An estimate of approximation of a matrix-valued function by an interpolation polynomial

Abstract

Let $A$ be a square complex matrix, $z_1$, ..., $z_{n}\in\mathbb C$ be (possibly repetitive) points of interpolation, $f$ be analytic in a neighborhood of the convex hull of the union of the spectrum of $A$ and the points $z_1$, ..., $z_{n}$, and $p$ be the interpolation polynomial of $f$, constructed by the points $z_1$, ..., $z_{n}$. It is proved that under these assumptions $$\Vert f(A)-p(A)\Vert\le\frac1{n!} \max_{t\in[0,1];\,μ\in\text{co}\{z_1,z_{2},\dots,z_{n}\}}\bigl\VertΩ(A)f^{(n)} \bigl((1-t)μ\mathbf1+tA\bigr)\bigr\Vert,$$ where $Ω(z)=\prod_{k=1}^n(z-z_k)$.