Restricted isometries in finite-dimensional ℓp-spaces
Abstract
For d,n∈ℤ≥1 and p∈[1,∞)∖{2}, we characterize all linear maps s:ℓdp→ℓnp for which there is E⊆ Xd such that |s(ξ)|p=|ξ|E|p for every ξ∈ℓdp. We call these maps \emph{restricted isometries}. Our Lamperti-type theorem gives a unique description of each such operator using a set transformation, a positive amplitude function, and a unimodular phase function. In particular, it includes noninjective and nonsurjective operators, even when the dimensions differ. We further characterize restricted isometries between finite ℓp-spaces with arbitrary positive atomic weights. Applications include formulas counting set transformations and uniform restricted isometries, as well as a geometric description of all restricted isometries with fixed column supports.
Related
- Pseudoconvex domains in complex Euclidean space with finite-dimensional Bergman spaces
- Rigidity of unit spheres in finite-dimensional real normed spaces
- Diagonalization of nonlinear functions in finite-dimensional spaces and the case of ℝ2
- Finite-dimensional normed spaces without small-norm hyperplane projections
- Strongly Pseudoconvex Domains with Finite Dimensional p-Bergman Spaces
BibTeX
@misc{dalfollodaley2026restricted,
title = {{Restricted isometries in finite-dimensional $\ell^p$-spaces}},
author = {Olivia Dalfollo-Daley and Alonso Delfín and Christina Fernández and Cameron Mars and Xintan Xia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32615},
eprint = {2610.32615},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32615},
}
RIS
TY - UNPB
TI - Restricted isometries in finite-dimensional ℓ^p-spaces
AU - Olivia Dalfollo-Daley
AU - Alonso Delfín
AU - Christina Fernández
AU - Cameron Mars
AU - Xintan Xia
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32615
UR - https://arxiv.org/abs/2610.32615
AB - For d,n∈ℤ_≥1 and p∈[1,∞)∖{2}, we characterize all linear maps s:ℓ_d^p→ℓ_n^p for which there is E⊆ X_d such that |s(ξ)|_p=|ξ|_E|_p for every ξ∈ℓ_d^p. We call these maps \emph{restricted isometries}. Our Lamperti-type theorem gives a unique description of each such operator using a set transformation, a positive amplitude function, and a unimodular phase function. In particular, it includes noninjective and nonsurjective operators, even when the dimensions differ. We further characterize restricted isometries between finite ℓ^p-spaces with arbitrary positive atomic weights. Applications include formulas counting set transformations and uniform restricted isometries, as well as a geometric description of all restricted isometries with fixed column supports.
ER -