Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series

2004 ◽  
Vol 76 (5/6) ◽  
pp. 673-681 ◽  
Author(s):  
M. I. D'yachenko
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Multiple Fourier Series ◽  
Partial Sums

scholarly journals Generalized localization for spherical partial sums of multiple Fourier series

2019 ◽  
Vol 489 (1) ◽  
pp. 7-10
Author(s):  
R. R. Ashurov
Keyword(s):  
Fourier Series ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Localization Principle ◽  
Open Set ◽  
Almost Everywhere

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - ​everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.

scholarly journals On the Localization of the Riesz Means of Multiple Fourier Series of Distributions

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
A. A. Rakhimov
Keyword(s):  
Fourier Series ◽  
Compact Support ◽  
Sufficient Conditions ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Riesz Means ◽  
Fourier Expansions

We study special partial sums of multiple Fourier series of distributions. We obtain sufficient conditions of summation of Riesz means of Fourier expansions of distributions with compact support.

scholarly journals Approximation by double Walsh polynomials

1992 ◽  
Vol 15 (2) ◽  
pp. 209-220 ◽  
Author(s):  
Ferenc Móricz
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Partial Sums ◽  
Lipschitz Classes ◽  
Sufficient Conditions For Convergence ◽  
By Products ◽  
De La Vallée Poussin

We study the rate of approximation by rectangular partial sums, Cesàro means, and de la Vallée Poussin means of double Walsh-Fourier series of a function in a homogeneous Banach spaceX. In particular,Xmay beLp(I2), where1≦p<∞andI2=[0,1)×[0,1), orCW(I2), the latter being the collection of uniformlyW-continuous functions onI2. We extend the results by Watari, Fine, Yano, Jastrebova, Bljumin, Esfahanizadeh and Siddiqi from univariate to multivariate cases. As by-products, we deduce sufficient conditions for convergence inLp(I2)-norm and uniform convergence onI2as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.

scholarly journals Summation of Multiple Fourier Series in Matrix Weighted -Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Morten Nielsen
Keyword(s):  
Fourier Series ◽  
Weighted Space ◽  
Multiple Fourier Series ◽  
Weighted Spaces ◽  
Partial Sums ◽  
Summation Methods ◽  
Matrix Weights ◽  
Muckenhoupt Condition ◽  
Jackson Kernel

This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted -spaces. We introduce a product Muckenhoupt condition for matrix weights and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space , , if and only if the weight satisfies the product Muckenhoupt condition. The same result is shown to hold true for other summation methods such as Cesàro and summation with the Jackson kernel.

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