TY - JOUR

T1 - Approximation by Entire Functions on a Countable Set of Continua. The Inverse Theorem

AU - Shirokov, N. A.

AU - Silvanovich, O. V.

N1 - Publisher Copyright: © 2021, Pleiades Publishing, Ltd.

PY - 2021/10

Y1 - 2021/10

N2 - Abstract: In approximation theory, statements in which functions from certain classes are approximated by functions from other fixed classes (for example, by polynomials, rational functions, harmonic functions, etc.) and the accuracy of approximation is measured in a certain scale are called direct approximation theorems. Statements where the smoothness class of the approximated function is derived from the known accuracy of approximation of this function by polynomials, rational functions, and harmonic functions are called inverse approximation theorems. It is usually said that some class of generally smooth functions is constructively described in terms of the approximation by polynomials, rational functions, harmonic functions, etc., if functions from this class can be approximated in the chosen scale of the approximation accuracy and if the accuracy of the approximation in this scale yields the belonging of the approximated function to the class under consideration. Since the constructive description of classes of functions is a high-priority area of investigation in approximation theory, there exists a tendency to add inverse statements to the existing direct theorems for some classes of functions. The authors have previously proved the direct theorem concerning the approximation of a set of analytic functions defined on a countable set of continua by entire functions of exponential type. This paper presents the inverse statement. Section 1 assembles definitions and formulations, and Section 2 provides a proof of the main result.

AB - Abstract: In approximation theory, statements in which functions from certain classes are approximated by functions from other fixed classes (for example, by polynomials, rational functions, harmonic functions, etc.) and the accuracy of approximation is measured in a certain scale are called direct approximation theorems. Statements where the smoothness class of the approximated function is derived from the known accuracy of approximation of this function by polynomials, rational functions, and harmonic functions are called inverse approximation theorems. It is usually said that some class of generally smooth functions is constructively described in terms of the approximation by polynomials, rational functions, harmonic functions, etc., if functions from this class can be approximated in the chosen scale of the approximation accuracy and if the accuracy of the approximation in this scale yields the belonging of the approximated function to the class under consideration. Since the constructive description of classes of functions is a high-priority area of investigation in approximation theory, there exists a tendency to add inverse statements to the existing direct theorems for some classes of functions. The authors have previously proved the direct theorem concerning the approximation of a set of analytic functions defined on a countable set of continua by entire functions of exponential type. This paper presents the inverse statement. Section 1 assembles definitions and formulations, and Section 2 provides a proof of the main result.

KW - approximation theory

KW - entire functions of exponential type

KW - Hölder classes

KW - inverse theorems

UR - http://www.scopus.com/inward/record.url?scp=85121418114&partnerID=8YFLogxK

U2 - 10.1134/S1063454121040178

DO - 10.1134/S1063454121040178

M3 - Article

AN - SCOPUS:85121418114

VL - 54

SP - 366

EP - 371

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -