Estimation of interpolation projectors using Legendre polynomials - Mikhail V. Nevskii (Моделирование и анализ информационных систем)
We give some estimates for the minimum projector norm under linear interpolation on a compact set in ${\mathbb R}^n$. Let $Π_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$, $Ω$ is a compactum in ${\mathbb R}^n$, $K={\rm conv}(Ω)$. We will assume that ${\rm vol}(K)>0$. Let the points $x^{(j)}\in Ω$, $1≤ j≤ n+1,$ be the vertices of an $n$-dimensional nondegenerate simplex. The interpolation projector $P:C(Ω)→ Π_1({\mathbb R}^n)$ with the nodes $x^{(j)}$ is defined by the equations $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$. By $\|P\|_Ω$ we mean the norm of $P$ as an operator from $C(Ω)$ to $C(Ω)$. By $θ_n(Ω)$ we denote the minimal norm $\|P\|_Ω$ of all operators $P$ with nodes belonging to $Ω$. By ${\rm simp}(E)$ we denote the maximal volume of the simplex with vertices in $E$. We establish the inequalities $χ_n^{-1}\left(\frac{{\rm vol}(K)}{{\rm simp}(Ω)}\right)≤ θ_n(Ω)≤ n+1.$ Here $χ_n$ 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 $γ\ge 1$ the volume of the set $\left\{x=(x_1,...,x_n)\in{\mathbb R}^n : \sum |x_j| +\left|1- \sum x_j\right|\leγ\right\}$ is equal to ${χ_n(γ)}/{n!}$. In the case when $Ω$ is an $n$-dimensional cube or an $n$-dimensional ball, the lower estimate gives the possibility to obtain the inequalities of the form $θ_n(Ω)\geqslant c\sqrt{n}$. Also we formulate some open questions.