JournalsrmiVol. 32, No. 1pp. 275–376

Fitting a Sobolev function to data I

  • Charles Fefferman

    Princeton University, United States
  • Arie Israel

    University of Texas at Austin, USA
  • Garving K. Luli

    University of California at Davis, USA
Fitting a Sobolev function to data I cover
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In this paper and two companion papers, we produce efficient algorithms to solve the following interpolation problem: Let m1m \geq 1 and p>n1p > n \geq 1. Given a finite set ERnE \subset \mathbb{R}^n and a function f:ERf: E \rightarrow \mathbb{R}, compute an extension FF of ff belonging to the Sobolev space Wm,p(Rn)W^{m,p}(\mathbb{R}^n) with norm having the smallest possible order of magnitude; secondly, compute the order of magnitude of the norm of FF. The combined running time of our algorithms is at most CNlogNC N \mathrm{log} N, where NN denotes the cardinality of EE, and CC depends only on mm, nn, and pp.

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Charles Fefferman, Arie Israel, Garving K. Luli, Fitting a Sobolev function to data I. Rev. Mat. Iberoam. 32 (2016), no. 1, pp. 275–376

DOI 10.4171/RMI/887