This paper surveys results on integral representations and discrete sum expansions of functions in $ L^2(\mathbb{R}) $ using coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, derived from translations and modulations of a single function, and affine coherent states, or wavelets, derived from translations and dilations of a single function. The paper discusses how any function in $ L^2(\mathbb{R}) $ can be represented as a sum or integral of these states. It reviews the work of I. Daubechies, A. Grossmann, and J. Morlet, and includes some results from the authors. The paper introduces the Gabor transform and wavelet transform as methods for time-dependent frequency analysis. The Gabor transform, named after D. Gabor, is based on the short-time Fourier transform, which divides a signal into short segments and computes Fourier coefficients. The wavelet transform, on the other hand, uses translations and dilations of a mother wavelet. Both transforms are deeply related through group representation theory. The paper discusses the continuous versions of these transforms, showing how they arise from representations of groups on $ L^2(\mathbb{R}) $. It also covers discrete versions of these transforms, developed by Daubechies, Grossmann, and Meyer, which use discrete lattices of translates and modulates or translations and dilates. These discrete transforms are based on the concept of Hilbert space frames. The paper then presents the Weyl–Heisenberg frame and affine frame, which are specific types of frames for $ L^2(\mathbb{R}) $. It discusses the properties of these frames and their applications in signal processing. The paper also introduces the Meyer wavelet, a smooth mother wavelet that generates an orthonormal basis for $ L^2(\mathbb{R}) $, and multiresolution analysis, a concept that has significant implications in both theoretical mathematics and signal processing. The paper concludes by summarizing the mathematical notations and definitions used, and provides an overview of frames in Hilbert spaces, which allow for discrete representations of functions in $ L^2(\mathbb{R}) $. It emphasizes the importance of frames in signal processing and the connection between frames and group representations. The paper also discusses the properties of the Gabor and wavelet transforms, their relationship to group representations, and their applications in signal analysis.This paper surveys results on integral representations and discrete sum expansions of functions in $ L^2(\mathbb{R}) $ using coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, derived from translations and modulations of a single function, and affine coherent states, or wavelets, derived from translations and dilations of a single function. The paper discusses how any function in $ L^2(\mathbb{R}) $ can be represented as a sum or integral of these states. It reviews the work of I. Daubechies, A. Grossmann, and J. Morlet, and includes some results from the authors. The paper introduces the Gabor transform and wavelet transform as methods for time-dependent frequency analysis. The Gabor transform, named after D. Gabor, is based on the short-time Fourier transform, which divides a signal into short segments and computes Fourier coefficients. The wavelet transform, on the other hand, uses translations and dilations of a mother wavelet. Both transforms are deeply related through group representation theory. The paper discusses the continuous versions of these transforms, showing how they arise from representations of groups on $ L^2(\mathbb{R}) $. It also covers discrete versions of these transforms, developed by Daubechies, Grossmann, and Meyer, which use discrete lattices of translates and modulates or translations and dilates. These discrete transforms are based on the concept of Hilbert space frames. The paper then presents the Weyl–Heisenberg frame and affine frame, which are specific types of frames for $ L^2(\mathbb{R}) $. It discusses the properties of these frames and their applications in signal processing. The paper also introduces the Meyer wavelet, a smooth mother wavelet that generates an orthonormal basis for $ L^2(\mathbb{R}) $, and multiresolution analysis, a concept that has significant implications in both theoretical mathematics and signal processing. The paper concludes by summarizing the mathematical notations and definitions used, and provides an overview of frames in Hilbert spaces, which allow for discrete representations of functions in $ L^2(\mathbb{R}) $. It emphasizes the importance of frames in signal processing and the connection between frames and group representations. The paper also discusses the properties of the Gabor and wavelet transforms, their relationship to group representations, and their applications in signal analysis.