On Equivalence of Anchored and ANOVA Spaces; Lower Bounds
Provides theoretical lower bounds for function space embeddings, relevant for high-dimensional approximation and numerical analysis.
This paper establishes lower bounds for the norms of embeddings between weighted Anchored and ANOVA spaces, showing polynomial growth in dimension for certain weight classes when p>1, and super-polynomial growth for product order-dependent weights.
We provide lower bounds for the norms of embeddings between $\boldsymbolγ$-weighted Anchored and ANOVA spaces of $s$-variate functions with mixed partial derivatives of order one bounded in $L_p$ norm ($p\in[1,\infty]$). In particular we show that the norms behave polynomially in $s$ for Finite Order Weights and Finite Diameter Weights if $p>1$, and increase faster than any polynomial in $s$ for Product Order-Dependent Weights and any $p$.