Continuous and discrete wavelet transforms. (English) Zbl 0683.42031

Summary: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in \(L^ 2({\mathbb{R}})\) in terms of coherent states. Two types of coherent states are considered: Weyl- Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called “wavelets”, which arise as translations and dilations of a single function. In each case it is shown how to represent any function in \(L^ 2({\mathbb{R}})\) as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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