Finite-order weights imply tractability of linear multivariate problems

Abstract

We study the minimal number n ( ɛ , d ) of information evaluations needed to compute a worst case ɛ -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 ( ɛ , d ) is bounded by a polynomial in both ɛ - 1 and d. Strong tractability means that n ( ɛ , d ) is bounded by a polynomial only in ɛ - 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.