Functions, spaces, and expansions. Mathematical tools in physics and engineering
type: Item
instance of: scholarly article
title: Functions, spaces, and expansions. Mathematical tools in physics and engineering
zbMATH Open document ID: 1231.46001
DOI: 10.1007/978-0-8176-4980-7
author: B. E. Eshmatov
published in: Applied and Numerical Harmonic Analysis
publication date: 2010-04-27T00:00:00Z
review text: This textbook aims at representing the mathematical tools of computational harmonic and Fourier analysis in a form that is suitable for students in applied mathematics, physics and engineering. For understanding the content one only needs some basic knowledge on linear algebra as well as on basic concepts of calculus and real analysis. The book is beautifully written. It contains also a lot of suitable exercises at the end of each section. Here is a short description of the chapters. Chapter 1 presents some mathematical background from linear algebra and real analysis that is needed throughout the book. Chapter 2 focuses on the analysis of vector spaces including simple topological concepts and approximation in vector spaces. In Chapter 3, the concept of Banach spaces is introduced with the sequence spaces \(\ell^{p}\) as a special example. Section 4 in devoted to Hilbert spaces. Particularly, the concept of orthogonality plays an important role for the decomposition of Hilbert spaces. The analysis of linear operators in Hilbert spaces covers the notion of adjoint operators, orthogonal projections and Bessel sequences. At the end of the chapter, also frames are introduced. Chapter 5 is devoted to \(L^{p}\) spaces, a very important class of Banach spaces in Fourier analysis. The Hilbert space \(L^{2}\) plays a special role here. Therefore, Chapter 6 is concerned with this Hilbert space on \(\mathbb R\) and on a finite interval, where in the latter case a function representation by Fourier expansions is possible. One main tool for analysing functions in \(L^{2}(\mathbb R)\), the Fourier transform, is introduced in Section 7. This chapter contains important assertions on the properties of the Fourier transform, convolutions and the Shannon sampling theorem. Chapter 8 gives an introduction to wavelet analysis, raising all important notions such as multiresolution analysis, scaling functions, vanishing moments, and orthogonal wavelets with compact support. Some technical…
Mathematics Subject Classification ID: 46-01
Mathematics Subject Classification ID: 42-01
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 5702304
zbMATH Keywords: Banach spaces
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: \(L^p\)-spaces
zbMATH Keywords: Fourier transform
zbMATH Keywords: wavelets
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: B-splines
zbMATH Keywords: orthogonal polynomials
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-8176-4980-7
OpenAlex ID: W2503902107
description: scientific article; zbMATH DE number 5702304