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

Abstract

This paper is concerned with rational Szegő quadrature formulas to approximate integrals of the form I μ (f) = ∫ π ―π f(e iθ )dμ(θ) by a formula such as I n (f) = Σ n k=1 λ k f(z k ), where the weights λ k are positive and the nodes z k 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 Szegő―Radau and rational Szegő―Lobatto quadrature formulas will be characterized.