Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

D-tensor paraproducts and its caricatures

Oluwadamilola Fasina

Abstract

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

D-tensor paraproducts and its caricatures

Abstract

We generalize the -tensor paraproduct decomposition result of [arXiv:2503.12629] to -tensors. In particular, we show that for , can be approximated by with the residual . Our theoretical findings are supported by a computational example for d=3.

Paper Structure

This paper contains 6 sections, 7 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. d-tensor paraproducts
  5. Paraproduct Caricatures
  6. Cone singularity in complex plane (d=3)

Key Result

Theorem 1.1

Suppose $A \in C^d(\mathbb{R})$, $f \in \Lambda_\alpha([0,1]^d), 0 < \alpha < \frac{1}{2}$, then for the operator $T: f \to A(f)$ we can approximate $A(f)$ with such that the multiscale tensor paraproduct transforms $T : f \to A(f)$ to where $\Delta_{(N_i)_{i=0}^d}(A,f) = A(f) - \tilde{A}_{(N_i)_{i=0}^d}(f) \in \Lambda_{2\alpha}([0,1]^2)$ is the residual containing twice the regularity of $f$.

Figures (1)

  • Figure 1: Cone singularity in complex plane: (a - c) $f, A(f), \tilde{A}_{(N_i)_{i=0}^d}(f)$ at t=15 (d- f) $f, A(f), \tilde{A}_{(N_i)_{i=0}^d}(f)$ at t=70, (g-i) $f, A(f), \tilde{A}_{(N_i)_{i=0}^d}(f)$ at t=115.

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 9 more