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 Model. Anal. Inform. Sist., 2018, Volume 25, Number 3, Pages 291–311 (Mi mais629)

Computational Geometry

On optimal interpolation by linear functions on an $n$-dimensional cube

M. V. Nevskii, A. Yu. Ukhalov

Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl, 150003, Russian Federation

Abstract: Let $n\in{\mathbb N}$, and let $Q_n$ be the unit cube $[0,1]^n$. By $C(Q_n)$ we denote the space of continuous functions $f:Q_n\to{\mathbb R}$ with the norm $\|f\|_{C(Q_n)}:=\max\limits_{x\in Q_n}|f(x)|,$ by $\Pi_1({\mathbb R}^n)$ — the set of polynomials of $n$ variables of degree $\leq 1$ (or linear functions). Let $x^{(j)},$ $1\leq j\leq n+1,$ be the vertices of $n$-dimnsional nondegenerate simplex $S\subset Q_n$. An interpolation projector $P:C(Q_n)\to \Pi_1({\mathbb R}^n)$ corresponding to the simplex $S$ is defined by equalities $Pf(x^{(j)})= f(x^{(j)})$. The norm of $P$ as an operator from $C(Q_n)$ to $C(Q_n)$ may be calculated by the formula $\|P\|=\max\limits_{x\in\mathrm{ver}(Q_n)} \sum\limits_{j=1}^{n+1} |\lambda_j(x)|$. Here $\lambda_j$ are the basic Lagrange polynomials with respect to $S,$ $\mathrm{ver}(Q_n)$ is the set of vertices of $Q_n$. Let us denote by $\theta_n$ the minimal possible value of $\|P\|$. Earlier, the first author proved various relations and estimates for values $\|P\|$ 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 may be written in the form $\frac{1}{4}\sqrt{n}$ $<\theta_n$ $<3\sqrt{n}$. If the nodes of the projector $P^*$ coincide with vertices of an arbitrary simplex with maximum possible volume, we have $\|P^*\|\asymp\theta_n$. When an Hadamard matrix of order $n+1$ exists, holds $\theta_n\leq\sqrt{n+1}$. In the paper, we give more precise upper bounds of numbers $\theta_n$ for $21\leq n \leq 26$. These estimates were obtained with the 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 numbers $\theta_n$ for a concrete $n$.

Keywords: $n$-dimensional simplex, $n$-dimensional cube, interpolation, projector, norm, numerical methods.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.12873.2018/12.1 This work was carried out within the framework of the state programme of the Ministry of Education and Science of the Russian Federation, project ¹ 1.12873.2018/12.1.

DOI: https://doi.org/10.18255/1818-1015-2018-3-291-311

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English version:
Automatic Control and Computer Sciences, 2018, 52:7, 828–842

UDC: 514.17+517.51+519.6

Citation: M. V. Nevskii, A. Yu. Ukhalov, “On optimal interpolation by linear functions on an $n$-dimensional cube”, Model. Anal. Inform. Sist., 25:3 (2018), 291–311; Automatic Control and Computer Sciences, 52:7 (2018), 828–842

Citation in format AMSBIB
\Bibitem{NevUkh18} \by M.~V.~Nevskii, A.~Yu.~Ukhalov \paper On optimal interpolation by linear functions on an $n$-dimensional cube \jour Model. Anal. Inform. Sist. \yr 2018 \vol 25 \issue 3 \pages 291--311 \mathnet{http://mi.mathnet.ru/mais629} \crossref{https://doi.org/10.18255/1818-1015-2018-3-291-311} \elib{http://elibrary.ru/item.asp?id=35144412} \transl \jour Automatic Control and Computer Sciences \yr 2018 \vol 52 \issue 7 \pages 828--842 \crossref{https://doi.org/10.3103%2FS0146411618070283} 

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This publication is cited in the following articles:
1. M. V. Nevskii, A. Yu. Ukhalov, “Lineinaya interpolyatsiya na evklidovom share v ${\mathbb R}^n$”, Model. i analiz inform. sistem, 26:2 (2019), 279–296
2. 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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