Summary: 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 \(f(x,y)\phi(x,y)\) over \([-\pi,\pi]\times [- \pi,\pi]\) exists and Parseval’s formula holds, where \(f(x,y)\) is (in Pringsheim’s sense) the limiting function of \(s_{mn}(x,y)\) and the generalized Fourier series of \(\phi\) has bounded one-sided partial sums at \((0,0)\). One of its consequences is that the \(c_{jk}\) are the Fourier coefficients of \(f\) in the sense of the improper Riemann integral. This implies that if \(f\) is Lebesgue integrable, then the double trigonometric series determining \(f\) is the Fourier series of \(f\). These results can be extended to any multiple trigonometric series. Our results not only extend the results of N. K. Bary [“A treatise on trigonometric series” (1964; Zbl 0129.280); p. 656] and R. P. Boas jun. [Duke Math. J. 18, 787-793 (1951; Zbl 0045.033)], but also generalize results of F. Móricz [J. Math. Anal. Appl. 154, No. 2, 452-465 (1991; Zbl 0724.42013); ibid. 165, No. 2, 419-437 (1992; Zbl 0756.42011)].