In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside [−1, 1] to arbitrary complex poles outside [−1, 1]. The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [−1, 1].

Abstract. In this paper we characterize rational Szegő quadrature formulas associated with Chebyshev weight functions, by giving explicit expressions for the corresponding para-orthogonal rational functions and weights in the quadratures. As an application, we give characterizations for Szegő quadrature formulas associated with rational modifications of Chebyshev weight functions. Some numerical experiments are finally presented.

Let µ be a positive bounded Borel measure on a subset I of the real line, and A = {α 1 ,. .. , α n } a sequence of arbitrary complex poles outside I. Suppose {ϕ 1 ,. .. , ϕ n } is the sequence of rational functions with poles in A orthonormal on I with respect to µ. First, we are concerned with reducing the number of different coefficients in the three term recurrence relation satisfied by these orthornormal rational functions. Next, we consider the case in which I = [−1, 1] and µ satisfies the Erdős-Turán condition µ > 0 a.e. on I (where µ is the Radon-Nikodym derivative of the measure µ with respect to the Lebesgue measure), to discuss the convergence of ϕ n+1 (x)/ϕ n (x) as n tends to infinity and to derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation. Finally, we give a strong convergence result for ϕ n (x) under the more restrictive condition that µ satisfies the Szegő condition (1 − x 2) −1/2 log µ (x) ∈ L 1 ([−1, 1]).

Let {ϕ n } be a sequence of rational functions with arbitrary complex poles, generated by a certain three-term recurrence relation. In this paper we show that under some mild conditions, the rational functions ϕ n form an orthonormal system with respect to a Hermitian positive-definite inner product.

In this paper we present formulas expressing the orthogonal rational functions associated with a rational modification of a positive bounded Borel measure on the unit circle, in terms of the orthogonal rational functions associated with the initial measure. These orthogonal rational functions are assumed to be analytic inside the closed unit disc, but the extension to the case of orthogonal rational functions analytic outside the open unit disc is easily made. As an application we obtain explicit expressions for the orthogonal rational functions associated with a rational modification of the Lebesgue measure.

We provide a fast algorithm to compute arbitrarily many nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [−1, 1]. This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-) orthogonal rational functions.

Abstract. Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss-Radau (m = 1) and Gauss-Lobatto (m = 2) quadrature formulas that approximate F{f }. These are quadrature formulas with n positive weights and with the n−m remaining nodes real and distinct, so that the quadrature is exact in a (2n−m)-dimensional space of rational functions.

In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among {α 1 , . . . , α n } ⊂ (C 0 ∪ {∞}), are not all real (unless α n is real), and hence, they are not suitable to construct a rational Gaussian quadrature rule (RGQ). For this reason, the zeros of a so-called quasi-ORF or a so-called para-ORF are used instead. These zeros depend on one single parameter τ ∈ (C ∪ {∞}), which can always be chosen in such a way that the zeros are all real and simple. In this paper we provide a generalized eigenvalue problem to compute the zeros of a quasi-ORF and the corresponding weights in the RGQ. First, we study the connection between quasi-ORFs, para-ORFs and ORFs. Next, a condition is given for the parameter τ so that the zeros are all real and simple. Finally, some illustrative and numerical examples are given. Mathematics Subject Classification (2000)42C05 · 65D32 · 65F15