An interpolation algorithm for orthogonal rational functions

摘要

Let A={α1,α2,…} be a sequence of numbers on the extended real line R̂=R∪{∞} and μ a positive bounded Borel measure with support in (a subset of) R̂. We introduce rational functions φn with poles {α1,…,αn} that are orthogonal with respect to μ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system {φn} is regular such that the orthogonal functions satisfy a three-term recurrence relation similar to the one for orthogonal polynomials.To compute the recurrence coefficients one can use explicit formulas involving inner products. We present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the Riesz–Herglotz–Nevanlinna transform Ωμ of the measure μ. Error bounds are derived and some examples serve as illustration.