Infinite-dimensional integration and the multivariate decomposition method

We further develop the Multivariate Decomposition Method (MDM) for the Lebesgue integration of functions of infinitely many variables x₁,x₂,x₃,… with respect to a corresponding product of a one dimensional probability measure. The method is designed for functions that admit a dominantly convergent decomposition f=∑_uf_u, where u runs over all finite subsets of positive integers, and for each u={i₁,…,i_k} the function f_u depends only on x_i1,…,x_ik. Although a number of concepts of infinite-dimensional integrals have been used in the literature, questions of uniqueness and compatibility have mostly not been studied. We show that, under appropriate convergence conditions, the Lebesgue integral equals the ‘anchored’ integral, independently of the anchor. For approximating the integral, the MDM assumes that point values of f_u are available for important subsets u, at some known cost. In this paper, we introduce a new setting, in which it is assumed that each f_u belongs to a normed space F_u, and that bounds B_u on ‖f_u‖_Fu are known. This contrasts with the assumption in many papers that weights γ_u, appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights γ_u were determined by minimizing an error bound depending on the B_u, the γ_u and the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper, only the bounds B_u are assumed to be known. We give two examples in which we specialize the MDM: in the first case, F_u is the |u|-fold tensor product of an anchored reproducing kernel Hilbert space; in the second case, it is a particular non-Hilbert space for integration over an unbounded domain.