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 a Euclidean ball in \({\mathbb R}^n\) given by the inequality \(\|x-x^{(0)}\|\leq R\), \(\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}\). Put \(B_n:=B(0,1)\). We mean by \(C(B)\) the space of continuous functions \(f:B\to{\mathbb R}\) with the norm \(\|f\|_{C(B)}:=\max_{x\in B}|f(x)|\) and by \(\Pi_1\left({\mathbb R}^n\right)\) the set of polynomials in \(n\) variables of degree \(\leq 1\), 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 \(Pf\left(x^{(j)}\right)=%f_j:=f\left(x^{(j)}\right).\) Denote by \(\|P\|_B\) the norm of \(P\) as an operator from \(C(B)\) into \(C(B)\). Let us define \(\theta_n(B)\) as minimal value of \(\|P\|\) under the condition \(x^{(j)}\in B\). In the paper, we obtain the formula to compute \(\|P\|_B\) 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 \(\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},\) where \(\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+\bigl|1-\frac{2t}{n+1}\bigr|\) \((0\leq t\leq n+1)\) and integer \(a\) has the form \(a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.\) For this projector, \(\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}\). The equality \(\|P\|_{B_n}=\sqrt{n+1}\) takes place if and only if \(\sqrt{n+1}\) is an integer number. We give the precise values of \(\theta_n(B_n)\) for \(1\leq n\leq 4\). To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.