Наближення класів $W^{r}_{\beta,\infty}$ узагальненими інтегралами Абеля – Пуассона

I. V. Kal'chuk; Yu. I. Kharkevych

doi:10.37863/umzh.v74i4.7164

I. V. Kal'chuk Lesya Ukrainka Volyn National University http://orcid.org/0000-0001-8822-3716
Yu. I. Kharkevych Lesya Ukrainka Volyn National University http://orcid.org/0000-0002-8577-5096

DOI: https://doi.org/10.37863/umzh.v74i4.7164

Keywords: generalized Abel-Poisson integral, Weil-Nagy classes, asymptotic equality, uniform metric

Abstract

UDC 517.5I

We study the approximative properties of generalized Abel–Poisson integrals $P_{\gamma}(\delta),$ $0<\gamma\leq2,$ on the Weil–Nagy classes $W^{r}_{\beta,\infty}$ under the condition $0<r\leq\gamma$ in the uniform metric.

References

A. I. Stepanec, Metody teorii priblizheniya, In-t matematiki NAN Ukrainy, Kiev (2002).

L. P. Falaleev, O priblizhenii funkcij obobshchennymi operatorami Abelya – Puassona, Sib. mat. zhurn., 42, № 4, 926 – 936 (2001). DOI: https://doi.org/10.1023/A:1010409901592

YA. S. Bugrov, Neravenstva tipa neravenstv Bernshtejna i ih primenenie k issledovaniyu differencial'nyh svojstv reshenij differencial'nyh uravnenij vysshego poryadka, Math. Cluj., 5, № 1, 5 – 25 (1963).

I. P. Natanson, O poryadke priblizheniya nepreryvnoj $2π$ -periodicheskoj funkcii pri pomoshchi ee integrala Puassona, Dokl. AN SSSR, 72, № 1, 11 – 14 (1950).

A. F. Timan, Tochnaya ocenka ostatka pri priblizhenii periodicheskih differenciruemyh funkcij integralami Puassona, Dokl. AN SSSR,74, № 1, 17 – 20 (1950).

I. V. Kal’chuk, Yu. I. Kharkevych, K. V. Pozharska, Asymptotics of approximation of functions by conjugate Poisson integrals, Carpat. Math. Publ., 12, № 1, 138 – 147 (2020), https://doi.org/10.15330/cmp.12.1.138-147 DOI: https://doi.org/10.15330/cmp.12.1.138-147

V. A. Baskakov, O nekotoryh svojstvah operatorov tipa operatorov Abelya – Puassona, Mat. zametki, 17, № 2, 169 – 180 (1975).

O. L. Shvai, K. V. Pozharska, On some approximation properties of Gauss – Weierstrass singular operators, J. Math. Sci., 260, № 5, 693 – 699 (2022), https://doi.org/10.31801/cfsuasmas.510560 DOI: https://doi.org/10.1007/s10958-022-05720-3

L. I. Bausov, O priblizhenii funkcij klassa $Z_{alpha}$ polozhitel'nymi metodami summirovaniya ryadov Fur'e, Uspekhi mat. nauk,16, № 3, 143 – 149 (1961).

L. I. Bausov, Linejnye metody summirovaniya ryadov Fur'e s zadannymi pryamougol'nymi matricami. I, Izv. vuzov,46, № 3, 15 – 31 (1965).

Yu. I. Kharkevych, Asymptotic expansions of upper bounds of deviations of functions of class $W^r$

from their generalized Poisson integrals, J. Automat. and Inform. Sci., 50, № 8, 38 – 49 (2018). DOI: https://doi.org/10.1615/JAutomatInfScien.v50.i8.40

I. V. Kal’chuk, Yu. I. Kharkevych, Complete asymptotics of the approximation of function from the Sobolev classes by the Poisson integrals, Acta Comment. Univ. Tartu. Math., 22, № 1, 23 – 36 (2018), https://doi.org/10.12697/acutm.2018.22.03 DOI: https://doi.org/10.12697/ACUTM.2018.22.03

Yu. I. Kharkevych, On approximation of the quasi-smooth functions by their Poisson type integrals, J. Automat. and Informat. Sci., 49, № 5, 30 – 36, (2017). DOI: https://doi.org/10.1615/JAutomatInfScien.v49.i10.80

I. V. Kal’chuk, Approximation of $(ψ , β)$-differentiable functions defined on the real axis by Weierstrass operators, Ukr. Math. J., 59, № 9, 1342 – 1363 (2007). DOI: https://doi.org/10.1007/s11253-007-0091-3

I. V. Kal’chuk, V. I. Kravets, U. Z. Hrabova, Approximation of the classes $W^r_{β, ∞}$ by three-harmonic Poisson integrals, J. Math. Sci., 246, № 1, 39 – 50 (2020), https://doi.org/10.37069/1810-3200-2019-16-3-4 DOI: https://doi.org/10.1007/s10958-020-04721-4

F. G. Abdullayev, Yu. I. Kharkevych, Approximation of the classes $C^ψ β H^α$ by biharmonic Poisson integrals, Ukr. Math. J., 72, № 1, 21 – 38 (2020). DOI: https://doi.org/10.1007/s11253-020-01761-6

Approximation of the classes $W^{r}_{\beta,\infty}$ by generalized Abel-Poisson integrals

Abstract

References