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This article is cited in 1 scientific paper (total in 1 paper)
Universal series and subsequences of functions
Sh. T. Tetunashviliab a Andrea Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Tbilisi, Georgia
b Georgian Technical University, Tbilisi, Georgia
Abstract:
Necessary and sufficient conditions for the existence of a universal series in any system of measurable functions are established. It is proved that if there exists a universal series in a system $\Phi$, then there exists a universal series in this system such that, for any measurable function $f(x)$, there exists a subsequence of partial sums $S_{m_k}(x)$ converging to $f(x)$ almost everywhere and such that the upper density of the subsequence of indices $(m_k)_{k=1}^{\infty}$ is $1$. Questions on the density of $(m_k)_{k=1}^{\infty}$ are also examined for general almost everywhere convergent subsequences of measurable functions $(U_{m_k}(x))_{k=1}^{\infty}$.
Bibliography: 7 titles.
Keywords:
system of measurable functions, universal series, density of a subsequence of natural numbers, upper density, lower density.
Received: 05.05.2017 and 16.10.2017
Citation:
Sh. T. Tetunashvili, “Universal series and subsequences of functions”, Sb. Math., 209:10 (2018), 1498–1532
Linking options:
https://www.mathnet.ru/eng/sm8965https://doi.org/10.1070/SM8965 https://www.mathnet.ru/eng/sm/v209/i10/p89
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Abstract page: | 418 | Russian version PDF: | 50 | English version PDF: | 20 | References: | 68 | First page: | 26 |
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