Complexity of weighted approximation over \(\mathbb{R}^d\) (Q1347852)

The authors study the complexity of the problem of approximating multivariate functions defined on \(\mathbb{R}^d\) and having all \(r\)th order partial derivatives continuous and uniformly bounded. Approximation algorithms use the value of function or its partial derivatives up to order \(r\) only. The function is recovered with small error (in a weighted \(L_q\) norm with a weight function). A set of necessary and sufficient conditions is derived in terms of the weight and the parameters \(q\) and \(r\) for the weighted approximation problem to have finite complexity. Also conditions guaranteeing that the complexity is of the same order as the complexity of the classical approximation problem over a finite domain are presented. It is found that the results of the paper also hold for weighted integration.