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2005/2006 Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series.
Ushangi Goginava, György Gát
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Real Anal. Exchange 31(2): 447-464 (2005/2006).

Abstract

The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double Walsh-Fourier series is of type $(H^{\#},L_{1})$ provided that the supremum in the maximal operator is taken over some special indices. The set of Walsh polynomials is dense in $ H^{\#}$, so by the well-known density argument we have that $t_{2^{n},2^{m}}f\left( x^{1},x^{2}\right) \rightarrow f\left( x^{1},x^{2}\right) $ a. e. as $m,n\rightarrow \infty$ for all $f\in H^{\#} (\supset L\log^{+} L$). We also prove the sharpness of this result. Namely, For all measurable function $\delta :[0,+\infty) \to [0,+\infty) , \, \lim_{t\to\infty}\delta(t)=0$ we have a function $f$ such as $f\in \llogld (L)$ and the two-dimensional Nörlund logarithmic means do not converge to $f$ a.e. (in the Pringsheim sense) on $I^2$.

Citation

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Ushangi Goginava. György Gát. "Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series.." Real Anal. Exchange 31 (2) 447 - 464, 2005/2006.

Information

Published: 2005/2006
First available in Project Euclid: 10 July 2007

zbMATH: 1103.42017
MathSciNet: MR2265786

Subjects:
Primary: 42C10

Rights: Copyright © 2005 Michigan State University Press

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