Smooth Interpolation of Data by Efficient Algorithms
Charles Fefferman · Applied and numerical harmonic analysis · 2012
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. 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.