Extension of a theorem of Ferenc!Lukacs from single to double conjugate series

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:259

Article

Móricz, F

关键词: Fourier series;    conjugate series;    rectangular partial sum;    rate of divergence;    function of bounded variation over a rectangle in the sense of Hardy and Krause;    sector limits of a function in two variables;    induced Borel measure;    criterion of nonatomic measure;   

DOI  :  10.1006/jmaa.2001.7432

来源: Elsevier

【 摘 要 】

A theorem of Ferenc Lukacs 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 log m. 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 [-pi, pi] X [-pi, pi] determine the atoms of the finite Borel measure induced by f. (C) 2001 Academic Press.

【 授权许可】

Free