Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces
Abstract
Let 𝔻 be the ridge dictionary generated by ReLUk on a bounded Lipschitz domain Ω⊂ℝd. We establish sharp embeddings between Besov spaces and the associated variation space ℒ1(𝔻) in the quasi-Banach range 0<p≤ 1. Specifically, Bsp,q(Ω)↪ ℒ1(𝔻) when s≥ k+d/p for 0<q≤ 1, and when s>k+d/p for 1<q≤∞. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for 0<p<1, ℒ1(𝔻)↪ Bk+1p,2(Ω), and both the smoothness k+1 and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow ReLUk networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.
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BibTeX
@misc{li2026sharp,
title = {{Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces}},
author = {Yuwen Li and Yupeng Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.00680},
eprint = {2609.00680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00680},
}
RIS
TY - UNPB
TI - Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces
AU - Yuwen Li
AU - Yupeng Wang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.00680
UR - https://arxiv.org/abs/2609.00680
AB - Let 𝔻 be the ridge dictionary generated by ReLU^k on a bounded Lipschitz domain Ω⊂ℝ^d. We establish sharp embeddings between Besov spaces and the associated variation space ℒ_1(𝔻) in the quasi-Banach range 0<p≤ 1. Specifically, B^s_p,q(Ω)↪ ℒ_1(𝔻) when s≥ k+d/p for 0<q≤ 1, and when s>k+d/p for 1<q≤∞. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for 0<p<1, ℒ_1(𝔻)↪ B^k+1_p,2(Ω), and both the smoothness k+1 and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow ReLU^k networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.
ER -