A qualitative uncertainty principle for functions generating a Gabor frame on LCA groups

Let G be a locally compact Abelian group. In this paper we study in which way the qualitative uncertainty principle is modified when we consider only functions f∈L²(G) which generate a Gabor frame associated with a uniform lattice K in G. This provides us with sharp lower bounds for the measure of the support of such functions and their Plancherel transforms.     

Ambiguity functions, Wigner distributions and Cohen's class for LCA groups

In this paper we construct a general class of time-frequency representations for LCA groups which parallel Cohen's class for the real line. For this, we generalize the notion of ambiguity function and Wigner distribution to the setting of general LCA groups in such a way that the Plancherel transform of the ambiguity function coincides with the Wigner distribution. Furthermore, properties of the general ambiguity function and Wigner distribution are studied. In detail we characterize those groups whose ambiguity functions and Wigner distributions vanish at infinity or are square-integrable. Finally, we explicitly construct Cohen's class for the group of p-adic numbers, p prime.     

Lehmer's problem for compact Abelian groups

We formulate Lehmer's Problem concerning the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components, then its Lehmer constant vanishes.

     

The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj,βj∈Aut(X), j=1,2,…,n, n?2, such that for all i≠j. Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L₂=β₁ξ₁+?+βnξn given L₁=α₁ξ₁+?+αnξn is symmetric, then each μj=γj∗ρj, where γj are Gaussian measures, and ρj are distributions supported in G.     

Measures with natural spectra on locally compact abelian groups

Every bounded regular Borel measure on noncompact LCA groups is a sum of an absolutely continuous measure and a measure with natural spectrum. The set of bounded regular Borel measures with natural spectrum on a nondiscrete LCA group whose Fourier-Stieltjes transforms vanish at infinity is closed under addition if and only if is compact.

     

Henstock type integral in harmonic analysis on zero-dimensional groups

A Henstock type integral is defined on compact subsets of a locally compact zero-dimensional abelian group. This integral is applied to obtain an inversion formula for the multiplicative integral transform.     

Weighted L p -algebras on groups

The space L _p (G), 1 > p < ∞, on a locally compact group G is known to be closed under convolution only if G is compact. However, the weighted spaces L _p (G, w) are Banach algebras with respect to convolution and natural norm under certain conditions on the weight. In the present paper, sufficient conditions for a weight defining a convolution algebra are stated in general form. These conditions are well known in some special cases. The spectrum (the maximal ideal space) of the algebra L _p (G,w) on an Abelian group G is described. It is shown that all algebras of this type are semisimple.     

Uncertainty principles for locally compact quantum groups

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.     

On a characterization theorem for locally compact abelian groups

The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish.     

Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group

Let $G$ be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group $\mathbb{H}\left(G\right)$ to be frames and Riesz bases in terms of the group Fourier transform.     

Galois groups of quantum group actions and regularity of fixed-point algebras

It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.

     

Extension and separation properties of positive definite functions on locally compact groups

Continuing earlier work, we investigate two related aspects of the set of continuous positive definite functions on a locally compact group . The first one is the problem of when, for a closed subgroup of , every function in extends to some function in . The second one is the question whether elements in can be separated from by functions in which are identically one on .

     

Spectral analysis for convolution operators on locally compact groups

We consider operators H_μ of convolution with measures μ on locally compact groups. We characterize the spectrum of H_μ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of H_μ, respectively. The proofs rely on commutator methods.     

Qualitative uncertainty principles for groups with finite dimensional irreducible representations

Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations.     

The structure of shift invariant spaces on a locally compact abelian group

We investigate shift invariant subspaces of L²(G), where G is a locally compact abelian group. We show, among other things, that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups

Let H be a locally compact group and K be a locally compact abelian group. Also let G=H× _τ K denote the semidirect product group of H and K, respectively. Then the unitary representation (U,L ²(K)) on G defined by is called the quasi regular representation. The properties of this representation in the case K=(ℝ ⁿ ,+), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i) , and (ii) the stabilizers H ^ω are compact for a.e. ; both are necessary for square integrability of U. Furthermore, we shall consider some sufficient conditions for the square integrability of U. Also, for the square integrability of subrepresentations of U, we will introduce a concrete form of the Duflo-Moore operator.      

Fixed point property and normal structure for Banach spaces associated to locally compact groups

In this paper we investigate when various Banach spaces associated to a locally compact group have the fixed point property for nonexpansive mappings or normal structure. We give sufficient conditions and some necessary conditions about for the Fourier and Fourier-Stieltjes algebras to have the fixed point property. We also show that if a -algebra has the fixed point property then for any normal element of , the spectrum is countable and that the group -algebra has weak normal structure if and only if is finite.