On Some Estimate for the Norm of an Interpolation Projector
Let $Q_n=[0,1]^n$ be the unit cube in ${\mathbb R}^n$ and let $C(Q_n)$ be a space of continuous functions $f:Q_n\to{\mathbb R}$ with the norm $\|f\|_{C(Q_n)}:=\max_{x\in Q_n}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ denote a set of polynomials in $n$ variables of degree $\leq 1$, i. e., a set of...
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Yaroslavl State University
2022-06-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/1648 |
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| author | Mikhail V. Nevskii |
| author_facet | Mikhail V. Nevskii |
| author_sort | Mikhail V. Nevskii |
| collection | DOAJ |
| description | Let $Q_n=[0,1]^n$ be the unit cube in ${\mathbb R}^n$ and let $C(Q_n)$ be a space of continuous functions $f:Q_n\to{\mathbb R}$ with the norm $\|f\|_{C(Q_n)}:=\max_{x\in Q_n}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ denote a set of polynomials in $n$ variables of degree $\leq 1$, i. e., a set of linear functions on ${\mathbb R}^n$. The interpolation projector $P:C(Q_n)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in Q_n$ is defined by the equalities $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right)$, $j=1,$ $\ldots,$ $ n+1$. Let $\|P\|_{Q_n}$ be the norm of $P$ as an operator from $C(Q_n)$ to $C(Q_n)$. If $n+1$ is an Hadamard number, then there exists a non-degenerate regular simplex having the vertices at vertices of $Q_n$. We discuss some approaches to get inequalities of the form $||P||_{Q_n}\leq c\sqrt{n}$ for the norm of the corresponding projector $P$. |
| format | Article |
| id | doaj-art-b1adcd9e5a4a433090cdc3155d5e768a |
| institution | DOAJ |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2022-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-b1adcd9e5a4a433090cdc3155d5e768a2025-08-20T03:22:03ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172022-06-012929210310.18255/1818-1015-2022-2-92-1031266On Some Estimate for the Norm of an Interpolation ProjectorMikhail V. Nevskii0P. G. Demidov Yaroslavl State UniversityLet $Q_n=[0,1]^n$ be the unit cube in ${\mathbb R}^n$ and let $C(Q_n)$ be a space of continuous functions $f:Q_n\to{\mathbb R}$ with the norm $\|f\|_{C(Q_n)}:=\max_{x\in Q_n}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ denote a set of polynomials in $n$ variables of degree $\leq 1$, i. e., a set of linear functions on ${\mathbb R}^n$. The interpolation projector $P:C(Q_n)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in Q_n$ is defined by the equalities $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right)$, $j=1,$ $\ldots,$ $ n+1$. Let $\|P\|_{Q_n}$ be the norm of $P$ as an operator from $C(Q_n)$ to $C(Q_n)$. If $n+1$ is an Hadamard number, then there exists a non-degenerate regular simplex having the vertices at vertices of $Q_n$. We discuss some approaches to get inequalities of the form $||P||_{Q_n}\leq c\sqrt{n}$ for the norm of the corresponding projector $P$.https://www.mais-journal.ru/jour/article/view/1648hadamard matrixregular simplexlinear interpolationprojectornorm |
| spellingShingle | Mikhail V. Nevskii On Some Estimate for the Norm of an Interpolation Projector Моделирование и анализ информационных систем hadamard matrix regular simplex linear interpolation projector norm |
| title | On Some Estimate for the Norm of an Interpolation Projector |
| title_full | On Some Estimate for the Norm of an Interpolation Projector |
| title_fullStr | On Some Estimate for the Norm of an Interpolation Projector |
| title_full_unstemmed | On Some Estimate for the Norm of an Interpolation Projector |
| title_short | On Some Estimate for the Norm of an Interpolation Projector |
| title_sort | on some estimate for the norm of an interpolation projector |
| topic | hadamard matrix regular simplex linear interpolation projector norm |
| url | https://www.mais-journal.ru/jour/article/view/1648 |
| work_keys_str_mv | AT mikhailvnevskii onsomeestimateforthenormofaninterpolationprojector |