Recurrence and asymptotics for orthonormal rational functions on an interval

Abstract

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]).

The existence and construction of rational Gauss-type quadrature rules

“…In this case the coefficient D k can be expressed in terms of inner products as follows (see [4,Thm. 3.7 and Thm.…”

A generalized eigenvalue problem for quasi-orthogonal rational functions

“…In [15,Sec. 3] the following three-term recurrence relation has been proved for nORFs ϕ k ∈ L k \L k−1 .…”

“…Finally, the coefficients E k and D k can also be expressed in terms of nORFs ϕ k as follows (see [15,Thm. 5.1]):…”

“…11.1] and [15]), and hence, the zeros of these ORFs are eigenvalues of a generalized eigenvalue problem (GEP) obtained by this threeterm recurrence relation (see e.g., [32]). In general, however, the zeros of ORFs are not all real so that they are not suitable to construct a rational Gaussian quadrature rule (see Lemma 12 further on).…”

Computing rational Gauss–Chebyshev quadrature formulas with complex poles: The algorithm

“…For further details on this topic in the case of all real poles outside the support of the measure, see e.g. [9]- [13], while some extensions to the case of arbitrarily complex poles outside the support of the measure can be found in [14,15].…”