On Optimal Interpolation by Linear Functions on n-Dimensional Cube

Abstract

Let \(n \in N\), and let \({{Q}_{n}}\) be the unit cube \({{[0,1]}^{n}}\). By \(C({{Q}_{n}})\) we denote the space of continuous functions \(f:{{Q}_{n}} \to R\) with the norm \({{\left\| f \right\|}_{{C({{Q}_{n}})}}}: = \mathop {max}\limits_{x \in {{Q}_{n}}} \left| {f(x)} \right|,\) by \({{\Pi }_{1}}\left( {{{R}^{n}}} \right)\) – the set of polynomials of \(n\) variables of degree \( \leqslant 1\) (or linear functions). Let \({{x}^{{(j)}}},\)\(1 \leqslant j \leqslant n + 1,\) be the vertices of an \(n\)-dimnsional nondegenerate simplex \(S \subset {{Q}_{n}}\). The interpolation projector \(P:C({{Q}_{n}}) \to {{\Pi }_{1}}({{R}^{n}})\) corresponding to the simplex \(S\) is defined by the equalities \(Pf\left( {{{x}^{{(j)}}}} \right) = f\left( {{{x}^{{(j)}}}} \right).\) The norm of \(P\) as an operator from \(C({{Q}_{n}})\) to \(C({{Q}_{n}})\) can be calculated by the formula \(\left\| P \right\| = \mathop {max}\limits_{x \in {\text{ver}}({{Q}_{n}})} \sum\nolimits_{j = 1}^{n + 1} {\left| {{{\lambda }_{j}}(x)} \right|} .\) Here \({{\lambda }_{j}}\) are the basic Lagrange polynomials with respect to \(S,\)\({\text{ver}}({{Q}_{n}})\) is the set of vertices of \({{Q}_{n}}\). Let us denote by \({{\theta }_{n}}\) the minimal possible value of \(\left\| P \right\|.\) Earlier the first author proved various relations and estimates for values \(\left\| P \right\|\) 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 can be written in the form \(\tfrac{1}{4}\sqrt n \)\( < {{\theta }_{n}}\)\( < 3\sqrt n .\) If the nodes of a projector \(P{\text{*}}\) coincide with vertices of an arbitrary simplex with maximum possible volume, then we have \(\left\| {P{\text{*}}} \right\| \asymp {{\theta }_{n}}.\) When an Hadamard matrix of order \(n + 1\) exists, holds \({{\theta }_{n}} \leqslant \sqrt {n + 1} .\) In the present paper, we give more precise upper bounds of \({{\theta }_{n}}\) for \(21 \leqslant n \leqslant 26\). These estimates were obtained with 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 \({{\theta }_{n}}\) for concrete \(n.\)