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 $n.$