Orthogonal Rational Functions with Poles on the Unit Circle

Abstract

Let {αn} be a sequence of (not necessarily distinct) points on the unit circle T= {z ∈ C: |z| = 1}. Set Ln =Span {1, 1/ω1, …, 1/ωn}, L = ∪∞n=0Ln, where we have used the notation ωn = ∏nk=1 (z − αk). Let M be a positive linear functional defined on the space L · L with M(R) real for functions that are real on T. Define 〈R, S〉 = M(R(z) S(1/z)) for R, S ∈ L. (In particular if M is given as M(R) = ∫π−πR(eiθ) dμ(θ) for some measure μ, then 〈R, S〉 = ∫π−πR(eiθ) S(eiθdμ(θ).) Let the orthogonal system {φn} be obtained from {1/ωn} by orthogonalization. Three-term recurrence relations, quadrature formulas, moment theory, and interpolation properties connected with the functional M and the system {φn} are discussed.