D-tensor paraproducts and its caricatures
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911454864605184 |
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| author | Fasina, Oluwadamilola |
| author_facet | Fasina, Oluwadamilola |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | D-tensor paraproducts and its caricatures Fasina, Oluwadamilola Analysis of PDEs Numerical Analysis 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. |
| title | D-tensor paraproducts and its caricatures |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2508.13322 |