# Analysis Seminar

#### Super-resolution, subspace methods, and minimum singular value of non-harmonic Fourier matrices

Speaker: Weilin LI, CIMS

Location: Warren Weaver Hall 1302

Date: Thursday, April 25, 2019, 11 a.m.

Synopsis:

This talk is concerned with the inverse problem of recovering a
discrete measure on the torus consisting of S atoms, given M consecutive
noisy Fourier coefficients. Super-resolution is involved when the distance
between two atoms is less than 1/M. We connect this problem to the minimum
singular value of non-harmonic Fourier matrices. New results for the latter
are presented, and as consequences, we derive results regarding the
information theoretic limit of super-resolution and the resolution limit of
subspace methods (namely, MUSIC and ESPRIT). These results rigorously
establish the super-resolution phenomena of these algorithms that were
empirically discovered long ago, and numerical results indicate that our
bounds are sharp or nearly sharp. Interesting connections to trigonometric
interpolation and uncertainty principles are also presented. Joint work
with John Benedetto, Albert Fannjiang, and Wenjing Liao.