Optimal approximation of elliptic problems by linear and nonlinear mappings III: Frames

  • Authors:

  • Stephan Dahlke;Erich Novak;Winfried Sickel

  • Affiliations:

  • Philipps-Universität Marburg, FB12 Mathematik und Informatik, Hans-Meerwein Straße, Lahnberge, 35032 Marburg, Germany;Friedrich-Schiller-Universität Jena, Mathematisches Institut, Ernst-Abbe-Platz 2, 07743 Jena, Germany;Friedrich-Schiller-Universität Jena, Mathematisches Institut, Ernst-Abbe-Platz 2, 07743 Jena, Germany

  • Venue:
  • Journal of Complexity
  • Year:
  • 2007

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Abstract

We study the optimal approximation of the solution of an operator equation A(u)=f by certain n-term approximations with respect to specific classes of frames. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space B"q^t(L"p(@W)) and @W@?R^d is a bounded Lipschitz domain; the error is always measured in the H^s-norm. We study the order of convergence of the corresponding nonlinear frame widths and compare it with several other approximation schemes. Our main result is that the approximation order is the same as for the nonlinear widths associated with Riesz bases, the Gelfand widths, and the manifold widths. This order is better than the order of the linear widths iff p