  Optimization-based problems have become of great interest for signal
approximation purposes, as they achieved good accuracy results while being
extremely flexible and versatile. In this work, we put our focus on the context
of periodic signals sampled with spatial measurements. The optimization
problems are penalized thanks to the total variation norm, using a specific
class of (pseudo-)differential operators to use well-chosen reconstruction
functions. We introduce three algorithms and their adaptation to this specific
use case. The first one is a discrete grid-based $\ell_1$ method, the second is
called CPGD, which relies on the estimation of discrete innovations within the
FRI framework, and the last one is the Frank-Wolfe algorithm. We put the
emphasis on that later algorithm as we underline its greedy behavior. We
consider a refined version of this algorithm, that leads to very encouraging
outputs. We pursue the algorithmic treatment with a theoretical analysis of the
optimization problem, based on convexity and duality theory. We draw away two
conditions that, if verified by the differential operator, would inform on the
shape of the solutions and even ensure uniqueness, in some cases. Moreover, we
demonstrate the general expression of periodic one-dimensional exponential
splines, as a first step to verify the latter conditions for exponential
operators.
