Chébli-Trimèche hypergroups and W-type spaces

The generalized Fourier transformation associated with Chébli-Trimèche hypergroups is investigated in some spaces of W-type introduced by Gelfand and Shilov. It is established that this transformation is an isomorphism from the space W_M,a onto the space W^M×,1/a, where the function M and the parameter a determine the growth of the testing functions in the first space, and M^× denotes the Young dual function of M. The translation operator and the convolution corresponding to this transform are also studied in this class of spaces.