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

This article presents a systematic study for abstract harmonic analysis aspects of wave-packet transforms over locally compact abelian (LCA) groups. Let HH be a locally compact group, let KK be an LCA group, and let θ:H→Aut(K)θ:H→Aut(K) be a continuous homomorphism. We introduce the abstract notion of the wave-packet group generated by θθ, 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.