Overview
- Combines an explanation of classical and modern approximation methods for Euclidean and spherical geometries
- Detailed explanations and illustrations included to optimize the understanding of topics
- Concentrates on the essentials for a course
- Uses examples of data sets to explain the tasks, challenges, advantages, and disadvantages of the methods presented
- First work that explicitly treats approximation methods on the ball
- Includes supplementary material: sn.pub/extras
Part of the book series: Applied and Numerical Harmonic Analysis (ANHA)
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Table of contents (12 chapters)
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Front Matter
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Basics
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Front Matter
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Approximation on the Sphere
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Front Matter
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Approximation on the 3D Ball
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Front Matter
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Back Matter
Keywords
About this book
Lectures on Constructive Approximation: Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball focuses on spherical problems as they occur in the geosciences and medical imaging. It comprises the author’s lectures on classical approximation methods based on orthogonal polynomials and selected modern tools such as splines and wavelets.
Methods for approximating functions on the real line are treated first, as they provide the foundations for the methods on the sphere and the ball and are useful for the analysis of time-dependent (spherical) problems. The author then examines the transfer of these spherical methods to problems on the ball, such as the modeling of the Earth’s or the brain’s interior. Specific topics covered include:
* the advantages and disadvantages of Fourier, spline, and wavelet methods
* theory and numerics of orthogonal polynomials on intervals, spheres, and balls
* cubic splines and splines based on reproducing kernels
* multiresolution analysis using wavelets and scaling functions
This textbook is written for students in mathematics, physics, engineering, and the geosciences who have a basic background in analysis and linear algebra. The work may also be suitable as a self-study resource for researchers in the above-mentioned fields.
Reviews
From the book reviews:
“This is a constructive approach to approximation by Fourier series (orthogonal polynomials), splines and wavelets. … The basis functions are illustrated with many color plots and the proofs are fully written out. … 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.” (Adhemar Bultheel, zbMATH, Vol. 1295, 2014)
“The textbook Lectures on constructive approximation teaches the basics and details of Fourier, spline, and wavelet methods on the real line, the sphere, and the ball. … The style of the book is clearly that of a textbook, since the author makes a great effort to make very complicated concepts comprehensible to the reader. Throughout the book, numerous numerical examples and graphical illustrations support the explanations. This book is appropriate for applied mathematicians and numerical analysts as well as for geoscientists and engineers.” (Willi Freeden, Mathematical Reviews, August, 2013)Authors and Affiliations
About the author
Dr. Volker Michel teaches at University of Siegen
Bibliographic Information
Book Title: Lectures on Constructive Approximation
Book Subtitle: Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball
Authors: Volker Michel
Series Title: Applied and Numerical Harmonic Analysis
DOI: https://doi.org/10.1007/978-0-8176-8403-7
Publisher: Birkhäuser Boston, MA
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Science+Business Media New York 2013
Hardcover ISBN: 978-0-8176-8402-0Published: 11 December 2012
eBook ISBN: 978-0-8176-8403-7Published: 12 December 2012
Series ISSN: 2296-5009
Series E-ISSN: 2296-5017
Edition Number: 1
Number of Pages: XVI, 326
Number of Illustrations: 2 b/w illustrations, 5 illustrations in colour
Topics: Approximations and Expansions, Special Functions, Fourier Analysis, Mathematical Methods in Physics, Numerical Analysis