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The generalized Fourier transformation associated with Che´bli–Trime`che 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 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. [Copyright &y& Elsevier]