  We present a new algorithm and the corresponding convergence analysis for the
regularization of linear inverse problems with sparsity constraints, applied to
a new generalized sparsity promoting functional. The algorithm is based on the
idea of iteratively reweighted least squares, reducing the minimization at
every iteration step to that of a functional including only $\ell_2$-norms.
This amounts to smoothing of the absolute value function that appears in the
generalized sparsity promoting penalty we consider, with the smoothing becoming
iteratively less pronounced. We demonstrate that the sequence of iterates of
our algorithm converges to a limit that minimizes the original functional.
