Mergelyan's theorem
Mergelyan's theorem is a famous result from complex analysis proved by the Armenian mathematician Sergei 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 is holomorphic, can be approximated uniformly on K with polynomials. Here, int 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.
Weierstrass and Runge's theorems were put forward in 1885, while Mergelyan's theorem dates from 1951. This rather large time difference is not surprising, as the proof of Mergelyan's theorem is based on a new powerful method created by Mergelyan. After Weierstrass and Runge, many mathematicians had been working on the same problem. The method of the proof suggested by Mergelyan is constructive, and remains the only known constructive proof of the result.
