A PhD thesis @ NALAG

TW 2004_02

Joris Van Deun
Orthogonal rational functions: asymptotic behaviour and computational aspects

Advisor(s): Adhemar Bultheel

Abstract

Orthogonal rational functions with fixed poles are a natural generalization of orthogonal polynomials. Most attention so far has gone to the case of orthogonality on the complex unit circle or on the real line, where many polynomial results have been generalized to the rational case. In this thesis we extend several of these results to the case of orthogonality on a finite interval. First we study different types of convergence. Ratio asymptotics are derived and then used to obtain the asymptotic behaviour of the recurrence coefficients. We conclude this part with some strong and weak-star convergence results. Next we briefly study the convergence of gaussian quadrature formulas on the interval and their relation with formulas on the unit circle. A substantial part of this thesis is devoted to computational aspects of orthogonal rational functions, both on an interval and on a halfline. We give some interpolation algorithms and generalize the method of modified moments to the case of rational functions. Next, a generally applicable and numerically stable algorithm is provided to compute the recurrence coefficients, together with computable error bounds. The case of poles close to the interval of integration is treated separately, because of the additional numerical difficulties. Finally, as a case study we generalize the well-known Chebyshev polynomials to the rational situation and present some related quadrature formulas which allow fast and efficient computation.