Stationary processes with pure point diffraction

Details Stationary processes with pure point diffraction Stationary processes with pure point diffraction

2011-11-15

arxiv.org/abs/1111.3617v1

Physics

Mathematical Physics

56 pages, 1 figure

Scientific paper

We consider the construction and classification of some new mathematical objects, called ergodic spatial stationary processes, on locally compact Abelian groups, which 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 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 a solution to the homometry problem for pure point diffraction. An ergodic spatial stationary process consists of a measure theoretical dynamical system and a mapping linking it with the ambient space in which diffracting density is supposed to exist. After introducing these processes we study their general properties and link pure point diffraction to almost periodicity. Given a pure point measure we show how to construct from it and a given set of phases a corresponding ergodic spatial stationary process. In fact we do this in two separate ways, each of which sheds its own light on the nature of the problem. The first construction can be seen as an elaboration of the Halmos--von Neumann theorem, lifted from the domain of dynamical systems to that of stationary processes. The second is a Gelfand construction obtained by defining a suitable Banach algebra out of the putative eigenfunctions of the desired dynamics.

Affiliated with

Also associated with

No associations

Rating

Stationary processes with pure point diffraction does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Stationary processes with pure point diffraction, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Stationary processes with pure point diffraction will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-709949