Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform

Abstract

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.

Linear independence and sets of uniqueness

“…For example, consider the Schauder basis property vs. the Muckenhoupt A pcondition (see [11]) or the use of the Carleson theorem in the recent proof of the ℓ 2 -linear independence characterization by S. Saliani ([12]). For the systematic treatment of such properties see [5] and for the extension to the general, LCA groups, case see [4]. In this paper we would like to establish yet another connection, as far as we know not yet observed, between the ℓ p -linear independence of the system of translates and the ℓ p -sets of uniqueness, where 1 p 2.…”

“…In the study of ψ , the fundamental role is played by the periodization function (see [4,5,14] and the numerous references within)…”

On quasi-greedy bases associated with unitary representations of countable groups

“…An abstract way to interpret dual integrability, which was pointed out in [9], is through Stone's theorem, see e.g. [3,Theorem 4.44].…”

“…However, the abstract approach in the present paper will show that the result in [13] is just one instance of a more general theory that applies to many other "popular" systems in harmonic analysis such as Gabor and Vilenkin systems. Our general result relies on the notion of a dual integrable representation introduced in [9].…”

“…We will use the terminology and notation presented in [9]. Suppose that G is a locally compact abelian (LCA) group.…”