Abstract

Let \({{Q}_{n}}{{ = [0,1]}^{n}}\) be the unit cube in \({{\mathbb{R}}^{n}}\) and let \(C({{Q}_{n}})\) be the space of continuous functions \(f:{{Q}_{n}} \to \mathbb{R}\) with the norm \({{\left| {\left| f \right|} \right|}_{{C({{Q}_{n}})}}}: = \mathop {\max }\nolimits_{x \in {{Q}_{n}}} \left| {f(x)} \right|.\) By \({{\Pi }_{1}}\left( {{{\mathbb{R}}^{n}}} \right)\) denote the set of polynomials of degree \( \leqslant 1\), i. e., the set of linear functions on \({{\mathbb{R}}^{n}}\). The interpolation projector \(P:C({{Q}_{n}}) \to {{\Pi }_{1}}({{\mathbb{R}}^{n}})\) with the nodes \({{x}^{{(j)}}} \in {{Q}_{n}}\) is defined by the equalities \(Pf\left( {{{x}^{{(j)}}}} \right) = f\left( {{{x}^{{(j)}}}} \right)\), \(j = 1,\) \( \ldots ,\) \(n + 1\). Let \({{\left| {\left| P \right|} \right|}_{{{{Q}_{n}}}}}\) be the norm of \(P\) as an operator from \(C({{Q}_{n}})\) to \(C({{Q}_{n}})\). If \(n + 1\) is an Hadamard number, then there exists a nondegenerate regular simplex having the vertices at vertices of \({{Q}_{n}}\). We discuss some approaches to get inequalities of the form \({{\left| {\left| P \right|} \right|}_{{{{Q}_{n}}}}} \leqslant c\sqrt n \) for the norm of the corresponding projector \(P\).

中文翻译：

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关于插值投影仪范数的一些估计

摘要

设\({{Q}_{n}}{{ = [0,1]}^{n}}\)为\({{\mathbb{R}}^{n}}\)中的单位立方体令\(C({{Q}_{n}})\)为连续函数空间\(f:{{Q}_{n}} \to \mathbb{R}\)，范数为\ ({{\left| {\left| f \right|} \right|}_{{C({{Q}_{n}})}}}: = \mathop {\max }\nolimits_{x \在 {{Q}_{n}}} \left| {f(x)} \right|.\)由\({{\Pi }_{1}}\left( {{{\mathbb{R} }^{n}}} \right)\)表示次数为\( \leqslant 1\)的多项式集合，即\({{\mathbb{R}}^{n}}上的线性函数集合\)。插值投影仪\(P:C({{Q}_{n}}) \ 到 {{\Pi }_{1}}({{\mathbb{R}}^{n}})\) {{Q}_{n}}\) 中的节点\({{x}^{{(j)}}} \)由等式\(Pf\left( {{{x}^{{(j )}}}} \right) = f\left( {{{x}^{{(j)}}}} \right)\) , \(j = 1,\) \( \ldots ,\) \ (n + 1\)。设\({{\left| {\left| P \right|} \right|}_{{{{Q}_{n}}}}}\) 为\(P\)作为运算符的范数从\(C({{Q}_{n}})\)到\(C({{Q}_{n}})\)。如果\(n + 1\)是哈达玛数，则存在一个非简并正则单纯形，其顶点位于\({{Q}_{n}}\)的顶点。我们讨论一些获得\({{\left| {\left| P \right|} \right|}_{{{{Q}_{n}}}}} \leqslant c\sqrt 形式的不等式的方法n \)为对应投影仪\(P\)的范数。