Stable splittings of Hilbert spaces of functions of infinitely many variables

Abstract

We present an approach to defining Hilbert spaces of functions depending on infinitely many variables or parameters, with emphasis on a weighted tensor product construction based on stable space splittings. The construction has been used in an exemplary way for guiding dimension- and scale-adaptive algorithms in application areas such as statistical learning theory, reduced order modeling, and information-based complexity. We prove results on compact embeddings, norm equivalences, and the estimation of <em>ϵ</em>-dimensions. A new condition for the equivalence of weighted ANOVA and anchored norms is also given.