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 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.