Approximation of infinitely differentiable multivariate functions is intractable

  • Authors:

  • Erich Novak;Henryk Woniakowski

  • Affiliations:

  • Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, 07740 Jena, Germany;Department of Computer Science, Columbia University, New York, NY 10027, USA and Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

  • Venue:
  • Journal of Complexity
  • Year:
  • 2009

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Abstract

We prove that L"~-approximation of C^~-functions defined on [0,1]^d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most @e@?(0,1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.