The distribution of the poles of the best approximating rational functions and the analytical properties of the approximated function

Abstract

The following theorem was proved in our paper in Math. USSR-Sb.9 (1969): Iff(z)∈H 2 and the poles of the rational functions of best approximation tend to infinity sufficiently quickly, thenf(z) is an entire function. In the present article we weaken the restrictions on the distribution of poles by assuming only that these poles have no finite limit point (Theorem 1). Some generalizations of this result are also given.