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 |
| Published: |
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| Summary: | 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. |
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