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Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups. (English) Zbl 1354.43004

Summary: 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 \(\theta:H\to\operatorname{Aut}(K)\) 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 illustrate applications of these techniques in the case of some well-known examples.

MSC:

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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