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|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.
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 (in particular Walsh, Keldysh, and Lavrentyev) 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.
- Lennart Carleson, Mergelyan's theorem on uniform polynomial approximation, Math. Scand., V. 15, (1964) 167–175.
- Dieter Gaier, Lectures on Complex Approximation, Birkhäuser Boston, Inc. (1987), ISBN 0-8176-3147-X.
- W. Rudin, Real and Complex Analysis, McGraw–Hill Book Co., New York, (1987), ISBN 0-07-054234-1.
- A. G. Vitushkin, Half a century as one day, Mathematical events of the twentieth century, 449–473, Springer, Berlin, (2006), ISBN 3-540-23235-4/hbk.