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There are a couple of methods with sparsity enhancing properties for regularization of inverse problems. This paper deals with the Tikhonov regularization with an ℓ 1 penalty and with the orthogonal matching pursuit. Conditions are derived, which ensure that the regularized solution has the exact support. These conditions are especially applicable in the case of ill-posed problems, where other conditions based on coherence or the restricted isometry property are usually not applicable.doi:10.4310/maa.2011.v18.n1.a7 fatcat:zcimt7b6mfg4nbmya3dg3ub4ci