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Dilation systems with two generators and the optimal Riesz bounds of a basis of ReLU networks
Abstract
For two $1$-periodic functions ψ1,ψ2∈ L2(0,1) with mean zero we characterize when {1}∪{ψ1(α· x),ψ2(α· x)}α∈ℤd, α≻0 is a Riesz basis of L2([0,1]d), in terms of a 2×2 matrix of Dirichlet series, and we compute its optimal bounds. The tool is a 2×2 matrix version of the theorem of Hedenmalm, Lindqvist and Seip for systems of dilations with two generators, proved through the reproducing kernel of the Hardy space of the infinite polydisk. In particular, we get the optimal bounds 2/3 and 3/2 for ψ1,ψ2 the triangle waves considered by Daubechies, DeVore, Foucart, Hanin and Petrova in L2(0,1), and the extended version in L2([0,1]d) defined by Schneider and Vybíral.
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BibTeX
@misc{antezana2026dilation,
title = {{Dilation systems with two generators and the optimal Riesz bounds of a basis of ReLU networks}},
author = {Jorge Antezana and Daniel Carando and Tomás Fernández Vidal and Melisa Scotti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41154},
eprint = {2610.41154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41154},
}
RIS
TY - UNPB
TI - Dilation systems with two generators and the optimal Riesz bounds of a basis of ReLU networks
AU - Jorge Antezana
AU - Daniel Carando
AU - Tomás Fernández Vidal
AU - Melisa Scotti
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41154
UR - https://arxiv.org/abs/2610.41154
AB - For two $1$-periodic functions ψ_1,ψ_2∈ L^2(0,1) with mean zero we characterize when {1}∪{ψ_1(α· x),ψ_2(α· x)}_α∈ℤ^d, α≻0 is a Riesz basis of L^2([0,1]^d), in terms of a 2×2 matrix of Dirichlet series, and we compute its optimal bounds. The tool is a 2×2 matrix version of the theorem of Hedenmalm, Lindqvist and Seip for systems of dilations with two generators, proved through the reproducing kernel of the Hardy space of the infinite polydisk. In particular, we get the optimal bounds 2/3 and 3/2 for ψ_1,ψ_2 the triangle waves considered by Daubechies, DeVore, Foucart, Hanin and Petrova in L^2(0,1), and the extended version in L^2([0,1]^d) defined by Schneider and Vybíral.
ER -