The Minimum Norm of a Projector under Linear Interpolation on a Euclidean Ball

Abstract

We prove the following proposition. Under linear interpolation on a Euclidean $n$-dimensional ball $B$, an interpolation projector whose nodes coincide with the vertices of a regular simplex inscribed into the boundary sphere has the minimum $C$-norm. This minimum norm ${\theta }_n(B)$ is equal to $\max \{\psi (a_n),\psi (a_n+1)\}$, where $\psi (t)=\frac{2\sqrt{n}}{n+1}(t(n+1-t){)}^{1/2}+\left|1-\frac{2t}{n+1}\right|$, $0\le t\le n+1$, and $a_n=\lfloor \frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\rfloor$. For any $n$, $\sqrt{n}\le {\theta }_n(B)\le \sqrt{n+1}.$ Moreover, ${\theta }_n(B)$ $=$ $\sqrt{n}$ only for $n=1$ and ${\theta }_n(B)=\sqrt{n+1}$ if and only if $\sqrt{n+1}$ is an integer.

7 pages

References in corpus (1)

• On Properties of a Regular Simplex Inscribed into a Ball

Cited by in corpus (2)

• On a Simplex Inscribed in a Ball
• Optimal Lagrange Interpolation Projectors and Legendre Polynomials