We consider linear inverse problems where the solution is assumed to have a sparse expansion on some orthonormal basis and we show that the usual quadratic regularizing penalties can be replaced by weighted lp-norm penalties (with p equal to or larger than one) on the coefficients of such expansions to enforce sparsity. To compute the corresponding regularized solutions we propose an iterative algorithm which amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove convergence and stability of this algorithm. Finally, we give an overview of the potential applications of the method.
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