On Optimal Interpolation by Linear Functions on an n-Dimensional Cube

Abstract

Let \(n\in{\mathbb N}\), and let \(Q_n\) be the unit cube \([0,1]^n\). By \(C(Q_n)\) we denote the space of continuous functions \(f:Q_n\to{\mathbb R}\) with the norm \(\|f\|_{C(Q_n)}:=\max\limits_{x\in Q_n}|f(x)|,\) by \(\Pi_1\left({\mathbb R}^n\right)\) --- the set of polynomials of \(n\) variables of degree \(\leq 1\) (or linear functions). Let \(x^{(j)},\) \(1\leq j\leq n+1,\) be the vertices of \(n\)-dimnsional nondegenerate simplex \(S\subset Q_n\). An interpolation projector \(P:C(Q_n)\to \Pi_1({\mathbb R}^n)\) corresponding to the simplex \(S\) is defined by equalities \(Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right).\) The norm of \(P\) as an operator from \(C(Q_n)\) to \(C(Q_n)\) may be calculated by the formula \(\|P\|=\max\limits_{x\in ver(Q_n)} \sum\limits_{j=1}^{n+1} |\lambda_j(x)|.\) Here \(\lambda_j\) are the basic Lagrange polynomials with respect to \(S,\) \(ver(Q_n)\) is the set of vertices of \(Q_n\). Let us denote by \(\theta_n\) the minimal possible value of \(\|P\|.\) Earlier, the first author proved various relations and estimates for values \(\|P\|\) and \(\theta_n\), in particular, having geometric character. The equivalence \(\theta_n\asymp \sqrt{n}\) takes place. For example, the appropriate, according to dimension \(n\), inequalities may be written in the form \linebreak \(\frac{1}{4}\sqrt{n}\) \(<\theta_n\) \(<3\sqrt{n}.\) If the nodes of the projector \(P^*\) coincide with vertices of an arbitrary simplex with maximum possible volume, we have \(\|P^*\|\asymp\theta_n.\)When an Hadamard matrix of order \(n+1\) exists, holds \(\theta_n\leq\sqrt{n+1}.\) In the paper, we give more precise upper bounds of numbers \(\theta_n\) for \(21\leq n \leq 26\). These estimates were obtained with the application of maximum volume simplices in the cube. For constructing such simplices, we utilize maximum determinants containing the elements \(\pm 1.\) Also, we systematize and comment the best nowaday upper and low estimates of numbers \(\theta_n\) for a concrete \(n.\)