A generalization of the rogosinski-rogosinski theorem
We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f j ⊂ K(f 1) ∀j≥2 and \(\mathop {\lim }\limits_{U_r } \sum\nolimits_{j = 1}^\infty {\alpha _j (x)f_j (x) = a} \) yield \(\mathop {\lim }\limits_{U_r } f_1 (x) = a.\) T...
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| Date: | 2000 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2000
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4410 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510539404279808 |
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| author | Dekanov, S. Ya. Mikhalin, G. A. Деканов, С. Я. Михалін, Г. О. |
| author_facet | Dekanov, S. Ya. Mikhalin, G. A. Деканов, С. Я. Михалін, Г. О. |
| author_sort | Dekanov, S. Ya. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:28:32Z |
| description | We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f j ⊂ K(f 1) ∀j≥2 and \(\mathop {\lim }\limits_{U_r } \sum\nolimits_{j = 1}^\infty {\alpha _j (x)f_j (x) = a} \) yield \(\mathop {\lim }\limits_{U_r } f_1 (x) = a.\) The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α j (x) ≡ αj, and the space L is the m-dimensional Euclidean space. |
| first_indexed | 2026-03-24T02:58:36Z |
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| id | umjimathkievua-article-4410 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:58:36Z |
| publishDate | 2000 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/25/ca5e59945386ae963f0b1cfc93773825.pdf |
| spelling | umjimathkievua-article-44102020-03-18T20:28:32Z A generalization of the rogosinski-rogosinski theorem Узагальнення однієї теореми Рогозинських Dekanov, S. Ya. Mikhalin, G. A. Деканов, С. Я. Михалін, Г. О. We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f j ⊂ K(f 1) ∀j≥2 and \(\mathop {\lim }\limits_{U_r } \sum\nolimits_{j = 1}^\infty {\alpha _j (x)f_j (x) = a} \) yield \(\mathop {\lim }\limits_{U_r } f_1 (x) = a.\) The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α j (x) ≡ αj, and the space L is the m-dimensional Euclidean space. Вказано необхідні і достатні умови, які накладаються на числові функції $αj(x), j ∈ N, x ∈ X,$ для того щоб з умов $K(f j ⊂ K(f 1) ∀j≥2$ і $\mathop {\lim }\limits_{U_r } \sum\nolimits_{j = 1}^\infty {\alpha _j (x)f_j (x) = a}$ випливала умова $\mathop {\lim }\limits_{U_r } f_1 (x) = a.$ Функції $f_j(x)$ рівномірно обмежені на множині $X$ і набувають значень з обмежено компактного простору $L$, a $K(f_j)$ — ядро функції $f_j$. Відома теорема Рогозинських випливає з доведених тверджень, коли $X = N, α_j (x) ≡ α_j$, а простір $L$ є $m$ -вимірним евклідовим простором. Institute of Mathematics, NAS of Ukraine 2000-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4410 Ukrains’kyi Matematychnyi Zhurnal; Vol. 52 No. 2 (2000); 220-227 Український математичний журнал; Том 52 № 2 (2000); 220-227 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4410/5524 https://umj.imath.kiev.ua/index.php/umj/article/view/4410/5525 Copyright (c) 2000 Dekanov S. Ya.; Mikhalin G. A. |
| spellingShingle | Dekanov, S. Ya. Mikhalin, G. A. Деканов, С. Я. Михалін, Г. О. A generalization of the rogosinski-rogosinski theorem |
| title | A generalization of the rogosinski-rogosinski theorem |
| title_alt | Узагальнення однієї теореми Рогозинських |
| title_full | A generalization of the rogosinski-rogosinski theorem |
| title_fullStr | A generalization of the rogosinski-rogosinski theorem |
| title_full_unstemmed | A generalization of the rogosinski-rogosinski theorem |
| title_short | A generalization of the rogosinski-rogosinski theorem |
| title_sort | generalization of the rogosinski-rogosinski theorem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4410 |
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