Tractability of Approximation and Integration for Weighted Tensor Product Problems over Unbounded Domains

Abstract

We study tractability and strong tractability of multivariate approximation and integration in the worst case deterministic setting. Tractability means that the number of functional evaluations needed to compute an ∈-approximation of the multivariate problem with d variables is polynomially bounded in ∈-1 and d. Strong tractability means that this minimal number is bounded independently of d by a polynomial in ∈-1. Both problems are considered for certain Sobolev spaces of functions denned over the whole space IRd. These spaces are characterized by a number of parameters: r is the smoothness of functions, λd,k is a space weight which measures the relative importance of the k th variable for d-variate functions, and a weight function ψ that monitors the behavior of the functions at infinity. The approximation and integration problems are defined in a weighted sense with respect to a probability density w and variances σd,k. We find conditions on the weights w and σ such that the approximation and integration are well defined. For the approximation problem, we consider two classes of functional evaluations: Aall consisting of all linear continuous functionals and Astd consisting of function evaluations. Of course, for integration we only consider Astd. Under natural assumptions on the weight functions w and σ, we prove that strong tractability holds iff sup\( \sup _{d \geqslant 1} \sum\nolimits_{k = 1}^d {\left( {\gamma d,k,\sigma _{d,k}^{2r - 1} } \right)^b } < \infty \) tractability holds iff \( {{\sup _{d \geqslant 1} \sum\nolimits_{k = 1}^d {\left( {\gamma d,k,\sigma _{d,k}^{2r - 1} } \right)^b } } \mathord{\left/ {\vphantom {{\sup _{d \geqslant 1} \sum\nolimits_{k = 1}^d {\left( {\gamma d,k,\sigma _{d,k}^{2r - 1} } \right)^b } } {ln\left( {d + 1} \right)}}} \right. \kern-\nulldelimiterspace} {ln\left( {d + 1} \right)}} < \infty \) .Here b can be any positive number for approximation in Aall, and b = 1 for approximation and integration in Astd.