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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 ${L}^{2}\left(ℝ\right)$ 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 ${L}^{2}\left(ℝ\right)$ 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.

##### MSC:
 42C40 Wavelets and other special systems 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type