On Some Estimate for the Norm of an Interpolation Projector

TL;DR

This paper investigates bounds on the norm of interpolation projectors on the unit cube, focusing on cases where the nodes form a regular simplex related to Hadamard numbers, aiming to establish inequalities of the form ||P||_{Q_n} ≤ c√n.

Contribution

The paper explores approaches to derive inequalities bounding the norm of interpolation projectors, specifically aiming for bounds proportional to √n, in the context of regular simplices and Hadamard numbers.

Findings

Established connections between regular simplices and Hadamard numbers.

Discussed methods to derive inequalities for the projector norm.

Proposed bounds of the form ||P||_{Q_n} ≤ c√n for certain configurations.

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 $\|f\|_{C(Q_n)}:=\max_{x\in Q_n}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ denote a set of polynomials of degree $\leq 1$, 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 $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right)$, $j=1,$ $\ldots,$ $ n+1$. Let $\|P\|_{Q_n}$ 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 $||P||_{Q_n}\leq c\sqrt{n}$ for the norm of the corresponding projector $P$.