  We introduce a general framework for the reconstruction of periodic
multivariate functions from finitely many and possibly noisy linear
measurements. The reconstruction task is formulated as a penalized convex
optimization problem, taking the form of a sum between a convex data fidelity
functional and a sparsity-promoting total variation based penalty involving a
suitable spline-admissible regularizing operator L. In this context, we
establish a periodic representer theorem, showing that the extreme-point
solutions are periodic L-splines with less knots than the number of
measurements. The main results are specified for the broadest classes of
measurement functionals, spline-admissible operators, and convex data fidelity
functionals. We exemplify our results for various regularization operators and
measurement types (e.g., spatial sampling, Fourier sampling, or
square-integrable functions). We also consider the reconstruction of both
univariate and multivariate periodic functions.
