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 $\psi(t)=\dfrac{2\sqrt{n}}{n+1}\Bigl(t(n+1-t)\Bigr)^{1/2}+ \left|1-\dfrac{2t}{n+1}\right|$, $0\leq t\leq n+1$, and $a_n=\left\lfloor\dfrac{n+1}{2}-\dfrac{\sqrt{n+1}}{2}\right\rfloor$. For any $n$, $\sqrt{n}\leq \theta_n(B)\leq \sqrt{n+1}.$ 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.