Lectures on constructive approximation. Fourier, spline, and wavelet methods on the real line, the sphere, and the ball
type: Item
instance of: scholarly article
title: Lectures on constructive approximation. Fourier, spline, and wavelet methods on the real line, the sphere, and the ball
zbMATH Open document ID: 1295.65003
DOI: 10.1007/978-0-8176-8403-7
author: V. Michel
published in: Applied and Numerical Harmonic Analysis
publication date: 2012-11-15T00:00:00Z
review text: This is a constructive approach to approximation by Fourier series (orthogonal polynomials), splines and wavelets. Each of these are introduced for the real line, the sphere, and the ball. The sphere is clearly the central and most extensive part. The first part on the real line introduces the necessary background for approximation theory (Hilbert space, orthogonal basis, Fourier series, (cubic) spline, (Haar) wavelet, and the classical theorems of approximation theory). This part is deliberately kept short because many textbooks treat this already. The second part is a more extensive discussion of spherical harmonics, and splines and wavelets on the sphere (and introduces along the road also reproducing kernel Hilbert spaces and Sobolev spaces). The third part is again shorter since on the ball, it basically suffices to add a radial dimension to the spherical discussion of Part two. The book is self-contained and at a relatively elementary level. Basic knowledge of linear algebra and analysis suffices. Concepts are introduced preferably in the simplest situation. The basis functions are illustrated with many color plots and the proofs are fully written out. It can be used as a textbook, although there are no exercises. Instead, every chapter ends with a summary in the form of a list of questions running over all the elements discussed: What is...? How to ...? Why is ...? etc. If the reader can answer all of these, he/she is sure to have understood the whole chapter. This is a clear introduction to subjects that are not easily found in other textbooks at this level. Obviously it is of interest for geophysical applications.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65-01
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 33C55
zbMATH DE Number: 6106785
zbMATH Keywords: constructive approximation
zbMATH Keywords: Fourier series
zbMATH Keywords: splines
zbMATH Keywords: wavelets
zbMATH Keywords: spherical approximation
zbMATH Keywords: spherical harmonics
zbMATH Keywords: textbook
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-8176-8403-7
OpenAlex ID: W2481653444
description: scientific article; zbMATH DE number 6106785