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

Abstract

By $B=B(x^{(0)};R)$ we denote the Euclidean ball in ${\mathbb{R}}^n$ given by the inequality $\Vert x-x^{(0)}\Vert \le R$. Here $x^{(0)}\in {\mathbb{R}}^n,R>0$, $\Vert x\Vert :={\left({\sum }_{i=1}^n{x}_i^2\right)}^{1/2}$. We mean by $C(B)$ the space of continuous functions $f:B\to \mathbb{R}$ with the norm $\Vert f{\Vert }_{C(B)}:={\max }_{x\in B}|f(x)|$ and by ${\Pi }_1\left({\mathbb{R}}^n\right)$ the set of polynomials in $n$ variables of degree $\le 1$, i.e., linear functions on ${\mathbb{R}}^n$. Let $x^{(1)},\dots ,x^{(n+1)}$ 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 $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ We obtain the formula to compute the norm of $P$ as an operator from $C(B)$ into $C(B)$ via $x^{(0)}$, $R$ and coefficients of basic Lagrange polynomials of $S$. In more details we study the case when $S$ is a regular simplex inscribed into $B_n=B(0,1)$.

17 pages, 6 figures, 1 table