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

TL;DR

This paper studies the asymptotic behavior of Frobenius-Padé approximants to Cauchy transforms of measures, establishing Szegő-type asymptotics along specific ray sequences under certain regularity conditions.

Contribution

It provides the first Szegő-type asymptotic analysis for ray sequences of Frobenius-Padé approximants with measures supported on real intervals and holomorphic Radon-Nikodym derivatives.

Findings

Established Szegő-type asymptotics for Frobenius-Padé approximants

Analyzed convergence along ray sequences where n/(n+m+1) approaches a positive constant

Extended classical asymptotic results to rational approximants of Cauchy transforms

Abstract

Let $\widehat\sigma$ be a Cauchy transform of a possibly complex-valued Borel measure $\sigma$ and $\{p_n\}$ be a system of orthonormal polynomials with respect to a measure $\mu$, $\mathrm{supp}(\mu)\cap\mathrm{supp}(\sigma)=\varnothing$. An $(m,n)$-th Frobenius-Pad\'e approximant to $\widehat\sigma$ is a rational function $P/Q$, $\mathrm{deg}(P)\leq m$, $\mathrm{deg}(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the linear form $Q\widehat\sigma-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Pad\'e approximants to $\widehat\sigma$ along ray sequences $\frac n{n+m+1}\to c>0$, $n-1\leq m$, when $\mu$ and $\sigma$ 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.