On Heyde’s Theorem for Probability Distributions on a Discrete Abelian Group

2018 ◽  
Vol 97 (1) ◽  
pp. 11-14
Author(s):  
G. M. Feldman
Keyword(s):  
Abelian Group ◽  
Probability Distributions ◽  
Discrete Abelian Group

scholarly journals HEYDE’S CHARACTERIZATION THEOREM FOR DISCRETE ABELIAN GROUPS

2010 ◽  
Vol 88 (1) ◽  
pp. 93-102 ◽  
Author(s):  
MARGARYTA MYRONYUK
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Random Variables ◽  
Characterization Theorem ◽  
Gaussian Distributions ◽  
Discrete Abelian Group

AbstractLet X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

scholarly journals The Spectral Radius Formula for Fourier–Stieltjes Algebras

2019 ◽  
Vol 63 (2) ◽  
pp. 269-275
Author(s):  
Przemysław Ohrysko ◽  
Maria Roginskaya
Keyword(s):  
Abelian Group ◽  
Spectral Radius ◽  
Haar Measure ◽  
Short Note ◽  
Measure Algebra ◽  
Stieltjes Transform ◽  
Locally Compact ◽  
Convolution Powers ◽  
Discrete Abelian Group ◽  
Spectral Radius Formula

AbstractIn this short note we first extend the validity of the spectral radius formula, obtained by M. Anoussis and G. Gatzouras, for Fourier–Stieltjes algebras. The second part is devoted to showing that, for the measure algebra on any locally compact non-discrete Abelian group, there are no non-trivial constraints among three quantities: the norm, the spectral radius, and the supremum of the Fourier–Stieltjes transform, even if we restrict our attention to measures with all convolution powers singular with respect to the Haar measure.

Continuous singular measures equivalent to their convolution squares

1976 ◽  
Vol 80 (2) ◽  
pp. 249-268 ◽  
Author(s):  
Gavin Brown ◽  
Edwin Hewitt
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Measure Algebra ◽  
Locally Compact ◽  
Character Group ◽  
Singular Measures ◽  
Continuous Measures ◽  
Discrete Abelian Group

Throughout this paper, G will denote a locally compact, non-discrete, Abelian group (subjected to various conditions) and X wi11 denote the character group of G. All terminology and notation are as in (7). The measure algebra M(G), as is known, is a very complicated entity. We address ourselves here to some novel peculiarities of the subspace Ms(G) of continuous measures in M(G) that are singular with respect to Haar measure λ.

scholarly journals Filter banks on discrete abelian groups

2018 ◽  
Vol 16 (04) ◽  
pp. 1850029 ◽  
Author(s):  
A. G. García ◽  
M. A. Hernández-Medina ◽  
G. Pérez-Villalón
Keyword(s):  
Abelian Group ◽  
Filter Bank ◽  
Filter Banks ◽  
Abelian Groups ◽  
Perfect Reconstruction ◽  
Frame Bounds ◽  
Theoretical Analyses ◽  
Discrete Abelian Group

In this work, we provide polyphase, modulation, and frame theoretical analyses of a filter bank on a discrete abelian group. Thus, multi-dimensional or cyclic filter banks as well as filter banks for signals in [Formula: see text] or [Formula: see text] spaces are studied in a unified way. We obtain perfect reconstruction conditions and the corresponding frame bounds in this particular context.

scholarly journals Spectral synthesis on discrete Abelian groups

2007 ◽  
Vol 143 (1) ◽  
pp. 103-120 ◽  
Author(s):  
M. LACZKOVICH ◽  
L. SZÉKELYHIDI
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Spectral Synthesis ◽  
Free Rank ◽  
Discrete Abelian Group ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractWe prove that spectral synthesis holds on a discrete Abelian group G if and only if the torsion free rank of G is finite.

scholarly journals The Relationship Between ϵ-Kronecker Sets and Sidon Sets

2016 ◽  
Vol 59 (3) ◽  
pp. 521-527 ◽  
Author(s):  
Kathryn Hare ◽  
L. Thomas Ramsey
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Dual Group ◽  
Sidon Sets ◽  
Sidon Set ◽  
Arithmetic Properties ◽  
The Fourier Transform ◽  
The Relationship ◽  
Discrete Abelian Group

AbstractA subset E of a discrete abelian group is called ϵ-Kronecker if all E-functions of modulus one can be approximated to within ϵ by characters. E is called a Sidon set if all bounded E-functions can be interpolated by the Fourier transform of measures on the dual group. As ϵ-Kronecker sets with ϵ < 2 possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.

ON GRADED -ALGEBRAS

2017 ◽  
Vol 97 (1) ◽  
pp. 127-132 ◽  
Author(s):  
IAIN RAEBURN
Keyword(s):  
Abelian Group ◽  
Dual Group ◽  
Graded Algebras ◽  
Discrete Abelian Group

We show that every topological grading of a $C^{\ast }$-algebra by a discrete abelian group is implemented by an action of the compact dual group.

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