INTEGRABILITY OF MULTIPLE TRIGONOMETRIC SERIES AND PARSEVALS FORMULA

[摘要] 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 off(x, y)phi(x, y) over [-pi, pi] X [-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 fis the Fourier series of f. 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 Moricz [J. Math. Anal, Appl. 154 (1991), 452-465; 165 (1992), 419-437] (C) 1994 Academic Press, Inc.

[发布日期] 1994-08-15