Optimal Lagrange Interpolation Projectors and Legendre Polynomials

Abstract

Let $K$ be a convex body in ${\mathbb R}^n$, and let $Π_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$. Given an $(n+1)$-element set $Y\subset K$ in general position, we let $P_Y$ denote the Lagrange interpolation projector $P_Y: C(K)\to Π_1({\mathbb R}^n)$ with nodes in $Y$. In this paper, we study upper and lower bounds for the norm of the optimal Lagrange interpolation projector, i.e., the projector with minimal operator norm where the minimum is taken over all $(n+1)$-element sets of interpolation nodes in $K$. We denote this minimal norm by $θ_n(K)$. Our main result, Theorem 5.2, provides an explicit lower bound for the constant $θ_n(K)$ for an arbitrary convex body $K\subset{\mathbb R}^n$ and an arbitrary $n\ge 1$. We prove that $θ_n(K)\ge χ_n^{-1}\left({{\rm vol}(K)}/{{\rm simp}(K)}\right)$ where $χ_n$ is the Legendre polynomial of degree $n$ and ${\rm simp}(K)$ is the maximum volume of a simplex contained in $K$. The proof of this result relies on a geometric characterization of the Legendre polynomials in terms of the volumes of certain 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 ${χ_n(γ)}/{n!}$. If $K$ is an $n$-dimensional ball, this approach leads us to the equivalence $θ_n(K) \asymp\sqrt{n}$ which is complemented by the exact formula for $θ_n(K)$. If $K$ is an $n$-dimensional cube, we obtain explicit efficient formulae for upper and lower bounds of the constant $θ_n(K)$; moreover, for small $n$, these estimates enable us to compute the exact values of this constant.