TY - JOUR

T1 - Approximation by Double Periodic Functions in the Classes $$C_{A}^{{r*}}$$

AU - Sintsova, K.A.

AU - Shirokov, N.A.

N1 - Export Date: 21 October 2024 Адрес для корреспонденции: Sintsova, K.A.; St. Petersburg State UniversityRussian Federation; эл. почта: kseniasintlead@gmail.com Адрес для корреспонденции: Shirokov, N.A.; St. Petersburg State UniversityRussian Federation; эл. почта: nikolai.shirokov@gmail.com Сведения о финансировании: Russian Science Foundation, RSF, 23-11-00171 Текст о финансировании 1: N. A. Shirokova was supported by Russian Science Foundation grant 23-11-00171.

PY - 2024/9/1

Y1 - 2024/9/1

N2 - Abstract: We consider approximations by polynomials of Weierstrass doubly periodic functions for functions that are analytic in a domain and continuous in its closure. This problem is closely related to the approximation by holomorphic bivariate polynomials of a function holomorphic in a domain on an elliptic curve. We assume that the length of the arc is commensurable with the length of the chord at the boundary of the domain on the plane. This condition can also be extended to the domain on an elliptic curve. The possibility of obtaining an approximation estimate that is consistent with the so-called inverse theorem, i.e., with restoring the smoothness of the function by the rate of approximation, was previously established for classes of functions analytic in the domain whose derivative of given order in the closure of the domain has the Hölder modulus of continuity with an order of less than one. The approximation method used earlier does not make it possible to study classes of analytic functions whose derivative of some order is bounded. In this paper, we use an alternative method of approximation by polynomials of Weierstrass doubly periodic functions for functions analytic in a domain for which the derivative of a given order is bounded in this domain. © Pleiades Publishing, Ltd. 2024. ISSN 1063-4541, Vestnik St. Petersburg University, Mathematics, 2024, Vol. 57, No. 3, pp. 345–352. Pleiades Publishing, Ltd., 2024. Russian Text The Author(s), 2024, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2024, Vol. 11, No. 3, pp. 508–516.

AB - Abstract: We consider approximations by polynomials of Weierstrass doubly periodic functions for functions that are analytic in a domain and continuous in its closure. This problem is closely related to the approximation by holomorphic bivariate polynomials of a function holomorphic in a domain on an elliptic curve. We assume that the length of the arc is commensurable with the length of the chord at the boundary of the domain on the plane. This condition can also be extended to the domain on an elliptic curve. The possibility of obtaining an approximation estimate that is consistent with the so-called inverse theorem, i.e., with restoring the smoothness of the function by the rate of approximation, was previously established for classes of functions analytic in the domain whose derivative of given order in the closure of the domain has the Hölder modulus of continuity with an order of less than one. The approximation method used earlier does not make it possible to study classes of analytic functions whose derivative of some order is bounded. In this paper, we use an alternative method of approximation by polynomials of Weierstrass doubly periodic functions for functions analytic in a domain for which the derivative of a given order is bounded in this domain. © Pleiades Publishing, Ltd. 2024. ISSN 1063-4541, Vestnik St. Petersburg University, Mathematics, 2024, Vol. 57, No. 3, pp. 345–352. Pleiades Publishing, Ltd., 2024. Russian Text The Author(s), 2024, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2024, Vol. 11, No. 3, pp. 508–516.

KW - analytic functions smooth in the closure of a domain

KW - polynomials

KW - Weierstrass doubly periodic functions

UR - https://www.mendeley.com/catalogue/b0ebe7e6-db30-351c-a891-c49dd169fdea/

U2 - 10.1134/s1063454124700195

DO - 10.1134/s1063454124700195

M3 - статья

VL - 57

SP - 345

EP - 352

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -