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 $$n.$$