This paper describes some new results on recursive l_1-minimizing by Kalman filtering. We consider the l_1-norm as an explicit constraint, formulated as a nonlinear observation of the state to be estimated. Interpretiing a sparse vector to be estimated as a state which is observed from erroneous (undersampled) measurements we can address time- and space-variant sparsity, any kind of a priori information and also easily address nonstationary error influences in the measurements available. Inherently in our approach we move slightly away from the classical RIP-based approaches to a more intuitive understanding of the structure of the nullspace which is implicitly related to the well understood engineering concepts of deterministic and stochastic observability in estimation theory