Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups

Abstract

This article presents a systematic study for abstract harmonic analysis aspects of wave-packet transforms over locally compact abelian (LCA) groups. Let $H$ be a locally compact group, let $K$ be an LCA group, and let $\theta :H\to Aut\left(K\right)$ be a continuous homomorphism. We introduce the abstract notion of the wave-packet group generated by $\theta$, and we study basic properties of wave-packet groups. Then we study theoretical aspects of wave-packet transforms. Finally, we will illustrate application of these techniques in the case of some well-known examples.