Convergence of the linear's average multiple Fourier series of continuous functions
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| Datum: | 1988 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch |
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Institute of Mathematics, NAS of Ukraine
1988
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/9175 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513433923878912 |
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| author | Zaderei , P. V. Задерей , П. В. |
| author_facet | Zaderei , P. V. Задерей , П. В. |
| author_sort | Zaderei , P. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-04-16T12:11:56Z |
| description | - |
| first_indexed | 2026-03-24T03:44:37Z |
| format | Article |
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| id | umjimathkievua-article-9175 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:44:37Z |
| publishDate | 1988 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c8/dc1626910786253b4b0112d5637cddc8.pdf |
| spelling | umjimathkievua-article-91752025-04-16T12:11:56Z Convergence of the linear's average multiple Fourier series of continuous functions Сходимость линейных средних кратных рядов Фурье непрерывных функций Zaderei , P. V. Задерей , П. В. - - Пусть $M =\{1,2,\dots,m\}$, $G=\{j_1,j_2,\dots,j_r\}\subset M$, $l_G=(l_{j_1},l_{j_2},\dots,l_{j_r})$, $\Lambda=\{\lambda_k^{(n)}\}=\{ \lambda_k\}$, $n=(n_1,n_2,\dots,n_m)$, $n_i=0,1,\dots,k_i=0,1,\dots, n_i-1$, $\Delta_1^{f_1}\lambda_k=\lambda_k-\lambda_{k_{M \setminus \{j_i\},k_{j_1}+i_{j_1}}}$, $\Delta_1^G\lambda_k=\Delta_1^{G\setminus \{i_1\}}(\Delta_1^{j_1}\lambda_k)= \Delta_1^{G\setminus \{i_r\}}(\Delta_1^{j_r}\lambda_k)$, $\nabla_{2l_{j_1}}^{j_1}\lambda_k=\lambda_{k_{M\setminus \{j_1\},k_{j_1}-l_{j_1}}}-\lambda_{k_{M\setminus \{j_1\},k_{j_1}+l_{j_1}}}$, $\Delta_{l_G}^G\lambda_k=\nabla_{l_{G\setminus \{j_1\}}(\nabla_{l_j}^{f_1},\lambda_k)}^{G\setminus \{j_1\}}$. Решается задача С. M. Никольского: доказывается, что если для матрицы $\Lambda=\{\lambda_k\}$ выполняется неравенство \[\sum_{DG\subset M}\sum_{k_i=1, i\in D}^{n_i-1}\sum_{k_j=2, j\in G}^{n_j-2}\sum_{k_s=0, s\in M\setminus (D\cup G)}^{n_s-1}\Pi_{i\in D}\frac{1}{k_i}|\sum_{l_j=1, j\in G}^{k_j,n_j}\nabla_{l_G}^G(\nabla_l^{M \setminus D}\lambda_{n_D-k_D,k_{M\setminus D}})\times \Pi_{j\in G}l_j^{-1}\leq C, \quad C=const,\] то для того чтобы для любой функции $F(x)\in C_{2\pi}$ в каждой точке $x$ (или равномерно по $x$) было справедливо равенство $lim_{n→\infty} U_n (f; x; \Lambda)=f(x)$, необходимо и достаточно выполнения условий $lim_{n→\infty} \lambda_k^{(n)}=1$ и $\sum_{k=1}^{n-1}|\lambda_{n-k}|\times \Pi_{i\in M}k_i^{-1}\leq C$. Здесь $T^m=(-\pi,\pi]^m$. \[U_n(f;x;\Lambda)=\frac{1}{\pi^m}\int_{T^m}f(x+t)H_n(t)dt,\] \[H_n(t)=\sum_{k=0}^{n-1}2^{-\nu_k}\lambda_k^{(n)}\Pi_{i\in M}cos k_ix_i.\] Institute of Mathematics, NAS of Ukraine 1988-06-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/9175 Ukrains’kyi Matematychnyi Zhurnal; Vol. 40 No. 4 (1988); 424-431 Український математичний журнал; Том 40 № 4 (1988); 424-431 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/9175/10396 Copyright (c) 1988 П. В. Задерей |
| spellingShingle | Zaderei , P. V. Задерей , П. В. Convergence of the linear's average multiple Fourier series of continuous functions |
| title | Convergence of the linear's average multiple Fourier series of continuous functions |
| title_alt | Сходимость линейных средних кратных рядов Фурье непрерывных функций |
| title_full | Convergence of the linear's average multiple Fourier series of continuous functions |
| title_fullStr | Convergence of the linear's average multiple Fourier series of continuous functions |
| title_full_unstemmed | Convergence of the linear's average multiple Fourier series of continuous functions |
| title_short | Convergence of the linear's average multiple Fourier series of continuous functions |
| title_sort | convergence of the linear's average multiple fourier series of continuous functions |
| topic_facet | - |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9175 |
| work_keys_str_mv | AT zadereipv convergenceofthelinear039saveragemultiplefourierseriesofcontinuousfunctions AT zaderejpv convergenceofthelinear039saveragemultiplefourierseriesofcontinuousfunctions AT zadereipv shodimostʹlinejnyhsrednihkratnyhrâdovfurʹenepreryvnyhfunkcij AT zaderejpv shodimostʹlinejnyhsrednihkratnyhrâdovfurʹenepreryvnyhfunkcij |