Abstract harmonic analysis of continuous wavelet transforms / Hartmut Führ

The book contains six chapters. The first one gives an overview of the topic and some basic background from functional analysis and representation theory. In Chapter 2, the representation-theoretic approach to continuous wavelet analysis is presented. The wavelet transform is built upon the general notion of coherent state systems, and the connection with the regular representation is made precise. Essential results for discrete series representations, in particular the Duflo-Moore theorem, are contained in Section 2.4. Some classical examples are discussed as well, the possible images of wavelet transforms are studied, and questions concerning discretization and sampling are addressed. Chapter 3 is devoted to the Plancherel theory for locally compact groups. This involves an introduction to direct integral decompositions of representations, the concepts of the dual and the quasi-dual, Type 1 representations, the Mackey machine which is needed for the description of the Plancherel measure in the non-unimodular case, and finally the Plancherel theorem in both the unimodular and the non-unimodular case. Chapter 4 contains the main results of the book, namely the solution of the existence and characterization problem of admissible vectors for Type 1 groups. The results are based on a pointwise Plancherel inversion formula (Theorem 4.15) which is one of the central new results of the book. Chapter 5 is devoted to group extensions as an interesting class of examples where the abstract admissibility conditions can be made more explicit. Chapter 6 is concerned with sampling theory for the Heisenberg group and Weyl-Heisenberg frames. (MathSciNet)