Mat. Nauk , 7 : 2 (1952) pp. 31–122
[3]	M.A. [M.A. Lavrent'ev] Lavrentieff, "Sur les fonctions d'une variable complexe représentables par des series de polynômes" , Hermann (1936)
[4]	M.V. Keldysh, "Sur la réprésentation par des séries de polynômes des fonctions d'une variable complexe dans des domaines fermés" Mat. Sb. , 16 : 3 (1945) pp. 249–258
[5]	A.G. Vitushkin, "The analytic capacity of sets in problems of approximation theory" Russian Math. Surveys , 22 : 6 (1967) pp. 139–200 Uspekhi Mat. Nauk , 22 : 6 (1967) pp. 141–199
[6]	, Some questions in approximation theory , Moscow (1963) (In Russian; translated from English)
[7]	T.W. Gamelin, "Uniform algebras" , Prentice-Hall (1969)

Comments

Another important forerunner of Mergelyan's theorem was the Walsh theorem: the case where $K$ is the closure of a Jordan domain (a set with boundary consisting of Jordan curves, cf. Jordan curve).

An interesting proof of Mergelyan's theorem, based on functional analysis, is due to L. Carlesson, see [a1].

For analogues of Mergelyan's theorem in $\mathbf C^n$, see [a2]. See also Approximation of functions of a complex variable.

References

[a1]	L. Carlesson, "Mergelyan's theorem on uniform polynomial approximation" Math. Scand. , 15 (1964) pp. 167–175
[a2]	E.M. Chirka, G.M. Khenkin, "Boundary properties of holomorphic functions of several complex variables" J. Soviet Math. , 5 : 5 (1976) pp. 612–687 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 4 (1975) pp. 13–142
[a3]	D. Gaier, "Lectures on complex approximation" , Birkhäuser (1987) (Translated from German)