Finite-order weights imply tractability of linear multivariate problems

We study the minimal number n (ε, d) of information evaluations needed to compute a worst case ε -approximation of a linear multivariate problem. This problem is defined over a weighted Hilbert space of functions f of d variables. One information evaluation of f is defined as the evaluation of a linear continuous functional or the value of f at a given point. Tractability means that n (ε, d) is bounded by a polynomial in both ε^-1 and d. Strong Tractability means that n (ε, d) is bounded by a polynomial only in ε^-1. We consider weighted reproducing kernel Hilbert spaces with finite-order weights. This means that each function of d variables is a sum of functions depending only on q* variables, where q* is independent of d. We prove that finite-order weights imply strong tractability or tractability of linear multivariate problems, depending on a certain condition on the reproducing kernel of the space. The proof is not constructive if one uses values of f.