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We study the super-resolution problem of recovering a periodic continuous-domain function from its low-frequency information. This means that we only have access to possibly corrupted versions of its Fourier samples up to a maximum cut-off frequency Kc. The reconstruction task is specified as an optimization problem with generalized total-variation regularization involving a pseudo-differential operator. Our special emphasis is on the uniqueness of solutions. We show that, for elliptic regularization operators (eg, the derivatives of any order), uniqueness is always guaranteed. To achieve this goal, we provide a new analysis of constrained optimization problems over Radon measures. We demonstrate that either the solutions are always made of Radon measures of constant sign, or the solution is unique. Doing so, we identify a general sufficient condition for the uniqueness of the solution of a constrained optimization problem with TV-regularization, expressed in terms of the Fourier samples.