Topologically invariant linear forms on spaces of abstract distributions on a locally compact abelian group

1992 ◽  
Vol 114 (1) ◽  
pp. 77-82
Author(s):  
Rodney Nillsen
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Linear Forms

scholarly journals THE SKITOVICH–DARMOIS THEOREM FOR LOCALLY COMPACT ABELIAN GROUPS

2010 ◽  
Vol 88 (3) ◽  
pp. 339-352 ◽  
Author(s):  
GENNADIY FELDMAN ◽  
PIOTR GRACZYK
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Wide Class ◽  
Random Variables ◽  
Independent Random Variables ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Linear Forms ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

AbstractAccording to the Skitovich–Darmois theorem, the independence of two linear forms of n independent random variables implies that the random variables are Gaussian. We consider the case where independent random variables take values in a second countable locally compact abelian group X, and coefficients of the forms are topological automorphisms of X. We describe a wide class of groups X for which a group-theoretic analogue of the Skitovich–Darmois theorem holds true when n=2.

scholarly journals On a group analogue of the Heyde theorem

2020 ◽  
Vol 32 (2) ◽  
pp. 307-318 ◽  
Author(s):  
Margaryta Myronyuk
Keyword(s):  
Abelian Group ◽  
Present Article ◽  
Gaussian Distribution ◽  
Compact Abelian Group ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Linear Forms

AbstractHeyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analogue of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

scholarly journals A singular convolution kernel without pseudo-periods

1981 ◽  
Vol 83 ◽  
pp. 1-4
Author(s):  
Jesper Laub
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Convolution Kernel ◽  
Locally Compact ◽  
Excessive Measures

Let G be a locally compact abelian group and N a non-zero convolution kernel on G satisfying the domination principle. We define the cone of N-excessive measures E(N) to be the set of positive measures ξ for which N satisfies the relative domination principle with respect to ξ. For ξ ∈ E(N) and Ω ⊆ G open the reduced measure of ξ over Ω is defined as.

scholarly journals Pure point/continuous decomposition of translation-bounded measures and diffraction

2018 ◽  
Vol 40 (2) ◽  
pp. 309-352
Author(s):  
JEAN-BAPTISTE AUJOGUE
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Diffraction Theory ◽  
Natural Generalization ◽  
Pure Point ◽  
Locally Compact Abelian Group ◽  
Uniqueness Property ◽  
Locally Compact ◽  
Continuous Decomposition

In this work we consider translation-bounded measures over a locally compact Abelian group$\mathbb{G}$, with a particular interest in their so-called diffraction. Given such a measure$\unicode[STIX]{x1D714}$, its diffraction$\widehat{\unicode[STIX]{x1D6FE}}$is another measure on the Pontryagin dual$\widehat{\mathbb{G}}$, whose decomposition into the sum$\widehat{\unicode[STIX]{x1D6FE}}=\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}+\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$of its atomic and continuous parts is central in diffraction theory. The problem we address here is whether the above decomposition of$\widehat{\unicode[STIX]{x1D6FE}}$lifts to$\unicode[STIX]{x1D714}$itself, that is to say, whether there exists a decomposition$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}+\unicode[STIX]{x1D714}_{\text{c}}$, where$\unicode[STIX]{x1D714}_{\text{p}}$and$\unicode[STIX]{x1D714}_{\text{c}}$are translation-bounded measures having diffraction$\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}$and$\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$, respectively. Our main result here is the almost sure existence, in a sense to be made precise, of such a decomposition. It will also be proved that a certain uniqueness property holds for the above decomposition. Next, we will be interested in the situation where translation-bounded measures are weighted Meyer sets. In this context, it will be shown that the decomposition, whether it exists, also consists of weighted Meyer sets. We complete this work by discussing a natural generalization of the considered problem.

Sign in / Sign up

Export Citation Format

Share Document