# Wavelet

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. Wavelets have important applications in mathematics, physics, and engineering.

## Quotes

* Wavelets are everywhere nowadays. Whether in signal or image processing, in astronomy, in fluid dynamics (turbulence), or in condensed matter physics, wavelets have found applications in almost every corner of physics. Furthermore, wavelet methods have become standard fare in applied mathematics, numerical analysis, and approximation theory.
** Jean-Pierre Antoine: (2003). "Review of Wavelet Transforms and Their Applications by Lokenath Debnath". Physics Today 56 (4). ISSN 0031-9228. DOI:10.1063/1.1580056.
* On the one hand, the concept of wavelets can be viewed as a synthesis of ideas which originated during the last twenty or thirty years in engineering (subbing coding), physics (coherent states, renormalization group), and pure mathematics (study of Calderón-Zygmund operators). As a conseuqence of these interdiscplinary origins, wavelets appeal to scientists and engineers of many different backgrounds. On the other hand, wavelets are a fairly simple mathematical tool with a great variety of possible applications.
** Ingrid Daubechies: Ten Lectures on Wavelets. SIAM. 1 June 1992. p. vii. ISBN 978-0-89871-274-2.
* Wavelets were developed independently by mathematicians, quantum physicists, electrical engineers and geologists, but collaborations among these fields during the last decade have led to new and varied applications. What are wavelets, and why might they be useful to you? The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers feel that using wavelets means adopting a whole new mind-set or perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions.
** A. Graps (Summer 1995)"An introduction to wavelets". IEEE Computational Science and Engineering 2 (2): 50-61. DOI:10.1109/99.388960.
* Wavelet theory is nowadays a very active field of approximation theory with a wide impact on signal analysis, high-performance imaging applications, and adaptive transversal filter theory. It is concerned with the modeling of univariate and multivariate signals with a set of specific signals. The specific signals are just the wavelets. Families of wavelets are used to approximate a given signal (with respect to the L$^2$ norm, say), and each element in the wavelet set is constructed from the same original window, the mother wavelet.
** Walter Schempp: (1994). "Book Review: Wavelet theory and its applications by Randy K. Young". Bulletin of the American Mathematical Society 30 (2): 277–283. DOI:10.1090/S0273-0979-1994-00463-X. (quote from p. 278)
* Wavelets were introduced at the beginning of the 'eighties by J. Morlet, a French geophysicist at Elf-Aquitaine, as a tool for signal analysis in view of applications for the analysis of seismic data. The numerical success of Morlet prompted A. Grossmann to make a more detailed study of the wavelet transform, which resulted in a paper giving the mathematical foundations (see Grossmann & Morlet ..., where the title of the paper still shows the name wavelets of constant shape. In 1985, the harmonic analyst Y. Meyer became aware of this theory and he recognised many classical results inside it. Meyer pointed out to Grossmann and Morlet that there was a connection between their signal analysis methods and existing, powerful techniques in the mathematical study of singular integral operators. Then Ingrid Daubechies became involved, and all this resulted in the first construction of a special type of frames (see Daubechies, Grossmann & Meyer ..), (the concept frame generalizes the concept basis in a Hilbert space). It was also the start of a cross-fertilization between the signal analysis applications and the purely mathematical aspects of techniques based on dilations and translations.
** Nico M. Temme: "Wavelets: First Steps". Wavelets: An Elementary Treatment of Theory and Applications. Series in Approximations and Decompositions – vol. 1, edited by Tom H. Koornwinder. January 1993. pp. 1–12. ISBN 9789810224868. (quote from p. 1)
* A Haar wavelet is the simplest type of wavelet. In discrete form, Haar wavelets are related to a mathematical operation called the Haar transform. The Haar transform serves as a prototype for all other wavelet transforms.
** James S. Walker: A Primer on Wavelets and Their Scientific Applications. 17 July 2019. p. 1. ISBN 9781420050011.