AN HARMONIC ANALYSIS FOR OPERATORS IN THE CASE OF A LOCALLY COMPACT ABELIAN GROUP: F. AND M. RIESZ THEOREMS

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)).

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A Factorization Theorem for the Fourier Transform of a Separable Locally Compact Abelian Group

Special Functions: Group Theoretical Aspects and Applications ◽

10.1007/978-94-010-9787-1_7 ◽

1984 ◽

pp. 261-269 ◽

Cited By ~ 5

Author(s):

Louis Auslander

Keyword(s):

Fourier Transform ◽

Abelian Group ◽

Compact Abelian Group ◽

Factorization Theorem ◽

Locally Compact Abelian Group ◽

Locally Compact ◽

The Fourier Transform

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A singular convolution kernel without pseudo-periods

Nagoya Mathematical Journal ◽

10.1017/s0027763000019383 ◽

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.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.38 ◽

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.

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When are dense subgroups of LCA groups closed under intersections?

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700016087 ◽

1994 ◽

Vol 49 (1) ◽

pp. 59-67

Author(s):

M.A. Khan

Keyword(s):

Abelian Group ◽

Positive Integer ◽

Compact Abelian Group ◽

Additive Group ◽

Locally Compact Abelian Group ◽

Torsion Subgroup ◽

Dense Subgroup ◽

Locally Compact ◽

Dense Subgroups ◽

Lca Groups

Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:(1) The intersection of any two dense subgroups of G is dense in G.(2) The intersection of all dense subgroups of G is dense in G.(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

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A Theorem on Algebras of Measures on Topological Groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500021908 ◽

1959 ◽

Vol 11 (4) ◽

pp. 195-206 ◽

Cited By ~ 13

Author(s):

J. H. Williamson

Keyword(s):

Banach Space ◽

Abelian Group ◽

Compact Abelian Group ◽

Locally Compact Abelian Group ◽

Topological Groups ◽

Finite Sets ◽

Locally Compact ◽

Borel Measures ◽

Bounded Complex ◽

Image Position

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined bythe supremum being over all finite sets of disjoint Borel subsets of G.

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