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

Abstract

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.