  Splines come in a variety of flavors that can be characterized in terms of
some differential operator L. The simplest piecewise-constant model corresponds
to the derivative operator. Likewise, one can extend the traditional notion of
total variation by considering more general operators than the derivative. This
leads us to the definition of the generalized Beppo-Levi space M, which is
further identified as the direct sum of two Banach spaces. We then prove that
the minimization of the generalized total variation (gTV) over M, subject to
some arbitrary (convex) consistency constraints on the linear measurements of
the signal, admits nonuniform L-spline solutions with fewer knots than the
number of measurements. This shows that non-uniform splines are universal
solutions of continuous-domain linear inverse problems with LASSO, L1, or
TV-like regularization constraints. Remarkably, the spline-type is fully
determined by the choice of L and does not depend on the actual nature of the
measurements.
