Estimation of interpolation projectors using Legendre polynomials
We give some estimates for the minimum projector norm under linear interpolation on a compact set in ${\mathbb R}^n$. Let $\Pi_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$, $\Omega$ is a compactum in ${\mathbb R}^n$, $K={\rm conv}(\Omega)$. We will assume that...
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Yaroslavl State University
2024-09-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/1880 |
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| author | Mikhail V. Nevskii |
| author_facet | Mikhail V. Nevskii |
| author_sort | Mikhail V. Nevskii |
| collection | DOAJ |
| description | We give some estimates for the minimum projector norm under linear interpolation on a compact set in ${\mathbb R}^n$. Let $\Pi_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$, $\Omega$ is a compactum in ${\mathbb R}^n$, $K={\rm conv}(\Omega)$. We will assume that ${\rm vol}(K)>0$. Let the points $x^{(j)}\in \Omega$, $1\leq j\leq n+1,$ be the vertices of an $n$-dimensional nondegenerate simplex. The interpolation projector $P:C(\Omega)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}$ is defined by the equations $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$. By $\|P\|_\Omega$ we mean the norm of $P$ as an operator from $C(\Omega)$ to $C(\Omega)$. By $\theta_n(\Omega)$ we denote the minimal norm $\|P\|_\Omega$ of all operators $P$ with nodes belonging to $\Omega$. By ${\rm simp}(E)$ we denote the maximal volume of the simplex with vertices in $E$. We establish the inequalities $\chi_n^{-1}\left(\frac{{\rm vol}(K)}{{\rm simp}(\Omega)}\right)\leq \theta_n(\Omega)\leq n+1.$ Here $\chi_n$ is the standardized Legendre polynomial of degree $n$. The lower estimate is proved using the obtained characterization of Legendre polynomials through the volumes of convex polyhedra. More specifically, we show that for every $\gamma\ge 1$ the volume of the set $\left\{x=(x_1,...,x_n)\in{\mathbb R}^n : \sum |x_j| +\left|1- \sum x_j\right|\le\gamma\right\}$ is equal to ${\chi_n(\gamma)}/{n!}$. In the case when $\Omega$ is an $n$-dimensional cube or an $n$-dimensional ball, the lower estimate gives the possibility to obtain the inequalities of the form $\theta_n(\Omega)\geqslant c\sqrt{n}$. Also we formulate some open questions. |
| format | Article |
| id | doaj-art-23712bcfb72841ffad5c838c5183e8e7 |
| institution | Kabale University |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-23712bcfb72841ffad5c838c5183e8e72025-08-20T04:00:19ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172024-09-0131331633710.18255/1818-1015-2024-3-316-3371417Estimation of interpolation projectors using Legendre polynomialsMikhail V. Nevskii0P.G. Demidov Yaroslavl State UniversityWe give some estimates for the minimum projector norm under linear interpolation on a compact set in ${\mathbb R}^n$. Let $\Pi_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$, $\Omega$ is a compactum in ${\mathbb R}^n$, $K={\rm conv}(\Omega)$. We will assume that ${\rm vol}(K)>0$. Let the points $x^{(j)}\in \Omega$, $1\leq j\leq n+1,$ be the vertices of an $n$-dimensional nondegenerate simplex. The interpolation projector $P:C(\Omega)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}$ is defined by the equations $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$. By $\|P\|_\Omega$ we mean the norm of $P$ as an operator from $C(\Omega)$ to $C(\Omega)$. By $\theta_n(\Omega)$ we denote the minimal norm $\|P\|_\Omega$ of all operators $P$ with nodes belonging to $\Omega$. By ${\rm simp}(E)$ we denote the maximal volume of the simplex with vertices in $E$. We establish the inequalities $\chi_n^{-1}\left(\frac{{\rm vol}(K)}{{\rm simp}(\Omega)}\right)\leq \theta_n(\Omega)\leq n+1.$ Here $\chi_n$ is the standardized Legendre polynomial of degree $n$. The lower estimate is proved using the obtained characterization of Legendre polynomials through the volumes of convex polyhedra. More specifically, we show that for every $\gamma\ge 1$ the volume of the set $\left\{x=(x_1,...,x_n)\in{\mathbb R}^n : \sum |x_j| +\left|1- \sum x_j\right|\le\gamma\right\}$ is equal to ${\chi_n(\gamma)}/{n!}$. In the case when $\Omega$ is an $n$-dimensional cube or an $n$-dimensional ball, the lower estimate gives the possibility to obtain the inequalities of the form $\theta_n(\Omega)\geqslant c\sqrt{n}$. Also we formulate some open questions.https://www.mais-journal.ru/jour/article/view/1880polynomial interpolationprojectornormesimatelegendre polynomials |
| spellingShingle | Mikhail V. Nevskii Estimation of interpolation projectors using Legendre polynomials Моделирование и анализ информационных систем polynomial interpolation projector norm esimate legendre polynomials |
| title | Estimation of interpolation projectors using Legendre polynomials |
| title_full | Estimation of interpolation projectors using Legendre polynomials |
| title_fullStr | Estimation of interpolation projectors using Legendre polynomials |
| title_full_unstemmed | Estimation of interpolation projectors using Legendre polynomials |
| title_short | Estimation of interpolation projectors using Legendre polynomials |
| title_sort | estimation of interpolation projectors using legendre polynomials |
| topic | polynomial interpolation projector norm esimate legendre polynomials |
| url | https://www.mais-journal.ru/jour/article/view/1880 |
| work_keys_str_mv | AT mikhailvnevskii estimationofinterpolationprojectorsusinglegendrepolynomials |