Lectures on constructive approximation. Fourier, spline, and wavelet methods on the real line, the sphere, and the ball
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.
scientific article; zbMATH DE number 6106785