Revista Matemática Iberoamericana

The Jet of an Interpolant on a Finite Set

Charles Fefferman and Arie Israel

Full-text: Open access

Abstract

We study functions $F \in C^m (\mathbb{R}^n)$ having norm less than a given constant $M$, and agreeing with a given function $f$ on a finite set $E$. Let $\Gamma_f (S,M)$ denote the convex set formed by taking the $(m-1)$-jets of all such $F$ at a given finite set $S \subset \mathbb{R}^n$. We provide an efficient algorithm to compute a convex polyhedron $\tilde{\Gamma}_f (S,M)$, such that $$ \Gamma_f (S,cM) \subset \tilde{\Gamma}_f (S,M) \subset \Gamma_f (S,CM), $ where $c$ and $C$ depend only on $m$ and $n$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 27, Number 1 (2011), 355-360.

Dates
First available in Project Euclid: 4 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1296828838

Mathematical Reviews number (MathSciNet)
MR834355

Zentralblatt MATH identifier
1217.49024

Subjects
Primary: 49K24 52A35: Helly-type theorems and geometric transversal theory

Keywords
interpolation jet algorithm Whitney extension theorem

Citation

Fefferman, Charles; Israel, Arie. The Jet of an Interpolant on a Finite Set. Rev. Mat. Iberoamericana 27 (2011), no. 1, 355--360. https://projecteuclid.org/euclid.rmi/1296828838


Export citation

References

  • 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. Assoc. Comput. Mach. 42 (1995), no. 1, 67-90.
  • Fefferman, C. and Klartag, B.: Fitting a $C^m$-smooth function to data. Part I, Ann. of Math. (2) 169 (2009), 315-346. Part II, Rev. Mat. Iberoam. 25 (2009), 49-273.
  • Fefferman, C.: Fitting a $C^m$-smooth function to data. III. Ann of Math. (2) 170 (2009), no. 1, 427-441.
  • 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 (electronic).
  • Malgrange, B.: Ideals of differentiable functions. Oxford University Press, London, 1967.
  • von Neumann, J.: First draft of a report on the EDVAC. Contract No. W-670-ORD-492, Moore School of Electrical Engineering, Univ. Pennsylvania, 1945. Reprinted in IEEE Ann. Hist. Comput. 15 (1993), no. 4, 27-75.
  • Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Math. Series 30. Princeton Univ. Press, Princeton, 1970.