Schur's algorithm, orthogonal polynomials, and convergence of Wall's continued fractions in L2(T)

A function f in the unit ball B of the Hardy algebra H-f on the unit disc D = {z is an element of C : \z\ < 1} is a non-exposed point of B (\.f\ < 1 a.e. on T = {zeta is an element of C: \zeta\ = 1}) iff lim(n) integral (T) \.f(n)\(2) dm = 0, where m is the Lebesgue measure on T and (f(n))(n greater than or equal to0) are the Schur functions of f. This result easily implies Rakhmanov's well-known theorem which states that lim(n)a(n) = 0 if sigma' > 0 a.e. on T. (a(n))(n greater than or equal to0) being the parameters of thr orthogonal polynomials (phi (n))(n greater than or equal to0) in L-2(d sigma). We prove that f(n)b(n) is the Schur function of the probability measure \phi (n)\(2)d sigma, which leads to an important formula relating \phi (n)\(2)sigma' to f(n) and b(n) = phi (n) phi (n)*. A probability measure sigma is called a Rakhmanov measure if (*) - lim(n) \phi (n)\(2) d sigma = dm. We show that a probability measure sigma with parameters (a(n))(n greater than or equal to0) is a Rakhmanov measure iff the a(n)'s satisfy the Mate-Nevai condition lim(n) a(n)a(n+kappa) = 0 for every kappa = 1, 2, ... Next, we prove that even approximants A(n):B-n of the Wall continued fraction for f converge in L-2(T) iff either f is an inner function or lim(n) a(n) = 0. This implies that measures satisfying lim(n) a(n)a(n+kappa) = 0, kappa = 1, 2, ... and (lim(n)) over bar \a(n)\ > 0 are all singular. (C) 2001 Academic Press.