D-tensor paraproducts and its caricatures

TL;DR

This paper extends tensor paraproduct decomposition techniques from 2-tensors to d-tensors, providing a method to approximate functions with controlled residuals and demonstrating the theory with a computational example.

Contribution

It generalizes the 2-tensor paraproduct decomposition to d-tensors, offering new approximation formulas and residual bounds for functions in Hölder spaces.

Findings

The approximation $ ilde{A}_{(N_i)}(f)$ converges to $A(f)$ with residuals in $ ext{H"older}^{2 ext{alpha}}$ space.

Theoretical results are validated through a computational example for 3-tensors.

The method provides a systematic way to approximate functions using tensor paraproducts.

Abstract

We generalize the $2$-tensor paraproduct decomposition result of [arXiv:2503.12629] to $d$-tensors. In particular, we show that for $A \in C^{d}(\mathbb{R}), f \in \Lambda_{\alpha}([0,1]^d)$, $A(f)$ can be approximated by $\tilde{A}_{(N_i)_{i=0}^d}(f) = (\sum_{\beta=1}^d A^{\beta}(P^{j_1,j_2, \ldots, j_d}(f)) \tilde{\mathbf{v}}^{\beta}(f) ) $ with the residual $\Delta_{(N_i)_{i=1}^d}(A,f) = \tilde{A}_{(N_i)_{i=1}^d}(f) - A(f) \in \Lambda_{2\alpha}([0,1]^d)$. Our theoretical findings are supported by a computational example for d=3.