Orthogonal Rational Functions: Quadrature, Recurrence and Rational Krylov
Orthogonal rational functions (ORFs) with prescribed poles are a natural generalization of orthogonal polynomials. Many results have already been generalized to the rational case. However, there are less cases in which explicit expressions are known for the ORFs. Moreover, the theory of orthogonality on a subset of the real line has so far been restricted to the case of rational functions with all real poles. In the first part of this thesis, we derive new explicit expressions for ORFs and extend existing expressions to the case of arbitrary complex poles. We then use these expressions to obtain equations for the nodes and weights in rational quadrature formulas associated with the Chebyshev weight functions on the unit circle and on the interval. In the second part, we generalize the three-term recurrence for ORFs on a subset of the real line to the case of arbitrary complex poles, and give a Favard-type theorem for rational functions generated by such a three-term recurrence. As an application, we study associated rational functions based on the three-term recurrence with shifted recurrence coefficients. Next, we prove a relation between ORFs on the unit circle and on the interval. To conclude this part, we then use this relation to study different types of convergence, and to derive asymptotic formulas for the recurrence coefficients, for ORFs on the interval. Finally, in the last part of this thesis we study the relation between ORFs and the rational Lanczos method for Hermitian matrices.