On weighted Hilbert spaces and integration of functions of infinitely many variables

摘要

We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables x_1, x_2.... ∈ D. The setting is based on a reproducing kernel k for functions on D, a family of non-negative weights γ_u, where u varies over all finite subsets of N, and a probability measure p on D. We consider the weighted superposition K =∑_uγ_uk_u of finite tensor products k_u of k. Under mild assumptions we show that If is a reproducing kernel on a properly chosen domain in the sequence space D~N, and that the reproducing kernel Hilbert space H(K) is the orthogonal sum of the spaces H(γ_uk_u). Integration on H(K) can be defined in two ways, via a canonical representer or with respect to the product measure p~N on D~N. We relate both approaches and provide sufficient conditions for the two approaches to coincide.