Integrability of Multiple Trigonometric Series and Parseval′s Formula

Abstract

Let s mn ( x, y) denote the rectangular partial sums of the double trigonometric series with the coefficients c jk . We prove that if the c jk 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 s mn ( 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 c jk 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]