Fix m,n≥1. Given an N-point set E⊂ℝn, we exhibit a list of O(N) subsets S1,S2,…,SL⊂E, each containing O(1) points, such that the following holds: Let f:E→ℝn. Suppose that, for each ℓ=1,…,L, there exists Fℓ∈Cm(ℝn) with norm ≤1, agreeing with f on Sℓ. Then there exists F∈Cm(ℝn), with norm O(1), agreeing with f on E.
We give an application to the problem of discarding outliers from the set E.