Mergelyan theorem
Mergelyan's theorem is a famous result from complex analysis proved by the Armenian mathematician Sergei Nikitovich Mergelyan in 1951. It states the following:
Let K be a compact subset of the complex plane C such that C∖K is connected. Then, every continuous function f : K C, such that the restriction f to int(K) is holomorphic, can be approximated uniformly on K with polynomials. Here, int(K) denotes the interior of K.
Mergelyan's theorem is the ultimate development and generalization of the Weierstrass approximation theorem and Runge's theorem. It gives the complete solution of the classical problem of approximation by polynomials.
In the case that C∖K is not connected, in the initial approximation problem the polynomials have to be replaced by rational functions. An important step of the solution of this further rational approximation problem was also suggested by Mergelyan in 1952. Further deep results on rational approximation are due to, in particular, A. G. Vitushkin.
...
Wikipedia