Convergence of ray sequences of Frobenius-Pad\'e approximants

A. I. Aptekarev ; A. I. Bogolyubskii ; M. Yattselev

Sbornik. Mathematics, Tome 208 (2017) no. 3, pp. 313-334

Résumé

Let $\widehat\sigma$ be a Cauchy transform of a possibly complex-valued Borel measure $\sigma$ and $\{p_n\}$ a system of orthonormal polynomials with respect to a measure $\mu$, where $\operatorname{supp}(\mu)\cap\operatorname{supp}(\sigma)=\varnothing$. An $(m,n)$th Frobenius-Padé approximant to $\widehat\sigma$ is a rational function $P/Q$, $\deg(P)\leq m$, $\deg(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the remainder function $Q\widehat\sigma-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Padé approximants to $\widehat\sigma$ along ray sequences $n/(n+m+1)\to c>0$, $n-1\leq m$, when $\mu$ and $\sigma$ 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. Bibliography: 30 titles.

Détail

Keywords: linear Padé-Chebyshev approximants, Padé approximants of orthogonal expansions, orthogonality, Markov-type functions, Riemann-Hilbert matrix problem.
Mots-clés : Frobenius-Padé approximants