The problem of super-resolution (SR) is to extract the fine details of a signal from incomplete and inaccurate measurements of its spectrum. It is an inverse problem of contemporary interest, arising in many fields: spectral estimation, sampling, direction of arrival and source localization, imaging beyond the diffraction limit, inverse scattering, exponential data analysis, to name a few. A major challenge for applied mathematics is to establish computational limits of resolution, and develop optimal inversion al­gorithms.

The course will introduce the students to mathematical theories of super-resolution, covering both classical techniques as well as modern approaches. List of topics (some of them time permitting):

1. Well-posed and ill-posed problems, conditioning.

2. Resolution in inverse problems: bandlimited systems, classical resolution limits. Computational vs. instrumental SR. Importance of priors.

3. Super-resolution by analytic continuation. A-priori stability estimates, regularization theory, truncated SVD as a universal method. Discretization. Iterative methods. Introduction to super-oscillations.

4. Super-resolution with sparsity: parametric deconvolution of spike trains by algebraic techniques. Fundamental computational limits. Connections to harmonic analysis. Vandermonde matrices on the unit circle. High-resolution algorithms. Sampling beyond the Nyquist rate. Optimization-based methods.

5. Inverse scattering and inverse diffraction problems.

The course will include a theoretical/computational project, possibly leading to an M.Sc. thesis.