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 Tr. Mat. Inst. Steklova, 2014, Volume 285, Pages 41–63 (Mi tm3547)

A weak generalized localization criterion for multiple Walsh–Fourier series with $J_k$-lacunary sequence of rectangular partial sums

S. K. Bloshanskayaa, I. L. Bloshanskiib

a National Engineering Physics Institute "MEPhI", Moscow, Russia
b Moscow State Region University, Moscow, Russia

Abstract: We obtain a criterion for the validity of weak generalized localization almost everywhere on an arbitrary set of positive measure $\mathfrak A$, $\mathfrak A\subset\mathbb I^N=\{x\in\mathbb R^N\colon0\leq x_j<1, j=1,2,…,N\}$, $N\geq3$ (in terms of the structure and geometry of the set $\mathfrak A$), for multiple Walsh–Fourier series (summed over rectangles) of functions $f$ in the classes $L_p(\mathbb I^N)$, $p>1$ (i.e., necessary and sufficient conditions for the convergence almost everywhere of the Fourier series on some subset of positive measure $\mathfrak A_1$ of the set $\mathfrak A$, when the function expanded in a series equals zero on $\mathfrak A$), in the case when the rectangular partial sums $S_n(x;f)$ of this series have indices $n=(n_1,…,n_N)\in\mathbb Z^N$ in which some components are elements of (single) lacunary sequences.

DOI: https://doi.org/10.1134/S0371968514020058

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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 285, 34–55

Bibliographic databases:

UDC: 517.5

Citation: S. K. Bloshanskaya, I. L. Bloshanskii, “A weak generalized localization criterion for multiple Walsh–Fourier series with $J_k$-lacunary sequence of rectangular partial sums”, Selected topics of mathematical physics and analysis, Collected papers. In commemoration of the 90th anniversary of Academician Vasilii Sergeevich Vladimirov's birth, Tr. Mat. Inst. Steklova, 285, MAIK Nauka/Interperiodica, Moscow, 2014, 41–63; Proc. Steklov Inst. Math., 285 (2014), 34–55

Citation in format AMSBIB
\Bibitem{BloBlo14} \by S.~K.~Bloshanskaya, I.~L.~Bloshanskii \paper A weak generalized localization criterion for multiple Walsh--Fourier series with $J_k$-lacunary sequence of rectangular partial sums \inbook Selected topics of mathematical physics and analysis \bookinfo Collected papers. In commemoration of the 90th anniversary of Academician Vasilii Sergeevich Vladimirov's birth \serial Tr. Mat. Inst. Steklova \yr 2014 \vol 285 \pages 41--63 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3547} \crossref{https://doi.org/10.1134/S0371968514020058} \elib{http://elibrary.ru/item.asp?id=21726841} \transl \jour Proc. Steklov Inst. Math. \yr 2014 \vol 285 \pages 34--55 \crossref{https://doi.org/10.1134/S0081543814040051} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000339949700005} \elib{http://elibrary.ru/item.asp?id=24048661} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84926365606} 

• http://mi.mathnet.ru/eng/tm3547
• https://doi.org/10.1134/S0371968514020058
• http://mi.mathnet.ru/eng/tm/v285/p41

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This publication is cited in the following articles:
1. Bloshanskaya S.K., Bloshanskii I.L., “Convergence and Localization in Orlicz Classes For Multiple Walsh-Fourier Series With a Lacunary Sequence of Rectangular Partial Sums”, J. Math. Anal. Appl., 435:1 (2016), 765–782
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