Waterman classes and spherical partial sums of double Fourier series

1995 ◽  
Vol 21 (1) ◽  
pp. 3-21 ◽  
Author(s):  
M. I. Dyachenko
Keyword(s):  
Fourier Series ◽  
Double Fourier Series ◽  
Partial Sums

Convergence of the Hausdorff Means of Double Fourier Series*

1968 ◽  
Vol 11 (4) ◽  
pp. 585-591 ◽  
Author(s):  
Fred Ustina
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Double Fourier Series ◽  
Partial Sums ◽  
Closed Region ◽  
Hausdorff Means

In this paper we prove that if {sm, n(x, y)} is the sequence of partial sums of the Fourier series of a function f(x, y), which is periodic in each variable and of bounded variation in the sense of Hardy-Krause in the period rectangle, then {sm, n(x, y)} converges uniformly to f(x, y) in any closed region D in which this function is continuous at every point. This result is then used to prove that the regular Hausdorff means of the Fourier series of such a function also converge uniformly in such a region.

Uniform estimates for families of singular integrals and double Fourier series

1986 ◽  
Vol 41 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Elena Prestini
Keyword(s):  
Fourier Series ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Double Fourier Series ◽  
Variable Coefficients ◽  
Partial Sums ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Almost Everywhere ◽  
The One

AbstractIt is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

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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