Convergence almost everywhere of Fourier series with respect to orthonormal systems of Franklin type

1982 ◽  
Vol 23 (2) ◽  
pp. 188-197
Author(s):  
V. N. Demenko
Keyword(s):  
Fourier Series ◽  
Convergence Almost Everywhere ◽  
Almost Everywhere

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
Keyword(s):  
Fourier Series ◽  
Multiple Fourier Series ◽  
Convergence Almost Everywhere ◽  
Almost Everywhere

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).

Function classes and convergence almost everywhere

2011 ◽  
Vol 18 (1) ◽  
pp. 177-189
Author(s):  
Vakhtang Tsagareishvili
Keyword(s):  
Fourier Series ◽  
Convergence Almost Everywhere ◽  
Function Classes ◽  
Almost Everywhere ◽  
The Family ◽  
Fourier Integrals

Abstract For a certain class of orthonormal systems (ONS) on [0, 1] there exists a family of functions from L 2(0, 1), independent from that class, such that the Fourier series with respect to each ONS of the class converges a.e. for any member of the family. Similar result holds for the summability of Fourier integrals.

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