On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric
In the paper, we investigate an asymptotic behavior of the sharp upper bounds in the integral metric of deviations of the Abel-Poisson integrals from functions from the class $L^{\psi}_{\beta, 1}$. The Abel-Poisson integrals are solutions of the partial differential equations of elliptic type with c...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2022-06-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/5678 |
| _version_ | 1797205808549724160 |
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| author | T.V. Zhyhallo Yu.I. Kharkevych |
| author_facet | T.V. Zhyhallo Yu.I. Kharkevych |
| author_sort | T.V. Zhyhallo |
| collection | DOAJ |
| description | In the paper, we investigate an asymptotic behavior of the sharp upper bounds in the integral metric of deviations of the Abel-Poisson integrals from functions from the class $L^{\psi}_{\beta, 1}$. The Abel-Poisson integrals are solutions of the partial differential equations of elliptic type with corresponding boundary conditions, and they play an important role in applied problems. The approximative properties of the Abel-Poisson integrals on different classes of differentiable functions were studied in a number of papers. Nevertheless, a problem on the respective approximation on the classes $L^{\psi}_{\beta,1}$ in the metric of the space $L$ remained unsolved. We managed to obtain the estimates for the values of approximation of $(\psi, \beta)$-differentiable functions from the unit ball of the space $L$ by the Abel-Poisson integrals. In some cases, we also write down asymptotic equalities for these quantities, that is we solve the Kolmogorov-Nikol'skii problem for the the Abel-Poisson integrals on the classes $L^{\psi}_{\beta,1}$ in the integral metric. |
| first_indexed | 2024-04-24T08:57:00Z |
| format | Article |
| id | doaj.art-db5e251b7112432b8c5a8a1af7ad39c0 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:00Z |
| publishDate | 2022-06-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-db5e251b7112432b8c5a8a1af7ad39c02024-04-16T07:10:59ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102022-06-0114122322910.15330/cmp.14.1.223-2294910On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metricT.V. Zhyhallo0Yu.I. Kharkevych1Lesya Ukrainka Volyn National University, 13 Voli avenue, 43025, Lutsk, UkraineLesya Ukrainka Volyn National University, 13 Voli avenue, 43025, Lutsk, UkraineIn the paper, we investigate an asymptotic behavior of the sharp upper bounds in the integral metric of deviations of the Abel-Poisson integrals from functions from the class $L^{\psi}_{\beta, 1}$. The Abel-Poisson integrals are solutions of the partial differential equations of elliptic type with corresponding boundary conditions, and they play an important role in applied problems. The approximative properties of the Abel-Poisson integrals on different classes of differentiable functions were studied in a number of papers. Nevertheless, a problem on the respective approximation on the classes $L^{\psi}_{\beta,1}$ in the metric of the space $L$ remained unsolved. We managed to obtain the estimates for the values of approximation of $(\psi, \beta)$-differentiable functions from the unit ball of the space $L$ by the Abel-Poisson integrals. In some cases, we also write down asymptotic equalities for these quantities, that is we solve the Kolmogorov-Nikol'skii problem for the the Abel-Poisson integrals on the classes $L^{\psi}_{\beta,1}$ in the integral metric.https://journals.pnu.edu.ua/index.php/cmp/article/view/5678kolmogorov-nikol'skii problemabel-poisson integral$(\psi, \beta)$-differentiable functionasymptotic equalityintegral metric |
| spellingShingle | T.V. Zhyhallo Yu.I. Kharkevych On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric Karpatsʹkì Matematičnì Publìkacìï kolmogorov-nikol'skii problem abel-poisson integral $(\psi, \beta)$-differentiable function asymptotic equality integral metric |
| title | On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric |
| title_full | On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric |
| title_fullStr | On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric |
| title_full_unstemmed | On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric |
| title_short | On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric |
| title_sort | on approximation of functions from the class l psi beta 1 by the abel poisson integrals in the integral metric |
| topic | kolmogorov-nikol'skii problem abel-poisson integral $(\psi, \beta)$-differentiable function asymptotic equality integral metric |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/5678 |
| work_keys_str_mv | AT tvzhyhallo onapproximationoffunctionsfromtheclasslpsibeta1bytheabelpoissonintegralsintheintegralmetric AT yuikharkevych onapproximationoffunctionsfromtheclasslpsibeta1bytheabelpoissonintegralsintheintegralmetric |