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 = {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 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 orthonormal rational functions. Next, we consider the case in which I = [- 1, 1] and μ satisfies the Erdos-Turan 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]).