Elementary $L^\infty$ error estimates for super-resolution de-noising

• DOI: 10.48550/arxiv.1702.03021
• Publication Date: 2017-01-01
• Authors: Li, Weilin
• Keywords: FOS: Computer and information sciences; Computer Science - Information Theory; Information Theory (cs.IT); 0101 mathematics; 01 natural sciences

This paper studies the problem of recovering a discrete complex measure on the torus from a finite number of corrupted Fourier samples. We assume the support of the unknown discrete measure satisfies a minimum separation condition and we use convex regularization methods to recover approximations of the original measure. We focus on two well-known convex regularization methods, and for both, we establish an error estimate that bounds the smoothed-out error in terms of the target resolution and noise level. Our $L^\infty$ approximation rate is entirely new for one of the methods, and improves upon a previously established $L^1$ estimate for the other. We provide a unified analysis and an elementary proof of the theorem.