Von-Neumann Lecture Course: Approximation Theory and Approximation Practice
A course consisting of eight 90-minute weekly lectures
beginning Monday 10 May 2010; 14:15–15:45, room MI 00.07.011.
This is a mathematics course for students and researchers
interested in numerical computation. Familiarity with Matlab
is essential. Some familiarity with approximation theory is
desirable but not essential.
The course is built on an unusual book being completed by the
lecturer with the title "Approximation Theory and Approximation
Practice: A 21st-Century Treatment in the Form of 32 Executable
Chebfun M-Files". It aims to teach both old and new ideas
of univariate approximation of functions in a fresh and
computational way, illustrating everything through the
system (
http://www.maths.ox.ac.uk/chebfun/). Both theorems
and algorithms will be emphasized: for the former, always
with reference to their originator whether in 1912 or 2004;
for the latter, always in a hands-on and exploratory fashion.
Topics to be treated include:
- Chebyshev points and interpolants
- Chebyshev polynomials and series
- Barycentric interpolation formula
- Weierstrass Approximation Theorem
- Analyticity and convergence rates
- The Gibbs phenomenon
- The Runge phenomenon
- Best approximation and the Remez algorithm
- Lebesgue constants
- Clenshaw-Curtis and Gauss quadrature
- Polynomial roots and colleague matrices
- Approximations based on a conformal map
- Rational functions
For Bachelor and Master students: The lecture has 3 ECTS. Successful attendance
depends on the written solutions for the exercises given as homework assignements each week.
Solutions to Assignments
Additional Reference
C. Runge, Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten 
