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:

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

中文翻译：

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整个函数在可数连续集上的逼近

摘要

在连续数G _n的可数集合E上定义的指数类型的整个函数的逼近问题，E = \（\ bigcup \ nolimits_ {n \ in \ mathbb {Z}} {{{G} _ {n}}} \）在本文中考虑。假设所有G _n成对不相交，并且位于实轴附近。还假设所有的G _n在某种意义上是可比较的，并且具有统一的平滑边界。函数f在每个G _n上独立定义，并以E和f ^（r）为界  具有满足条件的连续性ω

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

整个功能˚F _σ指数型≤σ然后构造成使得该函数的近似的估计如下˚F由功能˚F _σ是有效的：

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