Approximation by entire functions on a countable union of segments on the real axis

Abstract

We state 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, and the rate of approximation turns out to be better in a vicinity of ends of segments similary to the case of case of polynomial approximation of a function on one segment. Now, we consider a set E ⊂ R consisting of disjonct 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 on all of segments [an, bn] and satisfying the condition |f(x) − f(y)| ≤ c0|x − y|α, x, y ∈ [an, bn], 0 < α < 1. For σ ≥ 1, ξ = 1 σ , x ∈ [an, bn] we define a scale of approximation d1+ξ(x, [an, bn]) such that d1+ξ(x, [an, bn]) ξ(ξ2 + min(x − an)2, (bn − x)2) 1 2 . Then the main theorem states that there exists a constant c1 depending only on f and E such that we can find a function Fσ of exponential type ≤ σ which approximate a function f in a following way: |Fσ(x) − f(x)| ≤ c1dα 1+ξ(x, [an, bn]), x ∈ [an, bn]. Refs 4.