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We study the problem of recovering piecewise-polynomial periodic functions from their low-frequency information. This means that we only have access to possibly corrupted versions of the Fourier samples of the ground truth up to a maximum cutoff frequency K c. The reconstruction task is specified as an optimization problem with total-variation (TV) regularization (in the sense of measures) involving the M th order derivative regularization operator L= D M. The order M≥ 1 determines the degree of the reconstructed piecewise-polynomial spline, whereas the TV regularization norm, which is known to promote sparsity, guarantees a small number of pieces. We show that the solution of our optimization problem is always unique, which, to the best of our knowledge, is a first for TV-based problems. Moreover, we show that this solution is a periodic spline matched to the regularization operator L whose number of knots …