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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 4, Pages 205–224 (Mi izv8240)  

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

Summability of trigonometric Fourier series at $d$-points and a generalization of the Abel–Poisson method

R. M. Trigub

Donetsk National University

Abstract: We study the convergence of linear means of the Fourier series $\sum_{k=-\infty}^{+\infty}\lambda_{k,\varepsilon}\hat{f}_ke^{ikx}$ of a function $f\in L_1[-\pi,\pi]$ to $f(x)$ as $\varepsilon\searrow0$ at all points at which the derivative $(\int_0^xf(t) dt)'$ exists (i. e. at the $d$-points). Sufficient conditions for the convergence are stated in terms of the factors $\{\lambda_{k,\varepsilon}\}$ and, in the case of $\lambda_{k,\varepsilon}=\varphi(\varepsilon k)$, in terms of the condition that the functions $\varphi$ and $x\varphi'(x)$ belong to the Wiener algebra $A(\mathbb R)$. We also study a new problem concerning the convergence of means of the Abel–Poisson type, $\sum_{k=-\infty}^\infty r^{\psi(|k|)}\hat{f}_ke^{ikx}$, as $r\nearrow1$ depending on the growth of the function $\psi\nearrow+\infty$ on the semi-axis. It turns out that $\psi$ cannot differ substantially from a power-law function.

Keywords: Fourier series, Banach algebra of absolutely convergent Fourier integrals, multiplier, Abel–Poisson method.

DOI: https://doi.org/10.4213/im8240

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English version:
Izvestiya: Mathematics, 2015, 79:4, 838–858

Bibliographic databases:

UDC: 517.51
MSC: 42A24
Received: 10.04.2014

Citation: R. M. Trigub, “Summability of trigonometric Fourier series at $d$-points and a generalization of the Abel–Poisson method”, Izv. RAN. Ser. Mat., 79:4 (2015), 205–224; Izv. Math., 79:4 (2015), 838–858

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. M. Trigub, “Almost Everywhere Summability of Fourier Series with Indication of the Set of Convergence”, Math. Notes, 100:1 (2016), 139–153  mathnet  crossref  crossref  mathscinet  isi  elib
    2. F. Weisz, Convergence and summability of Fourier transforms and Hardy spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, 2017, xxii+435 pp.  crossref  mathscinet  zmath  isi
    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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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