Equivalence of Weighted Anchored and ANOVA Spaces of Functions with Mixed Smoothness of Order one in $L_p$

TL;DR

This paper establishes conditions under which weighted anchored and ANOVA function spaces with mixed smoothness are equivalent, with implications for integration and approximation in high-dimensional settings.

Contribution

It provides new criteria for the equivalence of weighted anchored and ANOVA spaces with mixed smoothness, including bounds on the norm equivalence constants.

Findings

Conditions for space equivalence are established.

Norm equivalence constants are uniformly or polynomially bounded in dimension.

Applications to high-dimensional integration and approximation are discussed.

Abstract

We consider $\gamma$-weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in a weighted $L_p$ norm with $1 \leq p \leq \infty$. The domain of the functions is $D^d$, where $D \subseteq \mathbb{R}$ is a bounded or unbounded interval. We provide conditions on the weights $\gamma$ that guarantee that anchored and ANOVA spaces are equal (as sets of functions) and have equivalent norms with equivalence constants uniformly or polynomially bounded in $d$. Moreover, we discuss applications of these results to integration and approximation of functions on $D^d$.