On a Geometric Approach to the Estimation of Interpolation Projectors

Suppose $\mathrm{\Omega }$ is a closed bounded subset of ${\mathbb{R}}^n,$ $S$ is an $n$-dimensional non-degenerate simplex, $\xi (\mathrm{\Omega };S):=\min \{\sigma \ge 1:\mathrm{\Omega }\subset \sigma S\}$. 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, ${\phi }_1(x)\equiv 1,$ ${\phi }_2(x)=x_1,\dots ,{\phi }_{n+1}(x)=x_n\mathrm{.}$ Put $\mathrm{\Pi }:=\mathrm{l}\mathrm{i}\mathrm{n}({\phi }_1,\dots ,{\phi }_d)\mathrm{.}$ The interpolation projector $P:C(\mathrm{\Omega })\to \mathrm{\Pi }$ with a set of nodes $x^{(1)},\dots ,x^{(d)}$ $\in \mathrm{\Omega }$ is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)\mathrm{.}$ Denote by $\mathrm{\parallel }P{\mathrm{\parallel }}_{\mathrm{\Omega }}$ the norm of $P$ as an operator from $C(\mathrm{\Omega })$ to $C(\mathrm{\Omega })$. Consider the mapping $T:{\mathbb{R}}^n\to {\mathbb{R}}^{d-1}$ of the form $T(x):=({\phi }_2(x),\dots ,{\phi }_d(x))\mathrm{.}$ We have the following inequalities: $\frac{1}{2}(1+\frac{1}{d-1})(\mathrm{\parallel }P{\mathrm{\parallel }}_{\mathrm{\Omega }}-1)+1$ $\le \xi (T(\mathrm{\Omega });S)\le \frac{d}{2}(\mathrm{\parallel }P{\mathrm{\parallel }}_{\mathrm{\Omega }}-1)+1.$ Here $S$ is the $(d-1)$-dimensional simplex with vertices $T\left(x^{(j)}\right)\mathrm{.}$ We discuss this and other relations for polynomial interpolation of functions continuous on a segment. The results of numerical analysis are presented.