and H. Wo'zniakowski 2
Abstract:
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 " \Gamma1 and d. Strong tractability means that this minimal number is bounded independently of d by a polynomial in " \Gamma1. Both problems are considered for certain Sobolev spaces of functions defined over the whole space IR d. These spaces are characterized by a number of parameters: r is the smoothness of functions, fl d;k is a space weight which measures the relative importance of the kth variable for d-variate functions, and a weight function / 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! and variances oe d;k. We find conditions on the weights! and / such that the approximation and integration are well defined. For the approximation problem, we consider two classes of functional evaluations: all consisting of all linear continuous functionals and std consisting of function evaluations. Of course, for integration we only consider std
