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

Abstract

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.

Cited by in corpus (1)

• Conditioning of restricted Fourier matrices and super-resolution of MUSIC