Fitting a $C^m$-smooth function to data, III

Abstract

Fix $m, n\geq 1$. Given an $N$-point set $E \subset {\mathbb R}^n$, we exhibit a list of $O(N)$ subsets $S_1,S_2,\ldots,S_L\subset E$, each containing $O(1)$ points, such that the following holds: Let $f: E\to {\mathbb R}^n$. Suppose that, for each $\ell=1,\ldots,L$, there exists $F_\ell \in C^m(\mathbb R^n)$ with norm $\le 1$, agreeing with $f$ on $S_\ell$. Then there exists $F\in C^m(\mathbb R^n)$, with norm $O(1)$, agreeing with $f$ on $E$.
We give an application to the problem of discarding outliers from the set $E$.