Multipoint Padé-type approximants. Exact rate of convergence
In this interesting paper, the authors consider sequences of interpolating rational functions to Markov-type analytic functions, that is functions of the form \[ f(z):= \int^1_{-1} {d\mu(x)\over z-x},\quad z\in\mathbb{C}\backslash[- 1,1], \] where \(\mu\) is a finite positive Borel measure on \([-1,1]\). The interpolation points \(\{\alpha_{n,j}\}\) are assumed to be symmetric with respect to the real line, and some of the poles of the interpolating rational functions may be taken as zeros of a monic polynomial \(B_n\) of degree \(m(n)\leq n\), which is of one sign in \((-1,1)\). The rest of the poles of the interpolating rationals are determined by interpolation conditions. Under the assumption that \(\mu'>0\) a.e. in \((-1,1)\) and under suitable asymptotic assumptions on the distribution of the \(\{\alpha_{n,j}\}\) and the zeros of \(B_n\), the authors obtain the exact asymptotic rate of convergence of the interpolants. Then applications are given to the convergence rates of approximate quadratures. The proofs involve sophisticated potential theory, and the results strengthen and sharpen earlier work of these authors as well as of other authors including A. Ambroladze, L. Karlberg, H. Wallin. This is essential reading for researchers in the convergence theory of rational interpolation.
scientific article; zbMATH DE number 1143073