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 $$\Vert f(A)-p(A)\Vert \le \frac{1}{n!}\underset{t\in [0,1];\mu \in \text{co}\{z_1,z_2,\dots ,z_n\}}{\max }\Vert \Omega (A)f^{(n)}((1-t)\mu 1+tA)\Vert ,$$ where $\Omega (z)={\prod }_{k=1}^n(z-z_k)$.

arXiv comment8 pages, 1 figure