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 Mat. Zametki, 2016, Volume 100, Issue 1, Pages 163–179 (Mi mz11124)

Almost Everywhere Summability of Fourier Series with Indication of the Set of Convergence

R. M. Trigub

Sumy State University

Abstract: In this paper, the following problem is studied. For what multipliers $\{\lambda_{k,n}\}$ do the linear means of the Fourier series of functions $f\in L_1[-\pi,\pi]$,
$$\sum_{k=-\infty}^\infty \lambda_{k,n}\widehat{f}_k e^{ikx}, \qquad where \widehat{f_k is the kth Fourier coefficient},$$
converge as $n\to \infty$ at all points at which the derivative of the function $\int_0^x f$ exists? In the case $\lambda_{k,n}=(1-|k|/(n+1))_+$, a criterion of the convergence of the $(C,1)$-means and, in the general case $\lambda_{k,n}=\phi(k/(n+1))$, a sufficient condition of the convergence at all such points (i.e., almost everywhere) are obtained. In the general case, the answer is given in terms of whether $\phi(x)$ and $x\phi'(x)$ belong to the Wiener algebra of absolutely convergent Fourier integrals. New examples are given.

Keywords: Fourier series, Lebesgue point, $d$-point, Wiener–Banach algebra, Szidon's inequality, Hardy–Littlewood inequality.

DOI: https://doi.org/10.4213/mzm11124

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English version:
Mathematical Notes, 2016, 100:1, 139–153

Bibliographic databases:

UDC: 517.518.4+517.443
Revised: 17.02.2016

Citation: R. M. Trigub, “Almost Everywhere Summability of Fourier Series with Indication of the Set of Convergence”, Mat. Zametki, 100:1 (2016), 163–179; Math. Notes, 100:1 (2016), 139–153

Citation in format AMSBIB
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\jour Math. Notes
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\vol 100
\issue 1
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• http://mi.mathnet.ru/eng/mz11124
• https://doi.org/10.4213/mzm11124
• http://mi.mathnet.ru/eng/mz/v100/i1/p163

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This publication is cited in the following articles:
1. F. Weisz, Convergence and summability of Fourier transforms and Hardy spaces, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, 2017, 435 pp.
2. D. V. Fufaev, “Summation of Fourier Series on the Infinite-Dimensional Torus”, Math. Notes, 103:6 (2018), 990–996
3. R. M. Trigub, “The Fourier transform of bivariate functions that depend only on the maximum of the absolute values of their variables”, Sb. Math., 209:5 (2018), 759–779
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