Locally Compact Abelian Research Articles

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

A Gabor orthonormal basis, on a locally compact Abelian (LCA) group A, is an orthonormal basis of L2(A) that consists of time-frequency shifts of some template f∈L2(A). It is well known that, on Rd, the elements of such a basis cannot have a good time-frequency localization. The picture is drastically different on LCA groups containing a compact open subgroup, where one can easily construct examples of Gabor orthonormal bases with f maximally localized, in the sense that the ambiguity function of f (i.e., the correlation of f with its time-frequency shifts) has support of minimum measure, compatibly with the uncertainty principle. In this note we find all the Gabor orthonormal bases with this extremal property. To this end, we identify all the functions in L2(A) that are maximally localized in the time-frequency space in the above sense – an issue that is open even for finite Abelian groups. As a byproduct, on every LCA group containing a compact open subgroup we exhibit the complete family of optimizers for Lieb's uncertainty inequality, and we also show previously unknown optimizers on a general LCA group.

Gabor frame multipliers and Parseval duals on the half real line

Recently, Gabor analysis on locally compact abelian (LCA) groups has become the focus of an active research. In practice, the time variable cannot be negative. The half real line [Formula: see text] is an LCA group under multiplication and the usual topology, with the Haar measure [Formula: see text]. This paper addresses Gabor frame multipliers and Parseval duals for [Formula: see text]. We introduce and characterize Gabor frame multipliers and Parseval Gabor frame multipliers based on Zak transform matrices. Our Zak transform matrix is essentially different from the conventional Zibulski–Zeevi matrix. It allows us to define Gabor frame generators by designing suitable matrix-valued functions of finite size. We also prove that an arbitrary Gabor frame [Formula: see text] admits a Parseval dual frame/tight dual frame whenever [Formula: see text] are rational numbers not greater than [Formula: see text].

Rationally sampled Gabor frames on the half real line

Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line R+=(0,∞) is an LCA group under multiplication and the usual topology, with the Haar measure dμ=dxx. This paper addresses rationally sampled Gabor frames for L2(R+,dμ). Given a function in L2(R+,dμ), we introduce a new Zak transform matrix associated with it, which is different from the conventional Zibulski-Zeevi matrix. It allows us to define a function by designing its Zak transform matrix. Using our Zak transform matrix method, we characterize and express complete Gabor systems, Bessel sequences, Gabor frames, Riesz bases and Gabor duals of an arbitrarily given Gabor frame for L2(R+,dμ), and prove the minimality of the canonical dual frames in some sense. Some examples are also provided to illustrate the generality of our theory.

Lieb's Inequality for Continuous Modulated Shearlet Transform on LCA Groups

We have proved a version of Lieb's inequality for continuous modulated shearlet transform on locally compact Abelian (LCA) groups. An alternative proof for the inequality has also been provided using Riesz-Thorin interpolation theorem.

The Stockwell transform on locally compact abelian groups

Let [Formula: see text] be a locally compact abelian (LCA) group and [Formula: see text] be a topological automorphism on [Formula: see text]. For [Formula: see text], the Stockwell transform on the locally compact group [Formula: see text] with respect to the automorphism [Formula: see text] is defined and denoted by [Formula: see text]. The inversion formula for the Stockwell transform [Formula: see text] is established and as a result it is concluded that [Formula: see text] is an isometry. Moreover, some properties of the Stockwell transform [Formula: see text] are examined. Finally, some examples are given to support our technical considerations.

Spline function generating Gabor systems on the half real line

Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line is an LCA group under multiplication and the usual topology. This paper addresses spline Gabor frames for , where is the corresponding Haar measure. We introduce the concept of spline functions on by ‐convolution and estimate their Gabor frame sets, that is, lattice sets such that spline generating Gabor systems are frames for . For an arbitrary spline Gabor frame with special lattices, we present its one dual Gabor frame window, which has the same smoothness as the initial window function. For a class of special spline Gabor Bessel sequences, we prove that they can be extended to a tight Gabor frame by adding a new window function, which has compact support and same smoothness as the initial windows. And we also demonstrate that two spline Gabor Bessel sequences can always be extended to a pair of dual Gabor frames with the adding window functions being compactly supported and having the same smoothness as the initial windows.

Homogeneous Banach Spaces as Banach Convolution Modules over M(G)

This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on (M(G),∥·∥M), the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over Rd. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space(B,∥·∥B) on G, such as (Lp(G),∥·∥p), for 1≤p<∞, or the Fourier–Stieltjes algebra FM(G), and in particular any Segal algebra is a Banach convolution module over (M(G),∥·∥M) in a natural way. Via the Haar measure we can then identify L1(G),∥·∥1 with the closure (of the embedded version) of Cc(G), the space of continuous functions with compact support, in (M(G),∥·∥M), and show that these homogeneous Banach spaces are essentialL1(G)-modules. Thus, in particular, the approximate units act properly as one might expect and converge strongly to the identity operator. The approach is in the spirit of Hans Reiter, avoiding the use of structure theory for LCA groups and the usual techniques of vector-valued integration via duality. The ultimate (still distant) goal of this approach is to provide a new and elementary approach towards the (extended) Fourier transform in the setting of the so-called Banach–Gelfand triple(S0,L2,S0′)(G), based on the Segal algebra S0(G). This direction will be pursued in subsequent papers.

Explicit construction of frames and pairs of dual frames on locally compact abelian groups

Locally compact abelian (LCA) groups form a natural setting for frame theory in signal processing: while classical frame theory was developed in the setting of \( L^2({\mathbb {R}}),\) concrete applications have to be implemented on \({\mathbb {R}}^d,\) two scenarios that are covered under the setup of LCA groups. Due to rapid progress over the past few years, it is known within the mathematical community that large parts of the standard frame theory for \( L^2({\mathbb {R}})\) can be carried over to the setting of LCA groups. We show that this also holds on the level of explicit constructions, which is a crucial issue for applications within signal processing; in fact we provide explicit constructions of frames and pairs of dual frames on LCA groups. Special attention will be given to the so-called elementary LCA groups, i.e., groups that are tensor products of \({\mathbb {R}},\) \({\mathbb {Z}},\) the torus group \({{\mathbb {T}}},\) and the finite group \({\mathbb {Z}}_N.\) The frames will have the form of a Fourier-like system, i.e., the Fourier transform of a generalized shift-invariant system. The constructions are based on a partition of unity; in particular we provide a general and flexible approach for constructing a partition of unity that makes it easy to control key features of the underlying functions, e.g., support size and continuity. The results apply to arbitrary LCA groups with a supply of “sufficiently fine” lattices, a condition that is satisfied for all elementary LCA groups. Concrete constructions of pairs of dual frames for several elementary LCA groups are provided as illustration. In the case of a system of functions generated by a single lattice, such as a Gabor system, the general results become particularly transparent.

Uniqueness theorem for Fourier transformable measures on LCA groups

We show that if points of supports of two discrete ”not very thick” Fourier transformable measures on locally compact abelian (LCA) groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete ”thick” measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.

Multivariate Gabor frames for operators in matrix-valued signal spaces over locally compact abelian groups

In this paper, we study multivariate Gabor frames in matrix-valued signal spaces over locally compact abelian (LCA) groups, where the lower frame condition depends on a bounded linear operator [Formula: see text] on the underlying matrix-valued signal space. This type of Gabor frame is also known as a multivariate [Formula: see text]-Gabor frame. By extending work of G[Formula: see text]vruta, we present necessary and sufficient conditions for the existence of [Formula: see text]-Gabor frames of multivariate matrix-valued Gabor systems. Some operators which can transform multivariate matrix-valued Gabor and [Formula: see text]-Gabor frames into [Formula: see text]-Gabor frames in terms of adjointable operators are discussed. Finally, we give a Paley–Wiener-type perturbation result for multivariate matrix-valued [Formula: see text]-Gabor frames.

A Sequential Approach to Mild Distributions

The Banach Gelfand Triple ( S 0 , L 2 , S 0 ′ ) ( R d ) consists of S 0 ( R d ) , ∥ · ∥ S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , ∥ · ∥ 2 and the dual space S 0 ′ ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , ∥ · ∥ S 0 and hence ( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra S 0 ( R d ) , ∥ · ∥ S 0 can be used to establish this natural identification.

Sampling and periodization of generators of Heisenberg modules

This paper considers generators of Heisenberg modules in the case of twisted group [Formula: see text]-algebras of closed subgroups of locally compact abelian (LCA) groups and how the restriction and/or periodization of these generators yield generators for other Heisenberg modules. Since generators of Heisenberg modules are exactly the generators of (multi-window) Gabor frames, our methods are going to be from Gabor analysis. In the latter setting, the procedure of restriction and periodization of generators is well known. Our results extend this established part of Gabor analysis to the general setting of LCA groups. We give several concrete examples where we demonstrate some of the consequences of our results. Finally, we show that vector bundles over an irrational noncommutative torus may be approximated by vector bundles for finite-dimensional matrix algebras that converge to the irrational noncommutative torus with respect to the module norm of the generators, where the matrix algebras converge in the quantum Gromov–Hausdorff distance to the irrational noncommutative torus.

Existence of quasicrystals and universal stable sampling and interpolation in LCA groups

We characterize all the locally compact abelian (LCA) groups that contain quasicrystals (a class of model sets). Moreover, we describe all possible quasicrystals in the group constructing an appropriate lattice associated with the cut and project scheme that produces it. On the other hand, if an LCA group G admits a simple quasicrystal, we prove that recent results of Meyer and Matei for the case of the n-dimensional Euclidean space can be extended to G. More precisely, we prove that simple quasicrystals are universal sets of stable sampling and universal sets of stable interpolation in generalized Paley-Wiener spaces.

Pairwise orthogonal frames generated by regular representations of LCA groups

Having potential applications in multiplexing techniques and in the synthesis of frames, orthogonality (or strongly disjointness) plays a significant role in frame theory (e.g. construction of new frames from existing ones, constructions related with duality, etc.). In this article, we study orthogonality of a pair of frames over locally compact abelian (LCA) groups. We start with the investigation of the dual Gramian analysis tools of Ron and Shen through a pre-Gramian operator over the set-up of LCA groups. Then we fiberize some operators associated with Bessel families generated by unitary actions of co-compact (not necessarily discrete) subgroups of LCA groups. Using this fiberization, we study and characterize a pair of orthogonal frames generated by the action of a unitary representation ρ of a co-compact subgroup Γ⊂G on a separable Hilbert space L2(G), where G is a second countable LCA group. Precisely, we consider frames of the form {ρ(γ)ψ:γ∈Γ,ψ∈Ψ} for a countable family Ψ in L2(G). We pay special attention to this problem in the context of translation-invariant space by assuming ρ as the action of Γ on L2(G) by left-translation. The representation of Γ acting on L2(G) by (left-)translation is called the (left-)regular representation of Γ. Further, we apply our results on co-compact Gabor systems over LCA groups. At this juncture, it is pertinent to note that the resulting characterization can be useful for constructing new frames by using various techniques including the unitary extension principle by Ron and Shen [24] and its recent extension to LCA groups by Christensen and Goh [7].

Splitting of Closed Subgroups of Locally Compact Abelian Groups

Let £ be the category of all locally compact abelian (LCA) groups. In this paper, we investigate the splitting of the identity component, the subgroup of all compact elements and the maximal torsion subgroup of a group £. A group £ will be called split full if every closed subgroup of G splits. In this paper, we give a necessary condition for an LCA group to be split full.

Riesz Wavelets, Tiling and Spectral Sets in LCA Groups

Let G be a locally compact abelian (LCA) group equipped with a Haar measure. A collection of measurable subsets $$\{\Omega _i\}_{i=1}^m$$ in $$\hat{G}$$ is called a Riesz wavelet collection if there are countable subsets $$A\subset \mathcal {A}ut(G)$$ and $$\Lambda _i\subset G$$ such that for $$\hat{\psi }_i:= 1_{\Omega _i}$$ , the family $$\begin{aligned} \mathcal {W}:=\cup _{i=1}^m\{\Delta (a)^{1/2} \psi _i(a(x)-\lambda ): \lambda \in \Lambda _i, a\in A\} \end{aligned}$$ is a Riesz basis for $$L^2(G)$$ . In this paper we show that if $$\Omega _i$$ ’s are Riesz spectral sets and $$\Omega =\cup _i \Omega _i$$ is a multiplicative tiling set for $$\hat{G}$$ , then $$\mathcal {W}$$ is a Riesz basis for $$L^2(G)$$ . The converse also holds if the unit element $$e\in G$$ belongs to all $$\Lambda _i$$ . As a result, if $$\Omega _i$$ ’s multi-tile $$\hat{G}$$ additively by lattice and $$\Omega $$ is a multiplicative tiling, then $$\mathcal {W}$$ is a Riesz basis for $$L^2(G)$$ . When $$m=1$$ , we show that the multiplicative tiling property of $$\Omega $$ is equivalent to the Riesz spectral property of $$\hat{\alpha }(\Omega )$$ , $$\alpha \in A$$ , provided that $$\mathcal {W}$$ is a Riesz basis.

Approximately dual and perturbation results for generalized translation-invariant frames on LCA groups

The results in this paper can be divided into three parts. First, we generalize the recent results of Benavente, Christensen and Zakowicz on approximately dual generalized shift-invariant frames on the real line to generalized translation-invariant (GTI) systems on locally compact abelian (LCA) groups. Second, we explain in detail how GTI frames can be realized as [Formula: see text]-frames. Finally, the known results on perturbation of [Formula: see text]-frames and results on perturbation of generalized shift-invariant systems are applied and extended to GTI systems on LCA groups.

Orthogonality of a pair of frames over locally compact abelian groups

In this article, we provide necessary and sufficient conditions for the orthogonality of two Bessel families when such families have the form of generalized translation invariant (GTI) systems over a second countable locally compact abelian (LCA) group G. The work is motivated by the utility of a recent notion given by Jakobsen and Lemvig on GTI systems in L2(G), and the concept of the orthogonality (or strongly disjointness) of a pair of frames studied by Balan, Han, and Larson. Consequently, we deduce similar results for several function systems including the case of TI systems, and GTI systems on compact abelian groups. We apply our results to the Bessel families having wave-packet structure (combination of wavelet as well as Gabor structure), and hence a characterization for pairwise orthogonal wave-packet frame systems over LCA groups is obtained. In addition, we relate the well-established theory from literature with our results by observing several deductions in the context of wavelet and Gabor systems over LCA groups with G=Rd,Zd, etc.

Sampling in Unitary Invariant Subspaces Associated to LCA Groups

In this paper a sampling theory for unitary invariant subspaces associated to locally compact abelian (LCA) groups is deduced. Working in the LCA group context allows to obtain, in a unified way, sampling results valid for a wide range of problems which are interesting in practice, avoiding also cumbersome notation. Along with LCA groups theory, the involved mathematical technique is that of frame theory which meets matrix analysis when appropriate dual frames are computed.

Abstract relative Gabor transforms over canonical homogeneous spaces of semidirect product groups with Abelian normal factor

This paper introduces a unified approach to the abstract notion of relative Gabor transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let [Formula: see text] be a locally compact group, [Formula: see text] be a locally compact Abelian (LCA) group, and [Formula: see text] be a continuous homomorphism. Let [Formula: see text] be the semi-direct product of [Formula: see text] and [Formula: see text] with respect to [Formula: see text], [Formula: see text] be the canonical homogeneous space of [Formula: see text], and [Formula: see text] be the canonical relatively invariant measure on [Formula: see text]. Then we present a unified harmonic analysis approach to the theoretical aspects of the notion of relative Gabor transform over the Hilbert function space [Formula: see text].