Geometric Estimates in Interpolation by Linear Functions on a Euclidean Ball

Abstract

Let $B_n$ be the Euclidean unit ball in ${\mathbb{R}}^n$ given by the inequality $\Vert x\Vert \le 1$, $\Vert x\Vert :={\left(\sum _{i=1}^n{x}_i^2\right)}^{\frac{1}{2}}$. By $C(B_n)$ we mean the space of continuous functions $f:B_n\to \mathbb{R}$ with the norm $\Vert f{\Vert }_{C(B_n)}:=\underset{x\in B_n}{\max }|f(x)|$. The symbol ${\Pi }_1\left({\mathbb{R}}^n\right)$ denotes the set of polynomials in $n$ variables of degree $\le 1$, i.e., the set of linear functions upon ${\mathbb{R}}^n$. Assume $x^{(1)},\dots ,x^{(n+1)}$ are the vertices of an $n$-dimensional nondegenerate simplex $S\subset B_n$. The interpolation projector $P:C(B_n)\to {\Pi }_1({\mathbb{R}}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ Denote by $\Vert P{\Vert }_{B_n}$ the norm of $P$ as an operator from $C(B_n)$ onto $C(B_n)$. We describe the approach in which $\Vert P{\Vert }_{B_n}$ can be estimated from below via the volume of $S$.

arXiv comment10 pages

References in corpus (1)

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

Cited by in corpus (3)

• On Properties of a Regular Simplex Inscribed into a Ball
• Estimates for Interpolation Projectors and Related Problems in Computational Geometry
• Optimal Lagrange Interpolation Projectors and Legendre Polynomials