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 Mat. Sb., 2014, Volume 205, Number 7, Pages 73–94 (Mi msb8303)

The convergence of double Fourier-Haar series over homothetic copies of sets

G. G. Oniani

Akaki Tsereteli State University, Kutaisi

Abstract: The paper is concerned with the convergence of double Fourier-Haar series with partial sums taken over homothetic copies of a given bounded set $W\subset \mathbb{R}_+^2$ containing the intersection of some neighbourhood of the origin with $\mathbb{R}_+^2$. It is proved that for a set $W$ from a fairly broad class (in particular, for convex $W$) there are two alternatives: either the Fourier-Haar series of an arbitrary function $f\in L([0,1]^2)$ converges almost everywhere or $L\ln^+L([0,1]^2)$ is the best integral class in which the double Fourier-Haar series converges almost everywhere. Furthermore, a characteristic property is obtained, which distinguishes which of the two alternatives is realized for a given $W$.
Bibliography: 12 titles.

Keywords: Fourier-Haar series, double series, lacunary series, convergence.

 Funding Agency Grant Number Shota Rustaveli National Science Foundation 31/48

DOI: https://doi.org/10.4213/sm8303

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English version:
Sbornik: Mathematics, 2014, 205:7, 983–1003

Bibliographic databases:

UDC: 517.52
MSC: 42B05, 42B08

Citation: G. G. Oniani, “The convergence of double Fourier-Haar series over homothetic copies of sets”, Mat. Sb., 205:7 (2014), 73–94; Sb. Math., 205:7 (2014), 983–1003

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8303
• https://doi.org/10.4213/sm8303
• http://mi.mathnet.ru/eng/msb/v205/i7/p73

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This publication is cited in the following articles:
1. G. Oniani, “On the convergence of sparse multiple series”, Proc. A. Razmadze Math. Inst., 167 (2015), 151–155
2. M. G. Plotnikov, Yu. A. Plotnikova, “Decomposition of dyadic measures and unions of closed $\mathscr{U}$-sets for series in a Haar system”, Sb. Math., 207:3 (2016), 444–457
3. M. G. Plotnikov, V. A. Skvortsov, “On various types of continuity of multiple diadic intervals”, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 32:2 (2016), 247–275
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