Approximation by entire functions on a countable set of continua. The inverse theorem

Abstract

The approximation theory contains many statements where the rate of approximation of a function by polynomials, rational functions and so on is measured with the help of a scale. The statements where some points on the relevant scale are associated with the smoothness of the approximated function are called direct theorems of the theory of approximation. The statements where the smoothness of the approximated function is derived from the points on the scale of approximation by polynomials, rational functions etc., are called inverse theorems of the theory of approximation. The class of functions is constructively discribed in terms of the rate of approximation by polynomials, rational functions etc., if the direct theorems correspond to the inverse theorems, i. e. the smoothness of the approximated function and the points on the scale of approximation have one-to-one correspondence for the class under consideration. The authors have stated earlier the direct theorem concerning approximation by entire functions of exponential type. We considered a set of functions defined on the countable set of mutually disjoint continua and found the rate of their approximation by those entire functions. The present paper contains the inverse theorem to the menthioned above direct one.

Keywords:

inverse theorems, theory of approximation, entire functions of exponential type, H¨older classes