Interpolation by Linear Functions on an nn-Dimensional Ball

By $B=B(x^{(0)};R)$ we denote the Euclidean ball in ${\mathbb R}^n$ given by the inequality $\|x-x^{(0)}\|\leq R$. Here $x^{(0)}\in{\mathbb R}^n, R>0$, $\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}$. 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 $\Pi_1\left({\mathbb R}^n\right)$ 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 $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ We obtain the formula to compute the norm of $P$ as an operator from $C(B)$ into $C(B)$ via $x^{(0)}$, $R$ and coefficients of basic Lagrange polynomials of $S$. In more details we study the case when $S$ is a regular simplex inscribed into $B_n=B(0,1)$.Comment: 17 pages, 6 figures, 1 tabl

Geometric Estimates in Interpolation by Linear Functions on a Euclidean Ball

Let $B_n$ be the Euclidean unit ball in ${\mathbb R}^n$ given by the inequality $\|x\|\leq 1$, $\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}$. By $C(B_n)$ we mean the space of continuous functions $f:B_n\to{\mathbb R}$ with the norm $\|f\|_{C(B_n)} := \max\limits_{x\in B_n}|f(x)|$. The symbol $\Pi_1\left({\mathbb R}^n\right)$ denotes the set of polynomials in $n$ variables of degree $\leq 1$, i.e., the set of linear functions upon ${\mathbb R}^n$. Assume $x^{(1)}, \ldots, x^{(n+1)}$ are the vertices of an $n$-dimensional nondegenerate simplex $S\subset B_n$. The interpolation projector $P:C(B_n)\to \Pi_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right) = f\left(x^{(j)}\right).$ Denote by $\|P\|_{B_n}$ the norm of $P$ as an operator from $C(B_n)$ onto $C(B_n)$. We describe the approach in which $\|P\|_{B_n}$ can be estimated from below via the volume of $S$.Comment: 10 page

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~continuous 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 \linebreak 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 a 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 $n=1,2,3,4$. Keywords: regular simplex, ball, linear interpolation, projector, normComment: 13 page

The Minimum Norm of a Projector under Linear Interpolation on a Euclidean Ball

We prove the following proposition. Under linear interpolation on a Euclidean $n$-dimensional ball $B$, an interpolation projector whose nodes coincide with the vertices of a regular simplex inscribed into the boundary sphere has the minimum $C$-norm. This minimum norm $\theta_n(B)$ is equal to $\max\{\psi(a_n),\psi(a_n+~1)\}$, where $\psi(t)=\dfrac{2\sqrt{n}}{n+1}\Bigl(t(n+1-t)\Bigr)^{1/2}+ \left|1-\dfrac{2t}{n+1}\right|$, $0\leq t\leq n+1$, and $a_n=\left\lfloor\dfrac{n+1}{2}-\dfrac{\sqrt{n+1}}{2}\right\rfloor$. For any $n$, $\sqrt{n}\leq \theta_n(B)\leq \sqrt{n+1}.$ Moreover, $\theta_n(B)$ $=$ $\sqrt{n}$ only for $n=1$ and $\theta_n(B)=\sqrt{n+1}$ if and only if $\sqrt{n+1}$ is an integer.Comment: 7 page

On a Geometric Approach to the Estimation of Interpolation Projectors

Suppose $\Omega$ is a closed bounded subset of ${\mathbb R}^n,$ $S$ is an $n$-dimensional non-degenerate simplex, $\xi(\Omega;S):=\min \left\{\sigma\geq 1: \, \Omega\subset \sigma S\right\}$. Here $\sigma S$ is the result of homothety of $S$ with respect to the center of gravity with coefficient $\sigma$. Let $d\geq n+1,$ $\varphi_1(x),\ldots,\varphi_d(x)$ be linearly independent monomials in $n$ variables, $\varphi_1(x)\equiv 1,$ $\varphi_2(x)=x_1,\ \ldots, \ \varphi_{n+1}(x)=x_n.$ Put $\Pi:={\rm lin}(\varphi_1,\ldots,\varphi_d).$ The interpolation projector $P: C(\Omega)\to \Pi$ with a set of nodes $x^{(1)},\ldots, x^{(d)}$ $\in \Omega$ is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ Denote by $\|P\|_{\Omega}$ the norm of $P$ as an operator from $C(\Omega)$ to $C(\Omega)$. Consider the mapping $T:{\mathbb R}^n\to {\mathbb R}^{d-1}$ of the form $T(x):=(\varphi_2(x),\ldots,\varphi_d(x)).$ We have the following inequalities: $\frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_{\Omega}-1\right)+1$ $\leq \xi(T(\Omega);S)\leq \frac{d}{2}\left(\|P\|_{\Omega}-1\right)+1.$ Here $S$ is the $(d-1)$-dimensional simplex with vertices $T\left(x^{(j)}\right).$ We discuss this and other relations for polynomial interpolation of functions continuous on a segment. The results of numerical analysis are presented.Comment: 13 pages, 2 figure

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 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 $P$

Microstructure and properties of hypoeutectic silumin treated by high-current pulsed electron beams

The structural-phase states, microhardness, and tribological properties of hypoeutectic silumin after electron-beam treatment are studied by the methods of contemporary physical materials science. The object of the study is hypoeutectic АК10М2Н-type silumin containing 87.88 wt.% of Al and 11.1 wt.% of Si as the base components.Методами сучасного фізичного матеріалознавства досліджено структурно-фазові стани, мікротвердість і трибологічні властивості доевтектичного силуміну після електронно-пучкового оброблення. Об єктом дослідження був доевтектичний силумін марки АК10М2Н із вмістом 87,88 ваг.% Al й 11,1 ваг.% Si як головних компонентів.Методами современного физического материаловедения исследованы структурнофазовые состояния, микротв рдость и трибологические свойства доэвтектического силумина после электронно-пучковой обработки. Объектом исследования являлся доэвтектический силумин марки АК10М2Н с содержанием 87,88 вес.% Al и 11,1 вес.% Si как главных компонентов

О гипотезе Лассака для выпуклого тела

In 1993 M. Lassak formulated (in the equivalent form) the following conjecture. If we can inscribe a translate of the cube $[0,1]^n$ into a convex body $C \subset R^n$, then $\sum_{i=1}^n \frac{1}{\omega_i} \geq 1$. Here $\omega_i$ denotes the width of $C$ in the direction of the ith coordinate axis. The paper contains a new proof of this statement for n = 2. Also we show that if a translate of $[0,1]^n$ can be inscribed into the n-dimensional simplex, then for this simplex holds $\sum_{i=1}^n \frac{1}{\omega_i} = 1$.В 1993 г. М. Лассак сформулировал (в эквивалентном виде) следующую гипотезу. Если в выпуклое тело $C \subset R^n$ можно вписать транслят куба $[0,1]^n$, то $\sum_{i=1}^n \frac{1}{\omega_i} \geq 1$. Здесь $\omega_i$ - ширина $C$ в направлении i-й координатной оси. В статье даётся новое доказательство этого утверждения для n = 2. Также мы показываем, что для n-мерного симплекса, в который можно вписать транслят $[0,1]^n$, справедливо $\sum_{i=1}^n \frac{1}{\omega_i} = 1$