Schauder and Riesz Bases for Higher-Dimensional Lebesgue Spaces Inspired by ReLU Neural Networks
Abstract
Motivated by the role of piecewise linear functions in ReLU neural networks, we study systems of dilated functions in the Lebesgue spaces Lq([0,1]d), q∈(1,∞). For a sequence τ∈ℓ1 we consider the operator Tτg(x)=∑k=1∞ τk g(kx) and the function system 𝒵τ obtained by applying it to the trigonometric system. We show that τ↦ Tτ is an isomorphism of algebras from ℓ1 with the Dirichlet convolution as multiplication onto an algebra of operators and connect the inverse of Tτ to the inverse of τ with respect to the Dirichlet convolution. We give explicit conditions on τ, under which the resulting system 𝒵τ is a Schauder basis of Lq([0,1]d) and, for q=2, a Riesz basis for which we estimate the Riesz constants in terms of τ. We characterise the Riesz bases of this type in terms of a sequence space of Dirichlet convolution multipliers, and we treat mixed systems of cosine and sine functions with different sequences. Finally, we apply these results to the system generated by two piecewise linear functions inspired by ReLU networks and improve the known lower Riesz constant to 2/3, independently of the dimension.
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BibTeX
@misc{billner2026schauder,
title = {{Schauder and Riesz Bases for Higher-Dimensional Lebesgue Spaces Inspired by ReLU Neural Networks}},
author = {Julian Billner and Cornelia Schneider},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14266},
eprint = {2610.14266},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14266},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Schauder and Riesz Bases for Higher-Dimensional Lebesgue Spaces Inspired by ReLU Neural Networks
AU - Julian Billner
AU - Cornelia Schneider
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14266
UR - https://arxiv.org/abs/2610.14266
AB - Motivated by the role of piecewise linear functions in ReLU neural networks, we study systems of dilated functions in the Lebesgue spaces L_q([0,1]^d), q∈(1,∞). For a sequence τ∈ℓ_1 we consider the operator T_τg(x)=∑_k=1^∞ τ_k g(kx) and the function system 𝒵^τ obtained by applying it to the trigonometric system. We show that τ↦ T_τ is an isomorphism of algebras from ℓ_1 with the Dirichlet convolution as multiplication onto an algebra of operators and connect the inverse of T_τ to the inverse of τ with respect to the Dirichlet convolution. We give explicit conditions on τ, under which the resulting system 𝒵^τ is a Schauder basis of L_q([0,1]^d) and, for q=2, a Riesz basis for which we estimate the Riesz constants in terms of τ. We characterise the Riesz bases of this type in terms of a sequence space of Dirichlet convolution multipliers, and we treat mixed systems of cosine and sine functions with different sequences. Finally, we apply these results to the system generated by two piecewise linear functions inspired by ReLU networks and improve the known lower Riesz constant to 2/3, independently of the dimension.
ER -