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 Trudy Inst. Mat. i Mekh. UrO RAN, 2008, Volume 14, Number 3, Pages 3–18 (Mi timm36)

On the almost everywhere convergence of sequences of multiple rectangular Fourier sums

N. Yu. Antonov

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: In the case when a sequence of $d$-dimensional vectors $\mathrm n_k=(n_k^1,n_k^2,…,n_k^d)$ with nonnegative integer coordinates satisfies the condition
$$n_k^j=\alpha_j m_k+O(1),\quad k\in\mathbb N,\quad1\le j\le d,$$
where $\alpha_1…\alpha_d>0$, à $m_k\in\mathbb N$, $\lim_{k\to\infty}m_k=\infty$, under some conditions on the function $\varphi\colon[0,+\infty)\to[0,+\infty)$, it is proved that, if the trigonometric Fourier series of any function from $\varphi(L)([-\pi,\pi))$ converges almost everywhere, then, for any $d\in\mathbb N$ and all $f\in\varphi(L)(\ln^+L)^{d-1}([-\pi,\pi)d)$, the sequence $S_{\mathrm{n}_k}(f,\mathrm x)$ of the rectangular partial sums of the multiple trigonometric Fourier series of the function $f$, as well as the corresponding sequences of partial sums of all of its conjugate series, converges almost everywhere.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2009, 264, suppl. 1, S1–S18

Bibliographic databases:

UDC: 517.518

Citation: N. Yu. Antonov, “On the almost everywhere convergence of sequences of multiple rectangular Fourier sums”, Trudy Inst. Mat. i Mekh. UrO RAN, 14, no. 3, 2008, 3–18; Proc. Steklov Inst. Math. (Suppl.), 264, suppl. 1 (2009), S1–S18

Citation in format AMSBIB
\Bibitem{Ant08} \by N.~Yu.~Antonov \paper On the almost everywhere convergence of sequences of multiple rectangular Fourier sums \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2008 \vol 14 \issue 3 \pages 3--18 \mathnet{http://mi.mathnet.ru/timm36} \elib{http://elibrary.ru/item.asp?id=11929741} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2009 \vol 264 \issue , suppl. 1 \pages S1--S18 \crossref{https://doi.org/10.1134/S0081543809050010} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000265511100001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-65349190777} 

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This publication is cited in the following articles:
1. N. Yu. Antonov, “On the growth rate of arbitrary sequences of double rectangular Fourier sums”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S14–S20
2. N. Yu. Antonov, “On almost everywhere convergence for lacunary sequences of multiple rectangular Fourier sums”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 43–59
3. Nikolai Yu. Antonov, “On $\Lambda$-convergence almost everywhere of multiple trigonometric Fourier series”, Ural Math. J., 3:2 (2017), 14–21
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