Chen, Chang-Pao; Lin, Chin-Cheng Integrability, mean convergence, and Parseval’s formula for double trigonometric series. (English) Zbl 0907.42009 Taiwanese J. Math. 2, No. 2, 191-212 (1998). 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. Reviewer: F.Móricz (Szeged) Cited in 1 Document MSC: 42B05 Fourier series and coefficients in several variables 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) Keywords:double trigonometric series; regular convergence; uniform convergence; weighted \(L^r\)-integrability; Parseval’s formula PDFBibTeX XMLCite \textit{C.-P. Chen} and \textit{C.-C. Lin}, Taiwanese J. Math. 2, No. 2, 191--212 (1998; Zbl 0907.42009) Full Text: DOI