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, $\xi (\Omega ;S):=\min \left\{\sigma \ge 1:\Omega \subset \sigma S\right\}$. Here $\sigma S$ is the result of homothety of $S$ with respect to the center of gravity with coefficient $\sigma$. Let $d\ge n+1,$ ${\phi }_1(x),\dots ,{\phi }_d(x)$ be linearly independent monomials in $n$ variables, ${\phi }_1(x)\equiv 1,$ ${\phi }_2(x)=x_1,\dots ,{\phi }_{n+1}(x)=x_n.$ Put $\Pi :=\mathrm{lin}({\phi }_1,\dots ,{\phi }_d).$ The interpolation projector $P:C(\Omega )\to \Pi$ with a set of nodes $x^{(1)},\dots ,x^{(d)}$ $\in \Omega$ is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ Denote by $\Vert P{\Vert }_{\Omega }$ 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 $T(x):=({\phi }_2(x),\dots ,{\phi }_d(x)).$ We have the following inequalities: $\frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\Vert P{\Vert }_{\Omega }-1\right)+1$ $\le \xi (T(\Omega );S)\le \frac{d}{2}\left(\Vert P{\Vert }_{\Omega }-1\right)+1.$ 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.

arXiv comment13 pages, 2 figures

References in corpus (1)

• Interpolation by Linear Functions on an $n$-Dimensional Ball