Radar, seismic and and wireless communication systems observe waves (hidden in noise) by sensor arrays. Those systems infer the originating (spatially-sparse) set of sources, within a minimum prescribed resolution and with as few sensors as feasible. These requirements lead to under-determined systems of equations. Compressed sensing is an active research field which treats the recovery of a set of sources from an under-determined system of equations exploiting sparsity. Naturally two main questions arise in this context. Firstly, what is the minimum number of equations/sensors for which reconstruction can be guaranteed and secondly, how to achieve efficient reconstruction regarding the sensors 'observations' Solutions to the first question provide insights into the design of linear measurements and use quantities like the restricted isometry property or coherence to describe well-behaved matrices, such as equiangular tight frames. Algorithms as answers to the second question mostly rely on the assumption, that the measurements were made in accordance to the reconstruction guarantees. Suppose that one has to work with existing data acquisition systems, the measurement matrix is given a priori and algorithms studied under too idealistic assumptions are prone to failure. This thesis shows that well known greedy algorithms like orthogonal matching pursuit are not suited for array processing problems. We devise an algorithm based on the generalized Least Absolute Shrinkage and Selection Operator (LASSO), which is a penalized least squares problem. The heuristically chosen l1 penalty term ensures strict convexity and that strong duality holds. The corresponding dual problem is interpretable as a weighted conventional beamformer acting on the residuals of the LASSO. Based on physical insights provided by the dual problem¿s solution, three procedures for single snapshot reconstruction and one for sequential online reconstruction are proposed and analysed. The sequential procedure assumes a weighted Laplace-like prior for the sources such that the maximum a posteriori source estimate at the current time step is the solution to a generalized LASSO problem. For the sequential implementation, the posterior distribution is fitted to the Laplace-like density by use of the dual solution.