Continuous and Discrete Wavelet Transforms

TLDR

The paper surveys the literature on wavelet transforms, focusing on the foundational work of Daubechies, Grossmann, and Morlet. It aims to review integral representations and discrete sum expansions of L²(ℝ) functions using coherent states. The study examines Weyl–Heisenberg and affine (wavelet) coherent states, showing how any L²(ℝ) function can be expressed as a sum or integral of these states. The authors include several of their own results within this survey.

Abstract

This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ 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 $L^2 ({\bf R})$ 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.