Lectures on constructive approximation: Fourier, spline, and wavelet methods on the real line, the sphere, and the ball / Volker Michel

Auteur: Michel, Volker (1969-) - AuteurType de document: Monographie Collection: Applied and numerical harmonic analysis Langue: anglaisPays: Etats UnisÉditeur: New York, NY : Birkhäuser, 2013Description: 1 vol. (xvi-326 p.) ; 24 cm ISBN: 9780817684020 ; rel. Note: 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 (zbMath)Bibliographie: Bibliogr. (pages 307-315). Index. Sujets MSC: 65D07 Numerical analysis -- Numerical approximation and computational geometry (primarily algorithms) -- Splines
42C40 Harmonic analysis on Euclidean spaces -- Nontrigonometric harmonic analysis -- Wavelets and other special systems
65T60 Numerical analysis -- Numerical methods in Fourier analysis -- Wavelets
65T40 Numerical analysis -- Numerical methods in Fourier analysis -- Trigonometric approximation and interpolation
41-01 Approximations and expansions -- Instructional exposition (textbooks, tutorial papers, etc.)
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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 (zbMath)

Bibliogr. (pages 307-315). Index

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