Tractability of integration in non-periodic and periodic weighted tensor product Hilbert spaces

  • Authors:

  • Ian H. Sloan;Henryk Woźniakowski

  • Affiliations:

  • School of Mathematics, University of New South Wales, Sydney 2052, Australia;Department of Computer Science, Columbia University, New York, New York and Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

  • Venue:
  • Journal of Complexity
  • Year:
  • 2002

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Abstract

We study strong tractability and tractability of multivariate integration in the worst case setting. This problem is considered in weighted tensor product reproducing kernel Hilbert spaces. We analyze three variants of the classical Sobolev space of non-periodic and periodic functions whose first mixed derivatives are square integrable. We obtain necessary and sufficient conditions on strong tractability and tractability in terms of the weights of the spaces. For the three Sobolev spaces periodicity has no significant effect on strong tractability and tractability. In contrast, for general reproducing kernel Hilbert spaces anything can happen: we may have strong tractability or tractability for the non-periodic case and intractability for the periodic one, or vice versa.