The wavelet transform
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. All these topics are described briefly. Chapter two gives the wavelet transforms on \(L^p\) and \(L^p(\mathbb{R}^n)\) spaces, Hilbert transforms, generalized Hilbert transforms and Riesz fractional integrals with their generalizations. The composition of wavelet transforms, the continuity of the composition operator, reconstruction formula for the composition of the wavelet transforms with their distributional extensions, Abelian theorems for the wavelet transform of functions are the major topics of chapter three. Chapters four and five deal with the study of the continuous wavelet on certain Gelfand-Shilov spaces of type S and of type W. The wavelet transforms of certain ultra-differentiable functions, the continuity and boundedness results for continuous wavelet transform are discussed in these chapters. The wavelet transform on the generalized Sobolev space \(B^\omega_{p,k}\) boundedness results in this space, and the wavelet transform with compactly supported wavelets, \(B^\omega_{p,k}\)-space are the main topics of chapter six. Included are some generalizations of approximation properties of the wavelet transforms on Sobolev spaces. Chapter seven is concerned with generalized translation operator and the corresponding convolution operator for a general integral transform. These ideas and results are used to develop a unified theory of convolutions for all common integral transforms. The generalized convolution is introduced in terms of the generalized translation operator \(\tau_z\) defined by \[ \tau_z f(y)= f^*(z,y)= \int^b_a D(x,y,z) f(x)\,dx, \] where \(D(x,y,z)\) is a basic generalized function such that \[ \psi(t,x)\psi(t,y)= \int^b_a D(x,y,z)\psi(t,z)\,dz. \] In some simple cases, \(D(x,y,z)\) is a measure and in other cases, it turns out to be a distribution. In this chapter, some important properties of \(D(x,y,z)\) and the generalized convolution operation are discussed. This is followed by special cases that consist of the Fourier, Laplace, Mellin, Hankel, Legendre, Laguerre, Chebyshev convolutions and the Sturm-Liouville convolution. Included are the convolutions for the continuous and the discrete wavelet transforms. The boundedness results for the basic function \(D(x,y,z)\) are proved. An approximation theorem involving the wavelet convolution is also proved. Most of the results of chapter seven are new and interesting. Chapter eight is devoted to the wavelet convolution product and the existence theorems involving the convolution product. It is shown that, under certain conditions, the product of two wavelet transforms is a wavelet transform. This is followed by the wavelet convolution in a generalized Sobolev space. An approximation theorem involving wavelet convolution is presented, and the wavelet convolution is used to approximate certain \(L^p\) functions. The last two chapters deal with asymptotic expansions of the wavelet transform for large values of the translation parameter and for large and small values of dilation parameter. Included are asymptotic expansions for the Morlet wavelet transform, the Mexican hat wavelet transform and the Haar wavelet transform. In summary, this book contains new excellent topics of interest that have been presented by the author. Many selected books, research monographs and research articles directly related to the subject matter have been included in the list of references so that they may serve to stimulate new interest and in future advanced study and research. However, this book is likely to be less successful as a graduate level textbook due to lack of worked examples and exercises. However, this can be used as a graduate level research seminar presentation in order to stimulate interest in advanced study and research in this modern and useful subject. The author does not include any applications to time-frequency signal analysis, image processing, computer vision, computer graphics, numerical analysis, pattern recognition, and biomedial engineering. All these topics of applications are, of course, not expected in such a short book. The publisher should have considered the price of this volume, as it is too expensive for a short book. Some minor points are noted for future revision/correction, which include transform and transformation used in the book interchangely, \(\text{sgn}(\xi)= 0\), where \(\xi= 0\), wavelet stands for small wave on page 9 and \(w\) in place of \(\omega\). In addition, the Legendre transform was originally introduced by Churchill in 1954, not by Stens and Wehruns, the dual Poisson-Laguerre transformation was originally introduced by Debnath in 1960, not by Cholewinski and Haimo. In spite of these minor points, this short publication represents a good work of the author who has already made some important contributions to the subject matter of the book. Finally, this is a short, but interesting book for advanced graduate students and researchers who would be interested in research and seminar presentation on the wavelets theory and wavelet transforms. I strongly recommend to advanced graduate students, researchers and professionals in mathematics, physics and engineering.
asymptotic expansions of the wavelet transform for small and large values of dilation and translation parameters
scientific article; zbMATH DE number 5623277