International audienceThe purpose of this correspondence is to generalize a result by Donoho and Huo and Elad and Bruckstein on sparse representations of signals in a union of two orthonormal bases for R^N. We consider general (redundant) dictionaries for R^N, and derive sufficient conditions for having unique sparse representations of signals in such dictionaries. The special case where the dictionary is given by the union of L \ge 2 orthonormal bases for R^N is studied in more detail. In particular, it is proved that the result of Donoho and Huo, concerning the replacement of the \ell^0 optimization problem with a linear programming problem when searching for sparse representations, has an analog for dictionaries that may be highly redundant

This paper appeared as technical report in 2003, see http://hal.inria.fr/inria-00564038/International audienceThe purpose of this paper is to study sparse representations of signals from a general dictionary in a Banach space. For so-called localized frames in Hilbert spaces, the canonical frame coefficients are shown to provide a near sparsest expansion for several sparseness measures. However, for frames which are not localized, this no longer holds true and sparse representations may depend strongly on the choice of the sparseness measure. A large class of admissible sparseness measures is introduced, and we give sufficient conditions for having a unique sparse representation of a signal from the dictionary w.r.t. such a sparseness measure. Moreover, we give sufficient conditions on a signal such that the simple solution of a linear programming problem simultaneously solves all the non-convex (and generally hard combinatorial) problems of sparsest representation of the signal w.r.t. arbitrary admissible sparseness measures

Abstract. A new construction of tight frames for L 2 (R d ) with flexible time-frequency localization is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on R d , a class of so-called decomposition spaces. The decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients. As examples of the general construction, new tight frames yielding decompositions of Besov space, anisotropic Besov spaces, α-modulation spaces, and anisotropic α-modulation spaces are considered. Finally, curvelet-type tight frames are constructed on R d , d ≥ 2.

The α-modulation spaces M s,α p,q (R d ), α ∈ [0, 1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed.

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