  Extending results of Kazhdan to the relative case, we relate harmonic
analysis over some spherical spaces G(F)/H(F), where F is a field of positive
characteristic, to harmonic analysis over the spherical spaces G(E)/H(E), where
E is a suitably chosen field of characteristic 0. One of the Ingredients of the
proof is a condition for finite generation of some modules over the Hecke
algebra. We apply our results to show that the pair (GL_{n+1},GL_n) is a strong
Gelfand pair for all local fields, and that the pair (GL_{n+k},GL_n x GL_k) is
a Gelfand pair for all local fields of odd characteristic.
