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This article is cited in 15 scientific papers (total in 15 papers)
Two-dimensional Waterman classes and $u$-convergence of Fourier series
M. I. Dyachenko M. V. Lomonosov Moscow State University
Abstract:
New results on the $u$-convergence of the double Fourier series of functions from Waterman classes are obtained. It turns out that none of the Waterman classes wider than $BV(T^2)$ ensures even the uniform boundedness of the $u$-sums of the double Fourier series of functions in this class. On the other hand, the concept of $u(K)$-convergence is introduced (the sums are taken over regions that are forbidden to stretch along coordinate axes) and it is proved that for functions $f(x,y)$ belonging to the class $\Lambda_{1/2}BV(T^2)$, where $\Lambda_a=\biggl\{\dfrac{n^{1/2}}{{(\ln(n+1))}^a}\biggr\}_{n=1}^\infty$, the corresponding $u(K)$-partial sums are uniformly bounded, while if $f(x,y)\in\Lambda_aBV(T^2)$, where $a<\frac12$, then the double Fourier series of $f(x,y)$ is $u(K)$-convergent everywhere.
Received: 28.10.1998
Citation:
M. I. Dyachenko, “Two-dimensional Waterman classes and $u$-convergence of Fourier series”, Sb. Math., 190:7 (1999), 955–972
Linking options:
https://www.mathnet.ru/eng/sm414https://doi.org/10.1070/sm1999v190n07ABEH000414 https://www.mathnet.ru/eng/sm/v190/i7/p23
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Abstract page: | 589 | Russian version PDF: | 246 | English version PDF: | 16 | References: | 70 | First page: | 1 |
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