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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

Full text: PDF file (750 kB)
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MSC: 65F60, 97N50
Received: 18.03.2019
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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}


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