scholarly journals Convergence almost everywhere of multiple Fourier series over cubes

2017 ◽  
Vol 370 (3) ◽  
pp. 1629-1659
Author(s):  
Mieczysław Mastyło ◽  
Luis Rodríguez-Piazza
Author(s):  
I. L. BLOSHANSKII

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).


2019 ◽  
Vol 489 (1) ◽  
pp. 7-10
Author(s):  
R. R. Ashurov

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.


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