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 Model. Anal. Inform. Sist., 2019, Volume 26, Number 2, Pages 279–296 (Mi mais679)

Computing methodologies and applications

Linear interpolation on a Euclidean ball in ${\mathbb R}^n$

M. V. Nevskii, A. Yu. Ukhalov

P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150003, Russian Federation

Abstract: For $x^{(0)}\in{\mathbb R}^n, R>0$, by $B=B(x^{(0)};R)$ we denote a Euclidean ball in ${\mathbb R}^n$ given by the inequality $\|x-x^{(0)}\|\leq R$, $\|x\|:=(\sum_{i=1}^n x_i^2)^{1/2}$. Put $B_n:=B(0,1)$. We mean by $C(B)$ the space of continuous functions $f:B\to{\mathbb R}$ with the norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|$ and by $\Pi_1({\mathbb R}^n)$ the set of polynomials in $n$ variables of degree $\leq 1$, 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 $Pf(x^{(j)})= f(x^{(j)}).$
Denote by $\|P\|_B$ the norm of $P$ as an operator from $C(B)$ into $C(B)$. Let us define $\theta_n(B)$ as minimal value of $\|P\|_B$ under the condition $x^{(j)}\in B$. In the paper, we obtain the formula to compute $\|P\|_B$ 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 $\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},$ where $\psi(t)=\frac{2\sqrt{n}}{n+1}(t(n+1-t))^{1/2}+ |1-\frac{2t}{n+1}|$ $(0\leq t\leq n+1)$ and integer $a$ has the form $a=\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\rfloor.$ For this projector, $\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}$. The equality $\|P\|_{B_n}=\sqrt{n+1}$ takes place if and only if $\sqrt{n+1}$ is an integer number. We give the precise values of $\theta_n(B_n)$ for $1\leq n\leq 4$. To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.

Keywords: $n$-dimensional simplex, $n$-dimensional ball, linear interpolation, projector, norm.

DOI: https://doi.org/10.18255/1818-1015-279-296

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UDC: 514.17+517.51+519.6
Revised: 21.02.2019
Accepted:25.02.2019

Citation: M. V. Nevskii, A. Yu. Ukhalov, “Linear interpolation on a Euclidean ball in ${\mathbb R}^n$”, Model. Anal. Inform. Sist., 26:2 (2019), 279–296

Citation in format AMSBIB
\Bibitem{NevUkh19} \by M.~V.~Nevskii, A.~Yu.~Ukhalov \paper Linear interpolation on a Euclidean ball in ${\mathbb R}^n$ \jour Model. Anal. Inform. Sist. \yr 2019 \vol 26 \issue 2 \pages 279--296 \mathnet{http://mi.mathnet.ru/mais679} \crossref{https://doi.org/10.18255/1818-1015-279-296} 

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This publication is cited in the following articles:
1. M. V. Nevskii, “Geometricheskie otsenki pri interpolyatsii na $n$-mernom share”, Model. i analiz inform. sistem, 26:3 (2019), 441–449
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