Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in L2(double-struck T sign)

Abstract

A function f in the unit ball ℬ of the Hardy algebra H∞ on the unit disc double-struck D sign = {z∈ℂ: |z| <1} is a non-exposed point of ℬ (|f|<1 a.e. on double-struck T sign = {ζ∈ℂ: |ζ| = 1}) iff limn∫double-struck T sign|fn|2dm= 0, where m is the Lebesgue measure on double-struck T sign and (fn)n≥o are the Schur functions of f. This result easily implies Rakhmanov's well-known theorem which states that limnan = 0 if σ′>0 a.e. on double-struck T sign, (an)n≥0 being the parameters of the orthogonal polynomials (φn)n≥0 in L2(dσ). We prove that fnbn is the Schur function of the probability measure |φn|2 dσ, which leads to an important formula relating \φn\2 σ′ to fn, and bn = φn|φ*n. A probability measure σ is called a Rakhmanov measure if (*) -limn|φn|2 dσ = dm. We show that a probability measure σ with parameters (an)n≥0 is a Rakhmanov measure iff the an's satisfy the Máté-Nevai condition limn anan+k= 0 for every k= 1, 2, .... Next, we prove that even approximants An/Bn of the Wall continued fraction for f converge in L2(double-struck T sign) iff either f is an inner function or limn an = 0. This implies that measures satisfying limn anan+k= 0, k = 1, 2, ..., and limn|an|>0 are all singular. © 2001 Academic Press.