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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
##### Keywords:
smoothing; interpolation; computer aided design
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##### References:
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