scholarly journals Gordon's conjectures 1 and 2: Pontryagin-van Kampen duality in the hyperfinite setting

2021 ◽  
Vol 13 ◽  
Author(s):  
Pavol Jan Zlatos

Using the ideas of E. I. Gordon we present and farther advancean approach, based on nonstandard analysis, to simultaneousapproximations of locally compact abelian groups and their dualsby (hyper)finite abelian groups, as well as to approximations ofvarious types of Fourier transforms on them by the discrete Fouriertransform. Combining some methods of nonstandard analysis andadditive combinatorics we prove the three Gordon's Conjectureswhich were open since 1991 and are crucial both in the formulationsand proofs of the LCA groups and Fourier transform approximationtheorems

2016 ◽  
Vol 15 (04) ◽  
pp. 1650074 ◽  
Author(s):  
Przemysław Górka ◽  
Tomasz Kostrzewa

In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of [Formula: see text] in terms of the Fourier transform.


2014 ◽  
Vol 13 (04) ◽  
pp. 1350143 ◽  
Author(s):  
PRZEMYSłAW GÓRKA

In this paper, we show the version of Pego's theorem on locally compact abelian groups. This theorem, [R. L. Pego, Compactness in L2 and the Fourier transform, Proc. Amer. Math. Soc.95 (1985) 252–254], gives a characterization of precompact sets of L2 in terms of the Fourier transform.


2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Osamu Hatori

Abstract.Suppose that there exists a surjective isometry between open subgroups of the groups of invertible elements in the measure algebras on LCA groups. We show that the LCA groups are topologically isomorphic to each other.


1981 ◽  
Vol 33 (3) ◽  
pp. 664-670 ◽  
Author(s):  
M. A. Khan

In [4], Edwin Hewitt denned a-rich LCA (i.e., locally compact abelian) groups and classified them by their algebraic structure. In this paper, we study LCA groups with some properties related to a-richness. We define an LCA group G to be power-rich if for every open neighbourhood V of the identity in G and for every integer n > 1, λ(nV) > 0, where nV = {nx ∈ G : x ∈ V} and λ is a Haar measure on G. G is power-meagre if for every integer n > 1, there is an open neighbourhood V of the identity, possibly depending on n, such that λ(nV) = 0. G is power-deficient if for every integer n > 1 and for every open neighbourhood V of the identity such that is compact, . G is dual power-rich if both G and Ĝ are power-rich. We define dual power-meagre and dual power-deficient groups similarly.


2011 ◽  
Vol 54 (3) ◽  
pp. 544-555 ◽  
Author(s):  
Nicolae Strungaru

AbstractIn this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups.


2021 ◽  
Vol 71 (2) ◽  
pp. 369-382
Author(s):  
Seyyed Mohammad Tabatabaie ◽  
AliReza Bagheri Salec

Abstract In this paper, we study convolution operators on an Orlicz space L Φ(G) commuting with left translations, where Φ is an N-function and G is a locally compact group. We also present some basic properties of the Fourier transform of a Φ-convolution operator in the context of locally compact abelian groups.


2015 ◽  
Vol 24 (13) ◽  
pp. 1541002 ◽  
Author(s):  
Rinat M. Kashaev

The combinatorial structure of Pachner moves in four dimensions is analyzed in the case of a distinguished move of the type (3,3) and few examples of solutions are reviewed. In particular, solutions associated to Pontryagin self-dual locally compact abelian groups are characterized with remarkable symmetry properties which, in the case of finite abelian groups, give rise to a simple model of combinatorial TQFT with corners in four dimensions.


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