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

This article is cited in 3 scientific papers (total in 3 papers)

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 $f\in L_1[-\pi,\pi]$,
$$ \sum_{k=-\infty}^\infty \lambda_{k,n}\widehat{f}_k e^{ikx}, \qquad where $\widehat{f_k$ is the $k$th Fourier coefficient}, $$
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
Received: 02.09.2015
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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  • http://mi.mathnet.ru/eng/mz/v100/i1/p163

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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. F. Weisz, Convergence and summability of Fourier transforms and Hardy spaces, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, 2017, 435 pp.  crossref  mathscinet  isi
    2. D. V. Fufaev, “Summation of Fourier Series on the Infinite-Dimensional Torus”, Math. Notes, 103:6 (2018), 990–996  mathnet  crossref  crossref  isi  elib
    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  mathnet  crossref  crossref  adsnasa  isi  elib
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