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

## TL;DR

This paper provides an upper bound estimate for the approximation error when representing a matrix-valued function by an interpolation polynomial, involving derivatives and the spectrum of the matrix.

## Contribution

It introduces a new matrix norm bound for the approximation error of matrix functions by interpolation polynomials, extending classical scalar polynomial approximation results.

## Key findings

- Derived an explicit error bound involving matrix derivatives and spectrum.
- Generalized scalar polynomial approximation bounds to matrix-valued functions.
- Provides a theoretical tool for analyzing polynomial approximations of matrix functions.

## 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 $$\Vert f(A)-p(A)\Vert\le\frac1{n!} \max_{t\in[0,1];\,\mu\in\text{co}\{z_1,z_{2},\dots,z_{n}\}}\bigl\Vert\Omega(A)f^{{(n)}} \bigl((1-t)\mu\mathbf1+tA\bigr)\bigr\Vert,$$ where $\Omega(z)=\prod_{k=1}^n(z-z_k)$.