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Journal of Function Spaces and Applications
Volume 2012, Article ID 582726, 15 pages
http://dx.doi.org/10.1155/2012/582726
Research Article

Convergence in Measure of Logarithmic Means of Quadratical Partial Sums of Double Walsh-Kaczmarz-Fourier Series

1Institute of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze Street 1, 0128 Tbilisi, Georgia
2Institute of Mathematics and Computer Sciences, College of Nyíregyháza, P.O. Box 166, Nyíregyháza 4400, Hungary

Received 31 January 2011; Accepted 3 April 2011

Academic Editor: Anna Kaminska

Copyright © 2012 Ushangi Goginava and Károly Nagy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿 log+ 𝐿(𝐼2), the set of the functions the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series of which converge in measure is of first Baire category.