On a geometric approach to the estimation of interpolation projectors

Abstract

Suppose $\Omega$ is a closed bounded subset of ${\mathbb R}^n,$ $S$ is an $n$-dimensional non-degenerate simplex, $\xi(\Omega;S):=\min \left\{\sigma\geq 1: \, \Omega\subset \sigma S\right\}$. Here $\sigma S$ is the result of homothety of $S$ with respect to the center of gravity with coefficient $\sigma$. Let $d\geq n+1,$ $\varphi_1(x),\ldots,\varphi_d(x)$ be linearly independent monomials in $n$ variables, $\varphi_1(x)\equiv 1,$ $\varphi_2(x)=x_1,\ \ldots, \ \varphi_{n+1}(x)=x_n.$ Put $\Pi:={\rm lin}(\varphi_1,\ldots,\varphi_d).$ The interpolation projector $P: C(\Omega)\to \Pi$ with a set of nodes $x^{(1)},\ldots, x^{(d)}$ $ \in \Omega$ is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ Denote by $\|P\|_{\Omega}$ the norm of $P$ as an operator from $C(\Omega)$ to $C(\Omega)$. Consider the mapping $T:{\mathbb R}^n\to {\mathbb R}^{d-1}$ of the form $T(x):=(\varphi_2(x),\ldots,\varphi_d(x)). $ We have the following inequalities: $ \frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_{\Omega}-1\right)+1$ $ \leq \xi(T(\Omega);S)\leq \frac{d}{2}\left(\|P\|_{\Omega}-1\right)+1. $ Here $S$ is the $(d-1)$-dimensional simplex with vertices $T\left(x^{(j)}\right).$ We discuss this and other relations for polynomial interpolation of functions continuous on a segment. The results of numerical analysis are presented.