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 Eurasian Math. J., 2020, Volume 11, Number 1, Pages 86–94 (Mi emj358)

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

V. G. Kurbatov, I. V. Kurbatova

Voronezh State University, 1 Universitetskaya Square, 394018 Voronezh, Russia

Abstract: Let $A$ be a square complex matrix; $z_1,…,z_n\in\mathbb{C}$ 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 $z_1,…,z_n$; and $p$ be the interpolation polynomial of $f$ constructed by the points $z_1,…,z_n$. It is proved that under these assumptions
$$||f(A)-p(A)||\leqslant \frac1{n!}\max_{t\in[0,1]\atop {\mu\in coż_1,z_2,…,z_n\}}}||\Omega(A)f^{(n)}((1-t)\mu\mathbf{1}+tA)||,$$
where $\Omega(z)=\prod_{k=1}^n(z-z_k)$ and the symbol $co$ means the convex hull.

Keywords and phrases: matrix function, polynomial interpolation, estimate.

 Funding Agency Grant Number Russian Foundation for Basic Research 19-01-00732_a Ministry of Education and Science of the Russian Federation 3.1761.2017/4.6 The first author was supported by the Ministry of Education and Science of the Russian Federation under state order No. 3.1761.2017/4.6. The second author was supported by the Russian Foundation for Basic Research under research project No. 19-01-00732 A.

DOI: https://doi.org/10.32523/2077-9879-2020-11-1-86-94

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MSC: 65F60, 97N50
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Citation: V. G. Kurbatov, I. V. Kurbatova, “An estimate of approximation of a matrix-valued function by an interpolation polynomial”, Eurasian Math. J., 11:1 (2020), 86–94

Citation in format AMSBIB
\Bibitem{KurKur20} \by V.~G.~Kurbatov, I.~V.~Kurbatova \paper An estimate of approximation of a matrix-valued function by an interpolation polynomial \jour Eurasian Math. J. \yr 2020 \vol 11 \issue 1 \pages 86--94 \mathnet{http://mi.mathnet.ru/emj358} \crossref{https://doi.org/10.32523/2077-9879-2020-11-1-86-94}