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 \(P\).