  We present a novel framework for the reconstruction of 1D composite signals
assumed to be a mixture of two additive components, one sparse and the other
smooth, given a finite number of linear measurements. We formulate the
reconstruction problem as a continuous-domain regularized inverse problem with
multiple penalties. We prove that these penalties induce reconstructed signals
that indeed take the desired form of the sum of a sparse and a smooth
component. We then discretize this problem using Riesz bases, which yields a
discrete problem that can be solved by standard algorithms. Our discretization
is exact in the sense that we are solving the continuous-domain problem over
the search space specified by our bases without any discretization error. We
propose a complete algorithmic pipeline and demonstrate its feasibility on
simulated data.
