On the approximation of entire functions by trigonometric polynomials

Abstract

Let a set B have the following properties: if z ε B, then z ± 2π ε B and the intersection of B with the vertical strip 0 ≤ Re x ≤ π is a closed and bounded set. In this paper we study the approximation of a continuous on B and 2π-periodic function f(z) by trigonometric polynomials Tn(z). We establish the necessary and sufficient conditions for the function f(z) to be entire and specify a formula for calculating its order. In addition, we describe some metric properties of periodic sets in a plane.