Extension of a Theorem of Ferenc Lukács from Single to Double Conjugate Series

A theorem of Ferenc Lukács states that if a periodic function f is integrable in the Lebesgue sense and has a discontinuity of the first kind at some point x, then the mth partial sum of the conjugate series of its Fourier series diverges at x at the rate of logm. The aim of the present paper is to...

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Veröffentlicht in:Journal of mathematical analysis and applications Jg. 259; H. 2; S. 582 - 595
1. Verfasser: MORICZ, Ferenc
Format: Journal Article
Sprache:Englisch
Veröffentlicht: San Diego, CA Elsevier Inc 15.07.2001
Elsevier
Schlagworte:
conjugate series
criterion of nonatomic measure
Exact sciences and technology
Fourier analysis
Fourier series
function of bounded variation over a rectangle in the sense of Hardy and Krause
induced Borel measure
Mathematical analysis
Mathematics
rate of divergence
rectangular partial sum
Sciences and techniques of general use
sector limits of a function in two variables
rectangular partial sum
rate of divergence
function of bounded variation over a rectangle in the sense of Hardy and Krause
induced Borel measure
sector limits of a function in two variables
Fourier series
conjugate series
criterion of nonatomic measure
Mathematical series
Rate of divergence
Conjugate series
Partial sum
Functional analysis
Borel measure
Ferenc Lukacs theorem
ISSN:0022-247X, 1096-0813
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Zusammenfassung:A theorem of Ferenc Lukács states that if a periodic function f is integrable in the Lebesgue sense and has a discontinuity of the first kind at some point x, then the mth partial sum of the conjugate series of its Fourier series diverges at x at the rate of logm. The aim of the present paper is to extend this theorem to the rectangular partial sum of the conjugate series of a double Fourier series when conjugation is taken with respect to both variables. We also consider functions of two variables which are of bounded variation over a rectangle in the sense of Hardy and Krause. As a corollary, we obtain that the terms of the Fourier series of a periodic function f of bounded variation over the square [−π,π]×[−π,π] determine the atoms of the finite Borel measure induced by f.
ISSN:0022-247X
1096-0813
DOI:10.1006/jmaa.2001.7432