scholarly journals Convergence and divergence of Fejér means of Fourier series on one and two-dimensional Walsh and Vilenkin groups

2008 ◽  
Vol 21 (3) ◽  
pp. 291-307
Author(s):  
György Gát
Keyword(s):  
Fourier Series ◽  
Harmonic Analysis ◽  
Pointwise Convergence ◽  
Point Of View ◽  
Two Dimensional ◽  
Fejér Means ◽  
Vilenkin Groups ◽  
Recent Developments ◽  
Convergence And Divergence

It is a highly celebrated issue in dyadic harmonic analysis the pointwise convergence of the Fej?r (or (C, 1)) means of functions on the Walsh and Vilenkin groups both in the point of view of one and two dimensional cases. We give a resume of the very recent developments concerning this matter, propose unsolved problems and throw a glance at the investigation of Vilenkin-like systems too. .

Pointwise convergence of Marcinkiewicz-Fejér means of two-dimensional Walsh-Fourier series

2012 ◽  
Vol 49 (2) ◽  
pp. 236-253
Author(s):  
Ushangi Goginava ◽  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Pointwise Convergence ◽  
Two Dimensional ◽  
Fejér Means

In this paper we characterize the set of convergence of the Marcinkiewicz-Fejér means of two-dimensional Walsh-Fourier series.

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.

scholarly journals Subsequences of triangular partial sums of double fourier series on unbounded Vilenkin groups

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 3769-3778
Author(s):  
György Gát ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Double Fourier Series ◽  
Partial Sums ◽  
Two Dimensional ◽  
Vilenkin Groups ◽  
Almost Everywhere

In 1987 Harris proved-among others-that for each 1 ? p < 2 there exists a two-dimensional function f ? Lp such that its triangular partial sums S?2A f of Walsh-Fourier series does not converge almost everywhere. In this paper we prove that subsequences of triangular partial sums S?nAMAf,nA ? {1,2, ...,mA-1} on unbounded Vilenkin groups converge almost everywhere to f for each function f ? L2.

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