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

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

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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• http://mi.mathnet.ru/eng/izv8240
• https://doi.org/10.4213/im8240
• http://mi.mathnet.ru/eng/izv/v79/i4/p205

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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
2. F. Weisz, Convergence and summability of Fourier transforms and Hardy spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, 2017, xxii+435 pp.
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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