stationary processes with pure point diffraction
Abstract.
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.
Outline
This paper is concerned with mathematics of diffraction. More specifically we are interested in the famous inverse problem for diffraction: given something that is putatively the diffraction of something, what are all the somethings that could have produced this diffraction.
Diffraction has been a mainstay in crystallography for almost a hundred years.
The discovery of aperiodic tilings and quasicrystals revived interest in the mathematics of diffraction, particularly since the types of diffraction images generated by such structures – essentially or exactly pure point diffraction together with symmetries not occurring in ordinary crystals –
had not been foreseen by mathematicians, crystallographers, or materials research scientists.
The diffraction of a structure is defined as the Fourier transform of its autocorrelation. The latter is a positivedefinite measure in the ambient space of the structure – typically , but mathematically any locally compact Abelian group – and the diffraction is then a centrally symmetric positive measure in the corresponding Fourier reciprocal space – either or generally as the case may be.
In this paper we examine three basic questions that arise from this theory. The first question asks whether every positive centrally symmetric measure is actually the diffraction of something. The second asks about the classification of the all various ‘somethings’ that have this given diffraction. The third asks what kinds of structures the ‘somethings’ can be. In the case of pure point diffraction (when the diffraction measure is a pure point measure) we can give a satisfactory answer to these problems.
We will be in the setting of locally compact Abelian groups. The difficulties of the questions, at least as we examine them here, have little to do with the generality of the setting. They are just as hard for . Indeed the second question is a very old one in crystallography, called the homometry problem and is part of the fundamental problem of diffraction theory, namely how to unravel the nature of a structure that has created a given diffraction pattern.
Our first question immediately raises the third question. Exactly what do we mean by a structure that can diffract? Typically diffractive structures are conceived of as discrete measures, for instance point measures describing the positions of atoms, vertices in tilings, etc., weighted by appropriate scattering strengths, or as continuous measures describing the (scattering) distribution of the material structure in space. More recently, starting with the work of Gouéré [17, 18] there has been a
shift to discussing the diffraction of certain point processes, which in effect are random point measures, and various distortions of these (see e.g. [2, 11, 30]). In this paper we are led to introduce new structures which we call a spatial stationary processes. We shall show that these always lead
to autocorrelation and diffraction measures and contain the typical theories of diffraction
discussed above
Let be a locally compact Abelian group and its dual group. We treat these groups in additive notation. We begin with an abstract notion of a stationary process for and then shift our attention to the more concrete notion of a spatial stationary process for (§2 and §3.3). As for terminology let us note that outside of the discussion of usual stochastic processes, the word ‘stochastic’ is not used in our discussion of processes and stationary processes.
A spatial stationary process basically consists of a invariant mapping
where is a probability space on which there is a measure preserving action of .
The idea here derives directly from the theory of stochastic point processes, and it is perhaps useful to discuss this briefly. We follow [23] here. A stochastic point process, say in , is a random variable on a probability space with values in the space of point measures on for which for all bounded subsets of . These are point measures and can be written in the form where can be a positive integer or infinity and the points are (not necessarily distinct) points of . Usually the space is far larger than needed. For one thing it allows points in to appear with multiplicities and for another it does not require any minimal separation between distinct points (a hard core condition) that would usually be imposed in crystallography or in tiling theories. We assume then that we are really only interested in some subset of permissible outcomes for the point process, and that almost surely the point process produces measures in . Typically we wish to be invariant under translation since the position of our point sets is not particularly relevant, and we assume that the law of the process (the probability measure induced on by the random variable) is invariant under translation (i.e. the process is stationary).
Now the point is this. If then we obtain a random variable on by , where is the measure indicated above. In effect can be viewed as a mapping from into mappings on . Under mild assumptions . As is often the case with random variables, we ignore the underlying probability space since we have everything we need from the probability space and the mapping . Then itself can be referred to as the stochastic point process.
In this paper is replaced by any locally compact Abelian group . Also is not assumed to have any particular form (e.g. a set of point sets, or a set of measures). We do not wish to assume any particular mathematical structure on the outcomes of the processes we are studying. Our point of view is that we will start with a measure in which we would like to show is the diffraction of something. But we do not know in advance what sort of mathematical objects could produce this diffraction (point sets, continuous distributions of density, Schwartz distributions, etc.). We derive information about only from the behaviour of the functions , , upon it. This is, physically, rather appropriate. In a physical situation one determines the structure of a physical object by measurements upon it and the results of measurements are ultimately all that one can know about it.
We are particularly interested in pure point diffraction. However, the concept of stationary processes seems interesting in its own right and we develop the theory of these a bit, independent of the question of pure point diffraction. It is even useful initially to ignore the fact that the resulting process is spatial, that is, it is supposed to result from some structure in . So at the beginning we often only require that we have a invariant mapping where is a Hilbert space with conjugation – we call these stationary processes. Assuming existence of second moments, we can then, for each such process, find an ‘autocorrelation measure’ on and a ‘diffraction measure’ on . The latter is always a centrally symmetric positive measure, and through it we can define continuous and pure point diffraction. We show how pure point and continuous diffraction are tied to notions of almost periodicity of . Also we show how to decompose a sum of spatial processes into subprocesses which are respectively pure point and continuous.
Coming to the pure point part of the paper, §6, we assume given a centrally symmetric positive pure point measure on , which can be expressed as a Fourier transform. We wish to create a probability space on which there is an ergodic action of and a continuous stationary process whose diffraction is .
Not surprisingly phase factors play a crucial role in this since it is phase information that is lost in the process of diffraction. There is a natural combinatorial type of group generated by the positions of the Bragg peaks. This group can be seen as an abeleanized homotopy group of an associated Cayley graph. It is the dual group of , or the dual group of a natural factor of this group, that classifies all the spatial processes on with diffraction , the classification being up to isomorphism or up to translational isomorphism respectively.
We call these phase factors phase forms. Background on phase forms is given in Section 7. In §10 we show how to use and a phase form to construct a spatial process whose diffraction is . Uniqueness is already shown in §9. How all these ingredients give a solution to the homometry problem is discussed in §11.
Our solution to the pure point problem is in some sense an elaboration of the famous Halmosvon Neumann result about realizing ergodic pure point dynamical systems in terms of the character theory of compact Abelian groups. There the objective was to classify pure pointedness at the level of the spectrum. In our case we wish to classify the diffraction, which amounts to not only knowing the spectrum but also the intensity of diffraction at each point of the spectrum. Our approach to the construction of a spatial process is to realize as the dual to the discrete group generated by the set of Bragg peaks, but at the same time to include all the phase information which derives from the Bragg intensities. In this context we mention the recent work of Robinson [46] on how to realize systems with a given group of eigenvalues via so called cutandproject schemes.
In §12 we take a closer look at the situation that the underlying group is compact. In this case all processes can be realized as measure processes under some weak assumption. We then specialize even further and in §13.3 discuss some results of Grünbaum and Moore on rational diffraction from one dimensional periodic structures. These pertain to the simple case that , but even so, the results are interesting and not easy to obtain.
Finally in §14 we sketch out a second approach to the construction of a spatial process, this time based on Gel’fand theory.
Summarizing:

Each pure point ergodic spatial process gives rise to a pure point diffraction measure and a phase form .

Spatial processes with the same pure point diffraction measure and the same phase form are naturally isomorphic up to translation.

The set of all possible phase forms associated with a given pure point measure form an Abelian group , and for each choice of , there is a ergodic spatial process with diffraction and phase form .
The paper has two main features, after the introduction of stationary and spatial processes: one is a development of a general theory of these types of processes and the other is a detailed study of pure point diffraction and examples of how it looks in special cases. Readers interested primarily in the pure point theory can skip sections on first reading if they so wish.
1. Notation
In this section we gather some notation used throughout.
. Let be a locally compact abelian group. We let denote the space of compactly supported complex valued continuous functions on . For compact , is the subspace of elements of whose support is in . For , is defined by . The translation action of on itself is denoted by : for all . This action determines, in the usual way, an action on . By we denote the (fixed) Haar measure on .
The convolution of two functions is defined to be the function given by
The dual group of is denoted by and the Fouriert ransform of an is denoted by i.e.
A measure on is called transformable if there exists a measure on with
for all . In this case is uniquely determined and called the Fourier transform of . In such a situation we call backward transformable. Unlike all other notation introduced in this section the concept of backward transformability is not standard. However, it is clearly very useful in our context as it is exactly the property shared by the diffraction measures we investigate.
A sequence of compact subsets of is called a van Hove sequence if for every compact
(1) 
Here, for compact , the “boundary” of is defined as
The existence of van Hove sequences for all compact locally compact Abelian groups is shown in [47, p. 249], see also Section 3.3 and Theorem (3.L) of [52, Appendix].
2. Stationary processes introduced
In this section we introduce our concept of stationary process on the locally compact Abelian group . To do so we first discuss some background on unitary representations. The concept of stationary process is somewhat more general than needed in the remainder of the paper. The reason for this is twofold: On the one hand this general treatment does not lead to more complicated proofs but rather makes proofs more transparent. On the other hand for a the study of mixed spectra this general concept may prove to be especially useful.
Let be a locally compact abelian group and let be a Hilbert space with
inner product . We assume linearity in the first variable
and conjugate linearity in the second.
A strongly continuous unitary representation of the group on is a map from into the bounded operators on such that is constant on for any , is the identity on , holds for all and is continuous for any . Then, obviously each is unitary (i.e. bijective and norm preserving).
An is then called an eigenfunction of with
eigenvalue if for every
. The closed subspace of spanned by all eigenfunctions is denoted by . The representation has pure point spectrum if , or equivalently if has an orthonormal basis consisting of eigenfunctions
By Stone’s theorem, compare [34, Sec. 36D], there exists a projectionvalued measure
with
(2) 
for all and , where is the measure on defined by . Then is called the spectral measure of . It is the unique measure on satisfying (2).
A mapping of a Hilbert space into itself is said to be a conjugation of if

is conjugatelinear and ;

for all , .
Definition 2.1.
Let be a locally compact abelian group. A stationary process
or stationary process on is a triple consisting of a Hilbert space with
conjugation, a strongly continuous unitary representation of on and a linear
map such that for all
. A stationary process is called ergodic if the eigenspace of for the eigenvalue is onedimensional.
The set of all stationary processes on forms a real vector space in the obvious way by taking linear combinations of mappings. There is also a canonical notion of isomorphism between stochastic processes: The processes and are called isomorphic if there exists an invertible unitary map which intertwines and and satisfies .
We will be interested in the spectral theory of stationary processes. An important notion is given next.
Definition 2.2.
A stationary process is said to have pure point spectrum if the representation of on has pure point spectrum.
Definition 2.3.
A stationary process is called a spatial stationary process if there is a probability space and an action such that is invariant under the action on through , with the usual inner product , and the action of is extended to an action on by for all , . The conjugation here is the natural one: complex conjugation of functions on . We write . The stationary spatial process is called full if the algebra generated by functions of the form , , , is dense in .
Remark 2.4.
In some sense the assumption of fullness is only a convention. More precisely, given any spatial stationary process we may create a full process with essentially the same properties by a variant of Gelfand construction. Details are given below in Theorem 5.4.
For the sake of economy, we have used the same notation, , for the action of on and the corresponding action of on . We often omit the word ‘stationary’ in the sequel, but we will always understand it to be in force.
The definition of stationary processes implicitly defines them as real processes: is real for all realvalued functions on in the sense that . In the case of spatial processes, all realvalued functions on are mapped by to realvalued functions on . One could relax the conditions to allow complexvalued processes, but we shall not do that here.
Definition 2.5.
Two stationary spatial processes and are spatially isomorphic if there is an invertible unitary mapping with

;

for all ;

for all and for all .
The map is then called spatial isomorphism between the processes.
The isomorphism class of is denoted by .
Remark 2.6.
(a) The first two points of the definition just say that the processes are isomorphic. The special requirement for spatial isomorphy is the third point. As shown in the next proposition, the spatial isomorphism are exactly those isomorphisms which are compatible with the module structure of in the sense that for all and .
(b) Our definition of spatial isomorphism parallels the notion of spectral isomorphism for measure preserving group actions on probability spaces. Our conditions on imply that spatial isomorphism implies conjugacy in the sense that there is a measure isomorphism between the Borel sets of and those of [53], Theorem 2.4. If in addition are complete separable metric spaces then this can be improved to being an isomorphism between and , that is a bijective intertwining mapping between two subsets of full measure in and respectively [53], Theorem 2.6.
Proposition 2.7.
Let and be spatially isomorphic processes with spatial isomorphy . Then, maps into and holds for all and . The corresponding statements hold for as well.
Proof.
It suffices to consider the statements on . By assumption we have that for the product belongs to and in fact equals (and similarly for ). By a simple limit argument, we then see that for any the equation must hold for any . This shows that the operator of multiplication by is defined on the whole of (as is onto). Furthermore, the graph of every multiplication operator can easily be seen to be closed. Thus, the closed graph theorem gives that the operator of multiplication by is a bounded operator on . This in turn yields that belongs to . ∎
3. Diffraction theory for stochastic processes
In this section we show how each stationary process with a second moment comes with a positive definite measure on called the autocorrelation, a positive measure on called the diffraction measure, and the diffractiontodynamics map . We then discuss how the autocorrelation comes about as a limit and characterize existence of a second moment.
3.1. Autocorrelation and diffraction
Let a stationary process be given.
We say that has a second moment (or is with second moment) if there is a measure on (necessarily unique if it exists) satisfying
(3) 
for all real valued .
Much of this paper is based on the assumption that the second moment measure exists.
Since is real, so is , and hence it is real valued
Proposition 3.1.
Let be a process. Then has a second moment if and only if there is a measure on satisfying
(4) 
for all . In the case that such a measure exists, it is unique and furthermore it is real (i.e. assigns real values to real valued functions), positive definite, and translation bounded.
Proof.
It suffices to restrict attention to realvalued and in as is invariant under a conjugation and hence both and are linear in and antilinear in . Let above. The assumption of stationarity of shows that is invariant under simultaneous translation of both of its two variables. Define the mapping
(5)  
For we define by . This change of variables defines a new measure via
Since , the translation invariance of translates to translation invariance of the second variable of . Thus for measurable sets we have for all . For fixed this is leads to a translational invariant measure on which is hence a multiple of Haar measure :
The mapping is a measure on , and this measure is the reduction of :
Now for ,
so finally
and is the desired measure, which we now denote by .
It is positive definite since . All positive definite measures are translation bounded [9]. Since is a real measure, one sees that is also a real measure. The denseness of the set of functions in shows that is unique.
Existence of a second moment immediately implies some continuity properties of .
Corollary 3.2.
Let be a stationary process with second moment. Let be given with . Then, for any compact , there exists a with
for all with support contained in . In particular the map is continuous (with respect to the norm on ). Also is continuous with respect to the sup norm on .
Proof.
For with support contained in the support of is contained in and . As , this easily gives the first statement. Now, the second statement follows by taking . Finally, for , . ∎
The continuity property of the preceding corollary allows one to extend the map . This is discussed next.
Corollary 3.3.
Let be a stationary process with second moment. The map can be uniquely extended to the vector space of all measurable functions on all of which vanish outside some compact set and are square integrable with respect to the Haar measure on . This extension (again denoted by ) is equivariant and satisfies
meaning in particular that the integral exists and is finite.
Proof.
It suffices to show that can be uniquely extended to for any open in with compact closure. Let such an be given.
Let be given. As, is open, we can then find a sequence in converging to in the sense of . By the previous corollary, we infer then that must be a Cauchysequence, whose limit does not depend on the choice of the approximating sequence . This shows that can be extended to .
By construction of and the definition of we have furthermore
Moreover, a direct application of CauchySchwartz inequality shows that converges to with respect to the supremum norm and has support contained in . This easily gives that
This finishes the proof. ∎
Remark 3.4.
Note that any bounded measurable function with compact support belongs to for any open containing the support. Thus, can in particular be extended to bounded measurable functions with compact support.
The above discussion shows that any stationary process with a second moment gives rise to a real positive definite measure . As is positive definite it is transformable i.e. its Fourier transform exists [9, 51], and since is a real measure, is centrally symmetric ( for all measurable sets ). The measures and lie at the heart of our investigation.
Definition 3.5.
Let be a stationary process with second moment. Then, is called the autocorrelation of the stationary process. Its Fourier transform is called the diffraction or diffraction measure of the stationary process.
Definition 3.6.
A stationary process with second moment is said to have pure point diffraction (resp. continuous diffraction) if the diffraction measure is a pure point measure (resp. continuous measure).
The definition of yields that it is a centrally symmetric positive measure and is the unique measure satisfying
(6) 
or equivalently, by linearizing this,
for all . Considering the Hilbert space with the corresponding inner product being written and using the definition of , we obtain for any ,
(7) 
This shows that the map is an isometry. We define an action of on by defining for all through
for all and all .
Remark 3.7.
We note that if is a stationary process on with second moment then for all , is another such process. The corresponding autocorrelation and diffraction measures are then scaled by .
3.2. The diffraction to dynamics map
Any process with a second moment comes with an isometric map from to . This allows one to transfer certain questions from to . Also, it means that for certain eigenvalues there are canonical eigenfunctions available. These topics are studied next.
Proposition 3.8.
Let be a stationary process with second moment. Then is dense in and there exists a unique isometric embedding
(8) 
with for each . This mapping is a map and furthermore, for all ,
Proof.
We begin with the denseness statement. Let be the closure of in . We wish to show that . As is a translation bounded measure, the space is dense in . Thus it suffices to show that belongs to . As the convolutions of elements from belong to as well, we infer that for all
for all . Now belongs to the set of continuous functions on vanishing at infinity. An application of the StoneWeierstrass theorem then shows that
for each . Fixing we can then use another application of the StoneWeierstrass theorem, to infer that
for all and . This yields as we intended to show.
The existence of now follows by defining it initially by for all , using the fact that it is an isometric mapping by (7), and then extending it to the closure which is .
As for the second statement, first we note that the action on is designed to make the mapping into a map, while is already a map, and second, for all , , from which
We finish using the denseness of the first part. ∎
Definition 3.9.
The map associated to a stationary process with second moment is called the diffraction to dynamics map.
The relevance of for spectral theoretic considerations comes from the following consequence of the previous proposition.
Theorem 3.10.
Let be a stationary process with second moment and the associated diffraction to dynamics map. Then, for any the spectral measure is given by
Proof.
As is a map and an isomorphism a short calculation gives
Thus, the (inverse) Fourier transform of is given by . By the characterization of the spectral measure in (2) the theorem follows. ∎
From the previous theorem it is clear that if has pure point spectrum then is discrete and the diffraction is pure point. However, it is not clear that pure point diffraction implies pure point spectrum but as we shall see, it is so: each implies the other in the case of spatial processes.
The diffraction to dynamics map connects pure point diffraction to eigenvectors of the action on . Later, we shall see that in the case of spatial processes, this allows us to make a complete correspondence between pure point diffraction and pure point dynamics.
For a pure point diffractive stationary processes with diffraction measure we define the set of its atoms by
The set is often called the Bragg spectrum of the process, and
its elements are sometimes called Bragg peaks
In the sequel we will often omit the brackets when dealing with one element sets of the form . In particular, we will set for .
Let be the characteristic function of the set i.e. is or according as equals or not. It is easy to see that these functions are the only possible eigenfunctions for our action of on . Define , , by . Then each is an eigenfunction in for the eigenvalue , in the sense that .
Lemma 3.11.
Let be a stationary process with second moment. Then, for all , , and and .
Proof.
Since is centrally symmetric, , which gives the first statement. For the second, using Prop. 3.8 we have . Then . ∎
We can use the preceding considerations to compute the map in the case of pure point diffraction.
Proposition 3.12.
Let be a stationary process with pure point diffraction and associated Bragg peaks . Let be the associated diffraction to dynamics map and , . Then
for all .
Proof.
By definition of we have . Obviously, and the claim follows. ∎
3.3. Spatial stationary processes: Twopoint correlation
In this section we show how the autocorrelation can be given a meaning that agrees with ‘classical’ twopoint correlation associated to stationary point processes.
Assume that we are given an ergodic spatial stationary process . As discussed in Corollary 3.3 we can assume that is defined on all measurable bounded functions with compact support. We also assume that has a countable basis of topology and is hence metrizable. Let be a van Hove sequence for (as discussed in Section 1). We assume that this is fixed once and for all.
Definition 3.13.
For we define the twopoint correlation or autocorrelation of at any as
whenever the limit exists.
This needs some comments: The limits in the definition are taken in the order indicated: first the inner and then the outer. is an open (or measurable) neighbourhood of in . The statement means that we take a nested descending sequence of such neighbourhoods, all within some fixed compact set , and that . We are using the notation to stand for the longer . The definition requires that we give meaning to as a measurable function on . This uses the extension of from to functions with compact support given in Corollary 3.3.
The intuition behind the definition is as follows. The twopoint correlation at for should look something like
where the right hand side arises by using the usual trick from van Hove sequences and the compactness of the support of . (That is, the difference of the two sides of the equation is due to the difference between and , which by the van Hove assumption is irrelevant in the limit.) Of course in our case is not a function of and the integrands do not make sense. But the inner integral on the righthand side is what should be and we can rewrite this ‘autocorrelation’ as
The term has no meaning. But Palm theory tells us how to go around this. We instead average over a small neighbourhood of , and this brings us to Definition 3.13.
Theorem 3.14.
Let be a locally compact group whose topology has a countable basis and let be an ergodic spatial stationary process on with second moment. The twopoint correlation of exists almost surely and its value at is .
Proof.
The function is measurable as a function of and the Birkhoff theorem says that almost surely
(9) 
meaning that the limit will exist and equal the right hand side. We shall prove that
(10) 
proving that the definition of Definition 3.13 works almost surely in for each and that does have the meaning of an autocorrelation.
This has to be made to work simultaneously for all in , which will be shown in the usual way from a countable basis of . This will prove Theorem 3.14.
Let and choose . Let be a measurable subset of with compact closure. Then by Prop. 3.3, is defined and is a measurable function on . By linearization we have
Since is uniformly continuous on , for any sufficiently small neighbourhood of , for all and for all . Then