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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 1, Pages 99–116 (Mi izv8048)  

This article is cited in 3 scientific papers (total in 3 papers)

On the convergence of multiple Haar series

G. G. Oniani

Akaki Tsereteli State University, Kutaisi

Abstract: We prove that the rectangular and spherical partial sums of the multiple Fourier–Haar series of an individual summable function may behave differently at almost every point, although it is known that they behave in the same way from the point of view of almost-everywhere convergence in the scale of integral classes: $L(\ln^+L)^{n-1}$ is the best class in both cases. We also find optimal additional conditions under which the spherical convergence of a multiple Fourier–Haar series (general Haar series, lacunary series) follows from its convergence by rectangles, and prove that these conditions are indeed optimal.

Keywords: multiple Haar series, convergence by rectangles, spherical convergence, lacunary series.

DOI: https://doi.org/10.4213/im8048

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English version:
Izvestiya: Mathematics, 2014, 78:1, 90–105

Bibliographic databases:

Document Type: Article
UDC: 517.52
MSC: 42C40, 40B05, 40F05
Received: 27.08.2012
Revised: 15.12.2012

Citation: G. G. Oniani, “On the convergence of multiple Haar series”, Izv. RAN. Ser. Mat., 78:1 (2014), 99–116; Izv. Math., 78:1 (2014), 90–105

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  • https://doi.org/10.4213/im8048
  • http://mi.mathnet.ru/eng/izv/v78/i1/p99

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. G. G. Oniani, “The convergence of double Fourier-Haar series over homothetic copies of sets”, Sb. Math., 205:7 (2014), 983–1003  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. S. B. Vakarchuk, A. N. Shchitov, “Estimates for the error of approximation of functions in $L_p^1$ by polynomials and partial sums of series in the Haar and Faber–Schauder systems”, Izv. Math., 79:2 (2015), 257–287  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. G. Oniani, “On the Convergence of Sparse Multiple Series”, Proceedings of a Razmadze Mathematical Institute, 167 (2015), 151–155  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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