Functions, spaces, and expansions. Mathematical tools in physics and engineering
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 details on the construction of orthogonal wavelets using the multiresolution analysis are summarized in Chapter 9. The last two chapters of this textbook are devoted to B-splines that form an important tool in numerical analysis, and to special functions, particularly orthogonal polynomials that play a role as eigenfunctions of differential operators. Here the author focuses on regular Sturm-Liouville problems.
scientific article; zbMATH DE number 5702304