### Abstract

Suppose we are given a finite subset E ⊂ R{double struck} ^{n} and a function f: E → R{double struck}. How to extend f to a C^{m} function F: R{double struck} ^{n} → R{double struck} with C^{m} norm of the smallest possible order of magnitude? In this paper and in [20] we tackle this question from the perspective of theoretical computer science. We exhibit algorithms for constructing such an extension function F, and for computing the order of magnitude of its C^{m} norm. The running time of our algorithms is never more than CN log N, where N is the cardinality of E and C is a constant depending only on m and n.

Original language | English (US) |
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Pages (from-to) | 315-346 |

Number of pages | 32 |

Journal | Annals of Mathematics |

Volume | 169 |

Issue number | 1 |

DOIs | |

State | Published - Dec 15 2009 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Fefferman, C., & Klartag, B. (2009). Fitting a C

^{m}-smooth function to data I.*Annals of Mathematics*,*169*(1), 315-346. https://doi.org/10.4007/annals.2009.169.315