Finite-order weights imply tractability of linear multivariate problems

  • Authors:

  • G. W. Wasilkowski;H. Woźniakowski

  • Affiliations:

  • Department of Computer Science, University of Kentucky, College of Engineering, 773 Anderson Hall, Lexington, KY;Department of Computer Science, Columbia University, New York and Institute of Applied Mathematics, University of Warsaw

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2004

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Abstract

We study the minimal number n(ε, d) of information evaluations needed to compute a worst case ε-approximation of a linear multivariate problem. This problem is defined over a weighted Hilbert space of functions f of d variables. One information evaluation of f is defined as the evaluation of a linear continuous functional or the value of f at a given point. Tractability means that n(ε, d) is bounded by a polynomial in both ε-1 and d. Strong tractability means that n(ε, d) is bounded by a polynomial only in ε-1. We consider weighted reproducing kernel Hilbert spaces with finite-order weights. This means that each function of d variables is a sum of functions depending only on q* variables, where q* is independent of d. We prove that finite-order weights imply strong tractability or tractability of linear multivariate problems, depending on a certain condition on the reproducing kernel of the space. The proof is not constructive if one uses values of f.