Szeg\H{o}-type asymptotics for ray sequences of Frobenius-Pad\’e approximants

Abstract

Let σ^ be a Cauchy transform of a possibly complex-valued Borel measure σ and {pn} be a system of orthonormal polynomials with respect to a measure μ, supp(μ)∩supp(σ)=∅. An (m,n)-th Frobenius-Pad'e approximant to σ^ is a rational function P/Q, deg(P)≤m, deg(Q)≤n, such that the first m+n+1 Fourier coefficients of the linear form Qσ^−P vanish when the form is developed into a series with respect to the polynomials pn. We investigate the convergence of the Frobenius-Pad'e approximants to σ^ along ray sequences nn+m+1→c>0, n−1≤m, when μ and σ are supported on intervals on the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the respective interval are holomorphic functions.