Perturbed Exponential Systems, Pattern-Dependent Shifted-Sinc Operators, and Matrix Symbols on Finite Unions of Intervals
Abstract
We study perturbed exponential systems and unions of shifted frequency lattices through shifted-sinc operators and finite-dimensional fiberizations. For the auxiliary binary local-shift model, we identify the exact pattern-dependent invariant ρδ,s=|P-T2δP+|, which gives the optimal squared synthesis bounds 1±ρδ,s. For periodic patterns, the synthesis operator reduces exactly to a single constant Vandermonde-type matrix, yielding finite-dimensional basis criteria and optimal bounds. We further prove that the worst binary lower bound is the infimum of its periodic counterparts; at the critical value δ=1/4, a step pattern is governed by a half-shifted Hilbert matrix and has vanishing lower bound, although every fixed periodic pattern remains invertible. We also combine exact bounds for structured patterns with a support-independent perturbation estimate, obtaining explicit stability neighborhoods. For finite unions of bounded intervals and finitely many shifted lattices, folding modulo one yields finitely many rectangular phase matrices whose singular values and ranks give the exact bounds and complete Riesz-sequence, frame, and Riesz-basis criteria. This framework also yields finite Hadamard constraints for periodic spectra and an exact reduction of rationally commensurate multirate sampling.
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BibTeX
@misc{vellucci2026perturbed,
title = {{Perturbed Exponential Systems, Pattern-Dependent Shifted-Sinc Operators, and Matrix Symbols on Finite Unions of Intervals}},
author = {Pierluigi Vellucci and Gohin Shaikh Samad},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31956},
eprint = {2610.31956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31956},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Perturbed Exponential Systems, Pattern-Dependent Shifted-Sinc Operators, and Matrix Symbols on Finite Unions of Intervals
AU - Pierluigi Vellucci
AU - Gohin Shaikh Samad
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31956
UR - https://arxiv.org/abs/2610.31956
AB - We study perturbed exponential systems and unions of shifted frequency lattices through shifted-sinc operators and finite-dimensional fiberizations. For the auxiliary binary local-shift model, we identify the exact pattern-dependent invariant ρ_δ,s=|P_-T_2δP_+|, which gives the optimal squared synthesis bounds 1±ρ_δ,s. For periodic patterns, the synthesis operator reduces exactly to a single constant Vandermonde-type matrix, yielding finite-dimensional basis criteria and optimal bounds. We further prove that the worst binary lower bound is the infimum of its periodic counterparts; at the critical value δ=1/4, a step pattern is governed by a half-shifted Hilbert matrix and has vanishing lower bound, although every fixed periodic pattern remains invertible. We also combine exact bounds for structured patterns with a support-independent perturbation estimate, obtaining explicit stability neighborhoods. For finite unions of bounded intervals and finitely many shifted lattices, folding modulo one yields finitely many rectangular phase matrices whose singular values and ranks give the exact bounds and complete Riesz-sequence, frame, and Riesz-basis criteria. This framework also yields finite Hadamard constraints for periodic spectra and an exact reduction of rationally commensurate multirate sampling.
ER -