scholarly journals A singular convolution kernel without pseudo-periods

1981 ◽  
Vol 83 ◽  
pp. 1-4
Author(s):  
Jesper Laub

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.

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta

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)).


2018 ◽  
Vol 40 (2) ◽  
pp. 309-352
Author(s):  
JEAN-BAPTISTE AUJOGUE

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.


1994 ◽  
Vol 49 (1) ◽  
pp. 59-67
Author(s):  
M.A. Khan

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.


1959 ◽  
Vol 11 (4) ◽  
pp. 195-206 ◽  
Author(s):  
J. H. Williamson

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.


1996 ◽  
Vol 120 (2) ◽  
pp. 322-344 ◽  
Author(s):  
W.W. Comfort ◽  
Salvador Hernández ◽  
F.Javier Trigos-Arrieta

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