Multipoint Schur algorithm and orthogonal rational functions, I: Convergence properties

Abstract

Classical Schur analysis is intimately related to the theory of orthogonal polynomials on the circle. We investigate the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szegő theory in the case that interpolation points may accumulate on the unit circle. This leads to a generalization of results in [22,10], and new types of asymptotics.

The Jacobi matrices approach to Nevanlinna–Pick problems

“…First, note that (8.6) is sufficient for the Nevanlinna-Pick problem in R 0 to be determinate but not necessary (see [27,Chapter IV,Example 4.2]). It should also be noted that, under the Szegö condition and the negation of the Blashcke type condition, the locally uniform convergence of multipoint diagonal Padé approximants for ϕ ∈ R[α, β] was proved in [46] (see also [6]). Now, we are also able to adapt Theorem 5.5 for the self-adjoint case.…”

How poles of orthogonal rational functions affect their Christoffel functions

“…In a recent paper [3], we proved the following asymptotics of rational Christoffel functions. 1]. Assume that for some η > 0, the poles {α j } satisfy for all j ≥ 1,…”

“…Note the key restriction that the poles stay away from [−1, 1]. In some results on asymptotics of orthogonal rational functions [1], such a restriction has been replaced by a Blaschke type assumption that…”

Baxter’s difference systems and orthogonal rational functions

“…There has been much interest in the theory of orthogonal rational functions (ORFs) on the unit circle, partly due to their connection to the multipoint Schur algorithm [1,9,21] and partly due to the fact that they are natural generalizations of orthogonal polynomials to the case where not all the poles are at infinity. ORFs on the complex unit circle were first introduced by Džrbašian in the 1960s [15], and have been studied extensively during the past few decades; e.g.…”

“…. , (1) which was a generalization of the system solved by Szegő orthogonal polynomials [19]. He showed that if the coefficients in the above system were l 1 summable then certain solutions to the above equations formed a system of polynomials bi-orthogonal with respect to a complex measure which is absolutely continuous with respect to Lebesgue measure.…”