Approximation by Entire Functions on a Countable Set of Continua

Abstract

The problem of approximation by entire functions of exponential type defined on a countable set E of continua G_n, E = $\bigcup\nolimits_{n \in \mathbb{Z}} {{{G}_{n}}}$ is considered in this paper. It is assumed that all G_n are pairwise disjoint and are situated near the real axis. It is also assumed that all G_n are commensurable in a sense and have uniformly smooth boundaries. A function f is defined independently on each G_n and is bounded on E and f ^(r) has a module of continuity ω which satisfies condition                                                                 

$$\int\limits_0^x {\frac{{\omega (t)}}{t}dt} + x\int\limits_x^\infty {\frac{{\omega (t)}}{{{{t}^{2}}}}dt} \leqslant c\omega (x).$$

An entire function F_σ of exponential type ≤σ is then constructed so that the following estimate of approximation of the function f   by functions F_σ is valid:                                              

$$\left| {f(z) - {{F}_{\sigma }}(z)} \right| \leqslant {{c}_{f}}{{\sigma }^{{ - r}}}\omega ({{\sigma }^{{ - r}}}),\quad z \in \mathbb{Z},\quad \sigma \geqslant 1.$$