On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Saturation on locally compact abelian groups

Let G be an arbitrary locally compact abelian group. It is the purpose of the present paper to establish saturation theorems for approximation processes generated by families (_t)_{t > 0} of complex bounded Radon measures on G and operating on a submodule of the Banach module L^p(G), L^p(G), over the convolution algebra. A basic tool is the Fourier transform method and, in the case p>1 for noncompact G, its interpretation in the context of the theory of quasimeasures on G.     

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.     

Frame multiresolution analysis and infinite trees in Banach spaces on locally compact abelian groups

We extend the concept of frame multiresolution analysis to a locally compact abelian group and use it to define certain weighted Banach spaces and the spaces of their antifunctionals. We define analysis and synthesis operators on these spaces and establish the continuity of their composition. Also, we prove a general result to characterize infinite trees in the above Banach spaces of antifunctionals. This paper paves the way for the study of corresponding problems associated with some other types of Banach spaceson locally compact abelian groups including modulation spaces.     

On a characterization of probability distributions on locally compact abelian groups

Summary In one of his recent papers, I. J.Kotlarski has proved the following result. IfX ₁,X ₂,X ₃ are three independent real random variables and if the characteristic function of the pair (Z ₁,Z ₂) whereZ ₁=X ₁-X ₂,Z ₂=X ₁-X ₃ does not vanish, then the distribution of (Z ₁,Z ₂) determines the distributions ofX ₁,X ₂,X ₃ up to a change of location. He extended this result to random variables taking values in a Hilbert space in another paper. Our aim in this paper is to extend the above result to probability distributions on locally compact abelian groups.     

On some algebraic properties of locally compact and weakly linearly compact topological Abelian groups

Locally compact and weakly linearly compact topological groups are studied. The notion of a weakly linearly compact topological Abelian group is a generalization of the notion of a weakly separable topological Abelian group, introduced by N. Ya. Vilenkin. Some algebraic properties of these groups are studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 41, Topology and Its Applications, 2006.     

Symplectic structures on locally compact abelian groups and polarizations

LetX be a locally compact abelian group and ω(.,.) a symplectic structure on it. A polarization for (X, ω) is a pair of totally isotropic closed subgroupsG, G* ofX such thatX =G.G* and ω(.,.) defines a dual pairing ofG andG*. In this paper we describe a class of such groups which always admit a polarization and also discuss their structure. Dedicated to the memory of Professor K G Ramanathan     

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.

     

Some equivalent multiresolution conditions on locally compact abelian groups

Conditions under which a function generates a multiresolution analysis are investigated. The definition of the spectral function of a shift invariant space is generalized from ℝ ⁿ to a locally compact abelian group and the union density and intersection triviality properties of a multiresolution analysis are characterized in terms of the spectral functions. Finally, all multiresolution analysis conditions are characterized in terms of the scaling and the spectral functions.     

Countably infinite quasi-convex sets in some locally compact abelian groups

We produce a class of countably infinite quasi-convex sets (sequences converging to zero) in the circle group T and in the group J₂ of 2-adic integers determined by sequences of integers satisfying a mild lacunarity condition. We also extend our results to the group R of real numbers. All these quasi-convex sets have a stronger property: Every infinite (necessarily) symmetric subset containing 0 is still quasi-convex.     

A topological Paley-Wiener property for locally compact groups

We investigate a certain topological Paley-Wiener property and show, for instance, that compact-free nilpotent groups and simply connected solvable groups share this property.