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

A theorem of Ferenc Lukacs states that if a periodic function f is integrab le 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 se ries 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 corollar y, 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 at oms of the finite Borel measure induced by f. (C) 2001 Academic Press.