This volume contains a systematic discussion of wavelet-type inversion formulae based on group representations, and their close connection to the Plancherel formula for locally compact groups. The connection is demonstrated by the discussion of a toy example, and then employed for two purposes: Mathematically, it serves as a powerful tool, yielding existence results and criteria for inversion formulae which generalize many of the known results. Moreover, the connection provides the starting point for a reasonably self-contained exposition of Plancherel theory. Therefore, the volume can also be read as a problem-driven introduction to the Plancherel formula.
Introduction.- Wavelet Transforms and Group Representations.- The Plancherel Transform for Locally Compact Groups.- Plancherel Inversion and Wavelet Transforms.- Admissible Vectors for Group Extension.- Sampling Theorems for the Heisenberg Group.- References.- Index.
From the reviews:
It has become evident that the one-dimensional continuous wavelet transform has a natural foundation in unitary representation theory. & also various multidimensional generalizations as well as the windowed Fourier transform can be treated within the general context of discrete series transformations. & The present book develops a unified theory in an even more general setting, going beyond the discrete series. & The book is very well written and can be recommended to anyone interested in wavelet analysis and/or representation theory. (Margit R?sler, Mathematical Reviews, Issue 2006 m)
This book deals with generalizations, valid for general locally compact groups and unitary representations on arbitrary Hilbert spaces. & The book is well written, and it contains a nice blend of the general analysis and the concrete examples in wavelet analysis and Gabor analysis. The book is strongly recommended for readers with interest in abstract harmonic anallý
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