Approximation methods I

Summer term 2017
Lecture
Tuesday, 8:15 - 9:55, MN 55
Friday, 8:15 - 9:55, MN 55
First Lecture: April 11, 2017

Exercises
Wednesday, 10:15 - 11:5, Seminarraum 2

Participants:
The lecture is suitable for students in mathematics, physics and computer science.

Assumptions to finish the lecture successfully:
There will be oral examinations (20 minutes) in Summer 2017.

Assumptions to get admission for the examination:
Attendance of the exercises, oral presentation of two exercise solutions, and 50 percent of the achievable points for homework.
With this lecture + exercises you can obtain 9 ECTS points.

Content:
This lecture Approximation Methods I is concerned with numerical methods for approximation and representation of functions/signals.

Introduction

1. Abstract approximation
1.1. Best approximation
1.2. Discrete approximation

2. Chebyshev-Approximation
2.1. Polynomials
2.2. Remez algorithm

3. Approximation in Euclidean spaces
3.1. Orthogonal systems and projectors
3.2. N-term approximation
3.3. Orthogonal systems in Hilbert spaces
3.4. Theorems of Weierstrass
3.5. Acceleration of convergence for Fourier series

4. Fast algorithms
4.1. Trigonometric interpolation
4.2. Fast Fourier transform (FFT)
4.3. Interpolation on Chebyshev knots
4.4. Discrete Cosine transform (DCT)
4.5. Barycentric interpolation
4.6. Clenshaw Curtis quadrature
4.7. Acceleration of convergence

5. Sampling
5.1. Cardinal interpolation
5.2. Fourier transform and band-limited functions
5.3. Best approximation with sinc functions
5.4. Sampling theorem
5.5. Shannon series and error estimates for sinc approximation
5.6. Aliasing

6. Translation invariant spaces
6.1. Projections and approximation order
6.2. Strang-Fix conditions
6.3. Spline spaces

7. The Haar wavelet
7.1. Haar Scaling function and multi-resolution analysis (MRA)
7.2. Haar wavelet
7.3. The fast Haar wavelet transform

8. Wavelet bases
8.1. Refinable functions and Strang-Fix conditions
8.2. Construction of wavelets
8.3. B-spline wavelets
8.4. Orthogonal wavelets
8.5. Scaling functions and wavelets from masks
8.6 Daubechies wavelets
8.7. Biorthogonal wavelets
8.8. The fast wavelet transform

Literature:
I. Daubechies: Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.
P. G. Nevai: Orthogonal polynomials, Amer. Math. Soc., 1979.
R. Schaback, H. Wendland: Numerische Mathematik, Springer, 2004.
R. Schaback: Approximationsverfahren I (http://num.math.uni-goettingen.de/schaback/teaching/AV_1.pdf)