Smooth interpolation of data by efficient algorithms

Research output: Chapter in Book/Report/Conference proceedingChapter

4 Scopus citations

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.

Original languageEnglish (US)
Title of host publicationExcursions in Harmonic Analysis
Subtitle of host publicationThe February Fourier Talks at the Norbert Wiener Center
PublisherBirkhauser Boston
Pages71-84
Number of pages14
Volume1
ISBN (Electronic)9780817683764
ISBN (Print)9780817683757
DOIs
StatePublished - Jan 1 2013

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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