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 a space of continuous functions $f:Q_n\to \mathbb{R}$ with the norm $\Vert f{\Vert }_{C(Q_n)}:={\max }_{x\in Q_n}|f(x)|.$ By ${\Pi }_1\left({\mathbb{R}}^n\right)$ denote a set of polynomials of degree $\le 1$, i.e., a 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 $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,$ $\dots ,$ $n+1$. Let $\Vert P{\Vert }_{Q_n}$ 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 $||P|{|}_{Q_n}\le c\sqrt{n}$ for the norm of the corresponding projector $P$.

arXiv comment12 pages

References in corpus (1)

• Interpolation by Linear Functions on an $n$-Dimensional Ball