The standard Fourier transform is an object that plays a central role in the field of harmonic analysis. In the past decades, several important generalizations of the Fourier transform have been introduced, such as the Dunkl transform, and the (k,a) generalized transform. They provide deep links with the representation theory of finite reflection groups on the one hand, and with the theory of minimal representations on the other hand.

The present project aims at a deeper understanding of these generalized Fourier transforms, by constructing explicit expressions for the associated integral kernels, by computing bounds for the integral kernels and by developing generalized translation and convolution operators for them.