On uniform convergence of some classes of infinite products
Abstract
We find necessary and sufficient conditions $$$\{ \alpha_k(x) \}$$$ must satisfy for the infinite product
$$\prod\limits_{k=1}^{\infty} \bigl[ 1 + \alpha_k(x) u_k(x) \bigr]$$
to converge uniformly under the condition that:
1) the series $$$\sum\limits_{k=1}^{\infty} |\Delta u_k(x)|$$$ converges uniformly; 2) $$$\sum\limits_{k=1}^{\infty} |\Delta u_k(x)| = O(1)$$$.
$$\prod\limits_{k=1}^{\infty} \bigl[ 1 + \alpha_k(x) u_k(x) \bigr]$$
to converge uniformly under the condition that:
1) the series $$$\sum\limits_{k=1}^{\infty} |\Delta u_k(x)|$$$ converges uniformly; 2) $$$\sum\limits_{k=1}^{\infty} |\Delta u_k(x)| = O(1)$$$.
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PDF (Русский)References
Slepenchuk K.M. "Nonlinear transformations of some classes of sequences (of products)", Izv. vuzov. Ser. Matem., 1961; 2. (in Russian)
DOI: https://doi.org/10.15421/248717
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Copyright (c) 1987 K.M. Slepenchuk
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ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
