Rational quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

Abstract

This paper is concerned with rational Szeg quadrature formulas to approximate integrals of the form Iμ(f) = ∫-ππ f(eiθdμ(θ) by a formula such as I n(f) = ∑k=1n λkf(z kwhere the weights λk are positive and the nodes zk are carefully chosen on the complex unit circle. It will be shown that, for a given set of poles, the quadrature formulas can be chosen to be exact in certain subspaces of rational functions of dimension 2n. Also, the problem where one node (Radau) or two nodes (Lobatto) are prefixed will be analysed and the corresponding rational Szego′-Radau and rational Szego″ at most-Lobatto quadrature formulas will be characterized. © 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.