Abstract:
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.
Keywords:
Frobenius-Padé approximants, linear Padé-Chebyshev approximants, Padé approximants of orthogonal expansions, orthogonality, Markov-type functions, Riemann-Hilbert matrix problem.
A. I. Aptekarev's research was supported by the Russian Science Foundation (grant no. 14-21-00025). A. I. Bogolyubskii's research was supported by the Russian Foundation for Basic Research (grant nos. 14-01-00604_а and 17-01-00614_a). M. L. Yattselev's research was supported by the Simons Foundation (grant #354538).
Citation:
A. I. Aptekarev, A. I. Bogolyubskii, M. Yattselev, “Convergence of ray sequences of Frobenius-Padé approximants”, Sb. Math., 208:3 (2017), 313–334