Let G be a complex linear algebraic group and rho: G --> GL(V) a finite dimensional rational representation. Assume that G is connected and reductive, and that V has an open G-orbit. Let f in C[V] be a non-zero relative invariant with character phi is an element of Hom(G, C-X), meaning that f circle rho(g) = phi(g)f for all g in G. Choose a non-zero relative invariant f(V) in C[V-V], with character phi(-1), for the dual representation rho(V): G --> GL(V-V). Roughly, the fundamental theorem of the theory of prehomogeneous vector spaces due to M. Sate says that the Fourier transform of \f\(s) equals \f(V)\(-s) up to some factors. The purpose of the present paper is to study a finite field analogue of Sate's theorem and to give a completely explicit description of the Fourier transform assuming that the characteristic of the base field F-q is large enough. Now \f\(s) is replaced by chi(f), with chi in Hom (F-q(X), C-X), and the factors involve Gauss sums, the Bernstein-Sato polynomial b(s) of f, and the parity of the split rank of the isotropy group at upsilon(V) is an element of V-V(F-q). We also express this parity in terms of the quadratic residue of the discriminant of the Hessian of log f(V)(upsilon(V)). Moreover we prove a conjecture of N. Kawanaka on the number of integer roots of b(s).