Revista Matemática Iberoamericana

Fitting a $C^m$-Smooth Function to Data II

Charles Fefferman and Bo'az Klartag

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Abstract

We exhibit efficient algorithms to perform the following task: Given a function $f$ defined on a finite subset $E \subset \mathbb R^n$, compute a $C^m$ function $F$ on $\mathbb R^n$, with a controlled $C^m$ norm, that approximates $f$ on the subset $E$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 25, Number 1 (2009), 49-273.

Dates
First available in Project Euclid: 12 March 2009

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1236864106

Mathematical Reviews number (MathSciNet)
MR2514338

Zentralblatt MATH identifier
1170.65006

Subjects
Primary: 65D05: Interpolation 65D17: Computer aided design (modeling of curves and surfaces) [See also 68U07]

Keywords
Algorithm interpolation approximation $C^m$-smoothness

Citation

Fefferman, Charles; Klartag, Bo'az. Fitting a $C^m$-Smooth Function to Data II. Rev. Mat. Iberoamericana 25 (2009), no. 1, 49--273. https://projecteuclid.org/euclid.rmi/1236864106


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References

  • Arya, S., Mount, D., Netanyahu, N., Silverman, R. and Wu, A.: An optimal algorithm for approximate nearest neighbor searching in fixed dimensions. J. ACM 45 (1998), no. 6, 891-923.
  • Bierstone, E., Milman, P. and Pawłucki, W.: Differentiable functions defined on closed sets. A problem of Whitney. Invent. Math. 151 (2003), no. 2, 329-352.
  • Bierstone, E., Milman, P. and Pawłucki, W.: Higher-order tangents and Fefferman's paper on Whitney's extension problem. Ann. of Math. (2) 164 (2006), no. 1, 361-370.
  • Brudnyi, A. and Brudnyi, Y.: Metric spaces with linear extensions preserving Lipschitz condition. Amer. J. Math. 129 (2007), no. 1, 217-314.
  • Brudnyi, Y.: On an extension theorem. Funk. Anal. i Prilzhen. 4 (1970), 97-98; English transl. in Func. Anal. Appl. 4 (1970), 252-253.
  • Brudnyi, Y. and Shvartsman, P.: The traces of differentiable functions to closed subsets of $\bR^n$. In Function Spaces (Poznán, 1989), 206-210. Teubner-Texte Math. 120. Teubner, Stuttgart, 1991.
  • Brudnyi, Y. and Shvartsman, P.: A linear extension operator for a space of smooth functions defined on closed subsets of $R^n$. Dokl. Akad. Nauk SSSR 280 (1985), 268-270. English transl. in Soviet Math. Dokl. 31 (1985), no. 1, 48-51.
  • Brudnyi, Y. and Shvartsman, P.: Generalizations of Whitney's extension theorem. Int. Math. Research Notices 3 (1994), 129-139.
  • Brudnyi, Y. and Shvartsman, P.: The Whitney problem of existence of a linear extension operator. J. Geom. Anal. 7 (1997), no. 4, 515-574.
  • Brudnyi, Y. and Shvartsman, P.: Whitney's extension problem for multivariate $C^1, w$ functions. Trans. Amer. Math. Soc. 353 (2001), no. 6, 2487-2512.
  • Callahan, P. B. and Kosaraju, S. R.: A decomposition of multidimensional point sets with applications to $k$-nearest-neighbors and $n$-body potential fields. J. ACM 42 (1995), no. 1, 67-90.
  • Fefferman, C.: Interpolation and extrapolation of smooth functions by linear operators. Rev. Mat. Iberoamericana 21 (2005), no. 1, 313-348.
  • Fefferman, C.: A sharp form of Whitney's extension theorem. Ann. of Math. (2) 161 (2005), 509-577.
  • Fefferman, C.: Whitney's extension problem for $C^m$. Ann. of Math. (2) 164 (2006), no. 1, 313-359.
  • Fefferman, C.: Whitney's extension problem in certain function spaces. (preprint).
  • Fefferman, C.: A generalized sharp Whitney theorem for jets. Rev. Mat. Iberoamericana 21 (2005), no. 2, 577-688.
  • Fefferman, C.: Extension of $C^m, \omega$ smooth functions by linear operators. Rev. Mat. Iberoamericana 25 (2009), no. 1, 1-48.
  • Fefferman, C.: $C^m$ extension by linear operators Ann. of Math. (2) 166 (2007), no. 3, 779-835.
  • Fefferman, C. and Klartag, B.: Fitting a $C^m$-smooth function to data I. Annals of Math. (2), to appear.
  • Fefferman, C.: Fitting a $C^m$-smooth function to data III. (preprint).
  • Glaeser, G.: Étude de quelques algèbres tayloriennes. J. Analyse Math. 6 (1958), 1-124.
  • Har-Peled, S. and Mendel, M.: Fast construction of nets in low-dimensional metrics, and their applications. SIAM J. Comput. 35 (2006), no. 5, 1148-1184.
  • Hartmanis, J. and Simon, J.: On the power of multiplication in random-access machines. Proc. 15th Annu. IEEE Sympos. Switching Automata Theory (1974), 13-23.
  • Kahan, S. and Snoeyinkm, J.: On the bit complexity of minimum link paths: Superquadratic algorithms for problem solvable in linear time. Comput. Geom. 12 (1999), no. 1, 33-44.
  • Knuth, D.: The Art of Computer Programming, Volume 2: Seminumerical Algorithms, $3^rd$ edition. Addison-Wesley, 1997.
  • Malgrange, B.: Ideals of Differentiable Functions. Oxford University Press, 1966.
  • Preparata, F. P. and Shamos, M. I.: Computational Geometry: An introduction, $2^nd$ edition. Texts and Monographs in Computer Science. Springer-Verlag, New York, 1985.
  • Schönhage, A.: On the power of random access machines. In Proc. 6th. Internat. Colloq. Automata Lang. Program., 520-529. Lecture Notes Comput. Sci, Vol. 71. Springer-Verlag, 1979.
  • Shvartsman, P.: Lipschitz selections of multivalued mappings and traces of the Zygmund class of functions to an arbitrary compact. Dokl. Acad. Nauk SSSR 276 (1984), 559-562; English transl. in Soviet Math. Dokl. 29 (1984), 565-568.
  • Shvartsman, P.: On traces of functions of Zygmund classes. Sibirskyi Mathem. J. 28 N5 (1987), 203-215; English transl. in Siberian Math. J. 28 (1987), 853-863.
  • Shvartsman, P.: Lipschitz selections of set-valued functions and Helly's theorem. J. Geom. Anal. 12 (2002), no. 2, 289-324.
  • Stein, E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, 1970.
  • Tarski, A.: A Decision Method for Elementary Algebra and Geometry. U. of California Press, 1951.
  • Von Neumann, J.: First draft of a report on the EDVAC. Contract No. W-670-ORD-492, Moore School of Electrical Engineering, Univ. of Penn., Philadelphia, 1945. Reprinted in IEEE Annals of the History of Computing 15 (1993), no. 4, 27-75.
  • Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36 (1934), 63-89.
  • Whitney, H.: Differentiable functions defined in closed sets I. Trans. Amer. Math. Soc. 36 (1934), 369-389.
  • Whitney, H.: Functions differentiable on the boundaries of regions. Ann. of Math. 35 (1934), 482-485.
  • Zobin, N.: Whitney's problem on extendability of functions and an intrinsic metric. Adv. Math. 133 (1998), no. 1, 96-132.
  • Zobin, N.: Extension of smooth functions from finitely connected planar domains. J. Geom. Anal. 9 (1999), no. 3, 489-509.