  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.
