On the summability of generalized Fourier series by Abel's method
Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 227-239
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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.
@article{SM_1985_50_1_a14,
author = {A. Yu. Petrovich},
title = {On the summability of generalized {Fourier} series by {Abel's} method},
journal = {Sbornik. Mathematics},
pages = {227--239},
year = {1985},
volume = {50},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1985_50_1_a14/}
}
A. Yu. Petrovich. On the summability of generalized Fourier series by Abel's method. Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 227-239. http://geodesic.mathdoc.fr/item/SM_1985_50_1_a14/
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[3] Petrovich A. Yu., “O predstavlenii monotonnoi funktsii trigonometricheskim ryadom”, Matem. zametki, 27 (1980), 691–699 | MR | Zbl