Super-resolution of generalized spikes and spectra of confluent Vandermonde matrices

Dmitry Batenkov, Nuha Diab*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study the problem of super-resolution of a linear combination of Dirac distributions and their derivatives on a one-dimensional circle from noisy Fourier measurements. Following numerous recent works on the subject, we consider the geometric setting of “partial clustering”, when some Diracs can be separated much below the Rayleigh limit. Under this assumption, we prove sharp asymptotic bounds for the smallest singular value of a corresponding rectangular confluent Vandermonde matrix with nodes on the unit circle. As a consequence, we derive matching lower and upper min-max error bounds for the above super-resolution problem, under the additional assumption of nodes belonging to a fixed grid.

Original languageEnglish
Pages (from-to)181-208
Number of pages28
JournalApplied and Computational Harmonic Analysis
Volume65
DOIs
StatePublished - Jul 2023

Funding

FundersFunder number
Lower Saxony - Israel
Volkswagen Foundation
Israel Science Foundation1792/20

    Keywords

    • Confluent Vandermonde matrix
    • Decimation
    • Dirac distributions
    • ESPRIT
    • Min-max error
    • Partial Fourier matrix
    • Smallest singular value
    • Sparse recovery
    • Super-resolution

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