Extension of a theorem of Fejer to double Fourier-Stieltjes series

A theorem of Fejer states that if a periodic function F is of bounded varia tion on the closed interval [0, 2 pi], then the nth partial sum of its form ally differentiated Fourier series divided by n converges to pi (-1)[F(x+0) -F(x-0)] at each point x. The generalization of this theorem for Fourier-St ieltjes series of (nonperiodic) functions of bounded variation is also well known. The aim of the present article is to extend these results to the (m, n)th r ectangular partial sum of double Fourier or Fourier-Stieltjes series of a f unction F(x, y) of bounded variation over the closed square [0, 2 pi] x [0, 2 pi] in the sense of Hardy and Krause. As corollaries, we also obtain the following results: (i) The terms of the Fourier or Fourier-Stieltjes series of F(x, y) determi ne the atoms of the (periodic) Borel measure induced by (an appropriate ext ension of) F, (ii) In the case of periodic functions F (x, y) of bounded variation, the c lass of double Fourier-Stieltjes series coincides with the class of series that can be obtained from their Fourier series by a formal termwise differe ntiation with respect to both x and y.