  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
$\mathrm{L} = \mathrm{D}^M$. The order $M \geq 1$ 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
$\mathrm{L}$ whose number of knots is upper-bounded by $2 K_c$. We then
consider the grid-based discretization of our optimization problem in the space
of uniform $\mathrm{L}$-splines. On the theoretical side, we show that any
sequence of solutions of the discretized problem converges uniformly to the
unique solution of the gridless problem as the grid size vanishes. Finally, on
the algorithmic side, we propose a B-spline-based algorithm to solve the
grid-based problem, and we demonstrate its numerical feasibility
experimentally. On both of these aspects, we leverage the uniqueness of the
solution of the original problem.
