plays an extremely important role, where kA is the adele ring of k, X is a nontrivial character of the additive group of kA, trivial on k, and dx is the canonical measure on k. For example, when f is a quadratic mapping, the sum Leekm g9f($) is closely related to the Eisenstein-Siegel series in the sense of Weil [12], and when m = 1 and f is arbitrary, the integral | f(9()dd is kA substantially the singular series in the sense of Ramanujan-Hardy-Littlewood [2, IX]. For an obvious reason, we propose to call the mapping p 9.Wfthe Gauss transform relative to f; when n = m and f is the identity, tqw is nothing but the Fourier transform ' of p. In case m = 1, it is also important to consider the integral