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

Fix m, n ≥ 1. Given an N-point set E ⊂ R{double-struck}ⁿ, we exhibit a list of O (N) subsets S₁, S₂,..., S_L ⊂ E, each containing O (1) points, such that the following holds: Let f: E → R{double-struck}ⁿ. Suppose that, for each ℓ = 1,..., L, there exists F_ℓ ∈ C^m (R{double-struck}ⁿ) with norm ≤ 1, agreeing with f on S_ℓ Then there exists F ∈ C^m (R{double-struck}ⁿ) with norm O (1), agreeing with f on E. We give an application to the problem of discarding outliers from the set E.