Integrability of Multiple Trigonometric Series and Parseval′s Formula

Let smn(x, y) denote the rectangular partial sums of the double trigonometric series with the coefficients cjk. We prove that if the cjk form a null sequence of bounded variation, then the improper Riemann integral of ƒ(x, y)φ(x, y) over [−π, π] × [−π, π] exists and Parseval′s formula holds, where ƒ(x, y) is (in Pringsheim′s sense) the limiting function of smn(x, y) and the generalized Fourier series of φ has bounded one-sided partial sums at (0, 0), One of its consequences is that the cjk are the Fourier coefficients of ƒ in the sense of the improper Riemann integral. This implies that if ƒ is Lebesgue integrable, then the double trigonometric series determining ƒ is the Fourier series of ƒ. These results can be extended to any multiple trigonometric series. Our results not only extend the results of Bary ["A Treatise on Trigonometric Series," 1964, p. 656] and Boas [Duke Math. J.18 (1951), 787-793], but also generalize Móricz [J. Math. Anal. Appl.154 (1991), 452-465; 165 (1992), 419-437]