Tractability of Multivariate Integration for Weighted Korobov Spaces: My 15 Year Partnership with Ian Sloan

Abstract

This paper is intended as a birthday present for Ian Sloan who celebrated his 70th birthday during MCQMC’08 in Montreal. In the first paper with Ian we studied multivariate integration for the unweighted Korobov spaces of smooth and periodic functions equipped with L ∞-type norms expressed in terms of Fourier coefficients. We proved that this problem is intractable and suffers from the curse of dimensionality. To break intractability, weighted Korobov spaces are studied in this paper. Product weights are mainly considered, and finite-order weights are only briefly mentioned. Necessary and sufficient conditions for strong polynomial tractability, polynomial tractability and weak tractability are presented. The necessary and sufficient conditions coincide only for weak tractability, whereas there is a gap between them for strong polynomial and polynomial tractability. In terms of the exponent of strong polynomial tractability, the lower and upper bounds differ at most by a factor of two. Nevertheless, these bounds prove that the exponent of strong polynomial tractability depends on the decay of weights.