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

V. G. Kurbatov ; I. V. Kurbatova

Eurasian mathematical journal, Tome 11 (2020) no. 1, pp. 86-94

Résumé

Let $A$ be a square complex matrix; $z_1,\dots,z_n\in\mathbb{C}$ be (possibly repetitive) points of interpolation; $f$ be a function analytic in a neighborhood of the convex hull of the union of the spectrum of $A$ and the points $z_1,\dots,z_n$; and $p$ be the interpolation polynomial of $f$ constructed by the points $z_1,\dots,z_n$. It is proved that under these assumptions $$ ||f(A)-p(A)||\leqslant \frac1{n!}\max_{t\in[0,1]\atop {\mu\in co\{z_1,z_2,\dots,z_n\}}}||\Omega(A)f^{(n)}((1-t)\mu\mathbf{1}+tA)||, $$ where $\Omega(z)=\prod_{k=1}^n(z-z_k)$ and the symbol $co$ means the convex hull.