A group representation related to the Stockwell transform

We obtain a group structure admitting an irreducible and integrable representation on a Hilbert space with the property that the corresponding wavelet transform coincides with the Stockwell transform. The group is constructed in a similar way to the Weyl-Heisenberg group but it is not unimodular and it contains the affine group as a subgroup. The obtained results are coherent with the fact that the Stockwell transform is a hybrid of the Gabor and the wavelet transforms. We consider triples of Hilbert spaces of suitable tempered distributions and show that a reconstruction formula can be obtained for this generalized functional setting for the Stockwell transform. We restrict then our considerations to the Stockwell transform as classically defined and to a particular choice of the triple of Hilbert spaces which permits to consider the Gaussian, and more generally arbitrary functions in L2 as windows for this transform, and for which we have an inversion formula to recover the signal from its Stockwell transform. Such a formula already appeared in previous papers but under conditions which excluded the Gaussian as an admissible window. In fact, one of the main results of the paper is this generalization to Gaussians. The analysis of the group structure related to the Stockwell transform is also faced in the last section. It is proved that the group is not unimodular and it contains a subgroup isomorphic to the Affine group. Hence, although the construction of this group resembles that of the Weyl-Heisenberg group, it is in fact closer to the affine group.