The wavelet transform
type: Item
instance of: scholarly article
title: The wavelet transform
zbMATH Open document ID: 1192.44004
author: R. S. Pathak
published in: Atlantis Studies in Mathematics for Engineering and Science
publication date: 2009-10-29T00:00:00Z
review text: Historically, the concept of ``ondelettes'' or ``wavelets'' originated from the study of time-frequency signal analysis, wave propagation, and sampling theory. One of the main reasons for the discovery of wavelets and wavelet transforms is that the Fourier transform analysis does not contain the local information of signals. So the Fourier transform cannot be used for analyzing signals in a joint time and frequency domain. In 1982, Jean Morlet, in collaboration with a group of French engineers, first introduced the idea of wavelets as a family of functions constructed by using translation and dilation of a single function, called the mother wavelet, for the analysis of nonstationary signals. However, this new concept can be viewed as the synthesis of various ideas originating from different disciplines including mathematics (Calderón-Zygmund operators and Littlewood-Paley theory), physics (coherent states in quantum mechanics and the renormalization group), and engineering (quadratic mirror filters, sideband coding in signal processing, and pyramidal algorithms in image processing). Wavelet analysis is an exciting new method for solving difficult problems in mathematics, physics, and engineering, with modern applications as diverse as wave propagation, data compression, image processing, pattern recognition, computer graphics, the detection of aircraft and submarines, and improvement in CAT scans and other medical image technology. Wavelets allow complex information such as music, speech, images, and patterns to be decomposed into elementary forms, called the fundamental building blocks, at different positions and scales and subsequently reconstructed with high precision. This book has ten short chapters, bibliography and index. The first chapter provides an overview of distribution theory, Fourier transforms and their basic properties. Included are the Hilbert transform, wavelets, the continuums wavelet transform and its basic properties and asymptotic expansions. …
reviewed by: Lokenath Debnath
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 44-02
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 44A35
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 65T50
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 46F12
zbMATH DE Number: 5623277
zbMATH Keywords: wavelets
zbMATH Keywords: wavelet transforms
zbMATH Keywords: generalized translation
zbMATH Keywords: convolution operators
zbMATH Keywords: unified theory of convolutions
zbMATH Keywords: composition of wavelet transforms
zbMATH Keywords: wavelet convolution product
zbMATH Keywords: asymptotic expansions of the wavelet transform for small and large values of dilation and translation parameters
zbMATH Keywords: monograph
zbMATH Keywords: Fourier transform
zbMATH Keywords: Hilbert transforms
zbMATH Keywords: Riesz fractional integrals
zbMATH Keywords: Gelfand-Shilov spaces
zbMATH Keywords: Sobolev space
zbMATH Keywords: Fourier convolution
zbMATH Keywords: Laplace convolution
zbMATH Keywords: Mellin convolution
zbMATH Keywords: Hankel convolution
zbMATH Keywords: Legendre convolution
zbMATH Keywords: Laguerre convolution
zbMATH Keywords: Chebyshev convolution
zbMATH Keywords: Sturm-Liouville convolution
zbMATH Keywords: discrete wavelet transforms
zbMATH Keywords: Morlet wavelet transform
zbMATH Keywords: Mexican hat wavelet transform
zbMATH Keywords: Haar wavelet transform
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description: scientific article; zbMATH DE number 5623277