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Approximation by Entire Functions on a Countable Set of Continua. The Inverse Theorem. / Shirokov, N. A.; Silvanovich, O. V.
в: Vestnik St. Petersburg University: Mathematics, Том 54, № 4, 10.2021, стр. 366-371.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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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 -
ID: 95016877