Linear Interpolation on a Euclidean Ball in ℝⁿ

Abstract—

For ${{x}^{{(0)}}} \in {{\mathbb{R}}^{n}},R > 0$, by $B = B({{x}^{{(0)}}};R)$ we denote the Euclidean ball in ${{\mathbb{R}}^{n}}$ given by the inequality $\left\| {x - {{x}^{{(0)}}}} \right\| \leqslant R$, $\left\| x \right\|: = \mathop {\left( {\sum\nolimits_{i = 1}^n x_{i}^{2}} \right)}\nolimits^{1/2}$. Put ${{B}_{n}}: = B(0,1)$. We mean by $C(B)$ the space of continuous functions $f:B \to \mathbb{R}$ with the norm ${{\left\| f \right\|}_{{C(B)}}}: = ma{{x}_{{x \in B}}}\left| {f(x)} \right|$ and by ${{\Pi }_{1}}\left( {{{\mathbb{R}}^{n}}} \right)$ the set of polynomials in $n$ variables of degree $\leqslant 1$, 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 $Pf\left( {{{x}^{{(j)}}}} \right) = f\left( {{{x}^{{\left( j \right)}}}} \right)$ from C(B) into C(B). Let us define ${{\theta }_{n}}(B)$ as minimal value of ${{\left\| P \right\|}_{B}}$ under the condition ${{x}^{{(j)}}} \in B$. In the paper we obtain the formula to compute ${{\left\| P \right\|}_{B}}$ 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 ${{\left\| P \right\|}_{{{{B}_{n}}}}} = \max\{ \psi (a),\psi (a + 1)\} ,$ where $\psi (t) = \tfrac{{2\sqrt n }}{{n + 1}}{{(t(n + 1 - t))}^{{1/2}}} + \left| {1 - \tfrac{{2t}}{{n + 1}}} \right|$ $(0 \leqslant t \leqslant n + 1)$ and integer $a$ has the form $a = \left\lfloor {\tfrac{{n + 1}}{2} - \tfrac{{\sqrt {n + 1} }}{2}} \right\rfloor .$ For this projector, $\sqrt n \leqslant {{\left\| P \right\|}_{{{{B}_{n}}}}} \leqslant \sqrt {n + 1}$. The equality ${{\left\| P \right\|}_{{{{B}_{n}}}}} = \sqrt {n + 1}$ takes place if and only if $\sqrt {n + 1}$ is an integer number. We give the precise values of ${{\theta }_{n}}({{B}_{n}})$ for $1 \leqslant n \leqslant 4$. To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.