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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.