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Fitting a \(C^m\)-smooth function to data. III. (English) Zbl 1175.65025
Summary: Fix \(m,n\geq 1\). Given an \(N\)-point set \(E\subset \mathbb R^n\), we exhibit a list of \(O(N)\) subsets \(S_1,S_2,\dots,S_L\subset E\), each containing \(O(1)\) points, such that the following holds: Let \(f : E\to \mathbb R^n\). Suppose that, for each \(\ell = 1,\dots,L\), there exists \(F_\ell\in C^m(\mathbb R^n)\) with norm \(\leq 1\), 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 \(E\).
[For part II see C.-L. Fefferman and B.Klartag, Rev. Mat. Iberoam. 25, No. 1, 49–273 (2009; Zbl 1170.65006)]

MSC:
65D10 Numerical smoothing, curve fitting
65D17 Computer-aided design (modeling of curves and surfaces)
65D05 Numerical interpolation
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