Kolloquium Wintersemester 2014/15, Institüt für Mathematik location:Osnabrück, Germany date:4 February 2015
Orthogonal polynomials on the real line satisfy a three term recurrence relation.
This relation can be written in matrix notation by using a tridiagonal matrix.
Similarly, orthogonal polynomials on the unit circle satisfy a Szegö recurrence relation that corresponds to an (almost) unitary Hessenberg matrix.
It turns out that orthogonal rational functions with prescribed poles satisfy a recurrence relation
that corresponds to diagonal plus semiseparable matrices. This leads to
efficient algorithms for computing the recurrence parameters for these orthogonal
rational functions by solving corresponding linear algebra problems.
In the first part of this talk we will study several of these connections between orthogonal functions
and matrix computations and give some numerical examples illustrating the numerical
behaviour of these algorithms.
In the second part of the talk we will use multivariate orthogonal polynomials as a tool
to find "good" points for polynomial interpolation for several planar regions, e.g., for the square,
the L-shape, the disk, ...