TY - JOUR
T1 - Single-exponential bounds for the smallest singular value of Vandermonde matrices in the sub-Rayleigh regime
AU - Batenkov, Dmitry
AU - Goldman, Gil
N1 - Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/11
Y1 - 2021/11
N2 - 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.
AB - 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.
KW - Condition number
KW - Nonuniform Fourier matrices
KW - Singular values
KW - Sub-Rayleigh resolution
KW - Super-resolution
KW - Vandermonde matrices with nodes on the unit circle
UR - http://www.scopus.com/inward/record.url?scp=85111853525&partnerID=8YFLogxK
U2 - 10.1016/j.acha.2021.07.003
DO - 10.1016/j.acha.2021.07.003
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AN - SCOPUS:85111853525
SN - 1063-5203
VL - 55
SP - 426
EP - 439
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
ER -