Abstract—

For \({{x}^{{(0)}}} \in {{\mathbb{R}}^{n}},R > 0\), by \(B = B({{x}^{{(0)}}};R)\) we denote the Euclidean ball in \({{\mathbb{R}}^{n}}\) given by the inequality \(\left\| {x - {{x}^{{(0)}}}} \right\| \leqslant R\), \(\left\| x \right\|: = \mathop {\left( {\sum\nolimits_{i = 1}^n x_{i}^{2}} \right)}\nolimits^{1/2} \). Put \({{B}_{n}}: = B(0,1)\). We mean by \(C(B)\) the space of continuous functions \(f:B \to \mathbb{R}\) with the norm \({{\left\| f \right\|}_{{C(B)}}}: = ma{{x}_{{x \in B}}}\left| {f(x)} \right|\) and by \({{\Pi }_{1}}\left( {{{\mathbb{R}}^{n}}} \right)\) the set of polynomials in \(n\) variables of degree \( \leqslant 1\), i. e., linear functions on \({{\mathbb{R}}^{n}}\). Let \({{x}^{{(1)}}}, \ldots ,{{x}^{{(n + 1)}}}\) be the vertices of \(n\)-dimensional nondegenerate simplex \(S \subset B\). The interpolation projector \(P:C(B) \to {{\Pi }_{1}}({{\mathbb{R}}^{n}})\) corresponding to \(S\) is defined by the equalities \(Pf\left( {{{x}^{{(j)}}}} \right) = f\left( {{{x}^{{\left( j \right)}}}} \right)\) from C(B) into C(B). Let us define \({{\theta }_{n}}(B)\) as minimal value of \({{\left\| P \right\|}_{B}}\) under the condition \({{x}^{{(j)}}} \in B\). In the paper we obtain the formula to compute \({{\left\| P \right\|}_{B}}\) making use of \({{x}^{{(0)}}}\), \(R\), and coefficients of basic Lagrange polynomials of \(S\). In more details we study the case when \(S\) is a regular simplex inscribed into \({{B}_{n}}\). In this situation, we prove that \({{\left\| P \right\|}_{{{{B}_{n}}}}} = \max\{ \psi (a),\psi (a + 1)\} ,\) where \(\psi (t) = \tfrac{{2\sqrt n }}{{n + 1}}{{(t(n + 1 - t))}^{{1/2}}} + \left| {1 - \tfrac{{2t}}{{n + 1}}} \right|\) \((0 \leqslant t \leqslant n + 1)\) and integer \(a\) has the form \(a = \left\lfloor {\tfrac{{n + 1}}{2} - \tfrac{{\sqrt {n + 1} }}{2}} \right\rfloor .\) For this projector, \(\sqrt n \leqslant {{\left\| P \right\|}_{{{{B}_{n}}}}} \leqslant \sqrt {n + 1} \). The equality \({{\left\| P \right\|}_{{{{B}_{n}}}}} = \sqrt {n + 1} \) takes place if and only if \(\sqrt {n + 1} \) is an integer number. We give the precise values of \({{\theta }_{n}}({{B}_{n}})\) for \(1 \leqslant n \leqslant 4\). To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.

中文翻译：

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ℝn中欧式球上的线性插值

摘要-

对于\（{{x} ^ {{（（0）}}}} \ in {{\ mathbb {R}} ^ {n}}，R> 0 \），通过\（B = B（{{x} ^ {{（0）}}}; R } \）表示由不等式\（\ left \ | {x-{ }给出的\（{{\ mathbb {R}} ^ {n}} \）中的欧几里得球{x} ^ {{（（0）}}}} \ right \ | \ leqslant R \），\（\ left \ | x \ right \ |：= \ mathop {\ left（{\ sum \ nolimits_ {i = 1} ^ n x_ {i} ^ {2}} \ right）} \ nolimits ^ {1/2} \）。放\（{{B} _ {n}}：= B（0,1）\）。我们用\（C（B）\）表示连续函数\（f：B \ to \ mathbb {R} \）的空间，其范数为\（{{\ left \ | f \ right \ |} __ {{ C（B）}}}：= ma {{x} _ {{x \ in B}}} \ left | {f（x）} \ right | \）和\（{{\ Pi} _ {1 }} \ left（{{{\ mathbb {R}} ^ {n}}} \ right）\）\（n \）度\（\ leqslant 1 \）变量中的多项式集，即\（{{\ mathbb {R}} ^ {n}} \）上的线性函数。令\（{{x} ^ {{（（1）}}}，\ ldots，{{x} ^ {{（n + 1）}}} \）是\（n \）维非退化单纯形的顶点\（S \ subset B \）。对应于\（S \）的插值投影仪\（P：C（B）\ to {{\ Pi __1}}（{{\ mathbb {R}} ^ {n}}）\）由等于\（Pf \ left（{{{x} ^ {{{（j）}}}} \ right）= f \ left（{{{x} ^ {{\ left（j \ right）}}}}} \ right）\）从C（B）到C（B）。让我们将\（{{\ theta} _ {n}}（B）\）定义为\（{{\ left \ | P \ right \ |} _ {B}} \）的最小值在条件\（{{x} ^ {{{（j）}}} \ in B \）中。在本文中，我们获得了使用\（{{x} ^ {{{（0）}}} \）来计算\（{{\ left \ | P \ right \ |} _ {B}} \）的公式，\（R \）和\（S \）的基本拉格朗日多项式的系数。更详细地讲，我们研究\（S \）是刻在\（{{B} _ {n}} \）\中的常规单纯形的情况。在这种情况下，我们证明\（{{\ left \ | P \ right \ |} _ {{{{{B} _ {n}}}}}} = \ max \ {\ psi（a），\ psi（ a + 1）\}，\）其中\（\ psi（t）= \ tfrac {{2 \ sqrt n}} {{n + 1}} {{（t（n + 1-t））} ^ { {1/2}}} + \ left | {1-\ tfrac {{2t}} {{n + 1}}} \ right | \） \（（0 \ leqslant t \ leqslant n +1）\）和整数\（a \）的格式为\（a = \ left \ lfloor {\ tfrac {{n + 1}} {2}-\ tfrac {{\ sqrt {n + 1}}} {2}} \ right \ rfloor。\）对于此投影机，\（\ sqrt n \ leqslant {{\ left \ | P \ right \ |} _ {{{{{{B} _ {n}}}}}} \ leqslant \ sqrt {n + 1 } \）。等式\（{{\ left \ | P \ right \ |} _ {{{{{B} _ {n}}}}} = \ sqrt {n + 1} \）仅在\（\ sqrt {n + 1} \）是一个整数。我们给出的精确值\（{{\ THETA} _ {N}}（{{B} _ {N}}）\）为\（1 \ leqslantÑ\ leqslant 4 \）。为了补充理论结果，我们提出了计算数据。我们还将讨论有关欧几里得球上插值的其他一些问题。