Linear interpolation on a Euclidean ball in ${\mathbb R}^n$

M. V. Nevskii ; A. Yu. Ukhalov

Modelirovanie i analiz informacionnyh sistem, Tome 26 (2019) no. 2, pp. 279-296

Résumé

For $x^{(0)}\in{\mathbb R}^n, R>0$, by $B=B(x^{(0)};R)$ we denote a Euclidean ball in ${\mathbb R}^n$ given by the inequality $\|x-x^{(0)}\|\leq R$, $\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}$. Put $B_n:=B(0,1)$. We mean by $C(B)$ the space of continuous functions $f:B\to{\mathbb R}$ with the norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|$ and by $\Pi_1\left({\mathbb R}^n\right)$ the set of polynomials in $n$ variables of degree $\leq 1$, i. e. linear functions on ${\mathbb R}^n$. Let $x^{(1)}, \ldots, x^{(n+1)}$ be the vertices of $n$-dimensional nondegenerate simplex $S\subset B$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right).$ Denote by $\|P\|_B$ the norm of $P$ as an operator from $C(B)$ into $C(B)$. Let us define $\theta_n(B)$ as minimal value of $\|P\|_B$ under the condition $x^{(j)}\in B$. In the paper, we obtain the formula to compute $\|P\|_B$ making use of $x^{(0)}$, $R$, and coefficients of basic Lagrange polynomials of $S$. In more details we study the case when $S$ is a regular simplex inscribed into $B_n$. In this situation, we prove that $\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},$ where $\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+ \bigl|1-\frac{2t}{n+1}\bigr|$ $(0\leq t\leq n+1)$ and integer $a$ has the form $a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.$ For this projector, $\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}$. The equality $\|P\|_{B_n}=\sqrt{n+1}$ takes place if and only if $\sqrt{n+1}$ is an integer number. We give the precise values of $\theta_n(B_n)$ for $1\leq n\leq 4$. To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.

Détail

Keywords: $n$-dimensional simplex, $n$-dimensional ball, linear interpolation, projector
Mots-clés : norm.