scholarly journals Convergence in Measure of Logarithmic Means of Double Walsh–Fourier Series

2005 ◽  
Vol 12 (4) ◽  
pp. 607-618
Author(s):  
György Gát ◽  
Ushangi Goginava ◽  
George Tkebuchava
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Convergence In Measure ◽  
Logarithmic Means

Abstract The main aim of this paper is to prove that the logarithmic means of the double Walsh–Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿log 𝐿(𝐼2), the set of functions for which quadratic logarithmic means of the double Walsh–Fourier series converge in measure is of first Baire category.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Ushangi Goginava ◽  
Károly Nagy
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Partial Sums ◽  
Convergence In Measure ◽  
Logarithmic Means

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.

Convergence in Measure of Partial Sums of Double Vilenkin–Fourier Series

2009 ◽  
Vol 16 (3) ◽  
pp. 507-516
Author(s):  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Partial Sums ◽  
Convergence In Measure

Abstract The main aim of this paper is to prove that the partial sums of double Vilenkin–Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿ln+ 𝐿, the set of functions whose quadratic partial sums of double Vilenkin–Fourier series converge in measure is of first Baire category.

scholarly journals Integrability of the Majorant of the Fourier Series Partial Sums with Respect to Bases

2005 ◽  
Vol 12 (1) ◽  
pp. 181-188
Author(s):  
George Tkebuchava
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Partial Sums

Abstract The majorant of Fourier series partial sums with respect to the system of functions formed by the product of 𝐿([0, 1]) space bases is considered. It is proved that in any Orlicz space wider than 𝐿(log+ 𝐿)𝑑([0, 1]𝑑), 𝑑 ≥ 1, the set of functions with such a majorant is integrable on [0, 1]𝑑 and has the first Baire category.

Uniform and 𝐿-Convergence of Logarithmic Means of Double Walsh–Fourier Series

2005 ◽  
Vol 12 (1) ◽  
pp. 75-88
Author(s):  
György Gát ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Function Space ◽  
Modulus Of Continuity ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Two Dimensional ◽  
Logarithmic Means ◽  
Convergence And Divergence ◽  
Necessary And Sufficient ◽  
Series Of Functions

Abstract We discuss some convergence and divergence properties of twodimensional (Nörlund) logarithmic means of two-dimensional Walsh–Fourier series of functions both in the uniform and in the Lebesgue norm. We give necessary and sufficient conditions for the convergence regarding the modulus of continuity of the function, and also the function space.

On the boundedness of the maximal operators of double Walsh-logarithmic means of Marcinkiewicz type

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Ushangi Goginava ◽  
Károly Nagy
Keyword(s):  
Fourier Series ◽  
Type Inequality ◽  
Maximal Operators ◽  
Partial Sums ◽  
Logarithmic Means ◽  
P Type

AbstractThe main aim of this paper is to investigate the (H p, L p)-type inequality for the maximal operators of Riesz and Nörlund logarithmic means of the quadratical partial sums of Walsh-Fourier series. Moreover, we show that the behavior of Nörlund logarithmic means is worse than the behavior of Riesz logarithmic means in our special sense.

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