Asymptotics for Minimal Blaschke Products and Best L1 Meromorphic Approximants of Markov Functions

Abstract

Let μ be a positive Borel measure with support suppμ = E ⊂ (−1, 1) and let $$\Delta_n = {{\rm inf}\atop {{B \in {\cal B}_n}}} \int_E \mid(x)\mid^2 d\mu(x)$$ where \({\cal B}_n\) is the collection of all Blaschke products of degree n. Denote by \(B_n \in \mathcal{B}_n\) a Blaschke product that attains the value Δn. We investigate the asymptotic behavior, as n → ∞, of the minimal Blaschke products Bn in the case when the measure μ with support E = [a, b] satisfies the Szeg’o condition: $$\int_a^b {\frac{{\log \left( {{{d\mu } \mathord{\left/ {\vphantom {{d\mu } {dx}}} \right. \kern-\nulldelimiterspace} {dx}}} \right)}} {{\sqrt {(x - a)(b - x)} }}dx > - \infty .} $$ At the same time, we shall obtain results related to the convergence of best L1 approximants on the unit circle to the Markov function $$f(z) = \frac{1} {{2\pi i}}\int_E {\frac{{d\mu (x)}} {{z - x}}}$$ by meromorphic functions of the form P/Q, where P belongs to the Hardy space H1 of the unit disk and Q is a polynomial of degree at most n. We also include in an appendix a detailed treatment of a factorization theorem for Hardy spaces of the slit disk, which may be of independent interest.