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Mat. Sb., 1991, Volume 182, Number 5, Pages 622–637 (Mi msb1314)  

This article is cited in 2 scientific papers (total in 2 papers)

Almost everywhere convergence of multiple Fourier series of monotonic functions

M. I. Dyachenko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Let $m$ be a natural number, $m\geqslant2$. Then we shall say that a function $f(\mathbf t)$ of period $2\pi$ in each variable is monotonic if there exist an open rectangular parallelepiped $(\mathbf a,\mathbf b)=\prod\limits_{j=1}^m(a_j,b_j)\subseteq [-\pi,\pi)^m$ and numbers $\gamma_1,…,\gamma_m$, each of which is either 0 or 1, such that $f(\mathbf t)=0$ for $\mathbf t\in [-\pi,\pi)^m\setminus(\mathbf a,\mathbf b)$, and if $\mathbf x,\mathbf y\in (\mathbf a,\mathbf b)$ and $(-1)^{\gamma_j}x_j\leqslant(-1)^{\gamma_j}y_j$ for $j=1,…,m$, then $f(\mathbf x)\geqslant f(\mathbf y)$. The main result of this paper is that the multiple trigonometric Fourier series of an integrable monotonic function is Pringsheim convergent almost everywhere, in particular at each point of continuity of $f(\mathbf t)$ in the interior of $(\mathbf a,\mathbf b)$.

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English version:
Mathematics of the USSR-Sbornik, 1992, 73:1, 11–25

Bibliographic databases:

UDC: 517.51
MSC: 42B05
Received: 25.12.1989

Citation: M. I. Dyachenko, “Almost everywhere convergence of multiple Fourier series of monotonic functions”, Mat. Sb., 182:5 (1991), 622–637; Math. USSR-Sb., 73:1 (1992), 11–25

Citation in format AMSBIB
\Bibitem{Dya91}
\by M.~I.~Dyachenko
\paper Almost everywhere convergence of multiple Fourier series of monotonic functions
\jour Mat. Sb.
\yr 1991
\vol 182
\issue 5
\pages 622--637
\mathnet{http://mi.mathnet.ru/msb1314}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1124100}
\zmath{https://zbmath.org/?q=an:0782.42013|0733.42008}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..73...11D}
\transl
\jour Math. USSR-Sb.
\yr 1992
\vol 73
\issue 1
\pages 11--25
\crossref{https://doi.org/10.1070/SM1992v073n01ABEH002532}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1992KA53500002}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Dyachenko M., “Rate of Convergence of Fourier-Series of Multivariable Monotonic Functions in Pringsheim Sense”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1992, no. 4, 60–68  mathscinet  isi
  • Математический сборник - 1991 Sbornik: Mathematics (from 1967)
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