Mathematical Sciences

Lund University

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.

Course Material

Course Start

Start date:

Reading periods:
Spring, second half