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Abstract Harmonic Analysis of Continuous Wavelet Transforms (Lecture Notes in Mathematics)

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This volume discusses a construction situated at the intersection of two different mathematical fields: Abstract harmonic analysis, understood as the theory of group representations and their decomposition into irreducibles on the one hand, and wavelet (and related) transforms on the other. In a sense the volume reexamines one of the roots of wavelet analysis: The paper [60] by Grossmann, Morlet and Paul may be considered as one of the initial sources of wavelet theory, yet it deals with a unitary representation of the affine group, citing results on discrete series representations of nonunimodular groups due to Duflo and Moore. It was also observed in [60] that the discrete series setting provided a unified approach to wavelet as well as other related transforms, such as the windowed Fourier transform.

We consider generalizations of these transforms, based on a representationtheoretic construction. The construction of continuous and discrete wavelet transforms, and their many relatives which have been studied in the past twenty years, involves the following steps: Pick a suitable basic element (the wavelet) in a Hilbert space, and construct a system of vectors from it by the action of certain prescribed operators on the basic element, with the aim of expanding arbitrary elements of the Hilbert space in this system. The associated wavelet transform is the map which assigns each element of the Hilbert space its expansion coefficients, i.e. the family of scalar products with all elements of the system. A wavelet inversion formula allows the reconstruction of an element from its expansion coefficients.

Continuous wavelet transforms, as studied in the current volume, are obtained through the action of a group via a unitary representation. Wavelet inversion is achieved by integration against the left Haar measure of the group. The key questions that are treated –and solved to a large extent– by means of abstract harmonic analysis are: Which representations can be used? Which vectors can serve as wavelets?

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