Convergence of ray sequences of Frobenius-Padé approximants

Abstract

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