The Jet of an Interpolant on a Finite Set

Abstract

We study functions $F\in C^m({\mathbb{R}}^n)$ having norm less than a given constant $M$, and agreeing with a given function $f$ on a finite set $E$. Let ${\Gamma }_f(S,M)$ denote the convex set formed by taking the $(m-1)$-jets of all such $F$ at a given finite set $S\subset {\mathbb{R}}^n$. We provide an efficient algorithm to compute a convex polyhedron ${\tilde{\Gamma }}_f(S,M)$, such that $$ \Gamma_f (S,cM) \subset \tilde{\Gamma}_f (S,M) \subset \Gamma_f (S,CM), $where$c$and$C$dependonlyon$m$and$n$.