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Mathematics of the USSR-Sbornik, 1979, Volume 35, Issue 4, Pages 527–539
DOI: https://doi.org/10.1070/SM1979v035n04ABEH001570
(Mi sm2615)
 

This article is cited in 14 scientific papers (total in 14 papers)

Convergence of Fourier series almost everywhere and in the $L$-metric

Sh. V. Kheladze
References:
Abstract: The following theorems are proved.
Theorem 1. There exists a constant $C>0$ such that for any function $f\in L(0,2\pi)$ there is a measurable function $F$ for which $|F|=|f|$, and
a) $\displaystyle\int_0^{2\pi}\sup_n|S_n(F)(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$,
b) $\displaystyle\int_0^{2\pi}\sup_n|{\widetilde{S}}_n(F)(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$,
c) $\displaystyle\int_0^{2\pi}|\widetilde{F}(x)|\,dx\leqslant C\int_0^{2\pi}|f(x)|\,dx$,
\noindent where $S_n(F)$ is a partial sum of the Fourier series of $F$, $\widetilde S_n(F)$ is a partial sum of the conjugate Fourier series, and $\widetilde F$ is the conjugate function to $F$.
\medskip Theorem 2. {\it For any function $f\in L(0,2\pi)$ and $\varepsilon>0$ there exists a measurable function $F$ such that $|F|=|f|$, $\mu\{x\in[0,2\pi):F(x)\ne f(x)\}<\varepsilon$ ($\mu$ is Lebesgue measure), and both the Fourier series of $F$ and its conjugate series converge almost everywhere and in the metric of $L$.}
Bibliography: 11 titles.
Received: 20.12.1977
Russian version:
Matematicheskii Sbornik. Novaya Seriya, 1978, Volume 107(149), Number 2(10), Pages 245–258
Bibliographic databases:
UDC: 517.51
MSC: Primary 42A20, 42A40; Secondary 42A04, 42A08
Language: English
Original paper language: Russian
Citation: Sh. V. Kheladze, “Convergence of Fourier series almost everywhere and in the $L$-metric”, Mat. Sb. (N.S.), 107(149):2(10) (1978), 245–258; Math. USSR-Sb., 35:4 (1979), 527–539
Citation in format AMSBIB
\Bibitem{Khe78}
\by Sh.~V.~Kheladze
\paper Convergence of Fourier series almost everywhere and in the $L$-metric
\jour Mat. Sb. (N.S.)
\yr 1978
\vol 107(149)
\issue 2(10)
\pages 245--258
\mathnet{http://mi.mathnet.ru/sm2615}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=512010}
\zmath{https://zbmath.org/?q=an:0404.42008}
\transl
\jour Math. USSR-Sb.
\yr 1979
\vol 35
\issue 4
\pages 527--539
\crossref{https://doi.org/10.1070/SM1979v035n04ABEH001570}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1979JJ04900006}
Linking options:
  • https://www.mathnet.ru/eng/sm2615
  • https://doi.org/10.1070/SM1979v035n04ABEH001570
  • https://www.mathnet.ru/eng/sm/v149/i2/p245
  • This publication is cited in the following 14 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
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    Abstract page:626
    Russian version PDF:127
    English version PDF:19
    References:77
     
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