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Mat. Zametki, 2008, Volume 84, Issue 3, Pages 334–347 (Mi mz4000)  

This article is cited in 5 scientific papers (total in 5 papers)

Weak Generalized Localization for Multiple Fourier Series Whose Rectangular Partial Sums Are Considered with Respect to Some Subsequence

I. L. Bloshanskii, O. V. Lifantseva

Moscow State Region University

Abstract: In this paper, we obtain the structural and geometric characteristics of some subsets of $\mathbb{T}^N=[-\pi,\pi]^N$ (of positive measure), on which, for the classes $L_p(\mathbb{T}^N)$, $p>1$, where $N\ge 3$, weak generalized localization for multiple trigonometric Fourier series is valid almost everywhere, provided that the rectangular partial sums $S_n(x;f)$  ($x\in\mathbb{T}^N$, $f\in L_p$) of these series have a “number” $n=(n_1,…,n_N)\in\mathbb Z_{+}^{N}$ such that some components $n_j$ are elements of lacunary sequences. For $N=3$, similar studies are carried out for generalized localization almost everywhere.

Keywords: multiple Fourier series, weak generalized localization, generalized localization, partial sum, lacunary sequence, Hölder's inequality, Orlicz class

DOI: https://doi.org/10.4213/mzm4000

Full text: PDF file (618 kB)
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English version:
Mathematical Notes, 2008, 84:3, 314–327

Bibliographic databases:

UDC: 517.5
Received: 14.06.2007

Citation: I. L. Bloshanskii, O. V. Lifantseva, “Weak Generalized Localization for Multiple Fourier Series Whose Rectangular Partial Sums Are Considered with Respect to Some Subsequence”, Mat. Zametki, 84:3 (2008), 334–347; Math. Notes, 84:3 (2008), 314–327

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. O. V. Lifantseva, “Necessary Conditions for the Weak Generalized Localization of Fourier Series with “Lacunary Sequence of Partial Sums””, Math. Notes, 86:3 (2009), 373–384  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Bloshanskii I.L., Lifantseva O.V., “Structural and Geometric Characteristics of Sets of Convergence and Divergence of Multiple Fourier Series with J (K) -Lacunary Sequence of Rectangular Partial Sums”, Anal. Math., 39:2 (2013), 93–121  crossref  mathscinet  zmath  isi  elib  scopus
    3. I. L. Bloshanskii, Z. N. Tsukareva, “Localization for Multiple Fourier Series with “$J_k$-Lacunary Sequence of Partial Sums” in Orlicz Classes”, Math. Notes, 95:1 (2014), 22–31  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. 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”, Proc. Steklov Inst. Math., 285 (2014), 34–55  mathnet  crossref  crossref  isi  elib  elib
    5. 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  crossref  mathscinet  zmath  isi  scopus
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