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Theorem on a convergence condition in the spaces
S. V. Lapin Kaluga Branch, N. È. Bauman Moscow Higher Technical School
Abstract:
For a given $\varphi$-function $\varphi(u)$, a condition on a $\varphi$-function $\psi(u)$ is found such that it is necessary and sufficient for the following to hold: $f_n(x)\to f(x)$ and $\|f_n(x)\|_\psi\le M$ ($1,2,\dots$) where $M>0$ is an absolute constant, then $\|f_n(x)-f(x)\|_\varphi\to0$ ($n\to\infty$). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.
Received: 16.06.1976
Citation:
S. V. Lapin, “Theorem on a convergence condition in the spaces”, Mat. Zametki, 21:5 (1977), 615–626; Math. Notes, 21:5 (1977), 346–352
Linking options:
https://www.mathnet.ru/eng/mzm7994 https://www.mathnet.ru/eng/mzm/v21/i5/p615
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Abstract page: | 182 | Full-text PDF : | 111 | First page: | 1 |
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