Time-frequency and time-scale methods. Adaptive decompositions, uncertainty principles, and sampling
type: Item
instance of: scholarly article
title: Time-frequency and time-scale methods. Adaptive decompositions, uncertainty principles, and sampling
zbMATH Open document ID: 1079.42027
published in: Applied and Numerical Harmonic Analysis
publication date: 2005-03-15T00:00:00Z
review text: Mathematical methods play a significant role in the development of signal and image processing. Among these methods, harmonic analysis takes a prominent place. Foundations for these applications have been in place a long time ago, let me just mention here the Nobel Prize winner Dennis Gabor and his method of analyzing signals in which time and frequency play symmetrical parts. This gave start to the so-called time-frequency methods in mathematics and engineering. Also, the time-scale methods can be seen already in the Whitney decomposition and in the famous theory of Calderón and Zygmund. But the real boom for time-frequency and time-scale methods started since the advent of the wavelet theory and the observation of the role that multiresolution algorithms play in information theory. In particular, studies devoted to wavelets and their generalizations and applications renewed the mathematicians' interests in the Gabor systems. Moreover, it has been observed again and again that the two theories are in many ways parallel to each other. Many texts have been written in the recent years on the two subjects. However, the book under review is written from a relatively novel perspective: its goal is to carefully and deeply study the similarities and differences between time-frequency and time-scale methods. Thus, one may view the present book as a continuation of the famous textbook of \textit{I. Daubechies} [``Ten lectures on wavelets'' (1992; Zbl 0776.42018)]. In my opinion, the two books nicely complement each other and both are a ``must have'' for everyone who is interested in a deeper understanding of the time-frequency and time-scale methods. Chapter 1 introduces wavelets associated with multiresolution analysis (MRA) and presents several constructions of MRA wavelets with certain regularity properties. Chapter 2 is devoted to the study of these aspects of wavelets which are potentially important in their applications to partial differential equations, like, e.g., me…
reviewed by: Wojciech Czaja
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42Cxx
Mathematics Subject Classification ID: 42B05
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42C30
Mathematics Subject Classification ID: 62D05
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81Q15
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 81Qxx
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 94A20
zbMATH DE Number: 2146007
zbMATH Keywords: time-frequency analysis
zbMATH Keywords: time-scale analysis
zbMATH Keywords: Fourier transform
zbMATH Keywords: windowed Fourier transform
zbMATH Keywords: Wigner distribution
zbMATH Keywords: Zak transform
zbMATH Keywords: wavelets
zbMATH Keywords: wavelet packets
zbMATH Keywords: Gabor systems
zbMATH Keywords: adaptive decomposition
zbMATH Keywords: uncertainty principles
zbMATH Keywords: Balian-Low theorem
zbMATH Keywords: sampling
zbMATH Keywords: Schrödinger operators
author: Jeffrey A. Hogan
author: Joseph Lakey
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2146007