Orthogonal Rational Functions

Abstract

Let {α 1 , α 2 , . . . } be a sequence of real numbers outside the interval [−1, 1] and µ a positive bounded Borel measure on this interval. We introduce rational functions ϕ n (x) with poles {α 1 , . . . , α n } orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of ϕ n+1 (x)/ϕ n (x) as n tends to infinity under certain assumptions on the measure and the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation satisfied by the orthonormal functions.

A matrix approach to the computation of quadrature formulas on the unit circle

“…(1 − q) · · · (1 − q n ) with φ n (z) explicitly given by (2). The results are displayed in Tables 1-5.…”

“…Their study, not only suffered a rapid development in the last decades giving rise to a theory of orthogonal Laurent polynomials on the real line (see e.g. [8], [14], [24], [28], [33] and [36]), but it was extended to an ampler context leading to a general theory of orthogonal rational functions (see [2]). …”

Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions

“…In fact, we will see that the theory of orthogonal rational functions on the unit circle T := {u ∈ C : |u| = 1} pointed out in [6] can be used to detect the required Schur parameters or the rational functions γ m and δ m directly. The details about this can be found in Section 5.…”

Asymptotic behavior of solutions of general three term recurrence relations

“…iii) Laurent orthogonal polynomials and orthogonal rational functions as considered in [14] and [6].…”