Chébli–Trimèche hypergroups and W-type spaces ✩

The generalized Fourier transformation associated with Chébli–Trimèche hypergroups is investigated in some spaces ofW-type introduced by Gelfand and Shilov. It is established that this transformation is an isomorphism from the space WM,a onto the space WM × ,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.  2002 Elsevier Science (USA). All rights reserved.