Stationary processes and pure point diffraction

2016 ◽  
Vol 37 (8) ◽  
pp. 2597-2642 ◽  
Author(s):  
DANIEL LENZ ◽  
ROBERT V. MOODY
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
General Setting ◽  
Pure Point ◽  
Stationary Processes ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Point Measure

We consider the construction and classification of some new mathematical objects, called ergodic spatial stationary processes, on locally compact abelian groups. These objects provide a natural and very general setting for studying diffraction and the famous inverse problems associated with it. In particular, we can construct complete families of solutions to the inverse problem from any given positive pure point measure that is chosen to be the diffraction. In this case these processes can be classified by the dual of the group of relators based on the set of Bragg peaks, and this gives an abstract solution to the homometry problem for pure point diffraction.

Embeddings of locally compact abelian p-groups in Hawaiian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanga Bavuma ◽  
Francesco G. Russo
Keyword(s):  
Algebraic Topology ◽  
Geometric Interpretation ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Hawaiian Earring ◽  
Path Connected

Abstract We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

Orderings in locally compact Abelian groups and the theorem of F. and M. Riesz

1983 ◽  
Vol 93 (3) ◽  
pp. 441-457 ◽  
Author(s):  
Edwin Hewitt ◽  
Shozo Koshi
Keyword(s):  
Harmonic Analysis ◽  
Abelian Groups ◽  
Complete Classification ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Background (1·1). Ordered Abelian groups have been studied for nearly a century. Since the early 1950's, it has been recognized that orderings in locally compact Abelian groups can play an important rôle in harmonic analysis on such groups. In this paper we study orderings, especially in topological Abelian groups with either topological or measure-theoretic properties, obtaining nearly a complete classification of such orderings. We then apply these results to determine the limitations of the celebrated theorem of F. and M. Riesz on such groups.

Positive Definite Measures with Discrete Fourier Transform and Pure Point Diffraction

2011 ◽  
Vol 54 (3) ◽  
pp. 544-555 ◽  
Author(s):  
Nicolae Strungaru
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Abelian Groups ◽  
Positive Definite ◽  
Pure Point ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

AbstractIn this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups.

scholarly journals On the Existence of an Extremal Function in the Delsarte Extremal Problem

2020 ◽  
Vol 17 (6) ◽  
Author(s):  
Marcell Gaál ◽  
Zsuzsanna Nagy-Csiha
Keyword(s):  
Extremal Problem ◽  
Extremal Function ◽  
General Setting ◽  
Function Class ◽  
Continuous Functions ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Extremal Function

AbstractThis paper is concerned with a Delsarte-type extremal problem. Denote by $${\mathcal {P}}(G)$$ P ( G ) the set of positive definite continuous functions on a locally compact abelian group G. We consider the function class, which was originally introduced by Gorbachev, $$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$ G ( W , Q ) G = f ∈ P ( G ) ∩ L 1 ( G ) : f ( 0 ) = 1 , supp f + ⊆ W , supp f ^ ⊆ Q where $$W\subseteq G$$ W ⊆ G is closed and of finite Haar measure and $$Q\subseteq {\widehat{G}}$$ Q ⊆ G ^ is compact. We also consider the related Delsarte-type problem of finding the extremal quantity $$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$ D ( W , Q ) G = sup ∫ G f ( g ) d λ G ( g ) : f ∈ G ( W , Q ) G . The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem $${\mathcal {D}}(W,Q)_G$$ D ( W , Q ) G . The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where $$G={\mathbb {R}}^d$$ G = R d . So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.

scholarly journals Integrability of Fourier transforms under an ergodic hypothesis

1978 ◽  
Vol 26 (2) ◽  
pp. 129-153
Author(s):  
Gavin Brown
Keyword(s):  
Fourier Transforms ◽  
General Setting ◽  
Locally Compact ◽  
Ergodic Hypothesis ◽  
Locally Compact Abelian Groups ◽  
Discrete Random Variables ◽  
Ergodic Condition ◽  
Compact Abelian Groups ◽  
Plancherel’S Theorem ◽  
Underlying Theme

AbstractThe object is to unify and complement some recent theorems of Hewitt and Ritter on the integrability of Fourier transforms, but the underlying theme is the ancient one that Plancherel's theorem is the “only” integrability constraint on Fourier transforms. The distinguishing feature of the results is that we restrict attention to positive measures (or functions) which satisfy an ergodic condition and whose transforms are positive. (In fact we employ sums of discrete random variables, a technique which seems to have been largely ignored in context.) The general setting is that of locally compact abelian groups but we are chiefly interested in the line or the circle, and it appears that the theorems are new for these classical groups.

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.

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