Approximation by entire functions on a countable union of segments on the real axis. 4. Inverse theorem

Abstract

A problem of a constructive description of functional classes in terms of a possible rate of approximation of its functions by means of functions chosen from a certain set is one of the leading problem of approximation theory for more than a century. It turned out that the non-uniformity of a rate of approximation due to the point of a set where a functional class is defined is a rather usual circumstance of those description. One of the possible test for approximation is a question whether the rate of it permits to recognise the functional class under consideration. We have investigated approximation of classes of smooth functions on a countable union segments on the real axis by means of entire functions of exponential type. The present paper is devoted to a proof of the so-called inverse theorem, i. e. to the finding out a scale of smoothness of functions with the help of a rate of its approximation by entire functions.