Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–Fourier Series

2018 ◽  
Vol 44 (1) ◽  
pp. 73-88
Author(s):  
G. Gát ◽  
U. Goginava
Keyword(s):  
Fourier Series ◽  
Partial Sums ◽  
Two Dimensional ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere

Almost Everywhere Convergence of (𝐶, α)-Means of Quadratical Partial Sums of Double Vilenkin–Fourier Series

2006 ◽  
Vol 13 (3) ◽  
pp. 447-462
Author(s):  
György Gát ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Maximal Operator ◽  
Weak Type ◽  
Partial Sums ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere

Abstract We prove that the maximal operator of the (𝐶, α)-means of quadratical partial sums of double Vilenkin–Fourier series is of weak type (1,1). Moreover, the (𝐶, α)-means of a function 𝑓 ∈ 𝐿1 converge a.e. to 𝑓 as 𝑛 → ∞.

scholarly journals Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres

2021 ◽  
Vol 73 (3) ◽  
pp. 291-307
Author(s):  
A. A. Abu Joudeh ◽  
G. G´at
Keyword(s):  
Fourier Series ◽  
Maximal Operator ◽  
Weak Type ◽  
Partial Sums ◽  
Cesàro Means ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Walsh Group ◽  
Cesáro Means

UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type . Moreover, the -means of the function converge a.e. to for , where is the Walsh group for some sequences .

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.

APPLICATIONS OF MULTI-PARAMETER MARTINGALES IN FOURIER ANALYSIS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 551-568
Author(s):  
FERENC WEISZ
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Two Dimensional ◽  
Almost Everywhere Convergence ◽  
Fejér Means ◽  
Almost Everywhere ◽  
L Log L

With the help of the theory of multi-parameter martingales we prove almost everywhere convergence of the Fejér means of two-dimensional Walsh–Fourier series of f ∈ L log L.

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