Che´bli–Trime&grave;che hypergroups and <f>W</f>-type spaces

The generalized Fourier transformation associated with Che´bli–Trime&grave;che hypergroups is investigated in some spaces of <f>W</f>-type introduced by Gelfand and Shilov. It is established that this transformation is an isomorphism from the space <f>WM,a</f> onto the space <f>WM×,1/a</f>, where the function <f>M</f> and the parameter <f>a</f> determine the growth of the testing functions in the first space, and <f>M×</f> denotes the Young dual function of <f>M</f>. The translation operator and the convolution corresponding to this transform are also studied in this class of spaces. [Copyright &y& Elsevier]