Continuous and Discrete Wavelet Transforms

. This paper is an expository survey of results on integral representations and discrete sum expansions of functions in L 2 (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 (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. Key words. frame, wavelet, coherent states, integral transform, Gabor transform, wavelet transform, Weyl--Heisenberg group, affine group AMS subject classifications. 42C15, 42A38 0. Introduction. The representation of a signal by means of its spectrum or Fourier transform is essential to solving many problems both in ...