Summer School Orthogonal polynomials and special functions location:Leganés, SP date:8-18 July 2004
The purpose of these lecture notes is to give a short introduction to the theory of orthogonal rational functions (ORF) on the unit circle. We start with the classical problem of linear prediction of a stochastic process to give a motivation for the study of Szegő’s problem and to show that in this context it will turn out that not as much the ORF but rather the reproducing kernels will play a central role. Another example of rational Krylov iteration shows that it might also be interesting to consider ORF on the real line, which we shall not discuss in these lectures.
In a second part we will show that most of the results of the scalar case thanslate easily to the case of matrix valued orthogonal rational functions (MORF).
There are however many aspects that are intimately related to these ideas that we do not touch upon like continued fractions, Nevanlinna-Pick interpolation, moment problems, and many other aspects of what is generally known as Schur analysis.
Lecture notes in Mathematics, vol. 1883, edited by Van Assche, Walter and Marcellán, Francisco