On Some Estimate for the Norm of an Interpolation Projector

Abstract

Let ${{Q}_{n}}{{ = [0,1]}^{n}}$ be the unit cube in ${{\mathbb{R}}^{n}}$ and let $C({{Q}_{n}})$ be the space of continuous functions $f:{{Q}_{n}} \to \mathbb{R}$ with the norm ${{\left| {\left| f \right|} \right|}_{{C({{Q}_{n}})}}}: = \mathop {\max }\nolimits_{x \in {{Q}_{n}}} \left| {f(x)} \right|.$ By ${{\Pi }_{1}}\left( {{{\mathbb{R}}^{n}}} \right)$ denote the set of polynomials of degree $\leqslant 1$, i. e., the set of linear functions on ${{\mathbb{R}}^{n}}$. The interpolation projector $P:C({{Q}_{n}}) \to {{\Pi }_{1}}({{\mathbb{R}}^{n}})$ with the nodes ${{x}^{{(j)}}} \in {{Q}_{n}}$ is defined by the equalities $Pf\left( {{{x}^{{(j)}}}} \right) = f\left( {{{x}^{{(j)}}}} \right)$, $j = 1,$ $\ldots ,$ $n + 1$. Let ${{\left| {\left| P \right|} \right|}_{{{{Q}_{n}}}}}$ be the norm of $P$ as an operator from $C({{Q}_{n}})$ to $C({{Q}_{n}})$. If $n + 1$ is an Hadamard number, then there exists a nondegenerate regular simplex having the vertices at vertices of ${{Q}_{n}}$. We discuss some approaches to get inequalities of the form ${{\left| {\left| P \right|} \right|}_{{{{Q}_{n}}}}} \leqslant c\sqrt n$ for the norm of the corresponding projector $P$.