  We consider linear inverse problems that are formulated in the continuous
domain. The object of recovery is a function that is assumed to minimize a
convex objective functional. The solutions are constrained by imposing a
continuous-domain regularization. We derive the parametric form of the solution
(representer theorems) for Tikhonov (quadratic) and generalized total-variation
(gTV) regularizations. We show that, in both cases, the solutions are splines
that are intimately related to the regularization operator. In the Tikhonov
case, the solution is smooth and constrained to live in a fixed subspace that
depends on the measurement operator. By contrast, the gTV regularization
results in a sparse solution composed of only a few dictionary elements that
are upper-bounded by the number of measurements and independent of the
measurement operator. Our findings for the gTV regularization resonates with
the minimization of the $l_1$ norm, which is its discrete counterpart and also
produces sparse solutions. Finally, we find the experimental solutions for some
measurement models in one dimension. We discuss the special case when the gTV
regularization results in multiple solutions and devise an algorithm to find an
extreme point of the solution set which is guaranteed to be sparse.
