Approximation by entire functions on a countable union of segments on the real axis. 2. Proof of the main theorem

Abstract

We proof in the present paper a theorem about approximation of a function defined on a countable union of segments of the real line by means of entire functions of exponential type. The approximating function is supposed to belong to a Holder class α, 0 < α < 1. The set E ⊂ R consists of disjoint segments [an, bn],−∞ < n < ∞ such that bn − an ≍ bk − ak for any n and k and an+1 − bn ≍ bn − an for any n. The function f defined on E supposed to be bounded by a constant M and satisfying the condition |f(x) − f(y)| ≤ c0|x − y| , x, y ∈ [an, bn], 0 < α < 1. The construction of an approximation of f by entire functions we begin from a construction of a series of domains D+ and D− depending of sequens {x′}n∈Z and {x′′}n∈Z : x′ n, x′′ n ∈ [an, bn]. We introduce a special conformed mappings of D+ and D− to themselves and use these mapping to form kernels Rk(z,w, ξ, ζ). Afterwards we combine a continuation of a function f from the set E to the whole complex plane C from our first part of this paper and kernels Rk(z,w, ξ, ζ). We get as a result a series of entire functions of exponential type depending on sequences {x′ n}n∈Z and {x′′ n}n∈Z. Then we use sequences {x′ n}n∈Z and {x′′ n}n∈Z to construct a final approximating entire function F . Refs 3.