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

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

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

Full text: PDF file (568 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.52
MSC: 42B05, 42B08
Received: 15.11.2013

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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    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. Oniani, “On the convergence of sparse multiple series”, Proc. A. Razmadze Math. Inst., 167 (2015), 151–155  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathscinet  zmath  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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