# 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 $z$.
In the case where $K$ has no interior points this result was proved by M.A. Lavrent'ev ; the corresponding theorem in the case where $K$ is a compact domain with a connected complement is due to M.V. Keldysh  (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 ). A complete solution of this problem (for compacta $K\subset\mathbf C$) was obtained in terms of analytic capacities (cf. Analytic capacity), .
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.