  The problem of the minimization of least squares functionals with $\ell^1$
penalties is considered in an infinite dimensional Hilbert space setting. While
there are several algorithms available in the finite dimensional setting there
are only a few of them which come with a proper convergence analysis in the
infinite dimensional setting. In this work we provide an algorithm from a class
which have not been considered for $\ell^1$ minimization before, namely a
proximal-point method in combination with a projection step. We show that this
idea gives a simple and easy to implement algorithm. We present experiments
which indicate that the algorithm may perform better than other algorithms if
we employ them without any special tricks. Hence, we may conclude that the
projection proximal-point idea is a promising idea in the context of
$\ell^1$-minimization.
