Asymptotic Uniqueness of Best Rational Approximants of Given Degree to Markov Functions in L 2 of the Circle

Abstract

We improve over a sufficient condition given in [8] for uniqueness of a nondegenerate critical point in best rational approximation of prescribed degree over the conjugate-symmetric Hardy space $\overline{\cal H}^2_{{\bf R}}$ of the complement of the disk. The improved condition connects to error estimates in AAK approximation, and is necessary and sufficient when the function to be approximated is of Markov type. For Markov functions whose defining measure satisfies the Szego condition, we combine what precedes with sharp asymptotics in multipoint Pade approximation from [43], [40] in order to prove uniqueness of a critical point when the degree of the approximant goes large. This lends perspective to the uniqueness issue for more general classes of functions defined through Cauchy integrals.