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Sbornik: Mathematics, 1999, Volume 190, Issue 7, Pages 955–972
DOI: https://doi.org/10.1070/sm1999v190n07ABEH000414
(Mi sm414)
 

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
References:
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
Bibliographic databases:
UDC: 517.52
MSC: Primary 42B05, 42B08; Secondary 26B30
Language: English
Original paper language: Russian
Citation: M. I. Dyachenko, “Two-dimensional Waterman classes and $u$-convergence of Fourier series”, Sb. Math., 190:7 (1999), 955–972
Citation in format AMSBIB
\Bibitem{Dya99}
\by M.~I.~Dyachenko
\paper Two-dimensional Waterman classes and $u$-convergence of Fourier series
\jour Sb. Math.
\yr 1999
\vol 190
\issue 7
\pages 955--972
\mathnet{http://mi.mathnet.ru//eng/sm414}
\crossref{https://doi.org/10.1070/sm1999v190n07ABEH000414}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1725211}
\zmath{https://zbmath.org/?q=an:0937.42006}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000084021300002}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0033240352}
Linking options:
  • https://www.mathnet.ru/eng/sm414
  • https://doi.org/10.1070/sm1999v190n07ABEH000414
  • https://www.mathnet.ru/eng/sm/v190/i7/p23
  • This publication is cited in the following 15 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник - 1992–2005 Sbornik: Mathematics
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    Abstract page:589
    Russian version PDF:246
    English version PDF:16
    References:70
    First page:1
     
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