Wavelet Transforms and Admissible Group Representations

Abstract

Discrete wavelet transforms arise naturally from the sampling of a continuous wavelet transform, which in turn arises from a square-integrable (or more generally, admissible) representation of a locally compact group. We show that discrete wavelet transforms arising from groups with an affine structure possess an analogous admissibility condition. In particular, we show that the group performing the role of translations must satisfy an admissibility property. Finally, we relate these results to several notions of multiresolution.