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 $$n.$$
Highlights
we denote the space of continuous functions
to the simplex S is defined by equalities P f x(j
us denote by θn the minimal possible value of P
Summary
G через ver(G) обозначается совокупность его вершин. x(12). Обозначим через λj многочлен из Π1(Rn), коэффициенты которого составляют j-й столбец A−1 : λj(x) = l1jx1 + . Мы называем λj базисными многочленами Лагранжа, соответствующими S. Через σS обозначим образ S при гомотетии относительно центра тяжести с коэффициентом σ. Через α(S) обозначим минимальное σ > 0, для которого Qn принадлежит трансляту симплекса σS. Что точка x ∈ ver(Qn) является μ-веpшиной куба Qn относительно симплекса S, если для пpоектоpа P : C(Qn) → Π1 (Rn) c узлаn+1 ми в веpшинах S выполняется P = |λj(x)| и сpеди чисел λj(x) имеется pовно j=1 μ отpицательных. Если для некотоpого μ имеется μ-веpшина Qn относительно. Равенство в (6) достигается тогда и только тогда, когда существует 1-вершина Qn относительно S и симплекс ξ(S)S описан вокруг Qn
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