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

Ushangi Goginava,Károly Nagy

doi:10.1155/2012/582726

Ushangi Goginava, Károly Nagy

Open Access

https://doi.org/10.1155/2012/582726

Copy DOI

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 ofL 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.

Highlights

We denote by L0 L0 I2 the Lebesgue space of functions that are measurable and finite almost everywhere on I2 0, 1 × 0, 1 . mes A is the Lebesgue measure of the set A ⊂ I2
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
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

Summary

Definitions and Notations

We denote by L0 L0 I2 the Lebesgue space of functions that are measurable and finite almost everywhere on I2 0, 1 × 0, 1 . mes A is the Lebesgue measure of the set A ⊂ I2. 1.24 of double Walsh-Kaczmarz series and prove Theorem 2.1 that is, for any Orlicz space, which is not a subspace of L log L I2 , the set of the functions where logarithmic means converges in measure is of first Baire category. From this result it follows that Corollary 2.2 in classes wider than L log L I2 there exists functions f for which logarithmic means tκn f of quadratical partial sums of double Walsh-Kaczmarz series diverge in measure. In question of convergence in measure logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series differ from the Marcinkiewicz means and are like the usual quadratical partial sums of double Walsh-Fourier series

Main Results

Auxiliary Results

Proof of the Theorem