  We characterize the solution of a broad class of convex optimization problems
that address the reconstruction of a function from a finite number of linear
measurements. The underlying hypothesis is that the solution is decomposable as
a finite sum of components, where each component belongs to its own prescribed
Banach space; moreover, the problem is regularized by penalizing some composite
norm of the solution. We establish general conditions for existence and derive
the generic parametric representation of the solution components. These
representations fall into three categories depending on the underlying
regularization norm: (i) a linear expansion in terms of predefined "kernels"
when the component space is a reproducing kernel Hilbert space (RKHS), (ii) a
non-linear (duality) mapping of a linear combination of measurement functionals
when the component Banach space is strictly convex, and, (iii) an adaptive
expansion in terms of a small number of atoms within a larger dictionary when
the component Banach space is not strictly convex. Our approach generalizes and
unifies a number of multi-kernel (RKHS) and sparse-dictionary learning
techniques for compressed sensing available in the literature. It also yields
the natural extension of the classical spline-fitting techniques in (semi-)RKHS
to the abstract Banach-space setting.
