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Smooth Interpolation of Data by Efficient Algorithms

  • C. Fefferman
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In 1934, Whitney (Trans. Am. Math. Soc. 36:63–89, 1934; Trans. Am. Math. Soc. 36:369–389, 1934; Ann. Math. 35:482–485, 1934) posed several basic questions on smooth extension of functions. Those questions have been answered in the last few years, thanks to the work of Bierstone et al. (Inventiones Math. 151(2):329–352, 2003), Brudnyi and Shvartsman (Int. Math. Res. Notices 3:129–139, 1994; J. Geomet. Anal. 7(4):515–574, 1997), Fefferman (Ann. Math. 161:509–577, 2005; Ann. Math. 164(1):313–359, 2006; Ann. Math. 166(3):779–835, 2007) and Glaeser (J. d’ Analyse Math. 6:1–124, 1958). The solution of Whitney’s problems has led to a new algorithm for interpolation of data, due to Fefferman and Klartag (Ann. Math. 169:315–346, 2009; Rev. Mat. Iberoam. 25:49–273, 2009). The new algorithm is theoretically best possible, but far from practical. We hope it can be modified to apply to practical problems. In this expository chapyer, we briefly review Whitney’s problems, then formulate carefully the problem of interpolation of data. Next, we state the main results of Fefferman and Klartag (Ann. Math. 169:315–346, 2009; Rev. Mat. Iberoam. 25:49–273, 2009) on efficient interpolation. Finally, we present some of the ideas in the proofs.

Keywords

Convex Subset Lipschitz Constant Interpolation Problem Interpolation Algorithm Approximate Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

C. Fefferman was Supported by NSF Grant No. DMS-09-01-040 and ONR Grant No. N00014-08-1-0678. The author is grateful to Frances Wroblewski for TeXing this chapter.

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Copyright information

© Birkhäuser Boston 2013

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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