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 = {α₁, …, α_n} a sequence of arbitrary ‘complex’ poles outside I. Suppose {φ₁, …, 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–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 – x²)^–1/2 log μ′(x) ∈ L¹([– 1, 1]).