Geodesic
The convergence of double Fourier-Haar series over homothetic copies of sets
Sbornik. Mathematics, Tome 205 (2014) no. 7, pp. 983-1003

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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
Mots-clés : convergence. 
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     author = {G. G. Oniani},
     title = {The convergence of double {Fourier-Haar} series over homothetic copies of sets},
     journal = {Sbornik. Mathematics},
     pages = {983--1003},
     publisher = {mathdoc},
     volume = {205},
     number = {7},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2014_205_7_a3/}
}
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G. G. Oniani. The convergence of double Fourier-Haar series over homothetic copies of sets. Sbornik. Mathematics, Tome 205 (2014) no. 7, pp. 983-1003. http://geodesic.mathdoc.fr/item/SM_2014_205_7_a3/