Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals
We introduce the notion of $\overline{\psi}$-integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) $L^{\overline{\psi}}$. We obtain integral representations of deviations of the trigonometric polynomials $U_{n(f;x;Λ)}$ generated...
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| Date: | 1997 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
1997
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5102 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511301760974848 |
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| author | Stepanets, O. I. Степанец, А. И. Степанец, А. И. |
| author_facet | Stepanets, O. I. Степанец, А. И. Степанец, А. И. |
| author_sort | Stepanets, O. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:24:43Z |
| description | We introduce the notion of $\overline{\psi}$-integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) $L^{\overline{\psi}}$.
We obtain integral representations of deviations of the trigonometric polynomials $U_{n(f;x;Λ)}$ generated by a given Λ-method for summing the Fourier series of functions $f ε L^{\overline{\psi}}$. On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets $L^{\overline{\psi}}$ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets $L^{\overline{\psi}}$, which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality. |
| first_indexed | 2026-03-24T03:10:43Z |
| format | Article |
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| id | umjimathkievua-article-5102 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:10:43Z |
| publishDate | 1997 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/31/7259cacf4bb2844d41609d500dc91031.pdf |
| spelling | umjimathkievua-article-51022020-03-18T21:24:43Z Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals Скорость сходимости рядов Фурье на классах $\overline{\psi}$-интегралов Stepanets, O. I. Степанец, А. И. Степанец, А. И. We introduce the notion of $\overline{\psi}$-integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) $L^{\overline{\psi}}$. We obtain integral representations of deviations of the trigonometric polynomials $U_{n(f;x;Λ)}$ generated by a given Λ-method for summing the Fourier series of functions $f ε L^{\overline{\psi}}$. On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets $L^{\overline{\psi}}$ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets $L^{\overline{\psi}}$, which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality. Вводиться поняття $\overline{\psi}$-інтегралів 2π-періодичних сумовиих функцій f, f ε L, на основі якого проводиться розбиття простору L на підмножини (класи) $L^{\overline{\psi}}$. Одержані інтегральні зображення відхилень тригонометричних поліномів $U_{n(f;x;Λ)}$, що породжуються даним Λ-методом підсумовування рядів Фур'є від функцій $f ε L^{\overline{\psi}}$, і на їх основі досліджується швидкість збіжності рядів Фур'є для функцій із множин $L^{\overline{\psi}}$ в рівномірній та інтегральних метриках. В цьому напрямі, зокрема, знайдені асимптотичні рівності для верхніх меж відхилень сум Фур'є на множинах $L^{\overline{\psi}}$ у які дають розв'язки задачі Колмогорова—Нікольського, а також одержано аналог відомої нерівності Лебега. Institute of Mathematics, NAS of Ukraine 1997-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5102 Ukrains’kyi Matematychnyi Zhurnal; Vol. 49 No. 8 (1997); 1069-1113 Український математичний журнал; Том 49 № 8 (1997); 1069-1113 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5102/6901 https://umj.imath.kiev.ua/index.php/umj/article/view/5102/6902 Copyright (c) 1997 Stepanets O. I. |
| spellingShingle | Stepanets, O. I. Степанец, А. И. Степанец, А. И. Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals |
| title | Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals |
| title_alt | Скорость сходимости рядов Фурье на классах $\overline{\psi}$-интегралов |
| title_full | Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals |
| title_fullStr | Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals |
| title_full_unstemmed | Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals |
| title_short | Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals |
| title_sort | rate of convergence of fourier series on the classes of $\overline{\psi}$-integrals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5102 |
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