# Lectures on Constructive Approximation

## Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball

• Volker Michel
Textbook

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

1. Front Matter
Pages i-xvi
2. Volker Michel
Pages 1-9
3. ### Basics

1. Front Matter
Pages 11-11
2. Volker Michel
Pages 13-30
3. Volker Michel
Pages 31-82
4. ### Approximation on the Sphere

1. Front Matter
Pages 83-83
2. Volker Michel
Pages 85-96
3. Volker Michel
Pages 97-143
4. Volker Michel
Pages 145-182
5. Volker Michel
Pages 183-238
6. Volker Michel
Pages 239-245
5. ### Approximation on the 3D Ball

1. Front Matter
Pages 247-247
2. Volker Michel
Pages 249-264
3. Volker Michel
Pages 265-287
4. Volker Michel
Pages 289-302
5. Volker Michel
Pages 303-305
6. Back Matter
Pages 307-328

### Introduction

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:

* 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.

### Keywords

Fourier transform Fourier transform Fourier transform orthogonal polynomials orthogonal polynomials orthogonal polynomials sphere sphere sphere splines splines splines wavelets wavelets wavelets

#### Authors and affiliations

• Volker Michel
• 1
1. 1., Department of MathematicsUniversity of SiegenSiegenGermany