Entire functions of order 1/2 in the approximation of functions on the semiaxis

Abstract

Let In be disjoint segments of half-axis [1;1), I1 = [1; b1]. We assume that In and their supplementary intervals are commensurable. The union of mentioned segments we denote thought E. Let 0 < α< 1. The class of complex-valued functions defined on E with r-th derivative belonging to a Holder class α and bounded by 1 is denoted by H_1^(r+α)(E). We consider the set Cσ of entire functions of the order 1/2 and of the type ≤σ which are bounded on [0;1). The main result of this paper is the following: Let f∈ 2 H_1^(r+α)(E). Then there exist constants c0 and c1, depending of f,E, r and α such that for any σ≥ 1 one can find a function Fσ ∈ Cσ such that the following inequality holds |f(x) − Fσ(x)|≤ min(c0∙1/σr+α(√((x-a_n)(b_n )-x)+n/σ)r+α,c1), x ∈ In, n ≥ 1.