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.

STRUCTURAL AND GEOMETRIC CHARACTERISTICS OF SETS OF CONVERGENCE AND DIVERGENCE OF MULTIPLE FOURIER SERIES OF FUNCTIONS WHICH EQUAL ZERO ON SOME SET

2004 ◽  
Vol 02 (02) ◽  
pp. 187-195 ◽  
Author(s):  
I. L. BLOSHANSKII
Keyword(s):  
Fourier Series ◽  
Positive Measure ◽  
Linear Subspace ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Geometric Characteristics ◽  
Convergence Almost Everywhere ◽  
Almost Everywhere ◽  
Dimensional Cube ◽  
Series Of Functions

Let E be an arbitrary set of positive measure in the N-dimensional cube TN=(-π,π)N⊂ℝN, N≥1, and let f(x)=0 on E. Let [Formula: see text] be some linear subspace of L1(TN). We investigate the behavior of rectangular partial sums of multiple trigonometric Fourier series of a function f on the sets E and TN\E depending on smoothness of the function f (i.e. of the space [Formula: see text]), and, as well, of structural and geometric characteristics of the set E (SGC(E)). Thus, we are describing pairs [Formula: see text]. It is convenient to formulate and investigate the posed question in terms of generalized localization almost everywhere (GL) and weak generalized localization almost everywhere (WGL). This means that for the multiple Fourier series of a function f, that equals zero on the set E, convergence almost everywhere is investigated on the set E (GL), or on some of its subsets E1⊂E, of positive measure (WGL).

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.

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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