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

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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.