Worst Case Complexity of Weighted Approximation and Integration over Rd

AbstractWe study the worst case complexity of weighted approximation and integration for functions defined over Rd. We assume that the functions have all partial derivatives of order up to r uniformly bounded in a weighted Lp-norm for a given weight function ψ. The integration and the error for approximation are defined in a weighted sense for another given weight ϱ. We present a necessary and sufficient condition on weight functions ϱ and ψ for the complexity of the problem to be finite. Under additional conditions, we show that the complexity of the weighted problem is proportional to the complexity of the corresponding classical problem defined over a unit cube and with ϱ=ψ=1. Similar results have been obtained recently for scalar functions (d=1) and for multivariate functions under restriction that ψ=1 and p=∞