Szegő-type asymptotics for ray sequences of Frobenius-Padé approximants

Abstract

Let $\hat{\sigma }$ 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-Padé approximant to $\hat{\sigma }$ is a rational function $P/Q$, $\mathrm{deg}(P)\le m$, $\mathrm{deg}(Q)\le n$, such that the first $m+n+1$ Fourier coefficients of the linear form $Q\hat{\sigma }-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Padé approximants to $\hat{\sigma }$ along ray sequences $\frac{n}{n+m+1}\to c>0$, $n-1\le m$, 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.