TY - JOUR

T1 - Approximation by Entire Functions on a Countable Set of Continua

AU - Shirokov, N. A.

AU - Silvanovich, O. V.

N1 - Funding Information: N. A. Shirokov acknowledges the support of the Russian Foundation for Basic Research (project no. 20-01-00209). Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - Abstract: The problem of approximation by entire functions of exponential type defined on a countable set E of continua Gn, E = $$\bigcup\nolimits_{n \in \mathbb{Z}} {{{G}_{n}}} $$ is considered in this paper. It is assumed that all Gn are pairwise disjoint and are situated near the real axis. It is also assumed that all Gn are commensurable in a sense and have uniformly smooth boundaries. A function f is defined independently on each Gn 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 Fσ of exponential type ≤σ is then constructed so that the following estimate of approximation of the function f by functions Fσ is valid: $$\left| {f(z) - {{F}_{\sigma }}(z)} \right| \leqslant {{c}_{f}}{{\sigma }^{{ - r}}}\omega ({{\sigma }^{{ - r}}}),\quad z \in \mathbb{Z},\quad \sigma \geqslant 1.$$

AB - Abstract: The problem of approximation by entire functions of exponential type defined on a countable set E of continua Gn, E = $$\bigcup\nolimits_{n \in \mathbb{Z}} {{{G}_{n}}} $$ is considered in this paper. It is assumed that all Gn are pairwise disjoint and are situated near the real axis. It is also assumed that all Gn are commensurable in a sense and have uniformly smooth boundaries. A function f is defined independently on each Gn 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 Fσ of exponential type ≤σ is then constructed so that the following estimate of approximation of the function f by functions Fσ is valid: $$\left| {f(z) - {{F}_{\sigma }}(z)} \right| \leqslant {{c}_{f}}{{\sigma }^{{ - r}}}\omega ({{\sigma }^{{ - r}}}),\quad z \in \mathbb{Z},\quad \sigma \geqslant 1.$$

KW - approximation

KW - entire functions of exponential type

KW - Hölder classes

UR - http://www.scopus.com/inward/record.url?scp=85090026775&partnerID=8YFLogxK

U2 - 10.1134/S1063454120030139

DO - 10.1134/S1063454120030139

M3 - Article

AN - SCOPUS:85090026775

VL - 53

SP - 329

EP - 335

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -