Schurʹs Algorithm, Orthogonal Polynomials, and Convergence of Wallʹs Continued Fractions in L2(T) Original Research Article

A function f in the unit ball B of the Hardy algebra H∞ on the unit disc D={z∈C : |z|<1} is a non-exposed point of B (|f|<1 a.e. on T={ζ∈C : |ζ|=1}) iff limn ∫T |fn|2 dm=0, where m is the Lebesgue measure on T and (fn)n⩾0 are the Schur functions of f. This result easily implies Rakhmanovʹs well-known theorem which states that limn an=0 if σ′>0 a.e. on T, (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 anʹs satisfy the Máté–Nevai condition limn anan+κ=0 for every κ=1, 2, … . Next, we prove that even approximants An/Bn of the Wall continued fraction for f converge in L2(T) iff either f is an inner function or limn an=0. This implies that measures satisfying limn anan+κ=0, κ=1, 2, …, and limn |an|>0 are all singular.