# Numerical analysis: Seminar

This course gives an introduction to approximation theory, where we also address computational aspects of the subject. Classical and more recent topics of approximation are combined. The course consists of a lecture part and a seminar part. The lecture part will be given the week before easter, 4-10 till 4-13, by Prof. Armin Iske, University of Hamburg, an internationally known specialist on approximation theory. The seminar part takes place the rest of the term with participants giving talks on selected topics.

Lectures start on monday, 17-4-10, 10:15 in room 228.

The course consists of five chapters as follows.

Best Approximations

Existence, uniqueness, direct and dual characterizations

Euclidean Approximation

Orthogonal bases and orthogonal projections, approximation by Fourier partial sums and orthogonal polynomials

Chebychev Approximation

Strongly unique best approximations, characterization and construction of Chebychev systems, Remez algorithm (construction, convergence and implementation)

Asymptotics

Weierstrass theorem, complete orthogonal systems, convergence of Fourier partial sums, Jackson theorems

Kernel-based Approximation

Positive definite functions, reproducing kernel Hilbert spaces, optimality and stability of kernel-based reconstruction schemes, update strategies, penalized least squares approximation

Supplementary material, i.e., lecture notes (from growing text book), exercise sheets and Matlab programs, will be provided. The course is hands-on, and the students are encouraged to participate in lively discussions.