Hilbert function space splittings on domains with infinitely many variables

TL;DR

This paper develops a framework for Hilbert spaces of functions with infinitely many variables, using weighted tensor products and stable space splittings, with applications in statistical learning, reduced order modeling, and complexity analysis.

Contribution

It introduces a novel approach to defining and analyzing infinite-variable Hilbert spaces with weighted tensor products and stable splittings, including new conditions for norm equivalence.

Findings

Proves compact embedding and norm equivalence results.

Provides estimates for epsilon-dimensions.

Introduces a new condition for weighted ANOVA and anchored norm equivalence.

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 $epsilon$-dimensions. A new condition for the equivalence of weighted ANOVA and anchored norms is also given.