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Mathematics of the USSR-Sbornik, 1985, Volume 50, Issue 1, Pages 227–239
DOI: https://doi.org/10.1070/SM1985v050n01ABEH002826
(Mi sm2288)
 

On the summability of generalized Fourier series by Abel's method

A. Yu. Petrovich
References:
Abstract: For $2\pi$-periodic functions $f$ that have, on $[-\pi,\pi]$, only the point 0 as a nonsummable singular point, we consider generalized Fourier series depending on an integer-valued function $N(x)$. It is shown that if $|x|^{\alpha(x)}f(x)\in L(-\pi,\pi)$, where $\alpha(x)$ is an even nonnegative function, nonincreasing on $(0,\pi]$, and $\alpha(x)=o(\ln\frac1x)$, $x\to+0$, then under a certain condition on $N(x)$ the generalized Fourier series is almost everywhere summable to $f(x)$ by the Abel method. The estimate $o(\ln\frac1x)$ and the hypothesis on $N(x)$ are, in a certain sense, definitive.
Bibliography: 3 titles.
Received: 03.04.1981
Bibliographic databases:
UDC: 517.51
MSC: 42A24
Language: English
Original paper language: Russian
Citation: A. Yu. Petrovich, “On the summability of generalized Fourier series by Abel's method”, Math. USSR-Sb., 50:1 (1985), 227–239
Citation in format AMSBIB
\Bibitem{Pet83}
\by A.~Yu.~Petrovich
\paper On the summability of generalized Fourier series by Abel's method
\jour Math. USSR-Sb.
\yr 1985
\vol 50
\issue 1
\pages 227--239
\mathnet{http://mi.mathnet.ru//eng/sm2288}
\crossref{https://doi.org/10.1070/SM1985v050n01ABEH002826}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=717677}
\zmath{https://zbmath.org/?q=an:0566.42005|0535.42009}
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    Russian version PDF:161
    English version PDF:19
    References:62
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