Convergence of logarithmic means of quadratic partial sums of double Fourier series

2013 ◽  
Vol 131 (1) ◽  
pp. 99-112 ◽  
Author(s):  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Double Fourier Series ◽  
Partial Sums ◽  
Logarithmic Means

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 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.

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.

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.

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