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Smooth Interpolation of Data by Efficient Algorithms

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Excursions in Harmonic Analysis, Volume 1

Part of the book series: Applied and Numerical Harmonic Analysis ((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.

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Notes

  1. 1.

    It is perhaps natural to use two different positive numbers \({M}_{1},{M}_{2}\) in the two inequalities in (1). However, since we are free to multiply \(\sigma (x)\) by our favorite positive constant, we lose no generality in taking a single M.

  2. 2.

    This unrealistic model of computation is subject to serious criticism[15]. In Fefferman\(\textendash \)Klartag[10, 11], we make a rigorous analysis of the roundoff error. That analysis is omitted in this expository chapter for the sake of simplicity.

  3. 3.

    For the rest of this chapter, whenever we refer to Theorem 5, we mean the generalized version in which the \({\Gamma }_{\mathcal{l}}\) are not specified, but merely assumed to satisfy a list of key properties.

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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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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, USA

    C. Fefferman

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  1. C. Fefferman
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Correspondence to C. Fefferman .

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Editors and Affiliations

  1. , Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, 20742-0001, Maryland, USA

    Travis D. Andrews

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Radu Balan

  3. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    Wojciech Czaja

  5. , Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Kasso A. Okoudjou

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Fefferman, C. (2013). Smooth Interpolation of Data by Efficient Algorithms. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 1. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8376-4_4

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