The first two worst erasure patterns in Fourier sampling at fixed codimension

For fixed integers 2 <= r <= ell, we retain M-ell consecutive frequencies of the unnormalized Fourier matrix of order M and erase any r sampling rows. For all sufficiently large M, we identify the first two minima of the squared least singular value over all erasure sets, including sets depending on M. The minimum occurs exactly at cyclically consecutive erasures. After excluding these, the minimum occurs at translates and reflections of {0,2,3,...,r}. Both values have explicit leading constants and relative error O(M^-2), and their leading ratio depends only on r. The results also determine the worst least-squares noise amplification and spectral condition number.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-24
DOI
https://doi.org/10.5281/zenodo.23611289
Primary Topic
Mathematical Analysis and Transform Methods
Type
preprint
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preprint

The first two worst erasure patterns in Fourier sampling at fixed codimension

Zihuan Xu
Zenodo (CERN European Organization for Nuclear Research)
Mathematical Analysis and Transform Methods
preprint

The first two worst erasure patterns in Fourier sampling at fixed codimension

Zihuan Xu
preprint en

Abstract

For fixed integers 2 <= r <= ell, we retain M-ell consecutive frequencies of the unnormalized Fourier matrix of order M and erase any r sampling rows. For all sufficiently large M, we identify the first two minima of the squared least singular value over all erasure sets, including sets depending on M. The minimum occurs exactly at cyclically consecutive erasures. After excluding these, the minimum occurs at translates and reflections of {0,2,3,...,r}. Both values have explicit leading constants and relative error O(M^-2), and their leading ratio depends only on r. The results also determine the worst least-squares noise amplification and spectral condition number.

Zenodo (CERN European Organization for Nuclear Research)
Capital Normal University (CN)
Mathematical Analysis and Transform Methods
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