On a Geometric Approach to the Estimation of Interpolation Projectors

TL;DR

This paper introduces a geometric approach to estimate the norm of polynomial interpolation projectors using the minimal homothety coefficient of a domain containing a simplex, with implications for polynomial interpolation analysis.

Contribution

It establishes new inequalities relating the projector norm to geometric parameters of the domain and simplex, providing a novel geometric perspective on interpolation error estimation.

Findings

Derived bounds for the interpolation projector norm using geometric measures.

Connected geometric domain properties with polynomial interpolation stability.

Presented numerical analysis results supporting the theoretical inequalities.

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.