Equivalence of anchored and ANOVA spaces via interpolation

TL;DR

This paper extends the known equivalence between weighted anchored and ANOVA function spaces across all p in [1,∞], using interpolation techniques to unify and generalize previous results.

Contribution

It generalizes the equivalence results to the entire range of p, providing a unified interpolation-based approach for weighted function spaces.

Findings

Equivalence of anchored and ANOVA spaces established for all p in [1,∞].

Interpolation techniques successfully extend previous p=1 and p=∞ results.

Results hold uniformly or polynomially in the dimension.

Abstract

We consider weighted anchored and ANOVA spaces of functions with first order mixed derivatives bounded in $L_p$. Recently, Hefter, Ritter and Wasilkowski established conditions on the weights in the cases $p=1$ and $p=\infty$ which ensure equivalence of the corresponding norms uniformly in the dimension or only polynomially dependent on the dimension. We extend these results to the whole range of $p\in [1,\infty]$. It is shown how this can be achieved via interpolation.