Hierarchical paraproducts

TL;DR

This paper extends paraproduct decompositions to finite and product spaces, enabling analysis of compositions with less smooth functions, with applications in graph signal processing.

Contribution

It introduces a hierarchical paraproduct framework for finite and product spaces, generalizing previous continuous-space results and facilitating separation of singular and smooth components.

Findings

Constructed partition trees for finite and product spaces.

Developed paraproducts with controlled regularity properties.

Achieved bounds on the residual terms in the decompositions.

Abstract

We outline an extension of paraproduct decompositions for compositions of the form $A(f)$ where $A \in C^{d}(\mathbb{R}), f \in \Lambda_{\alpha}([0,1]^d)$ developed in [arXiv:2503.12629] and [arXiv:2508.13322] to settings where $(A \in C^1(\mathbb{R}),f \in \Lambda_{\alpha}(X))$ and $ (A \in C^2(\mathbb{R}),f \in \Lambda_{\alpha}(X \times Y))$. To do so, we construct partition trees on $X$ and $X \times Y$ such that analysis with respect to scale is sensible. We obtain results resembling those of [arXiv:2503.12629] and [arXiv:2508.13322], but with the finite sets $X$ and $X \times Y $ as support. In particular we construct the paraproduct $\Pi_{A',A''}^{L,S}: f \to \tilde{A}_{L,S}(f) + \Delta_{L,S}(A,f)$ such that $\Delta_{L,S}(A,f) \in \Lambda_{2\alpha}(X \times Y)$ and $\lVert \Delta_{L,S}(A,f) \rVert_{\Lambda_{2\alpha}(X \times Y)} \leq C_A \lVert f \rVert_{\Lambda_{\alpha}(X \times Y)}$. Analogous results are obtained when the support is just one finite set, $X$. This extension is motivated by situations where one wishes to separate the singular and smooth components of such compositions in graph signal processing environments.