  In this article a unified approach to iterative soft-thresholding algorithms
for the solution of linear operator equations in infinite dimensional Hilbert
spaces is presented. We formulate the algorithm in the framework of generalized
gradient methods and present a new convergence analysis. As main result we show
that the algorithm converges with linear rate as soon as the underlying
operator satisfies the so-called finite basis injectivity property or the
minimizer possesses a so-called strict sparsity pattern. Moreover it is shown
that the constants can be calculated explicitly in special cases (i.e. for
compact operators). Furthermore, the techniques also can be used to establish
linear convergence for related methods such as the iterative thresholding
algorithm for joint sparsity and the accelerated gradient projection method.
