Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means
We indicate criteria for the coincidence of the Knopp kernels K(f) K(A f), and K (R f) of bounded functions f(t); here, $$R_f \left( t \right) = \frac{1}{{P\left( x \right)}}\int\limits_{\left[ {0;\left. t \right)} \right.} {f\left( x \right)dP and A_f \left( t \right)} = \frac{1}{{\int_0^\infty {...
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| Date: | 1998 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
1998
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4792 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510961746575360 |
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| author | Usenko, E. G. Усенко, Є. Г. |
| author_facet | Usenko, E. G. Усенко, Є. Г. |
| author_sort | Usenko, E. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:14:27Z |
| description | We indicate criteria for the coincidence of the Knopp kernels K(f) K(A f), and K (R f) of bounded functions f(t); here, $$R_f \left( t \right) = \frac{1}{{P\left( x \right)}}\int\limits_{\left[ {0;\left. t \right)} \right.} {f\left( x \right)dP and A_f \left( t \right)} = \frac{1}{{\int_0^\infty {e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} t}} \right. \kern-\nulldelimiterspace} t}} dP} }}\int\limits_0^\infty {f\left( x \right)} e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} t}} \right. \kern-\nulldelimiterspace} t}} dP$$ . In Particular, we prove that K(f) = K(A f) ⇔ K(f) = K(R f). |
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| id | umjimathkievua-article-4792 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:05:19Z |
| publishDate | 1998 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/79/b03efcac428ccf924efe0ac1234be279.pdf |
| spelling | umjimathkievua-article-47922020-03-18T21:14:27Z Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means Критерії співпадання ядра функції з ядрами її інтегральних середніх Рісса та Абеля Usenko, E. G. Усенко, Є. Г. We indicate criteria for the coincidence of the Knopp kernels K(f) K(A f), and K (R f) of bounded functions f(t); here, $$R_f \left( t \right) = \frac{1}{{P\left( x \right)}}\int\limits_{\left[ {0;\left. t \right)} \right.} {f\left( x \right)dP and A_f \left( t \right)} = \frac{1}{{\int_0^\infty {e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} t}} \right. \kern-\nulldelimiterspace} t}} dP} }}\int\limits_0^\infty {f\left( x \right)} e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} t}} \right. \kern-\nulldelimiterspace} t}} dP$$ . In Particular, we prove that K(f) = K(A f) ⇔ K(f) = K(R f). Вказано критерії співпадання ядер $K(f) K(A_f)$, і $K (R f)$ у розумінні Кноппа обмежених функцій $f(t)$ $$R_f (t) = \frac{1}{P(x)}\int\limits_{[0; t)}f(x)dP \text{ та } A_f(t) = \frac1{\int_0^{\infty}e^{-x/t}dP} \int\limits_0^{\infty}f(x)e^{-x/t}dP.$$ Зокрема, доведено, що $K(f) = K(A f) ⇔ K(f) = K(R-f).$ Institute of Mathematics, NAS of Ukraine 1998-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4792 Ukrains’kyi Matematychnyi Zhurnal; Vol. 50 No. 12 (1998); 1712–1714 Український математичний журнал; Том 50 № 12 (1998); 1712–1714 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4792/6284 https://umj.imath.kiev.ua/index.php/umj/article/view/4792/6285 Copyright (c) 1998 Usenko E. G. |
| spellingShingle | Usenko, E. G. Усенко, Є. Г. Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means |
| title | Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means |
| title_alt | Критерії співпадання ядра функції з ядрами її інтегральних середніх Рісса
та Абеля |
| title_full | Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means |
| title_fullStr | Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means |
| title_full_unstemmed | Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means |
| title_short | Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means |
| title_sort | criteria for the coincidence of the kernel of a function with the kernels of its riesz and abel integral means |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4792 |
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