Power-Rich and Power-Deficient LCA Groups

1981 ◽  
Vol 33 (3) ◽  
pp. 664-670 ◽  
Author(s):  
M. A. Khan
Keyword(s):  
Haar Measure ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Open Neighbourhood ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Lca Group ◽  
Lca Groups ◽  
Dual Power

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.

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

2021 ◽  
Vol 13 ◽  
Author(s):  
Pavol Jan Zlatos
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Nonstandard Analysis ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Lca Groups

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

On the Upper Majorant Property for Locally Compact Abelian Groups

1978 ◽  
Vol 30 (5) ◽  
pp. 915-925
Author(s):  
M. Rains
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Let G be a compact abelian group and form the spaces LP(G) with respect to the normalized Haar measure on G.

On the Approximation of Functions on Locally Compact Abelian Groups

1999 ◽  
Vol 6 (4) ◽  
pp. 379-394
Author(s):  
D. Ugulava
Keyword(s):  
Haar Measure ◽  
Exponential Type ◽  
Entire Functions ◽  
Abelian Groups ◽  
Approximation Of Functions ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Hausdorff Group

Abstract Questions of approximative nature are considered for a space of functions 𝐿𝑝(𝐺, μ), 1 ≤ 𝑝 ≤ ∞, defined on a locally compact abelian Hausdorff group 𝐺 with Haar measure μ. The approximating subspaces which are analogs of the space of exponential type entire functions are introduced.

scholarly journals New criteria for equivalence of locally compact abelian groups

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Osamu Hatori
Keyword(s):  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Surjective Isometry ◽  
Compact Abelian Groups ◽  
Invertible Elements ◽  
Lca Groups ◽  
New Criteria

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.

Scaling sets and generalized scaling sets on Cantor dyadic group

2020 ◽  
Vol 18 (04) ◽  
pp. 2050019
Author(s):  
Prasadini Mahapatra ◽  
Divya Singh
Keyword(s):  
Abelian Groups ◽  
Real Case ◽  
Locally Compact ◽  
Wavelet Sets ◽  
Locally Compact Abelian Groups ◽  
Dyadic Group ◽  
Compact Abelian Groups ◽  
Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

Singular measures with absolutely continuous convolution squares

1966 ◽  
Vol 62 (3) ◽  
pp. 399-420 ◽  
Author(s):  
Edwin Hewitt ◽  
Herbert S. Zuckerman
Keyword(s):  
Abelian Groups ◽  
Natural Order ◽  
Continuous Measure ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Singular Measures ◽  
Compact Abelian Groups ◽  
Nikodym Derivative

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such thatfor every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

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