On Properties of a Regular Simplex Inscribed into a Ball
Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon $...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Yaroslavl State University
2021-06-01
|
| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/1487 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849338883609722880 |
|---|---|
| author | Mikhail Viktorovich Nevskii |
| author_facet | Mikhail Viktorovich Nevskii |
| author_sort | Mikhail Viktorovich Nevskii |
| collection | DOAJ |
| description | Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$. |
| format | Article |
| id | doaj-art-f7dde44bee2e4b76a6ff269134e59c75 |
| institution | Kabale University |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2021-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-f7dde44bee2e4b76a6ff269134e59c752025-08-20T03:44:17ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172021-06-0128218619710.18255/1818-1015-2021-2-186-1971130On Properties of a Regular Simplex Inscribed into a BallMikhail Viktorovich Nevskii0P. G. Demidov Yaroslavl State UniversityLet $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$.https://www.mais-journal.ru/jour/article/view/1487simplexballlinear interpolationprojectornorm |
| spellingShingle | Mikhail Viktorovich Nevskii On Properties of a Regular Simplex Inscribed into a Ball Моделирование и анализ информационных систем simplex ball linear interpolation projector norm |
| title | On Properties of a Regular Simplex Inscribed into a Ball |
| title_full | On Properties of a Regular Simplex Inscribed into a Ball |
| title_fullStr | On Properties of a Regular Simplex Inscribed into a Ball |
| title_full_unstemmed | On Properties of a Regular Simplex Inscribed into a Ball |
| title_short | On Properties of a Regular Simplex Inscribed into a Ball |
| title_sort | on properties of a regular simplex inscribed into a ball |
| topic | simplex ball linear interpolation projector norm |
| url | https://www.mais-journal.ru/jour/article/view/1487 |
| work_keys_str_mv | AT mikhailviktorovichnevskii onpropertiesofaregularsimplexinscribedintoaball |