Single-exponential bounds for the smallest singular value of Vandermonde matrices in the sub-Rayleigh regime

Dmitry Batenkov*, Gil Goldman

*Corresponding author for this work

Research output: Contribution to journalLetterpeer-review

Abstract

Following recent interest by the community, the scaling of the minimal singular value of a Vandermonde matrix with nodes forming clusters on the length scale of Rayleigh distance on the complex unit circle is studied. Using approximation theoretic properties of exponential sums, we show that the decay is only single exponential in the size of the largest cluster, and the bound holds for arbitrary small minimal separation distance. We also obtain a generalization of well-known bounds on the smallest eigenvalue of the generalized prolate matrix in the multi-cluster geometry. Finally, the results are extended to the entire spectrum.

Original languageEnglish
Pages (from-to)426-439
Number of pages14
JournalApplied and Computational Harmonic Analysis
Volume55
DOIs
StatePublished - Nov 2021

Keywords

  • Condition number
  • Nonuniform Fourier matrices
  • Singular values
  • Sub-Rayleigh resolution
  • Super-resolution
  • Vandermonde matrices with nodes on the unit circle

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