  We consider the problem of reconstructing an infinite set of sparse,
finite-dimensional vectors, that share a common sparsity pattern, from
incomplete measurements. This is in contrast to the work [17], where the single
vector signal can be infinite-dimensional, and [28], which extends the
aforementioned work to the joint sparse recovery of finite number of
infinite-dimensional vectors. In our case, to take account of the joint
sparsity and promote the coupling of nonvanishing components, we employ a
convex relaxation approach with mixed norm penalty $\ell_{2,1}$. This paper
discusses the computation of the solutions of linear inverse problems with such
relaxation by a forward-backward splitting algorithm. However, since the
solution matrix possesses infinitely many columns, the arguments of [17] no
longer apply. As such, we establish new strong convergence results for the
algorithm, in particular when the set of jointly sparse vectors is infinite.
