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

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
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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8048
• https://doi.org/10.4213/im8048
• http://mi.mathnet.ru/eng/izv/v78/i1/p99

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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
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
3. G. Oniani, “On the Convergence of Sparse Multiple Series”, Proceedings of a Razmadze Mathematical Institute, 167 (2015), 151–155
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