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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Vydáno v:	Journal of mathematical analysis and applications Ročník 259; číslo 2; s. 582 - 595
Hlavní autor:	MORICZ, Ferenc
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	San Diego, CA Elsevier Inc 15.07.2001 Elsevier
Témata:	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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Popis
Shrnutí:	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