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 Trudy Inst. Mat. i Mekh. UrO RAN, 2015, Volume 21, Number 4, Pages 30–45 (Mi timm1227)

On almost everywhere convergence for lacunary sequences of multiple rectangular Fourier sums

N. Yu. Antonov

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

Abstract: Let a sequence of $d$-dimensional vectors $\mathbf{n}_k=(n_k^1, n_k^2,\ldots,n_k^d)$ with positive integer coordinates satisfy the condition $n_k^j=\alpha_j m_k+O(1), k \in {\mathbb N}, 1 \le j \le d,$\; where $\alpha _1>0,$ $\ldots,\alpha _d>0,$ and $\{ m_k \} _{k=1}^{\infty }$ is an increasing sequence of positive integers. Under some conditions on a function $\varphi :[0,+\infty ) \to [0,+\infty )$, it is proved that, if the sequence of Fourier sums $S_{m_k}(g,x)$ converges almost everywhere for any function $g \in \varphi (L) ([0 , 2\pi ))$, then, for any $d \in {\mathbb N}$ and $f \in \varphi (L)(\ln ^+L)^{d-1}([0 , 2\pi ) ^d)$, the sequence $S_{\mathbf {n}_k} (f,\mathbf x)$ of rectangular partial sums of the multiple trigonometric Fourier series of the function $f$ and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.

Keywords: multiple trigonometric fourier series, convergence almost everywhere.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2017, 296, suppl. 1, 43–59

Bibliographic databases:

UDC: 517.518

Citation: N. Yu. Antonov, “On almost everywhere convergence for lacunary sequences of multiple rectangular Fourier sums”, Trudy Inst. Mat. i Mekh. UrO RAN, 21, no. 4, 2015, 30–45; Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 43–59

Citation in format AMSBIB
\Bibitem{Ant15} \by N.~Yu.~Antonov \paper On almost everywhere convergence for lacunary sequences of multiple rectangular Fourier sums \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2015 \vol 21 \issue 4 \pages 30--45 \mathnet{http://mi.mathnet.ru/timm1227} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3468428} \elib{http://elibrary.ru/item.asp?id=25300982} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2017 \vol 296 \issue , suppl. 1 \pages 43--59 \crossref{https://doi.org/10.1134/S0081543817020055} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000403678000005}