Integrability, mean convergence, and Parseval's formula for double trigonometric series (Q1392427)

The authors consider the double trigonometric series \[ \sum^\infty_{j=-\infty} \sum^\infty_{k= -\infty} c_{jk} e^{i(jx+ ky)}\tag{\(*\)} \] whose coefficients \(c_{jk}\) satisfy the conditions of bounded variation of order \((p,0)\), \((0,p)\), and \((p,p)\) with respect to the weight \((|\overline j| |\overline k|)^{p- 1}\) for some \(1<p<\infty\). They prove (i) regular convergence, (ii) uniform convergence, (iii) weighted \(L^r\)-integrability and weighted \(L^r\)-convergence of the symmetric rectangular partial sums of series \((*)\); (iv) the validity of Parseval's formula. These results generalize a number of earlier results by various authors.