By a "reproducing" method for H = L 2 (R n) we mean the use of two countable families {e α : α ∈ A}, {f α : α ∈ A}, in H, so that the first "analyzes" a function h ∈ H by forming the inner products {< h, e α >: α ∈ A}, and the second "reconstructs" h from this information: h = α∈A < h, e α > f α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that "unifies" all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on R n. Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices. Math Subject Classifications. 42C15, 42C40.

We just published a paper showing that the properties of the shift invariant spaces, f , generated by the translates by Z n of an f in L 2 (R n ) correspond to the properties of the spaces L 2 (T n , p), where the weight p equals [f ,f ]. This correspondence helps us produce many new properties of the spaces f . In this paper we extend this method to the case where the role of Z n is taken over by locally compact abelian groups G, L 2 (R n ) is replaced by a separable Hilbert space on which a unitary representation of G acts, and the role of L 2 (T n , p) is assumed by a weighted space L 2 ( b G, w), where b G is the dual group of G. This provides many different extensions of the theory of wavelets and related methods for carrying out signal analysis.2010 Mathematics Subject Classification: 42C40, 43A65, 43A70.

Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in: El acceso a la versión del editor puede requerir la suscripción del recurso Access to the published version may require subscriptionRiesz and frame systems generated by unitary actions of discrete groupsOctober 6, 2014 AbstractWe characterize orthonormal bases, Riesz bases and frames which arise from the action of a countable discrete group Γ on a single element ψ of a given Hilbert space H. As Γ might not be abelian, this is done in terms of a bracket map taking values in the L 1 -space associated to the group von Neumann algebra of Γ. Our result generalizes recent work for LCA groups in [26]. In many cases, the bracket map can be computed in terms of a noncommutative form of the Zak transform.

Abstract. We study closed subspaces of L 2 (X ), where (X , µ) is a σ-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group Γ on X . We provide a complete description for these spaces in terms of range functions and a suitable generalized Zak transform. As an application of our main result, we prove a characterization of frames and Riesz sequences in L 2 (X ) generated by the action of the unitary representation under consideration on a countable set of functions in L 2 (X ). Finally, closed subspaces of L 2 (G), for G being an LCA group, that are invariant under translations by elements on a closed subgroup Γ of G are studied and characterized. The results we obtain for this case are applicable to cases where those already proven in [5,7] are not.

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The final publication is available at Springer via http://dx.doi.org/10.1007/s00365-013-9209-zWe show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C log N. We show with two examples that this bound is attained for quasi-greedy democratic basesFirst and second authors supported by Grant MTM2010-16518 (Spain). Third author supported by a travel grant from Simons Foundation, and by a COR grant from University of California Syste

We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases.