On a Geometric Approach to the Estimation of Interpolation Projectors

Suppose $$\Omega $$ is a closed bounded subset of $${{\mathbb{R}}^{n}},$$ $$S$$ is an $$n$$ -dimensional nondegenerate simplex, $$\xi (\Omega ;S): = \min \left\{ {\sigma \geqslant 1:\Omega \subset \sigma S} \right\}$$ . Here $$\sigma S$$ is the result of homothety of $$S$$ with respect to the center of gravity with coefficient $$\sigma $$ . Let $$d \geqslant n + 1,$$ $${{\varphi }_{1}}(x), \ldots ,{{\varphi }_{d}}(x)$$ 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 $$\Pi : = {\text{lin}}({{\varphi }_{1}}, \ldots ,{{\varphi }_{d}}).$$ The interpolation projector $$P:C(\Omega ) \to \Pi $$ with a set of nodes $${{x}^{{(1)}}}, \ldots ,{{x}^{{(d)}}}$$ $$ \in \Omega $$ is defined by the equalities $$Pf\left( {{{x}^{{(j)}}}} \right) = f\left( {{{x}^{{(j)}}}} \right).$$ Denote by $${{\left\| P \right\|}_{\Omega }}$$ 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: $$\frac{1}{2}\left( {1 + \frac{1}{{d - 1}}} \right)\left( {{{{\left\| P \right\|}}_{\Omega }} - 1} \right) + 1$$ $$ \leqslant \xi (T(\Omega );S) \leqslant \frac{d}{2}\left( {{{{\left\| P \right\|}}_{\Omega }} - 1} \right) + 1.$$ Here $$S$$ is the $$(d - 1)$$ -dimensional simplex with vertices $$T({{x}^{{(j)}}}).$$ We discuss this and other relations for polynomial interpolation of functions continuous on a segment. The results of numerical analysis are presented.