On a Geometric Approach to the Estimation of Interpolation Projectors

Abstract

Suppose $Ω$ is a closed bounded subset of ${\mathbb R}^n,$ $S$ is an $n$-dimensional non-degenerate simplex, $ξ(Ω;S):=\min \left\{σ\geq 1: \, Ω\subset σS\right\}$. Here $σS$ is the result of homothety of $S$ with respect to the center of gravity with coefficient $σ$. Let $d\geq n+1,$ $φ_1(x),\ldots,φ_d(x)$ be linearly independent monomials in $n$ variables, $φ_1(x)\equiv 1,$ $φ_2(x)=x_1,\ \ldots, \ φ_{n+1}(x)=x_n.$ Put $Π:={\rm lin}(φ_1,\ldots,φ_d).$ The interpolation projector $P: C(Ω)\to Π$ with a set of nodes $x^{(1)},\ldots, x^{(d)}$ $ \in Ω$ is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ Denote by $\|P\|_Ω$ the norm of $P$ as an operator from $C(Ω)$ to $C(Ω)$. Consider the mapping $T:{\mathbb R}^n\to {\mathbb R}^{d-1}$ of the form $T(x):=(φ_2(x),\ldots,φ_d(x)). $ We have the following inequalities: $ \frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_Ω-1\right)+1$ $ \leq ξ(T(Ω);S)\leq \frac{d}{2}\left(\|P\|_Ω-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.