# Diffraction as a unitary representation and the orthogonality of measures
with respect to the reflected Eberlein convolution

Daniel Lenz  Mathematisches Institut, Friedrich Schiller Universität Jena,
07743 Jena, Germany<EMAIL_ADDRESS>http://www.analysis-lenz.uni-
jena.de and  Nicolae Strungaru  Department of Mathematical Sciences, MacEwan
University
10700 – 104 Avenue, Edmonton, AB, T5J 4S2, Canada
and
Institute of Mathematics “Simon Stoilow”
Bucharest, Romania<EMAIL_ADDRESS>https://sites.google.com/macewan.ca/nicolae-strungaru/home

###### Abstract.

We discuss how the diffraction theory of a single translation bounded measure
or a family of such measures can be understood within the framework of unitary
group representations. This allows us to prove an orthogonality feature of
measures whose diffractions are mutually singular. We apply this to study
dynamical systems, the refined Eberlein decomposition and validity of a
Bombieri–Taylor type result in a rather general context. Along the way we also
use our approach to (re)prove various characterisations of pure point
diffraction.

## Introduction

This article is concerned with mathematical diffraction theory. A core object
in mathematical diffraction theory is the autocorrelation of a measure. This
autocorrelation is an averaged quantity. The theory can be developed on
arbitrary locally compact Abelian groups and this is how we will proceed
below. In the case of the integers the autocorrelation deals with means of the
form

$\lim_{N}\frac{1}{N}\sum_{k=1}^{N}f(k)\overline{f(k-j)}=:\gamma_{f}(j)$

for a bounded function $f$ on ${\mathbb{Z}}$ and $j\in{\mathbb{Z}}$.

As was observed recently, mathematical diffraction theory can conveniently be
phrased with the help of the reflected Eberlein convolution [20].
Specifically, the autocorrelation of a measure is the reflected Eberlein
convolution of the measure and itself. As was also noted in [20], this
reflected Eberlein convolution provides a certain inner product like structure
to the space of measures.

The starting point of this article is the realization that the reflected
Eberlein convolution is not only somewhat similar to an inner product, but
that one can rather construct a proper Hilbert space together with a unitary
representation of the underlying group out of the reflected Eberlein
convolution. This allows us to study orthogonality with respect to the
reflected Eberlein convolution. Our main result gives orthogonality of
measures when their diffractions are mutually singular. Having established the
unitary representation we obtain this result rather directly from Stone
theorem. For bounded functions $f,g$ on the integers the result gives that

$\lim_{N}\frac{1}{N}\sum_{k=1}^{N}f(k)\overline{g(k)}=0$

must necessarily hold whenever the diffraction measures $\widehat{\gamma_{f}}$
of $f$ and $\widehat{\gamma_{g}}$ of $g$ are mutually singular.

The main result a allows for a variety of applications. One application
concerns what is sometimes called the Bombieri–Taylor conjecture. Another
application concerns orthogonality of dynamical dynamical systems $(X,G,m)$
and $(X^{\prime},G,m^{\prime})$. If for such systems the spectral measures of
$f\in C(X)$ and $f^{\prime}\in C(X^{\prime})$ are mutually singular, then the
functions $t\to f(tx)$ and $t\mapsto f^{\prime}(tx^{\prime})$ are orthogonal
with respect to the reflected Eberlein convolution for almost surely all $x\in
X$ and $x^{\prime}\in X^{\prime}$. A third application concerns what is known
as refined Eberlein decomposition. Besides these applications of the
orthogonality result we can also use the underlying unitary representation to
(re)prove various characterizations of pure point diffraction. This gives in
particular a new and unifying perspective on results achieved during the last
twenty years.

The article is organized as follows: In Section 1 we present the basic setting
of locally compact Abelian groups and the associated notions needed in the
remaining part of the article. Section 2 then features the reflected Eberlein
convolution and its basic properties. The reflected Eberlein convolution is
defined via a limit and we discuss existence of this limit in Section 3. The
construction of the unitary representation, first out of functions, and, then
out of measures admitting a reflected Eberlein convolution is done in Section
4. The mentioned characterisation of pure point diffractions is then derived
in Section 5, while our main result on orthogonality is proven in Section 5.
The subsequent three sections then discuss the mentioned applications to
Bombieri–Taylor conjecture and to dynamical systems as well as an application
to the refined Eberlein decomposition. The final section is devoted to the
reflected Eberlein convolution with a Besicovitch almost periodic measure. We
characterize in particular those measures which have vanishing reflected
Eberlein convolution with all Besicovitch almost periodic measures.

##  1\. The setting

Throughout this paper $G$ is a locally compact Abelian group (LCAG). The group
operations are written additively and the neutral element is denoted by $0$.
Integration of $f$ with respect to Haar measure is denoted by
$\int_{G}f\mbox{d}s$. The Haar measure of a measurable subset $A$ of $G$ is
also denoted by $|A|$.

We denote by $C_{\mathsf{u}}(G)$ the space of uniformly continuous bounded
functions on $G$ and by $C_{\mathsf{c}}(G)$ the subspace of
$C_{\mathsf{u}}(G)$ consisting of compactly supported continuous functions.
The space $C_{\mathsf{u}}(G)$ is equipped with the supremum norm
$\|\cdot\|_{\infty}$ defined by $\|f\|_{\infty}:=\sup\\{|f(t)|:t\in G\\}$. The
support $\mbox{\rm supp}(\varphi)$ of $\varphi\in C_{\mathsf{c}}(G)$ is the
smallest compact set outside of which $\varphi$ vanishes. The spaces
$C_{\mathsf{u}}(G)$ and $C_{\mathsf{c}}(G)$ admit the following operators

$f^{\dagger}(x)=f(-x)\mbox{ and }\tilde{f}(x)=\overline{f(-x)}\,.$

Moreover, each $t\in G$ induces a translation operator on these spaces via

$\tau_{t}f(x)=f(x-t)\,.$

A _Radon measure_ on $G$ is a linear functional
$\mu:C_{\mathsf{c}}(G)\to{\mathbb{C}}$ with the property that for each compact
set $K\subseteq G$ there exists $C_{K}\geq 0$ such that all functions
$\varphi\in C_{\mathsf{c}}(G)$ whose support is contained in $K$ satisfy

$\left|\mu(\varphi)\right|\leq C_{K}\|\varphi\|_{\infty}.$

We will often write $\int_{G}\varphi(t)\mbox{d}\mu(t):=\mu(\varphi)$.

The operators ${\,}^{\dagger},\tilde{\,},\tau_{t}$ extend naturally to
measures via

$\mu^{\dagger}(\varphi)=\mu(\varphi^{\dagger})\,;\,\tilde{\mu}(\varphi)=\overline{\mu(\tilde{\varphi})}\,;\,(\tau_{t}\mu)(\varphi)=\mu(\tau_{-t}\varphi)\,.$

To any Radon measure $\mu$ there exists a unique positive measure $\nu$ and a
measurable $h:G\longrightarrow{\mathbb{C}}$ with $|h|=1$ such that

$\mu(\varphi)=\int\varphi h\mbox{d}\nu$

holds for all $\varphi\in C_{\mathsf{c}}(G)$ [24, Thm. 6.5.6]. The measure
$\nu$ is called the total variation of $\mu$ and henceforth denoted by
$|\mu|$. The measure $\mu$ is called finite if $|\mu|(G)<\infty$ holds.

Whenever $A$ is a Borel subset of $G$ we define the restriction $\mu|_{A}$ of
$\mu$ to $A$ to be the measure satisfying

$\mu|_{A}(\varphi):=\int_{A}\varphi h\mbox{d}|\mu|$

for all $\varphi\in C_{\mathsf{c}}(G)$.

The dual group $\widehat{G}$ of $G$ is the set of all continuous group
homomorphisms $\xi:G\longrightarrow\mathbb{T}$. Here, $\mathbb{T}$ is the
group of complex numbers with modulus $1$ (equipped with multiplication). The
dual group $\widehat{G}$ is a locally compact Abelian group in a natural way.
The Fourier transform $\widehat{f}$ of $f\in L^{1}(G)$ is the function on
$\widehat{G}$ with

$\widehat{f}(\xi)=\int_{G}\overline{\xi(t)}f(t)\mbox{d}t.$

Moreover, whenever $\sigma$ is a finite positive measure on $\widehat{G}$, we
can define the inverse Fourier transform $\widecheck{\sigma}$ of $\sigma$ as
the function on $G$ given by

$\widecheck{\sigma}(t)=\int_{\hat{G}}\xi(t)\mbox{d}\sigma(\xi)\,.$

The convolution $\varphi\ast\psi$ of $\varphi,\psi\in C_{\mathsf{c}}(G)$ is
the function on $G$ defined by

$\varphi\ast\psi(t):=\int_{G}\varphi(s)\psi(t-s)\mbox{d}s\,.$

The convolution $\mu\ast\varphi$ between a measure $\mu$ and a function
$\varphi\in C_{\mathsf{c}}(G)$ is the function on $G$ defined by

$\mu\ast\varphi(t)=\int_{G}\varphi(t-s)\mbox{d}\mu(s)=\mu((\tau_{t}\varphi)^{\dagger})\,.$

It is easy to see that

$\tau_{t}(\mu\ast\varphi)=(\mu\ast\tau_{t}\varphi)=(\tau_{t}\mu)\ast\varphi\,$

for all $t\in G$, $\varphi\in C_{\mathsf{c}}(G)$ and any measure $\mu$.

The convolution $\mu\ast\nu$ between two finite measures is the measure given
by

$\mu\ast\nu(\varphi)=\int_{G}\int_{G}\varphi(s+t)\mbox{d}\mu(s)\mbox{d}\nu(t)\qquad\forall\varphi\in
C_{\mathsf{c}}(G)\,.$

A measure $\gamma$ is called positive definite if

$\gamma\ast\varphi\ast\widetilde{\varphi}(0)\geq 0$

holds for all $\varphi\in C_{\mathsf{c}}(G)$. This is equivalent to
$\gamma\ast\varphi\ast\widetilde{\varphi}$ being a continuous positive
definite function for all $\varphi\in C_{\mathsf{c}}(G)$ [6].

Every positive definite measure $\gamma$ admits a (unique) positive measure
$\widehat{\gamma}$ on $\widehat{G}$ such that

$\gamma\ast\varphi\ast\widetilde{\varphi}(0)=\int_{\widehat{G}}|\widehat{\varphi}(\xi)|^{2}\mbox{d}\widehat{\gamma}(\xi)$

for all $\varphi\in C_{\mathsf{c}}(G)$. The measure $\widehat{\gamma}$ is
called the Fourier transform of $\gamma$ (see [1, 6, 23] for details). From
the definition and polarisation we easily find

$\gamma\ast\varphi\ast\widetilde{\psi}(0)=\int_{\widehat{G}}\widehat{\varphi}(\xi)\overline{\widehat{\psi}(\xi)}\mbox{d}\widehat{\gamma}(\xi)$

for all $\varphi,\psi\in C_{\mathsf{c}}(G)$. Applying this with
$\psi=\tau_{-t}\varphi$ we obtain the following statement.

######  Proposition 1.1.

Let $\gamma$ be a positive definite measure on $G$. Then,

$\gamma\ast\varphi\ast\widetilde{\varphi}(t)=\int_{\widehat{G}}\xi(t)|\widehat{\varphi}|^{2}(\xi)\mbox{d}\widehat{\gamma}(\xi)$

holds for all $t\in G$ and $\varphi\in C_{\mathsf{c}}(G)$. ∎

A measure $\mu$ is called _translation bounded_ if for one (any) non-empty
open $V$ with compact closure

$\|\mu\|_{V}:=\sup_{t\in G}|\mu|(t+V)<\infty$

holds. This is equivalent to $\mu\ast\varphi$ being bounded for all
$\varphi\in C_{\mathsf{c}}(G)$ [1]. Note that for a translation bounded $\mu$
the function $\mu\ast\varphi$ belongs to $C_{\mathsf{u}}(G)$ [1] (compare
appendix for further discussion of translation bounded measures).

Let us complete the section by briefly discussing $\sigma$-compactness,
metrisability and second countability of $G$.

######  Remark 1.2 (Second countable LCAG).

  * (a)

If $X$ is any locally compact Hausdorff space, and $B$ is any basis of open
sets for the topology, then the set

$B_{c}:=\\{U\in B:\bar{U}\mbox{ is compact }\\}$

is also a basis of open sets for the topology. This immediately implies that
any second countable locally compact Hausdorff space is $\sigma$-compact.

  * (b)

Any locally compact group is a normal topological space. Therefore, by the
Urysohn Metrisation Theorem, any second countable LCAG is metrisable.

  * (c)

If $G$ is a metrisable LCAG and $K\subseteq G$ is compact, it is easy to show
that there exists a countable dense set $D\subseteq G$. In particular, any
$\sigma$-compact and metrisable group has a countable dense subset. This
combined with metrisability gives that any $\sigma$-compact and metrisable
LCAG is second countable.

  * (d)

Points (a-c) imply that a LCAG $G$ is second countable if and only if $G$ is
$\sigma$-compact and metrisable.

  * (e)

By [26, Thm. 4.2.7] and Pontryagin duality (see [26, Thm. 4.2.11]) we have the
following equivalences:

    * –

$G$ is $\sigma$-compact if and only if $\widehat{G}$ is metrisable.

    * –

$G$ is metrisable if and only if $\widehat{G}$ is $\sigma$-compact.

  * (f)

By (d) and (e) the group $G$ is second countable if and only if $\widehat{G}$
is second countable.

##  2\. The reflected Eberlein convolution

In this section we discuss the reflected Eberlein convolution. This discussion
is essentially taken from [20]. However, as [20] makes the additional
assumptions that the group is $\sigma$-compact with countable basis of
topology some (small) adjustments are necessary. In order to ease the approach
for the reader we present a rather complete treatment in this section.

Let $A$ be a subset of $G$. Then, for each compact set $K\subseteq G$ the
$K$-boundary $\partial^{K}A$ of $A$ is defined as

$\partial^{K}A:=\bigl{(}\overline{A+K}\setminus
A\bigr{)}\cup\bigl{(}(\left(G\backslash A\right)-K)\cap\overline{A}\bigr{)}$

A net $(A_{i})_{i\in I}$ of open subsets of $G$ with compact closure is called
a van Hove net if

$\lim_{i}\frac{|\partial^{K}A_{i}|}{|A_{i}|}=0$

holds for any compact $K\subset G$. Note that every LCAG admits a van Hove net
(see for example [25, Prop. 5.10]).

With the help of a van Hove net we can define an averaged version of the
convolution as follows.

######  Definition 2.1 (Reflected Eberlein convolution of measures).

Let ${\mathcal{A}}$ be a van Hove net. Let translation bounded measures $\mu$
and $\nu$ be given. If the limit

$\lim_{i}\frac{1}{|A_{i}|}(\mu|_{A_{i}})\ast\widetilde{(\nu|_{A_{i}})}$

exists in the vague topology it is called the reflected Eberlein convolution
of $\mu$ and $\nu$ with respect to the van Hove net ${\mathcal{A}}$ and
denoted by $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$.

Let us now list the basic properties of the reflected Eberlein convolution of
measures.

######  Lemma 2.2.

Let $\mu,\nu$ be translation bounded measures and let ${\mathcal{A}}$ be a van
Hove net. Then, the following assertions hold:

  * (a)

There exists an index $i_{0}$ and a vaguely compact set
$X\subseteq{\mathcal{M}}^{\infty}(G)$ such that, for all $i\geq i_{0}$ we have
$\frac{1}{|A_{i}|}(\mu|_{A_{i}})\ast\widetilde{(\nu|_{A_{i}})}\in X$. In
particular, the net
$\frac{1}{|A_{i}|}(\mu|_{A_{i}})\ast\widetilde{(\nu|_{A_{i}})}$ always admits
a convergent subnet.

  * (b)

Assume that $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$
exists. Then,
$\text{\textlquill}\tau_{t}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists
for all $t\in G$ and

$\text{\textlquill}\tau_{t}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}=\text{\textlquill}\mu,\tau_{-t}\nu\text{\textrquill}_{{\mathcal{A}}}=\tau_{t}\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$

holds. In particular,
$\text{\textlquill}\tau_{t}\mu,\tau_{t}\mu\text{\textrquill}_{{\mathcal{A}}}=\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$
holds.

  * (c)

Assume that $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$
exists. Then, $\text{\textlquill}\nu,\mu\text{\textrquill}_{{\mathcal{A}}}$
exists and satisfies

$\text{\textlquill}\nu,\mu\text{\textrquill}_{{\mathcal{A}}}=\widetilde{\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}}\,.$

  * (d)

Assume that $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ and
$\text{\textlquill}\sigma,\nu\text{\textrquill}_{{\mathcal{A}}}$ exist. Then,
for all $a,b\in{\mathbb{C}}$ also
$\text{\textlquill}a\mu+b\sigma,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists
and

$\text{\textlquill}a\mu+b\sigma,\nu\text{\textrquill}_{{\mathcal{A}}}=a\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}+b\text{\textlquill}\sigma,\nu\text{\textrquill}_{{\mathcal{A}}}$

holds.

  * (e)

If $\text{\textlquill}\mu,\mu\text{\textrquill}_{{\mathcal{A}}}$ exists then
it is positive definite.

###### Proof.

(a) This is discussed in the appendix.

(b), (c), (d) are straightforward.

(e) The measure
$\frac{1}{|A_{i}|}(\mu|_{A_{i}})\ast\widetilde{(\mu|_{A_{i}})}$ is positive
definite as for any $\varphi\in C_{\mathsf{c}}(G)$ we find

$\frac{1}{|A_{i}|}(\mu|_{A_{i}})\ast\widetilde{(\mu|_{A_{i}})}\ast\varphi\ast\widetilde{\varphi}(0)=\frac{1}{|A_{i}|}\int|\psi(s)|^{2}\mbox{d}s\geq
0$

with $\psi:=\mu|_{A_{i}}\ast\varphi$. Taking the limit we then find the
desired statement (e). ∎

######  Definition 2.3 (Autocorrelation and diffraction of $\mu$).

Let $\mu$ be a translation bounded measure and let ${\mathcal{A}}$ be a van
Hove net. If it exists, the measure
$\text{\textlquill}\mu,\mu\text{\textrquill}_{{\mathcal{A}}}$ is called the
autocorrelation of $\mu$ and denoted by $\gamma_{\mu}$. Then, the positive
measure
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}$
is called the _diffraction measure of $\mu$_.

Let us also recall the concept of Fourier–Bohr coefficients.

######  Definition 2.4 (Fourier–Bohr coefficient).

Let $\mu$ be a translation bounded measure, let ${\mathcal{A}}$ be a van Hove
net and let $\chi\in\widehat{G}$. If it exists, the

$\lim_{i}\frac{1}{|A_{i}|}\int_{A_{i}}\overline{\chi(t)}\mbox{d}\mu(t)\,.$

is called the _Fourier–Bohr coefficient_ of $\mu$ and denoted by
$a_{\chi}^{\mathcal{A}}(\chi)$.

Fourier–Bohr coefficients can be expressed via the reflected Eberlein
convolution as follows. The proof is straightforward.

######  Proposition 2.5.

Let $\mu$ be a translation bounded measure, let ${\mathcal{A}}$ be a van Hove
net and let $\chi\in\widehat{G}$. Then, the Fourier–Bohr coefficient
$a_{\chi}^{\mathcal{A}}(\chi)$ exists if and only if
$\text{\textlquill}\mu,\chi\text{\textrquill}_{{\mathcal{A}}}$ exists.
Moreover, in this case
$\text{\textlquill}\mu,\chi\text{\textrquill}_{{\mathcal{A}}}$ is the
absolutely continuous measure with density function
$a_{\chi}^{\mathcal{A}}(\mu)\chi$. ∎

We now turn to functions and their means. This will provide a further
understanding of the Eberlein convolution.

######  Definition 2.6 (The mean $M_{\mathcal{A}}$).

Let ${\mathcal{A}}$ be a van Hove net and $f$ a bounded measurable function on
$G$. If the limit

$\lim_{i}\frac{1}{|A_{i}|}\int_{A_{i}}f(t)\mbox{d}t\,,$

exists, it is called the _mean_ of $f$ with respect to ${\mathcal{A}}$ and
denote by $M_{{\mathcal{A}}}(f)$.

Clearly, the mean is $G$-invariant, i.e. for all bounded measurable function
on $G$ such that $M_{{\mathcal{A}}}(f)$ exists, and all $t\in G$, the mean
$M_{{\mathcal{A}}}(\tau_{t})f$ exists and satisfies

$M_{{\mathcal{A}}}(\tau_{t}f)=M_{{\mathcal{A}}}(f)\,.$

Taking the mean is similar to integration with respect to a (probability)
measure. Accordingly, the mean allows one to define analogues of convolution.

######  Definition 2.7 (Reflected Eberlein convolution of functions).

Let ${\mathcal{A}}$ be a van Hove net. Let $f,g\in C_{\mathsf{u}}(G)$ be
given. If for each $t\in G$ the mean $M_{\mathcal{A}}(f\overline{\tau_{t}g})$
exists, the function

$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}:G\longrightarrow{\mathbb{C}},t\mapsto
M_{\mathcal{A}}(f\overline{\tau_{t}g}),$

is called the reflected Eberlein convolution of $f$ and $g$ with respect to
${\mathcal{A}}$.

We note that the reflected Eberlein convolution of $f$ and $g$ is given by

$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}(t)=\lim_{i}\frac{1}{|A_{i}|}\int_{A_{i}}f(s)\overline{g(s-t)}\mbox{d}s\,.$

Hence, the reflected Eberlein convolution can be understood as an averaged
version of the convolution of $f$ and $\tilde{g}$. The relationship between
the reflected Eberlein convolution of measures and of functions is discussed
in the subsequent lemma.

######  Lemma 2.8.

Let ${\mathcal{A}}$ be a van Hove net. Let $\mu,\nu$ be translation bounded
measures. Then, the following statements are equivalent:

  * (i)

The reflected Eberlein convolution
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists.

  * (ii)

For all $\varphi,\psi\in C_{\mathsf{c}}(G)$ the reflected Eberlein convolution
$\text{\textlquill}\mu\ast\varphi,\nu\ast\psi\text{\textrquill}_{{\mathcal{A}}}$
exists.

  * (iii)

For all $\varphi,\psi\in C_{\mathsf{c}}(G)$ the mean
$M_{{\mathcal{A}}}(\mu\ast\varphi\,\cdot\,\overline{\nu\ast\psi})$ exists.

If one of the equivalent conditions (i), (ii) and (iii) is valid the
equalities

$\text{\textlquill}\mu\ast\varphi,\nu\ast\psi\text{\textrquill}_{{\mathcal{A}}}(t)=\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}\ast\varphi\ast\tilde{\psi}(t)=M_{{\mathcal{A}}}(\mu\ast\varphi\;\overline{\nu\ast\tau_{t}\psi})$

hold for all $\varphi,\psi\in C_{\mathsf{c}}(G)$ and all $t\in G$.

###### Proof.

We first discuss the equivalence between (ii) and (iii): By definition we have

$\text{\textlquill}\mu\ast\varphi,\nu\ast\psi\text{\textrquill}_{{\mathcal{A}}}(t)=M_{{\mathcal{A}}}(\mu\ast\varphi\;\overline{\tau_{t}(\nu\ast\psi}))=M_{{\mathcal{A}}}(\mu\ast\varphi\overline{\nu\ast\tau_{t}\psi})$

(if the corresponding limits exist). This easily gives the desired
equivalence. We now turn to proving the equivalence between (i) and (iii).
Along the way we will establish the given formulae: For $A\subset G$ and
$f:G\longrightarrow{\mathbb{C}}$ we write $f|_{A}$ for the functions which
agrees with $f$ on $A$ and is zero outside of $A$.

We start by comparing $\mu|_{A}\ast\varphi$ and $\mu\ast\varphi|_{A}$ for
$\varphi\in C_{\mathsf{c}}(G)$:

Let $K$ be a compact set containing the neutral element of $G$ and the support
of $\varphi$.

For $t\in G$ with $t\notin A+K$ we find

$(\mu|_{A}\ast\varphi)(t)=0=(\mu\ast\varphi)|_{A}(t)\,.$

For $t\in G$ with $t-K\subset A$ we find

$(\mu|_{A}\ast\varphi)(t)=\int_{A}\varphi(t-s)\mbox{d}\mu(s)=\int\varphi(t-s)\mbox{d}\mu=(\mu\ast\varphi)|_{A}(t)\,.$

Any remaining $t\in G$ belongs to $\partial^{K}A$. So, we can summarize that
$\mu|_{A}\ast\varphi$ and $\mu\ast\varphi|_{A}$ agree outside of
$\partial^{K}A$. From this together with

$\mu|_{A}\ast\widetilde{\nu|_{A}}|\ast\varphi\ast\widetilde{\psi}(0)=\int_{G}(\mu|_{A}\ast\varphi)(s)\overline{(\nu|_{A}\ast\psi)}(s)\mbox{d}s$

and

$\int_{A}\mu\ast\varphi(s)\overline{\nu\ast\psi(s)}\mbox{d}s=\int_{G}(\mu\ast\varphi)|_{A}(s)\overline{\nu\ast(\psi|_{A})(s)}ds$

we obtain from the van Hove property

$\frac{1}{|A_{i}|}\left|\mu|_{A_{i}}\ast\widetilde{\nu|_{A_{i}}}|\ast\varphi\ast\widetilde{\psi}(0)-\int_{A_{i}}\mu\ast\varphi(s)\overline{\nu\ast\psi}(s)\mbox{d}s\right|\to
0.$

Hence, existence of the mean
$M_{{\mathcal{A}}}(\mu\ast\varphi\overline{\nu\ast\psi})$ for all
$\varphi,\psi\in C_{\mathsf{c}}(G)$ is equivalent to existence of the limit

$\lim_{i}\frac{1}{|A_{i}|}\mu|_{A_{i}}\ast\widetilde{\nu|_{A_{n}i}}\ast\varphi\ast\widetilde{\psi}(0)\,.$

The latter in turn is equivalent to vague convergence of
$\frac{1}{|A_{i}|}\mu|_{A_{i}}\ast\widetilde{\nu|_{A_{i}}}|$ (as discussed in
the appendix). This finishes the proof. ∎

As it is both instructive and already interesting we point out the following
situation which is covered by our setting.

######  Example 2.9 (The case of ${\mathbb{Z}}$).

Let $G={\mathbb{Z}}$ be the group of integers (with addition). Then,
${\mathcal{A}}$ with $A_{n}=\\{1,\ldots,n\\}$ is a canonical choice of van
Hove sequence. We can identify the measures $\mu$ on ${\mathbb{Z}}$ with the
function - again denoted by $\mu$ \- with $\mu(k):=\mu(\\{k\\})$. Then,
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists if and
only if for each $j\in{\mathbb{Z}}$ the limit

$\lim_{n}\frac{1}{n}\sum_{k=1}^{n}\mu(k)\overline{\nu(k-j)}$

exists.

##  3\. Existence of the reflected Eberlein convolution

. The reflected Eberlein convolution is defined as a limit. So, a natural
question concerns existence of this limit. Here we discuss how existence of
the limit can always be achieved if one replaces the original van Hove net by
a subnet and we also discuss how we can work with van Hove sequences instead
of nets under suitable countability assumptions on the topology.

We start with the following result. The result is certainly well-known. We
include a proof for the convenience of the reader in the appendix.

######  Theorem 3.1 (Existence of universal van Hove net).

Every LCAG admits a van Hove net ${\mathcal{A}}$ with the following
properties:

  * (a)

For all $f\in L^{\infty}(G)$ and $\chi\in\widehat{G}$ the Fourier–Bohr
coefficient $a_{\chi}^{\mathcal{A}}(f)$ exists.

  * (b)

For all $f,g\in L^{\infty}(G)$,
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists.

  * (c)

For all $\mu,\nu\in{\mathcal{M}}^{\infty}(G)$,
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists.

  * (d)

For all $\mu\in{\mathcal{M}}^{\infty}(G)$, the autocorrelation
$\gamma=\text{\textlquill}\mu,\mu\text{\textrquill}_{{\mathcal{A}}}$ exists
with respect to ${\mathcal{A}}$.

Furthermore, any van Hove net ${\mathcal{B}}$ has a subnet ${\mathcal{A}}$
with these properties. ∎

We next discuss how van Hove nets can be replaced by van Hove sequences under
certain circumstances.

First note that a van Hove sequence exists if and only if the group $G$ is
$\sigma$-compact [29, Prop. B.6]. In this case, we can therefore replace van
Hove nets by van Hove sequences in the definition of the Eberlein convolution
of measures and functions. Then, the results in the subsequent part of this
article all hold with van Hove net replaced by van Hove sequence. Note,
however, we will still need to use van Hove nets in the proofs whenever we
apply (a) of Lemma 2.2. The reason is that in general the convergent subnet
appearing in (a) of that lemma will not be a subsequence.

We now turn to the case that the group is second countable. Then it is
$\sigma$-compact and we can therefore work with van Hove sequences. Moreover,
the subnet appearing in (a) of Lemma 2.2 can be chosen as a subsequence.
Hence, all subsequent considerations of this article (statements and proofs)
are valid with van Hove net replaced by van Hove sequence for second countable
locally compact Abelian groups. Moreover, in this case we have a - so to speak
- countable analogue of the above result on universal van Hove sequences.
Specifically, whenever ${\mathcal{B}}$ is a van Hove sequence and $M$ is any
set of translation bounded measures, for any pair $\mu,\nu$ of measures in
$M$, the reflected Eberlein convolution
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{C}}}$ exists along
some subsequence ${\mathcal{C}}$ of ${\mathcal{B}}$. If $M$ is countable, then
a standard diagonalisation argument (or Tychonoff’s Theorem) shows that given
any van Hove sequence ${\mathcal{A}}$, there exists some subsequence
${\mathcal{C}}$ such that, that for all $\mu,\nu\in M$ the reflected Eberlein
convolution $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{C}}}$
exists.

In the context of working with van Hove sequences instead of van Hove nets we
also record the following three results.

######  Lemma 3.2.

Let $f\in L^{\infty}(G),g\in C_{\mathsf{u}}(G)$ and let $(A_{i})_{i\in I}$ be
a van Hove net. Then, the set of $t\in G$ for which the net

$\frac{1}{|A_{i}|}\int_{A_{i}}f(s)\overline{g(s-t)}\mbox{d}s.$

converges is closed in $G$.

###### Proof.

For any $i\in I$ we define

$F_{i}:G\longrightarrow{\mathbb{C}},F_{i}(t)=\frac{1}{|A_{i}|}\int_{A_{i}}f(s)\overline{g(s-t)}\mbox{d}s\,.$

Then, the family $(F_{i})$ is uniformly equicontinuous. Indeed, by uniform
continuity of $g$ , for any $\varepsilon>0$ we can find a neighborhood $U$ of
$e\in G$ with $U=-U$ and $|g(s)-g(s^{\prime})|\leq\varepsilon$ whenever
$s-s^{\prime}\in U$ holds and this gives

$|F_{i}(t)-F_{i}(t^{\prime})|\leq\varepsilon\|f\|_{\infty}$

for all $i\in I$ whenever $t-t^{\prime}\in U$ holds. As $(F_{i})$ is uniformly
equicontinuous, the set of $t$ where $(F_{i}(t))$ is a Cauchy-net is closed.
This gives the desired statement. ∎

We can now prove the following result. The corresponding result for measures
is [20, Thm. 4.15].

######  Corollary 3.3.

Assume that $G$ is $\sigma$ compact and has a countable dense set $D\in G$.
Let $(A_{n})_{n}$ be a van Hove sequence in $G$. Then, for all $f\in
L^{\infty}(G),g\in C_{\mathsf{u}}(G)$ there exists a subsequence
${\mathcal{B}}$ of ${\mathcal{A}}$ such that
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{B}}}$ exists.

######  Remark 3.4.

We note that the assumptions of the corollary are satisfied in second
countable groups.

###### Proof.

Let $t_{m}$ be an enumeration of $D$. Since

$\frac{1}{|A_{n}|}\int_{A_{n}}f(s)\overline{g(s-t_{1})}\mbox{d}s$

is a bounded sequence in ${\mathbb{C}}$, there exists some increasing sequence
$k(n,1)$ of natural numbers such that

$\frac{1}{|A_{k(n,1)}|}\int_{A_{k(n,1)}}f(s)\overline{g(s-t_{1})}\mbox{d}s$

converges.

Inductively, by the same argument, for each $j\geq 2$ we can construct a
subsequence $k(n,j)$ of $k(n,j-1)$ such that

$\frac{1}{|A_{k(n,j)}|}\int_{A_{k(n,j)}}f(s)\overline{g(s-t_{j})}\mbox{d}s$

converges.

A standard diagonalisation argument gives a subsequence ${\mathcal{B}}$ of
${\mathcal{A}}$ such that

$\frac{1}{|B_{n}|}\int_{B_{n}}f(s)\overline{g(s-t_{j})}\mbox{d}s$

converges for all $t\in D$. The claim follows now from Lemma 3.2. ∎

##  4\. Construction of a unitary representation

Let a van Hove net ${\mathcal{A}}$ be given. We will be interested in subsets
$F$ of functions in $C_{\mathsf{u}}(G)$ with the property that
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists for all
$f,g\in F$. In this section we show that any such set gives rise to a unitary
representation on a suitably defined Hilbert space. This will then be applied
to sets $M$ of translation bounded measures with the property that
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists for all
$\mu,\nu\in M$.

We first characterize what existence of all reflected Eberlein convolutions
means for a set of functions.

######  Lemma 4.1.

Let $F\subseteq C_{\mathsf{u}}(G)$ and ${\mathcal{A}}$ be a van Hove net.
Then, the following assertions are equivalent:

  * (i)

For all $f,g\in F$ the reflected Eberlein convolution
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists.

  * (ii)

There exists set $B\subseteq C_{\mathsf{u}}(G)$ with the following properties:

    * –

$F\subseteq B$.

    * –

$B$ is $G$-invariant.

    * –

For all $f,g\in B$ the mean $M_{{\mathcal{A}}}(f\overline{g})$ exists.

###### Proof.

(ii)$\Longrightarrow$(i): For all $f,g\in F$ and $t\in G$ we have
$f,\tau_{t}g\in B$ and hence

$M(f\overline{\tau_{t}g})$

exists. This shows that $\text{\textlquill}f,g\text{\textrquill}$ exists.

(i)$\Longrightarrow$(ii): Define

$B_{F}:=\\{\tau_{t}f:f\in F,t\in G\\}\,.$

We show that $B_{F}$ has the desired properties.

Let $f,g\in B_{F}$. Then, exist some $f_{1},g_{1}\in F$ and $t,s\in G$ so that

$f=\tau_{t}f_{1}\,;\,g=\tau_{s}g_{1}\,.$

Then, by invariance of the mean and the definition of the reflected Eberlein
convolution we have

$M_{{\mathcal{A}}}(f\bar{g})=M_{{\mathcal{A}}}(\tau_{t}f_{1}\overline{\tau_{s}g_{1}})=M_{{\mathcal{A}}}(f_{1}\overline{\tau_{s-t}g_{1}})=\text{\textlquill}f_{1},g_{1}\text{\textrquill}_{{\mathcal{A}}}(s-t)$

exists for all $f,g\in B_{F}$.

It is clear that $F\subseteq B_{F}$ and, by definition $B_{F}$, is $G$
invariant. ∎

######  Remark 4.2.

We note that in the proof of Lemma 4.1, the set $B_{F}$ is the smallest $B$
satisfying the stated conditions.

Consider now a van Hove net ${\mathcal{A}}$ and let $F\subseteq
C_{\mathsf{u}}(G)$ be given such that
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists for all
$f,g\in F$. We then define $H_{F}$ to be the linear span of translates of
elements from $F$, i.e.

$H_{F}:=\mbox{Span}\\{\tau_{t}f:t\in G,f\in F\\}\,.$

From the assumption on existence of the reflected Eberlein convolutions we
then find that

$\langle f,g\rangle:=M_{A}(f\overline{g})$

exists for all $f,g\in H_{F}$. Clearly, $\langle\cdot,\cdot\rangle$ gives a
semi-inner product on $H_{F}$. We denote by ${\mathcal{H}}_{F}$ the Hilbert
space completion of $(H_{F},\langle\cdot,\cdot\rangle)$. By a slight abuse of
notation we denote the inner product on ${\mathcal{H}}_{F}$ by
$\langle\cdot,\cdot\rangle$ as well. Whenever $F$ consists just of a single
element $f$ we write ${\mathcal{H}}_{f}$ instead of ${\mathcal{H}}_{\\{f\\}}$.

Clearly, if $F$ and $F^{\prime}$ are subsets of $C_{\mathsf{u}}(G)$ with
$F\subseteq F^{\prime}$ there is a canonical isometric embedding
${\mathcal{H}}_{F}\hookrightarrow{\mathcal{H}}_{F^{\prime}}$ extending the
embedding $F\hookrightarrow F^{\prime},f\mapsto f$.

######  Theorem 4.3 (The unitary representation $T^{F,{\mathcal{A}}}$ induced
by $F$).

Let ${\mathcal{A}}$ be a van Hove net. Let $F\subseteq C_{\mathsf{u}}(G)$ be
such that $\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists for
all $f,g\in F$. Then, for every $t\in G$, the translation operator
$\tau_{t}:C_{\mathsf{u}}(G)\to C_{\mathsf{u}}(G)$ induces a unitary operator

$T_{t}=T_{t}^{F,{\mathcal{A}}}:{\mathcal{H}}_{F}\longrightarrow{\mathcal{H}}_{F}\,.$

The family $T=T^{F,{\mathcal{A}}}=(T_{t}^{F,{\mathcal{A}}})_{t\in G}$ is a
representation of $G$, i.e. satisfies

$T_{0}=\mbox{identity}\mbox{ and }T_{t+s}=T_{t}\circ T_{s}$

for all $t,s\in G$. This representation is strongly continuous, i.e. the map

$G\longrightarrow{\mathcal{H}}_{F},t\mapsto T_{t}f\,,$

is continuous for each $f\in{\mathcal{H}}_{F}$.

###### Proof.

Set $H:=H_{F}$ and ${\mathcal{H}}:={\mathcal{H}}_{F}$.

We show first that $\tau_{t}$ gives rise to a unique unitary operator: Let
$t\in G$ be arbitrary. Then, for all $f,g\in H_{F}$ we have

$\langle\tau_{t}f,\tau_{t}g\rangle=M(\tau_{t}f\overline{\tau_{t}g})=M(f\bar{g})=\langle
f,g\rangle$ (4.1)

by invariance of the mean. This means that $\tau_{t}:H\to
H\subseteq{\mathcal{H}}$ is a bounded operator and hence it can uniquely be
extended to a bounded operator

$T_{t}:{\mathcal{H}}\to{\mathcal{H}}\,.$

Now, (4.1) and the denseness of $H$ in ${\mathcal{H}}$ imply that $T_{t}$ is
inner product preserving. Moreover,

$\tau_{t}\circ\tau_{-t}=\mbox{identity}$

carries to $T_{t}$ and hence each $T_{t}$ is onto. This shows that $T_{t}$ is
an unitary operator for each $t$.

From $\tau_{t}\circ\tau_{s}=\tau_{t+s}$ and $\tau_{0}=\mbox{identity}$ we
immediately infer that $T$ is a representation.

It remains to show the statement on strong continuity: We first consider $f\in
H$. Let $t\in G$ be fixed. Then, with respect to the semi-norm $\|\cdot\|$
induced by $\langle.,.\rangle$ we have

$\|T_{s}f-T_{t}f\|^{2}=M_{{\mathcal{A}}}(|\tau_{s}f-\tau_{t}f|^{2})\leq\|\tau_{s}f-\tau_{t}f\|_{\infty}^{2}=\|\tau_{s-t}f-f\|_{\infty}^{2}\,.$

Now, as any $f\in C_{\mathsf{u}}(G)$ is uniformly continuous, the term
$\|\tau_{s-t}f-f\|_{\infty}$ goes to zero for $s\to t$. This gives the desired
continuity for $f\in H$. The case of general $f\in{\mathcal{H}}$ can then be
treated by denseness of $H$ in ${\mathcal{H}}$ as follows: Consider
$h\in{\mathcal{H}}$. Let $\varepsilon>0$. Then, there exists some $f\in H$
such that $\|f-h\|<\varepsilon$. By the above, there exists some open set
$0\in U\subseteq G$ such that $t\in U$ implies
$\|\tau_{t}f-f\|<\frac{\varepsilon}{3}$. Then , for all $t\in U$ we have

$\displaystyle\|T_{t}h-h\|$
$\displaystyle\leq\|T_{t}h-T_{t}f\|+\|T_{t}f-f\|+\|f-h\|=\|h-f\|+\|\tau_{t}f-f\|+\|f-h\|$
$\displaystyle<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}=\varepsilon\,.$

This finishes the proof. ∎

By Stone Theorem and the previous theorem there exists a (unique) map

$E:\mbox{Borel sets of $\widehat{G}$}\longrightarrow\mbox{ Projections on
${\mathcal{H}}_{F}$}$

with

  * •

$E(\emptyset)=0$, (Non-degenerate);

  * •

$E(\cup_{n}B_{n})=\lim_{N}\oplus_{n=1}^{N}E(B_{n})$ whenever $B_{n}$,
$n\in\mathbb{N}$, are mutually disjoint measurable subsets of $\widehat{G}$,
($\sigma$-additive);

  * •

$\langle f,T_{t}f\rangle=\widecheck{\varrho_{f}}(t)$ for all $t\in G$.

Here, the measure $\varrho_{f}$ is defined by

$\varrho_{f}(B)=\langle f,E(B)f\rangle$

and called the spectral measure of $f$. It is uniquely determined by

$\langle f,T_{t}f\rangle=\widecheck{\varrho_{f}}(t)$

for all $t\in G$. As $E$ takes values in the projections we have

$\varrho_{f}(B)=\langle f,E(B)f\rangle=\langle
E(B)f,E(B)f\rangle=\|E(B)f\|^{2}$

for all $f\in{\mathcal{H}}_{F}$ and any measurable $B\subset\widehat{G}$. It
is well-known that $E$ satisfies

$E(A)E(B)=E(A\cap B)$

for all measurable subsets $A,B\subset\widehat{G}$ as well as
$E(\widehat{G})=\mbox{Id}$. Indeed, the first equality follows easily from the
additivity property of $E$. The second equality follows as $E(\widehat{G})$ is
a projection with

$\|E(\widehat{G})f\|^{2}=\varrho_{f}(\widehat{G})=\int_{\widehat{G}}1\mbox{d}\varrho_{f}=\langle
f,T_{0}f\rangle=\|f\|^{2}$

for all $f$.

######  Remark 4.4.

Some people prefer the version of the Stone Theorem where translation appear
in the first (the linear) component, i.e.

$\langle T_{t}f,f\rangle=\widecheck{\varrho_{f}}(t)\,.$

To use this version, one would need to replace the operator $T_{t}$ induced by
the left shift $\tau_{t}(f)=f(t-\cdot)$ with the operator $S_{t}=T_{-t}$
induced by the right shift on functions. This change would induce a reflection
of the spectral measure.

We gather a few properties of spectral measures next. This clarifies in
particular the relationship between a spectral measure and the reflected
Eberlein convolution.

As usual we write $\mu\perp\nu$ whenever $\mu$ and $\nu$ are positive measures
on the same measurable space $X$ for which there exist measurable disjoint
subsets $A,B$ of $X$ with $\mu(X\setminus A)=0=\nu(X\setminus B)$. The
measures $\mu$ and $\nu$ are then called mutually singular.

######  Lemma 4.5 (Properties of spectral measures).

Consider a van Hove net ${\mathcal{A}}$ and $F\subseteq C_{\mathsf{u}}(G)$
such that $\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists for
any $f,g\in F$ and let $T=T^{F,{\mathcal{A}}}$ be the associated unitary
representation. Then,

  * (a)

$\varrho_{f}=\varrho_{T_{s}f}$ for any $f\in{\mathcal{H}}$ and $s\in G$.

  * (b)

The spectral measure $\varrho_{f}$ of $f\in F$ satisfies

$\widecheck{\varrho_{f}}(t)=\text{\textlquill}f,f\text{\textrquill}_{{\mathcal{A}}}(t)\qquad\mbox{
for all }t\in G\,.$

  * (c)

Let $f,g\in F$ be given. Then
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}=0$ if and only if
${\mathcal{H}}_{f}\perp{\mathcal{H}}_{g}$.

  * (d)

Let $f\in{\mathcal{H}}$ be given. Then $f=E(A)f$ whenever
$A\subset\widehat{G}$ satisfies $\varrho_{f}(\widehat{G}\setminus A)=0$.

  * (e)

If $f,g\in{\mathcal{H}}$ satisfy $\varrho_{f}\perp\varrho_{g}$ then $\langle
f,g\rangle=0$.

  * (f)

If $f,g\in F$ satisfy $\varrho_{f}\perp\varrho_{g}$ then
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}=0$.

###### Proof.

(a) The characteristic feature of $\varrho_{T_{s}f}$ is that

$\int_{\widehat{G}}\xi(t)\mbox{d}\varrho_{T_{s}f}(\xi)=\langle
T_{s}f,T_{t}T_{s}f\rangle$

holds for all $t\in G$. The characteristic feature of $\varrho_{f}$ is that

$\int_{\widehat{G}}\xi(t)\mbox{d}\varrho_{f}(\xi)=\langle f,T_{t}f\rangle$

holds for all $t\in G$. As $T$ is unitary, we have $\langle
T_{s}f,T_{t}T_{s}f\rangle=\langle f,T_{t}f\rangle$. Hence, the two spectral
measures agree.

(b) For all $f\in F$ we have

$\widecheck{\varrho_{f}}(t)=\langle f,T_{t}f\rangle\stackrel{{\scriptstyle
f\in
F}}{{=\joinrel=\joinrel=\joinrel=}}M(f\overline{\tau_{t}f})=\text{\textlquill}f,f\text{\textrquill}_{{\mathcal{A}}}(t)\,.$

(c) For all $f,g\in F$ we have

$\text{\textlquill}f,g\text{\textrquill}(t)=M(f\overline{\tau_{t}g})\stackrel{{\scriptstyle
f,g\in F}}{{=\joinrel=\joinrel=\joinrel=}}\langle f,T_{t}g\rangle\,.$

The claim follows immediately.

(d) We have $f=E(\widehat{G})f=E(A)f+E(\widehat{G}\setminus A)f$. Now, by
assumption on $A$, we also have

$\|E(\widehat{G}\setminus A)f\|^{2}=\varrho_{f}(\widehat{G}\setminus A)=0\,.$

Put together this yields $f=E(A)f$.

(e) Let $A,B$ be disjoint measurable subsets of $\widehat{G}$ with

$\varrho_{f}(\widehat{G}\setminus A)=0=\varrho_{g}(\widehat{G}\setminus B)\,,$

Then, $f=E(A)f$ and $g=E(B)g$ by (d). Hence, we can calculate

$\langle f,g\rangle=\langle E(A)f,E(B)g\rangle=\langle
E(B)E(A)f,g\rangle=0\,,$

where we used $E(A)E(B)=E(A\cap B)=E(\emptyset)=0$ to obtain the last
equality.

(f) By definition we have
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}(s)=\langle
f,T_{s}g\rangle$. Now, by (a) the spectral measure of $T_{s}g$ agrees with the
spectral measure of $g$ and the desired statement follows from (e). ∎

The converse of (e) is not necessarily true as shown in the next example:

######  Example 4.6 (Converse of (e) does not hold).

Let $f$ be

$f(x)=\left\\{\begin{array}[]{cc}-1&\mbox{ if }x<-1\\\ x&\mbox{ if }-1\leq
x\leq 1\\\ 1&\mbox{ if }x>1\end{array}\right.\,.$

and let $g=1$ be the constant function $1$. Set

$F:=\\{\tau_{t}f:t\in{\mathbb{R}}\\}\cup\\{g\\}\,,$

which is $G$ invariant. Then, by Lemma 4.5 we have

$\widecheck{\varrho_{f}}(t)=\text{\textlquill}f,f\text{\textrquill}_{{\mathcal{A}}}(t)=M(f\overline{\tau_{t}f})=1\qquad\mbox{
for all }t\in{\mathbb{R}}\,,$

with the last equality following from the fact that

$f(x)\overline{\tau_{t}f(x)}=1\mbox{ for all }x\notin[-|t|-1,|t|+1]\,.$

Also, for all $t\in G$ we have

$\widecheck{\varrho_{g}}(t)=\text{\textlquill}g,g\text{\textrquill}_{{\mathcal{A}}}(t)=M(g\overline{\tau_{t}g})=1\qquad\mbox{
for all }t\in{\mathbb{R}}\,.$

It follows that

$\sigma_{f}=\sigma_{g}=\delta_{0}\,.$

On another hand, for all $t\in{\mathbb{R}}$ we have

$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}(t)=M(f\overline{\tau_{t}g})=M(f)=0\,.$

######  Remark 4.7 (Orthogonality of ${\mathcal{H}}_{f}$ and
${\mathcal{H}}_{g}$).

Let $F\subseteq C_{\mathsf{u}}(G)$ be given such that
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists for all
$f,g\in F$. Then we have

${\mathcal{H}}_{\\{f,g\\}}={\mathcal{H}}_{f}+{\mathcal{H}}_{g}\subseteq{\mathcal{H}}_{F}\,.$

In many situations, ${\mathcal{H}}_{f}$ and ${\mathcal{H}}_{g}$ are not
disjoint subspaces, and then the sum above is not an orthogonal sum. Indeed,
for example, when $g=\tau_{t}f$ for some $t\in G$ we trivially have
${\mathcal{H}}_{f}={\mathcal{H}}_{g}$. However, Theorem 4.5 gives the
following implications

$\bigl{(}\sigma_{f}\perp\sigma_{g}\bigr{)}\Longrightarrow\bigl{(}\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}=0\bigr{)}\Longleftrightarrow\bigl{(}{\mathcal{H}}_{f}\perp{\mathcal{H}}_{g}\bigr{)}\,.$

Of course, if ${\mathcal{H}}_{f}\perp{\mathcal{H}}_{g}$ then we trivially have
${\mathcal{H}}_{\\{f,g\\}}={\mathcal{H}}_{f}\oplus{\mathcal{H}}_{g}$.

We also note the following example.

######  Example 4.8 (The space of Besicovitch-2-functions).

Let $F=\widehat{G}$ be the dual group of $G$. Then, the linear span of $A$ is
invariant under translations and for all $\xi,\eta\in\widehat{G}$ the mean
$M_{\mathcal{A}}(\xi\overline{\eta})$ exists and equals $0$ if $\xi\neq\eta$
and equals $1$ if $\xi=\eta$. Thus, the elements of $\widehat{G}$ form an
orthonormal basis of ${\mathcal{H}}_{\widehat{G}}$. The space
${\mathcal{H}}_{\widehat{G}}$ is usually denoted by
$Bap^{2}_{{\mathcal{A}}}(G)$, and is called the space of Besicovitch 2-almost
periodic functions.

We now turn to an application of the preceding considerations to measures.

Let $M\subseteq{\mathcal{M}}^{\infty}(G)$ be given and let ${\mathcal{A}}$ be
van Hove net and assume that for all $\mu,\nu\in M$ the reflected Eberlein
convolution $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$
exists. Let us note here in passing that this implies that the autocorrelation
$\gamma_{\mu}=\text{\textlquill}\mu,\mu\text{\textrquill}_{{\mathcal{A}}}$ of
each $\mu\in M$ exists with respect to ${\mathcal{A}}$. Define for $M$ the set

$F_{M}:=\\{\mu\ast\varphi:\mu\in M,\varphi\in C_{\mathsf{c}}(G)\\}\subset
C_{\mathsf{u}}(G)\,.$

By the assumption on $M$ and Lemma 2.8 the reflected Eberlein convolution
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists then for all
$f,g\in F_{M}$. So, we can apply the preceding considerations. In particular,
$H_{M}:=H_{F_{M}}$ carries a unique semi inner product with

$\langle\mu\ast\varphi,\nu\ast\psi\rangle=M_{{\mathcal{A}}}(\mu\ast\varphi\;\overline{\nu\ast\psi})$

for all $\varphi,\psi\in C_{\mathsf{c}}(G)$. The Hilbert space completion of
$H_{M}$ will be denoted by ${\mathcal{H}}_{M}$.

If $\mu$ is a translation bounded measure such that
$\text{\textlquill}\mu,\mu\text{\textrquill}_{{\mathcal{A}}}$ exists we can
apply the previous considerations to $M=\\{\mu\\}$. In this case, we simply
write ${\mathcal{H}}_{\mu}:={\mathcal{H}}_{\\{\mu\\}}$. Note that then the set

$C:=\\{\mu\ast\varphi:\varphi\in C_{\mathsf{c}}(G)\\}$

is already a subspace of $C_{\mathsf{u}}(G)$ and invariant under translations.
Hence, it is dense in ${\mathcal{H}}_{\mu}$. Note also that for $\mu$ in some
subset $M$ of translation bounded measures such that
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists for all
$\mu,\nu\in M$, the space ${\mathcal{H}}_{\mu}$ can be simply identified with
the Hilbert subspace of ${\mathcal{H}}_{M}$ arising as the closure of
$\\{\mu\ast\varphi:\varphi\in C_{\mathsf{c}}(G)\\}$.

######  Theorem 4.9.

Let $M$ be a set of translation bounded measures and ${\mathcal{A}}$ a van
Hove net with the property that for all $\mu,\nu\in M$ the reflected Eberlein
convolution $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$
exists. There exists a unique family $T:=T^{M}:=T^{F_{M},{\mathcal{A}}}$ of
unitary operators on ${\mathcal{H}}_{M}$ with

$T_{t}(\mu\ast\varphi)=\mu\ast(\tau_{t}\varphi)$

for all $\mu\in M$ and $\varphi\in C_{\mathsf{c}}(G)$. For all $\varphi\in
C_{\mathsf{c}}(G)$ and $\mu\in M$ we have

$\varrho_{\varphi\ast\mu}=\left|\widehat{\varphi}\right|^{2}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}\,.$

###### Proof.

By construction and Theorem 4.3, the family $T$ is a unitary representation
with

$T_{t}(\mu\ast\varphi)=\tau_{t}(\mu\ast\varphi)=\mu\ast(\tau_{t}\varphi)$

for all $\mu\in M$ and $\varphi\in C_{\mathsf{c}}(G)$.

It remains to show the statement on the spectral measure: By Lemma 4.5(b),
Lemma 2.8, the definition of $\gamma_{\mu}$ and Proposition 1.1 we have

$\displaystyle\widecheck{\varrho_{\mu\ast\varphi}}(t)$
$\displaystyle=\text{\textlquill}\mu\ast\varphi,\mu\ast\varphi\text{\textrquill}_{{\mathcal{A}}}(t)=\text{\textlquill}\mu,\mu\text{\textrquill}_{{\mathcal{A}}}\ast\varphi\ast\widetilde{\varphi}(t)$
$\displaystyle=\gamma_{\mu}\ast\varphi\ast\widetilde{\varphi}(t)=\int_{\widehat{G}}\xi(t)|\widehat{\varphi}(\xi)|^{2}\mbox{d}\widehat{\gamma_{\mu}}(\xi)\,.$

As the spectral measure is uniquely determined by its inverse Fourier
transform we infer the desired statement. ∎

######  Remark 4.10 (Off-diagonal spectral measures).

The result on the spectral measures in the previous theorem can be seen as the
‘diagonal’ part of a more general statement and be complemented by an ‘off
diagonal’ part. Specifically, consider the situation of the theorem. Then, by
polarisation we can write
$\gamma_{\mu,\nu}:=\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$
as a linear combination

$\gamma_{\mu,\nu}=\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}=\frac{1}{4}\sum_{k=0}^{3}i^{k}\text{\textlquill}\mu+i^{k}\nu,\mu+i^{k}\nu\text{\textrquill}_{{\mathcal{A}}}\,.$

Also we can define the measure

$\widehat{\gamma_{\mu,\nu}}:=\frac{1}{4}\sum_{k=0}^{3}i^{k}\widehat{\gamma_{\mu+i^{k}\nu}}\,.$

Then, by construction and Lemma 2.8, we have

$\int_{\widehat{G}}\widehat{\varphi}(\xi)\overline{\widehat{\psi}(\xi)}\mbox{d}\widehat{\gamma_{\mu,\nu}}(\xi)=\gamma_{\mu,\nu}\ast\varphi\ast\widetilde{\psi}(0)=\langle\mu\ast\varphi,\nu\ast\psi\rangle$

holds for all $\varphi,\psi\in C_{\mathsf{c}}(G)$. Replacing $\psi$ by
$\tau_{-t}\psi$ we can argue as in the proof of Proposition 1.1 to obtain

$\int_{\widehat{G}}\xi(t)\widehat{\varphi}(\xi)\overline{\widehat{\psi}(\xi)}\mbox{d}\widehat{\gamma_{\mu,\nu}}(\xi)=\langle\mu\ast\varphi,T_{t}(\nu\ast\psi)\rangle$

for all $t\in G$. In this sense,
$\widehat{\varphi}\overline{\widehat{\psi}}\widehat{\gamma_{\mu,\nu}}$ can be
considered as spectral measure to the pair $(\mu\ast\varphi,\nu\ast\psi)$.

We can also characterize orthogonality of subspaces generated by different
measures.

######  Theorem 4.11 (Orthogonality).

Let $M\subseteq{\mathcal{M}}^{\infty}(G)$ be any set and ${\mathcal{A}}$ be
any van Hove sequence with the property that for all $\mu,\nu\in M$ the
reflected Eberlein convolution
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists. Then,
for $\mu,\nu\in{\mathcal{M}}$, the following are equivalent:

  * (i)

$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}=0$.

  * (ii)

$\langle\mu\ast\varphi,\nu\ast\psi\rangle=0$ for all $\varphi,\psi\in
C_{\mathsf{c}}(G)$.

  * (iii)

${\mathcal{H}}_{\mu\ast\varphi}\perp{\mathcal{H}}_{\nu\ast\psi}$ for all
$\varphi,\psi\in C_{\mathsf{c}}(G)$.

  * (iv)

${\mathcal{H}}_{\mu}\perp{\mathcal{H}}_{\nu}$.

###### Proof.

(iv)$\Longrightarrow$(iii): This is clear as ${\mathcal{H}}_{\mu\ast\varphi}$
is a subspace of ${\mathcal{H}}_{\mu}$ and ${\mathcal{H}}_{\nu\ast\psi}$ is a
subspace of ${\mathcal{H}}_{\nu}$.

(iii)$\Longrightarrow$(ii): This is obvious.

(ii)$\Longrightarrow$(iv): By definition, the set
$\\{\mu\ast\varphi:\varphi\in C_{\mathsf{c}}(G)\\}$ is dense in
${\mathcal{H}}_{\mu}$ and the set $\\{\nu\ast\varphi:\varphi\in
C_{\mathsf{c}}(G)\\}$ is dense in ${\mathcal{H}}_{\nu}$. This easily gives the
implication (ii)$\Longrightarrow$(iv).

(ii)$\Longleftrightarrow$(i) By Lemma 2.8 and the definition of
$\langle\cdot,\cdot\rangle$, condition (ii) is equivalent to
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}\ast\varphi\ast\widetilde{\psi}(0)=0$
for all $\varphi,\psi\in C_{\mathsf{c}}(G)$. This in turn is equivalent to
(i). ∎

##  5\. Diffraction as spectral measure

Starting from a measure $\mu$ with autocorrelation $\gamma$ we can now
construct explicitly a unitary representation of $G$ on the Hilbert space
${\mathcal{H}}_{\mu}$ whose spectrum is exactly $\widehat{\gamma}$. We then go
on and use this to characterize pure point diffraction and then discuss
orthogonality features with respect to this Hilbert space.

######  Theorem 5.1 (Diffraction as a spectral measure).

Let $\mu$ be a translation bounded measure and let ${\mathcal{A}}$ be a van
Hove net such that the autocorrelation $\gamma_{\mu}$ of $\mu$ exists with
respect to ${\mathcal{A}}$. Then, there exists a unique unitary map

$\Theta:{\mathcal{H}}_{\mu}\longrightarrow
L^{2}(\widehat{G},\widehat{\gamma_{\mu}})$

such that $\Theta(\mu\ast\varphi)=\widehat{\varphi}$ for all $\varphi\in
C_{\mathsf{c}}(G)$. This map satisfies

$\Theta\circ T_{t}=Z_{t}\circ\Theta\,,$

for all $t\in G$, where $Z_{t}$ is the operator on
$L^{2}(\widehat{G},\widehat{\gamma_{\mu}})$ defined by

$(Z_{t}f)(\xi)=\xi(t)f(\xi)\,.$

###### Proof.

By definition, $H_{\mu}=\\{\mu\ast\varphi:\varphi\in C_{\mathsf{c}}(G)\\}$ is
a dense subspace of ${\mathcal{H}}_{\mu}$.

Uniqueness of $\Theta$ follows immediately as $H_{\mu}$ is dense. As for
existence we note that for $\varphi\in C_{\mathsf{c}}(G)$ we have by Theorem
4.9

$\|\mu\ast\varphi\|^{2}=\langle\mu\ast\varphi,T_{0}\mu\ast\varphi\rangle=\int_{\widehat{G}}\mbox{d}\sigma_{\mu\ast\varphi}(\xi)=\int_{\widehat{G}}|\widehat{\varphi}(\xi)|^{2}\mbox{d}\widehat{\gamma_{\mu}}(\xi)\,.$

Combined with the denseness of $H_{\mu}$ this shows that there exists a unique
isometric $\Theta$ mapping $\mu\ast\varphi$ to $\widehat{\varphi}$ for
$\varphi\in C_{\mathsf{c}}(G)$. Now, the set of $\widehat{\varphi}$,
$\varphi\in C_{\mathsf{c}}(G)$, is dense in
$L^{2}(\widehat{G},\widehat{\gamma_{\mu}})$ [1, Prop. 2.20]. So, $\Theta$ is
an isometry with dense range and, hence, a unitary map.

The equality $\Theta\circ T_{t}=Z_{t}\circ\Theta$ is clear on $H_{\mu}$ and
then follows by denseness on ${\mathcal{H}}_{\mu}$. ∎

######  Remark 5.2 (Understanding $\Theta$).

  * (a)

The map $\Theta$ diagonalizes the action $T$ in the sense that it transforms
it into multiplication operators.

  * (b)

Let $\gamma$ be a positive definite measure on $G$. Then, $\gamma$ induces
semi-inner product on $C_{\mathsf{c}}(G)$ by

$\langle\varphi,\psi\rangle_{\gamma}:=\gamma\ast\varphi\ast\widetilde{\psi}(0)\,.$

We can then form the Hilbert space completion
$({\mathcal{H}}_{\gamma},\langle\cdot,\cdot\rangle)$ of $C_{\mathsf{c}}(G)$
equipped with $\langle\cdot,\cdot\rangle_{\gamma}$. This space can naturally
be identified with $L^{2}(\widehat{G},\widehat{\gamma})$ via the unique
unitary extension $U_{\gamma}$ of the map

$C_{\mathsf{c}}(G)\longrightarrow
L^{2}(\widehat{G},\widehat{\gamma}),\varphi\mapsto\widehat{\varphi}\,.$

Indeed, a proof can be given by mimicking the arguments given in the proof of
the previous theorem.

Furthermore, if $\gamma=\gamma_{\mu}$ is the autocorrelation of a translation
bounded measure $\mu$, we can naturally identify
${\mathcal{H}}_{\gamma_{\mu}}$ with ${\mathcal{H}}_{\mu}$ via the unique
unitary extension $V_{\mu}$ of the map

$C_{\mathsf{c}}(G)\longrightarrow
C_{\mathsf{u}}(G),\varphi\mapsto\mu\ast\varphi\,.$

Then, $\Theta$ is just given as the composition
$U_{\gamma_{\mu}}\circ(V_{\mu})^{-1}$.

######  Remark 5.3 (Relating ${\mathcal{H}}_{\mu}$ to the literature).

  * (a)

In this part of the remark we relate the above approach to the Hilbert-space
$H_{\mu}$ to the approach via processes given in [16, 10]: Whenever $\mu$ is a
translation bounded measure with autocorrelation $\gamma$ we can define

$N:C_{\mathsf{c}}(G)\longrightarrow{\mathcal{H}}_{\mu},\varphi\mapsto\mu\ast\varphi$

and $N^{\prime}:C_{\mathsf{c}}(G)\longrightarrow
L^{2}(\widehat{G},\widehat{\gamma}),\varphi\mapsto\widehat{\varphi}$. Then,
$(N,{\mathcal{H}}_{\mu},T)$ is a process in the sense of [16] by Theorem 4.9
and so is $(N^{\prime},L^{2}(\widehat{G},\widehat{\gamma}))$ (by [16]). The
previous theorem can be understood as saying that these two processes are
equivalent (spatially isomorphic).

  * (b)

The Hilbert space ${\mathcal{H}}_{\mu}$ can be alternately be understood the
following way:

Let

$\Omega_{\mu}:=\overline{\\{\tau_{t}\mu:t\in G\\}}$

be the hull of $\mu$. Here, the closure is taken in the vague topology. Let
$m$ be an invariant measure of on $\Omega_{\mu}$ and define

$N:C_{\mathsf{c}}(G)\longrightarrow
L^{2}(\Omega_{\mu},m),N_{\varphi}(\omega):=\omega\ast\varphi(0)\,.$

Let $\mathcal{S}(m)$ be the closure of the range of $N$ in
$L^{2}(\Omega_{\mu},m)$. Assume now that $\mu$ is generic for $m$, where
generic means that

$\frac{1}{|A_{i}|}\int_{A_{i}}f(\tau_{t}\mu)\mbox{d}t\to\int_{\Omega_{\mu}}f(\omega)\mbox{d}m(\omega)$

holds for all continuous $f$ on $\Omega_{\mu}$. Then, the map
$\mu\ast\varphi\to N_{\varphi}$ can be extended uniquely to a unitary map
between ${\mathcal{H}}_{\mu}$ and $\mathcal{S}(m)$.

  * (c)

Part (b) of this remark raises the question whether any $\mu$ that admits an
autocorrelation along $(A_{i})$ also admits a measure $m$ on its hull such
that $\mu$ is generic with respect to this $m$ and $\mathcal{S}(m)$ can be
identified with $\mathcal{H}_{\mu}$ as in (b). If $G$ is second countable and
$(A_{n})$ is a van Hove sequence, this can be shown to hold after one replaces
$(A_{n})$ by a suitable subsequence $(B_{n})$. Indeed, by second countability
of $G$ the hull $\Omega_{\mu}$ is metrisable and compact. Hence,
$C(\Omega_{\mu})$ is second countable. Therefore, we can chose a subsequence
$(B_{n})$ of $(A_{n})$ such that

$\frac{1}{|B_{n}|}\int_{B_{n}}f(\tau_{t}\mu)dt$

converges for all $f\in C(\Omega_{\mu})$. With $m$ defined by

$m(f):=\lim_{n}\frac{1}{|B_{n}|}\int_{B_{n}}f(\tau_{t}\mu)dt$

we then have found a suitable measure.

In particular, Remark 5.3 (c) has the following important consequence (see [3]
for definitions).

######  Proposition 5.4 (Diffractions of measures are diffractions of
dynamical systems).

Let $G$ be a second countable LCAG, let $\mu$ be a translation bounded measure
on $G$ and let ${\mathcal{A}}$ be a van Hove sequence such that the
autocorrelation $\gamma$ of $\mu$ exists with respect to ${\mathcal{A}}$.
Then, there exists a $G$-invariant measure $m$ on $\Omega_{\mu}$ such that
$\gamma$ is the autocorrelation measure of $(\Omega_{\mu},G,m)$.

In particular, $\widehat{\gamma}$ is the diffraction measure of
$(\Omega_{\mu},G,m)$.

###### Proof.

Let $B_{n}$ and $m$ be as in Remark 5.3 (c). Since, $\mu$ is generic for $m$,
[3, Lemma 7] gives that $\gamma$ is the autocorrelation of
$(\Omega_{\mu},G,m)$. ∎

######  Remark 5.5.

For ergodic measures the converse also holds.

Let $G$ be any second countable LCAG, let $\mu$ be a translation bounded
measure on $G$, let $m$ be a $G$-invariant probability measure and let
$\gamma$ be the autocorrelation of $(\Omega_{\mu},G,m)$.

If $\omega\in\Omega_{\mu}$ and ${\mathcal{A}}$ is a van Hove sequence such
that $\omega$ is generic for $m$ with respect to ${\mathcal{A}}$, then
$\gamma$ is the autocorrelation of $\omega$.

In particular, when $m$ is ergodic, and ${\mathcal{A}}$ is a van Hove sequence
along which Birkhoff ergodic theorem holds, then there exists elements
$\omega\in\Omega_{\mu}$ such that $\gamma$ is the autocorrelation of $\omega$
with respect to ${\mathcal{A}}$ (compare [3]).

The preceding considerations allow us to (re)prove various characterisations
of pure point diffraction. Recall that a subset $S$ of $G$ is relatively dense
if there exists a compact $K\subset G$ with $S+K=G$ and that a bounded
continuous function $f:G\longrightarrow{\mathbb{C}}$ is Bohr-almost periodic
if for any $\varepsilon>0$ the set

$\\{t\in G:\|f-\tau_{t}f\|_{\infty}\leq\varepsilon\\}$

is relatively dense.

######  Corollary 5.6 (Characterization of pure point diffraction).

Let $\mu$ be a translation bounded measure with autocorrelation $\gamma$ and
associated unitary representation $T$ on
$({\mathcal{H}}_{\mu},\langle\,,\,\rangle)$. Then, the following assertions
are equivalent:

  * (i)

The unitary representation $T$ has pure point spectrum.

  * (ii)

The diffraction measure $\widehat{\gamma}$ is a pure point measure.

  * (iii)

The autocorrelation is $\gamma$ is strongly almost periodic, i.e. for any
$\varphi\in C_{c}(G)$ the function $\gamma\ast\varphi$ is Bohr almost
periodic.

  * (iv)

For any $\varphi\in C_{c}(G)$ the function $t\mapsto\langle
T_{t}\mu\ast\varphi,\mu\ast\varphi\rangle$ is Bohr-almost periodic.

  * (v)

For any $\varphi\in C_{c}(G)$ and any $\varepsilon>0$ the set

$\\{t\in G:|\langle
T_{t}\mu\ast\varphi,\mu\ast\varphi\rangle-\langle\mu\ast\varphi,\mu\ast\varphi\rangle|\leq\varepsilon\\}$

is relatively dense.

  * (vi)

The measure $\mu$ is mean almost periodic, i.e. for each $\varepsilon>0$ the
set

$\\{t\in G:\|T_{t}\mu\ast\varphi-\mu\ast\varphi\|\leq\varepsilon\\}$

is relatively dense.

###### Proof.

By the Thm. 5.1, $T$ has pure point spectrum if and only if $Z_{\cdot}$ has
pure point spectrum. This is turn is easily seen to be equivalent to
$\widehat{\gamma}$ being a pure point measure. In this way, the equivalence
between (i) and (ii) follows from Thm. 5.1. Now, clearly (ii) is equivalent to
each spectral measure

$\varrho_{\mu\ast\varphi}=|\widehat{\varphi}|^{2}\widehat{\gamma}$

being pure point. This, in turn is just equivalent to (iv) by Wiener lemma
(see e.g. [21, 23]) Now, the equivalence between (iv), (v) and (vi) is
standard for unitary representations (see e.g. [21] as well). It remains to
show the equivalence between (iii) and (iv). Now, by what we have shown

$\gamma\ast\varphi\ast\widetilde{\varphi}(t)=\langle
T_{t}\mu\ast\varphi,\mu\ast\varphi\rangle$

holds for all $\varphi\in C_{c}(G)$. Hence, (iv) is equivalent to almost
periodicity of

$t\mapsto\gamma\ast\varphi\ast\widetilde{\varphi}$

for all $\varphi\in C_{c}(G)$. By polarisation this is equivalent to Bohr-
almost periodicity of

$t\mapsto\gamma\ast\varphi\ast\widetilde{\psi}$

for all $\varphi,\psi\in C_{c}(G)$. This, in turn, can easily be seen to be
equivalent to (iii). ∎

######  Remark 5.7.

The equivalence between (ii) and (iii) has first been shown by Baake / Moody
[5]. The equivalence between (ii) and (vi) is contained in recent work of the
authors with Spindeler [18]. Our proof of these equivalences as well as the
other equivalences are new (as they are based on the unitary representation
$T$ which has not been considered before).

##  6\. An orthogonality result

We now turn to our main result on orthogonality.

######  Theorem 6.1 (Orthogonality with respect to the twisted Eberlein
convolution).

Let $\mu,\nu$ be translation bounded measures and ${\mathcal{A}}$ a van Hove
net such that the autocorrelations $\gamma_{\mu}$ of $\mu$ and $\gamma_{\nu}$
of $\nu$ exist with respect to ${\mathcal{A}}$. If
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}\perp\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}$
holds then $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$
exists and satisfies
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}=0.$

###### Proof.

By the compactness statement in (a) of Lemma 2.2 it suffices to show
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{B}}}=0$ whenever
${\mathcal{B}}$ is a subnet of ${\mathcal{A}}$ such that
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{B}}}$ exist.
Therefore, without loss of generality we can assume that
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists.

Let $M:=\\{\mu,\nu\\}$. By Theorem 4.9, we have an unitary representation of
$G$ on the space ${\mathcal{H}}_{\mu,\nu}:={\mathcal{H}}_{M}$ and, for all
$\varphi,\psi\in C_{\mathsf{c}}(G)$ we have

$\varrho_{\varphi\ast\mu}=\left|\widehat{\varphi}\right|^{2}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}\mbox{
and
}\varrho_{\psi*\nu}=\left|\widehat{\psi}\right|^{2}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}\,.$

Since
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}\perp\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}$
we get $\varrho_{\mu\ast\varphi}\perp\varrho_{\nu\ast\psi}$. Then, (e) of
Lemma 4.5 implies

$0=\langle\mu\ast\varphi,\nu\ast\psi\rangle$

for all $\varphi,\psi\in C_{\mathsf{c}}(G)$. Given this the desired statement
now follows from Theorem 4.11. ∎

######  Corollary 6.2.

Let $\mu,\nu$ be translation bounded measures and ${\mathcal{A}}$ a van Hove
net such that the autocorrelations $\gamma_{\mu}$ of $\mu$ and $\gamma_{\nu}$
of $\nu$ exist with respect to ${\mathcal{A}}$. If
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}\perp\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}$
then for all $a,b\in{\mathbb{C}}$ the measure $a\mu+b\nu$ has diffraction

$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{a\mu+b\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{a\mu+b\nu}}{\tmpbox}=|a|^{2}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}+|b|^{2}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}\,.$

###### Proof.

This is proven exactly like Pythagoras’ theorem in inner product spaces.
Indeed, by the preceding theorem we find for the autocorrelation measures the
following:

$\displaystyle\gamma_{a\mu+b\nu}$
$\displaystyle=\text{\textlquill}a\mu+b\nu,a\mu+b\nu\text{\textrquill}_{{\mathcal{A}}}=|a|^{2}\text{\textlquill}\mu,\mu\text{\textrquill}_{{\mathcal{A}}}+a\bar{b}\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}+b\bar{a}\text{\textlquill}\nu,\mu\text{\textrquill}_{{\mathcal{A}}}+|b|^{2}\text{\textlquill}\nu,\nu\text{\textrquill}_{{\mathcal{A}}}$
$\displaystyle=|a|^{2}\gamma_{\mu}+|b|^{2}\gamma_{\nu}\,.$

Taking the Fourier transforms we get the claim. ∎

######  Remark 6.3.

Validity of $\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}=0$
does not necessarily imply that
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}$
and
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}$
are mutually singular. Indeed, similarly to Example 4.6 we can show that the
measures

$\mu=\delta_{{\mathbb{Z}}}\,;\,\nu=\sum_{m\in{\mathbb{Z}}}\mbox{sgn}(m)\delta_{m}$

satisfy with respect to $A_{n}=[-n,n]$:

$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}=0\mbox{ and
}\widehat{\gamma_{\mu}}=\widehat{\gamma_{\nu}}=\delta_{{\mathbb{Z}}}\,.$

##  7\. Application: The point part of the diffraction and Bombieri–Taylor
type results

In this section we discuss consequences of the main orthogonality result to
diffraction theory. This sheds a new light on what is sometimes known as
Bombieri–Taylor conjecture (compare remark at the end of this section).

######  Proposition 7.1.

Let $\mu$ be a translation bounded measure and let ${\mathcal{A}}$ be a van
Hove net. Assume that the autocorrelation $\gamma_{\mu}$ exists with respect
to ${\mathcal{A}}$. Then, for all $\chi\in\widehat{G}$ for which
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}(\\{\chi\\})=0$
the Fourier–Bohr coefficient $a_{\chi}^{\mathcal{A}}(\mu)$ exists and
satisfies

$a_{\chi}^{\mathcal{A}}(\mu)=0\,.$

###### Proof.

Let $\nu:=\chi\theta_{G}$. Then, $\gamma_{\nu}$ exists with respect to
${\mathcal{A}}$ and

$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}=\delta_{\chi}\,.$

Therefore, by Theorem 6.1 the reflected Eberlein convolution
$\text{\textlquill}\mu,\chi\text{\textrquill}_{{\mathcal{A}}}$ exists and is
zero. The claim follows now from Proposition 2.5. ∎

######  Remark 7.2 (Converse of previous proposition fails).

The converse of the previous proposition is not true. Indeed, let

$\mu:=\sum_{m\in{\mathbb{Z}}}\mbox{sgn}(m)\delta_{m}\,.$

and let $A_{n}=[-n,n]$. Then, it is clear that

$a_{0}^{{\mathcal{A}}}(\mu)=0\,.$

On another hand, the autocorrelation $\gamma_{\mu}$ exists with respect to
$A_{n}$ and

$\gamma_{\mu}=\delta_{{\mathbb{Z}}}\,.$

It follows that

$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}(\\{0\\})=1\,.$

The proposition can be used to study existence of the Fourier-Bohr
coefficients. This is carried out next.

Recall first that a set $A$ is called locally countable if $A\cap K$ is
countable for all compact sets $K\subseteq G$. If $G$ is $\sigma$-compact,
then $A\subseteq G$ is locally countable if and only if it is countable.

######  Proposition 7.3.

Let $\mu\in{\mathcal{M}}^{\infty}(G)$ and ${\mathcal{A}}$ a van Hove net such
that the autocorrelation $\gamma$ of $\mu$ exists with respect to
${\mathcal{A}}$. Then, there exists some locally countable set
$A\subseteq\widehat{G}$ such that for all $\chi\notin A$ the Fourier–Bohr
coefficient $a_{\chi}^{{\mathcal{A}}}(\mu)$ exists and is zero. In particular,
the set of $\chi\in\widehat{G}$ for which the following limit does not exist
is locally countable:

$\lim_{i}\frac{1}{|A_{i}|}\int_{A_{i}}\overline{\chi(t)}\mbox{d}\mu(t)\,.$

###### Proof.

Since $\widehat{\gamma}$ is a measure, the set

$A=\\{\chi:\widehat{\gamma}(\\{\chi\\})\neq 0\\}$

is locally countable. The first claim now follows from Proposition 7.1. The
’In particular’ statement now is an immediate consequence. ∎

######  Corollary 7.4 (Existence of Fourier-Bohr-coefficients along a
subsequence).

Assume that $G$ is second countable. Let $\mu\in{\mathcal{M}}^{\infty}(G)$ and
${\mathcal{A}}$ a van Hove sequence. Then, there exists a subsequence
${\mathcal{B}}$ of ${\mathcal{A}}$ along which the autocorrelation $\gamma$
and all Fourier–Bohr coefficients $a_{\chi}^{{\mathcal{B}}}(\mu)$ exist.

###### Proof.

Pick first a sub sequence ${\mathcal{A}}^{\prime}=\\{A_{k_{n}}\\}$ of
${\mathcal{A}}$ along which the autocorrelation $\gamma$ exists. Since
$\widehat{G}$ is $\sigma$-compact, by Proposition 7.3 the Fourier–Bohr
coefficients exist outside a countable set $D$ of characters.

Let $\chi_{1},\chi_{2},\ldots,\chi_{n},\ldots$ be an enumeration of $D$. By
boundedness, there exists a subsequence $k(1,n)$ of $k_{n}$ such that the
following limit exists.

$\lim_{n}\frac{1}{|A_{k(1,n)}|}\int_{A_{k(1,n)}}\overline{\chi_{1}(t)}\mbox{d}\mu(t)$

Inductively we can now construct a subsequence $k(m+1,n)$ of $k(m,n)$ along
which

$\lim_{n}\frac{1}{|A_{k(n+1,n)}|}\int_{A_{k(m+1,n)}}\overline{\chi_{m+1}(t)}\mbox{d}\mu(t)$

exists.

A standard diagonalisation argument proves the claim. ∎

We now turn to another consequence (or rather a reformulation) of Proposition
7.1.

######  Corollary 7.5 (Bombieri–Taylor type result).

Let $\mu$ be a translation bounded measure and assume that the autocorrelation
$\gamma_{\mu}$ exists with respect to ${\mathcal{A}}$. If

$\lim_{i}\frac{1}{|A_{i}|}\int_{A_{i}}\overline{\chi(t)}\mbox{d}\mu(t)=0$

does not hold, then

$\widehat{\gamma_{\mu}}(\\{\chi\\})\neq 0\,.$

###### Proof.

Assume by contradiction that $\widehat{\gamma_{\mu}}(\\{\chi\\})=0$. Then, by
Proposition 7.1 we have

$\lim_{n}\frac{1}{|A_{n}|}\int_{A_{n}}\overline{\chi(t)}\mbox{d}\mu(t)=0\,.$

∎

An immediate consequence of the preceding corollary is the following.

######  Corollary 7.6.

Let $\mu$ be a translation bounded measure and ${\mathcal{A}}$ a van Hove
sequence. Assume that the autocorrelation $\gamma_{\mu}$ of $\mu$ exists with
respect to ${\mathcal{A}}$ and that
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}$
is a continuous measure. Then, all Fourier–Bohr coefficients $a_{\chi}(\mu)$
exist and satisfy

$a_{\chi}^{{\mathcal{A}}}(\mu)=0\qquad\mbox{ for all }\chi\in\widehat{G}\,.$

∎

The preceding results have shown how non-vanishing of the Fourier–Bohr
coefficient implies non-vanishing of the pure point diffraction component. We
now turn to converse type of statements. These converses tend to need some
extra uniformity assumption.

######  Lemma 7.7.

Let $G$ be second countable and let ${\mathcal{A}}$ be a van Hove sequence.
Let $\mu$ be a translation bounded measure and assume that the autocorrelation
$\gamma_{\mu}$ exists with respect to ${\mathcal{A}}$. If

$\widehat{\gamma_{\mu}}(\\{\chi\\})\neq 0\,.$

then there exists some $t_{n}\in G$ such that

$\lim_{n}\frac{1}{|A_{n}|}\int_{t_{n}+A_{n}}\overline{\chi(t)}\mbox{d}\mu(t)\neq
0\,.$

###### Proof.

Assume by contradiction that $t_{n}\in G$ such that

$\lim_{n}\frac{1}{|A_{n}|}\int_{t_{n}+A_{n}}\overline{\chi(t)}\mbox{d}\mu(t)=0\,.$

Then, by [18], the Fourier–Bohr coefficient $a_{\chi}(\mu)$ exists uniformly.
Therefore, by [31], the CPP holds, that is

$\widehat{\gamma_{\mu}}(\\{\chi\\})=|a_{\chi}(\mu)|^{2}=0\,.$

∎

Combining all results in this section, we get the following consequence.

######  Corollary 7.8.

Let $G$ be second countable, let $\mu$ be a translation bounded measure on $G$
and let ${\mathcal{A}}$ be a fixed van Hove sequence.

Let $\Gamma$ be the set of all vague cluster points of the sequences

$\frac{1}{|A_{n}|}(\mu|_{t_{n}+A_{n}}\ast\widetilde{\mu|_{t_{n}+A_{n}}})$

for all choices of $t_{n}\in G$.

Let $A$ be the set of all vague cluster points of the sequences

$\frac{1}{|A_{n}|}\int_{t_{n}+A_{n}}\overline{\chi(t)}\mbox{d}\mu(t)$

for all choices of $t_{n}\in G$.

Then, for $\chi\in\widehat{G}$, the following are equivalent:

  * (a)

$\widehat{\gamma}(\\{\chi\\})=0$ for all $\gamma\in\Gamma$.

  * (b)

$a_{\chi}=0$ for all $a_{\chi}\in A$.

  * (c)

The Fourier–Bohr coefficient $a_{\chi}(\mu)$ exists uniformly and is zero.

######  Remark 7.9.

Let $\varOmega_{\mu}$ be the hull of $\mu$ (compare (b) of Remark 5.3). The
set $\Gamma$ represents the set of all autocorrelations of elements
$\omega\in\varOmega_{\mu}$ calculated along subsequences of $A_{n}$. Same way,
$A$ represents the set of all Fourier–Bohr coefficients of
$\omega\in\varOmega_{\mu}$ calculated along subsequences of $A_{n}$.

Therefore, Corollary 7.8 says that, for a fixed $\chi\in\widehat{G}$, the
following are equivalent:

  * (a)

$\chi$ is not a Bragg peak for any $\omega\in\varOmega_{\mu}$.

  * (b)

$a_{\chi}^{{\mathcal{A}}}(\omega)=0$ for all $\omega\in\varOmega_{\mu}$.

  * (c)

$a_{\chi}^{{\mathcal{A}}}(\omega)=0$ uniformly for all
$\omega\in\varOmega_{\mu}$.

We also have the following characterisation of absence of point spectrum in
the diffraction.

######  Proposition 7.10.

Let $G$ be second countable, let $\mu\in{\mathcal{M}}^{\infty}(G)$ be a
measure and ${\mathcal{A}}=\\{A_{n}\\}$ be a van Hove sequence such that for
all sequences $t_{n}\in G$, every cluster points of

$\frac{1}{|A_{n}|}\mu|_{t_{n}+A_{n}}\ast\widetilde{\mu|_{t_{n}+A_{n}}}$

has continuous Fourier transform. Then,

  * (a)

The Fourier–Bohr coefficients $a_{\chi}(\mu)$ exist uniformly and
$a_{\chi}(\mu)=0$.

  * (b)

If $\eta$ is the autocorrelation of $\mu$ with respect to some van Hove
sequence, then $\widehat{\eta}$ is continuous.

###### Proof.

(a) Fix $t_{n}\in G$. Since

$\frac{1}{|A_{n}|}\int_{t_{n}+A_{n}}\overline{\chi(t)}\mbox{d}\mu(t)$

is bounded, it converges to $0$ if and only if $0$ is the only cluster point
of this sequence.

Pick a subsequence $(k_{n})$ such that

$c=\lim_{n}\frac{1}{|A_{k_{n}}|}\int_{t_{k_{n}}+A_{k_{n}}}\overline{\chi(t)}\mbox{d}\mu(t)\,,$

exists. Now, by translation boundedness, there exists some subsequence $l_{n}$
of $k_{n}$ such that the autocorrelation $\gamma$ of $\mu$ exists along
$t_{l_{n}}+A_{l_{n}}$. By the condition in the statement, $\widehat{\gamma}$
is continuous, and hence, by Corollary 7.6 we get $c=0$.

This shows that the Fourier–Bohr coefficient $a_{\chi}(\mu)$ exists and is $0$
for all translates of $A_{n}$, and hence it exists uniformly by [18].

(b) Since the Fourier–Bohr coefficients exist uniformly and are 0, they exist
uniformly and are 0 for all choices of van Hove sequence [18]. Therefore, the
CPP holds for $\eta$ by [14, 31] (compare [11, 12] for $G={\mathbb{R}}^{d}$).
This shows that $\widehat{\eta}$ is continuous. ∎

######  Remark 7.11 (Bombieri–Taylor).

A substantial part of the recent interest in diffraction theory comes from the
discovery of quasicrystals by Shechtman [28], which was later honored with a
nobel prize. The characteristic feature of quasicrystals is pure point
diffraction together with symmetries which exclude periods. Accordingly, a key
point in the theoretical investigation is the study of pure point diffraction.
Here, a particular issue is to compute the atoms of $\widehat{\gamma_{\mu}}$
(assuming that $\widehat{\gamma_{\mu}}$ is a pure point measure). From the
very beginning the idea was that the atoms of $\widehat{\gamma_{\mu}}$ are
exactly those $\xi$ where the Fourier–Bohr coefficient does not vanish.
Indeed, this is assumed in large parts of the physics literature. In the
mathematics literature this is sometimes known as Bombieri–Taylor conjecture
(after [7, 8] where this was assumed without any reasoning). An even stronger
condition found in many places is that
$\widehat{\gamma_{\mu}}(\\{\xi\\}=|a_{\xi}|^{2}$ holds (see for example [9,
13, 14, 18, 19]). This condition is known as consistent phase property. Our
treatment above provides the most general treatment of Bombieri–Taylor
conjecture so far. In particular, it is not restricted to examples coming from
(uniquely) ergodic dynamical systems. However, it does not prove the
consistent phase property.

##  8\. Application: Existence of the refined Eberlein decomposition of an
arbitrary measure

One open problem in diffraction theory is the following question:

######  Question 8.1.

For which $\omega\in{\mathcal{M}}^{\infty}(G)$ and van Hove net
${\mathcal{A}}$ such that $\gamma$ exists (with respect to ${\mathcal{A}}$)
can we find some measures
$\omega_{dpp},\omega_{dac},\omega_{dsc}\in{\mathcal{M}}^{\infty}(G)$ with the
following properties:

  * (a)

$\omega=\omega_{dpp}+\omega_{dac}+\omega_{dsc}$.

  * (b)

The autocorrelations $\eta_{dpp},\eta_{dsc},\eta_{dac}$ of
$\omega_{dpp},\omega_{dac},\omega_{dsc}$ exist with respect to ${\mathcal{A}}$
and

$\widehat{\eta_{dpp}}=\left(\widehat{\gamma}\right)_{pp}\mbox{ and
}\widehat{\eta_{dac}}=\left(\widehat{\gamma}\right)_{ac}\mbox{ and
}\widehat{\eta_{dsc}}=\left(\widehat{\gamma}\right)_{sc}\,.$

We will refer to any such decomposition as _refined Eberlein decomposition of
$\omega$_.

We are not able to answer this question here but we note that our main result
(Theorem 6.1) has the following consequence:

######  Corollary 8.2.

Let $\mu,\nu,\omega$ be translation bounded measures and ${\mathcal{A}}$ a van
Hove net such that the autocorrelations $\gamma_{\mu}$ of $\mu$,
$\gamma_{\nu}$ of $\nu$ and $\gamma_{\omega}$ of $\omega$ exist with respect
to ${\mathcal{A}}$. If
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}$
is pure point,
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}$
is absolutely continuous and
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\omega}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\omega}}{\tmpbox}$
of $\omega$ is singular continuous, then

$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}=\text{\textlquill}\mu,\omega\text{\textrquill}_{{\mathcal{A}}}=\text{\textlquill}\nu,\omega\text{\textrquill}_{{\mathcal{A}}}=0\,.$

In particular, for all $a,b,c\in{\mathbb{C}}$ the autocorrelation of
$a\mu+b\nu+c\omega$ exists with respect to ${\mathcal{A}}$ and

$\left(\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{a\mu+b\nu+c\omega}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{a\mu+b\nu+c\omega}}{\tmpbox}\right)_{pp}=|a|^{2}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\mu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\mu}}{\tmpbox}\mbox{
and
}\left(\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{a\mu+b\nu+c\omega}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{a\mu+b\nu+c\omega}}{\tmpbox}\right)_{ac}=|b|^{2}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\nu}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\nu}}{\tmpbox}\mbox{
and
}\left(\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{a\mu+b\nu+c\omega}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{a\mu+b\nu+c\omega}}{\tmpbox}\right)_{sc}=|c|^{2}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\gamma_{\omega}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\gamma_{\omega}}{\tmpbox}\,.$

∎

This allows us to give the following characterisation for existence of an
Eberlein decomposition.

######  Theorem 8.3.

Let $\omega$ be a translation bounded measure whose autocorrelation $\gamma$
exists with respect to ${\mathcal{A}}$. Then, the following are equivalent:

  * (i)

The refined Eberlein decomposition of $\omega$ exists with respect to
${\mathcal{A}}$.

  * (ii)

There exists some measures $\omega_{1},\omega_{2},\omega_{3}$ such that
$\omega=\omega_{1}+\omega_{2}+\omega_{3}$, the autocorrelations of
$\gamma_{1},\gamma_{2},\gamma_{3}$ of $\omega_{1},\omega_{2},\omega_{3}$
exists with respect to ${\mathcal{A}}$ and
$\widehat{\gamma_{1}},\widehat{\gamma_{2}},\widehat{\gamma_{3}}$ are pure
point, absolutely continuous and singular continuous, respectively.

###### Proof.

(i) $\Longrightarrow$ (ii) is obvious.

(ii) $\Longrightarrow$ (i) is Corollary 8.2. ∎

Let us note here in passing that this result simply says that to get a refined
Eberlein decomposition we only need to write our measure $\omega$ as the sum
of three measures of pairwise distinct spectral purity. If this is the case,
then the diffractions of three measures give us exactly the Lebesgue
decomposition of the diffraction of the initial measure.

##  9\. Application: Orthogonality of dynamical systems

In this section we assume that $G$ is not only a locally compact Abelian group
but also $\sigma$-compact. Let $\lambda$ be the Haar measure on $G$. A
dynamical system over $G$ is a triple $(X,\alpha,m)$ consisting of a compact
space $X$, a continuous action $\alpha$ of $G$ on $X$ and an $\alpha$
invariant probability measure $m$. The dynamical system is called ergodic if
any $\alpha$-invariant measurable set has measure in $\\{0,1\\}$. Whenever
$(X,\alpha,m)$ is an ergodic dynamical system and ${\mathcal{A}}$ is a van
Hove sequence, we say that Birkhoff theorem holds along ${\mathcal{A}}$ if for
any integrable $f$ the equality

$\lim_{n\to\infty}\frac{1}{|A_{n}|}\int_{A_{n}}f(\alpha_{t}x)\mbox{d}t=\int_{X}f(x)\mbox{d}m$

holds for $m$ almost every $x\in X$. Any ergodic dynamical system admits a van
Hove sequence along which Birkhoff ergodic theorem holds [22].

Any dynamical system comes with a unitary representation $T=T^{X}$ of $G$ on
$L^{2}(X,m)$ given by $T_{t}f=f\circ\alpha_{t}$. The spectral measure of $f\in
L^{2}(X,m)$ is denoted by $\varrho_{f}$.

For $f:X\longrightarrow{\mathbb{C}}$ and $x\in X$ we define
$f_{x}:G\longrightarrow{\mathbb{C}}$ by

$f_{x}(t):=f(\alpha_{t}x)\,.$

We now provide a variant of [15]:

######  Proposition 9.1.

Assume that $G$ is $\sigma$-compact and has a dense countable set. Consider an
ergodic dynamical system $(X,\alpha,m)$. Let ${\mathcal{A}}$ be a van Hove
sequence along which the Birkhoff ergodic theorem holds. Then, for any $f\in
C(X)$ and almost every $x\in X$ the reflected Eberlein convolution
$\text{\textlquill}f_{x},f_{x}\text{\textrquill}_{{\mathcal{A}}}$ exists and

$\text{\textlquill}f_{x},f_{x}\text{\textrquill}_{{\mathcal{A}}}(t)=\widecheck{\sigma_{f}}(t)\,.$

∎

###### Proof.

Let $D=\\{t_{n}:n\in\mathbb{N}\\}$ be a countable dense set in $G$. Then,
there exists some set $X_{n}\subseteq X$ of full measure such that, for all
$x\in X_{n}$ we have

$\lim_{i}\frac{1}{|A_{i}|}\int_{A_{i}}f_{x}(s)\overline{f_{x}(s-t_{n})}\mbox{d}s=\langle
f,T_{t_{n}}f\rangle=\widecheck{\sigma_{f}}(t_{n})\,.$

Then $Y=\bigcap_{n}X_{n}$ has full measure in $X$ and for all $x\in Y$ and all
$n$ we have

$\lim_{i}\frac{1}{|A_{i}|}\int_{A_{i}}f_{x}(s)\overline{f_{x}(s-t_{n})}\mbox{d}s=\langle
f,T_{t_{n}}f\rangle=\widecheck{\sigma_{f}}(t_{n})\,.$

Then, $\text{\textlquill}f,f\text{\textrquill}_{{\mathcal{A}}}$ exists by
Lemma 3.2 and for all $n$ we have

$\text{\textlquill}f_{x},f_{x}\text{\textrquill}_{{\mathcal{A}}}(t_{n})=\widecheck{\sigma_{f}}(t_{n})\,.$

The claim follows now by continuity and denseness of $D$. ∎

Given this proposition our main result on orthogonality has the following
immediate consequence.

######  Corollary 9.2.

Assume that $G$ is $\sigma$-compact and has a dense countable set. Let
$(X,\alpha,m)$ and $(X^{\prime},\alpha^{\prime},m^{\prime})$ be ergodic
dynamical systems. Let ${\mathcal{A}}$ be a van Hove sequence along which the
Birkhoff ergodic theorem holds (for both systems). Let $f\in L^{2}(X,m)$ and
$g\in L^{2}(X^{\prime},m^{\prime})$ be given with $\sigma_{f}\perp\sigma_{g}$.
Then, for $m\times m^{\prime}$ almost every $(x,x^{\prime})\in X\times
X^{\prime}$ we have

$\text{\textlquill}f_{x},g_{x^{\prime}}\text{\textrquill}_{{\mathcal{A}}}=0\,.$

∎

As an immediate consequence, we get the following.

######  Corollary 9.3.

In the situation of Lemma 9.2, assume furthermore that $(X,\alpha,m)$ has pure
point spectrum (i.e. all spectral measures are pure point measures) and
$(X,\alpha,m^{\prime})$ is weak mixing (i.e. the spectral measures of all
$f\perp 1$ are continuous). Let $f\in C(X)$ and $g\in C(X^{\prime})$ be given.
Then, for $m\times m^{\prime}$ almost every $(x,x^{\prime})$ the equality

$\displaystyle\text{\textlquill}f_{x},g_{x^{\prime}}\text{\textrquill}_{{\mathcal{A}}}=\langle
1,g\rangle\langle f,1\rangle 1_{G}$
$\displaystyle=\left(\int_{X^{\prime}}\overline{g}\mbox{d}m^{\prime}\right)\left(\int_{X}f\mbox{d}m\right)1_{G}$

holds.

###### Proof.

We can decompose $g=\langle g,1\rangle 1+h$ with $h\perp 1$. By $h\perp 1$
and, as $(X^{\prime},\alpha^{\prime},m^{\prime})$ is weakly mixing, the
spectral measure $\sigma_{h}$ is continuous. Hence,
$\gamma_{h_{x^{\prime}}}=\sigma_{h}$ is continuous for almost every
$x^{\prime}\in X^{\prime}$. Now, the result follows directly from the
preceding corollary. ∎

##  10\. Existence of the reflected Eberlein convolution for Besicovitch
almost periodic measures

In this section we discuss existence of the reflected Eberlein convolution for
Besicovitch almost periodic measures. In particular, we determine the
orthogonal complement of such measures.

Throughout this section we consider a $\sigma$-compact locally compact Abelian
group $G$ and we let a van Hove sequence ${\mathcal{A}}$ be given.

A translation bounded measure $\mu$ is called Besicovitch almost periodic if
$\mu\ast\varphi$ belongs to the space of Besicovitch 2- almost periodic
functions discussed above (in Example 4.8) for all $\varphi\in
C_{\mathsf{c}}(G)$.

Besicovitch almost periodic measures admit many reflected Eberlein
convolutions. In fact, we have the following characterization.

######  Theorem 10.1.

Let $G$ be a $\sigma$-compact LCAG. Then, the following statements are
equivalent for the translation bounded $\nu$.

  * (i)

The reflected Eberlein convolution
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists for all
Besicovitch almost periodic measures $\mu\in{\mathcal{M}}^{\infty}(G)$.

  * (ii)

The Fourier–Bohr coefficients $a_{\xi}(\nu)$ exist for all
$\xi\in\widehat{G}$.

Moreover, for $\mu,\nu$ satisfying conditions (i) and (ii) the measure
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ is the unique
strongly almost periodic measure satisfying the generalized consistency phase
property:

$a_{\xi}(\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}})=a_{\xi}(\mu)\overline{a_{\xi}(\nu)}\qquad\mbox{
for all }\xi\in\widehat{G}\,.$

###### Proof.

(i) $\Longrightarrow$ (ii) follows immediately from the fact that
$\xi\theta_{G}$ is Besicovitch almost periodic.

(ii) $\Longrightarrow$ (i) and the last claim follows from [20, Proposition
4.38] (compare [20, Proposition 3.26]). ∎

From the previous result and the fact that a strongly almost periodic measure
is zero if and only if all its Fourier–Bohr coefficients are 0 (see for
example [23, Corollary 4.6.10]) we obtain the following result on
orthogonality.

######  Corollary 10.2 (Orthogonal complement of Besicovitch almost periodic
measures).

Let $\nu$ be a translation bounded measure. Then
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}=0$ for all
Besicovitch almost periodic $\mu$ if and only if all Fourier–Bohr coefficients
$a_{\xi}^{\mathcal{A}}(\nu)$ of $\nu$ exist and vanish.

##  Appendix A Some background on translation bounded measures

Let $V\subset G$ be an open subset of $G$ with compact closure. Let $C>0$. We
denote by ${\mathcal{M}}_{C,V}(G)$ the set of those measures $\mu$ with

$\|\mu\|_{V}:=\sup_{t\in G}|\mu|(t+V)\leq C\,.$

Let now $W$ be another open relative compact set. Then, we can cover
$\overline{W}$ by finitely many translates of $V$. Hence, there exists a $D>0$
with

${\mathcal{M}}_{C,V}(G)\subset{\mathcal{M}}_{D,W}(G)\,.$

In particular,

${\mathcal{M}}^{\infty}(G):=\bigcup_{C>0}{\mathcal{M}}_{C,V}(G)$

is independent of $V$ open and relatively compact. The elements of
${\mathcal{M}}^{\infty}(G)$ are called translation bounded measures.

We recall the following statement of [3].

######  Proposition A.1 (Compactness of ${\mathcal{M}}_{C,V}(G)$).

For any $C>0$ and $V\subset G$ open with compact closure the set
${\mathcal{M}}_{C,V}(G)$ is compact.

Moreover, if $G$ is second countable, the vague topology is metrisable on
${\mathcal{M}}_{C,V}(G)$.∎

We next gather the following (well-known) characterizations of vague
convergence of measures.

######  Lemma A.2 (Characterization of vague convergence).

For a net $(\mu_{i})_{i\in I}\in{\mathcal{M}}_{C,V}(G)$ the following
statements are equivalent:

  * (i)

$\mu_{i}$ converges vaguely to some $\mu\in{\mathcal{M}}_{C,V}(G)$.

  * (ii)

$\mu_{i}(\varphi)$ converges for any $\varphi\in C_{\mathsf{c}}(G)$.

  * (iii)

$\mu_{i}\ast\varphi(0)$ converges for any $\varphi\in C_{\mathsf{c}}(G)$.

  * (iv)

$\mu_{i}\ast\varphi\ast\widetilde{\psi}(0)$ converges for any $\varphi,\psi\in
C_{\mathsf{c}}(G)$.

###### Proof.

(i)$\Longrightarrow$(iv): This is clear as

$\mu_{i}\ast\varphi\ast\widetilde{\psi}(0)=\mu_{i}(\varphi^{\dagger}\ast\overline{\psi})$

and $\varphi^{\dagger}\ast\overline{\psi}\in C_{\mathsf{c}}(G)$.

(iv)$\Longrightarrow$ (iii):

Let $\varepsilon>0$ be given. Let $W$ be an arbitrary open relatively compact
set containing the support of $\varphi$. Chose $D>0$ with
${\mathcal{M}}_{C,V}\subset{\mathcal{M}}_{D,W}$. Chose $\psi\in
C_{\mathsf{c}}(G)$ such that $\varphi\ast\widetilde{\psi}$ is supported in $W$
and

$\|\varphi-\varphi\ast\widetilde{\psi}\|_{\infty}\leq\varepsilon\,.$

Then

$|\mu_{i}\ast\varphi\ast\widetilde{\psi}-\mu_{i}\ast\varphi\|_{\infty}\leq
2\varepsilon D\,.$

This easily gives the desired implication.

(iii)$\Longrightarrow$(ii): For any measure $\mu$ and $\varphi\in
C_{\mathsf{c}}(G)$ we have $\mu(\varphi^{\dagger})=\mu\ast\varphi(0)$. This
easily gives the desired implication.

(ii)$\Longrightarrow$(i): We can define
$\mu:C_{\mathsf{c}}(G)\longrightarrow{\mathbb{C}},\mu(\varphi)=\lim_{i}\mu_{i}(\varphi)$.
Then, $\mu$ is linear as it is a pointwise limit of linear functionals.

Next, fix a compact set $K\subseteq G$ and let $U$ be a pre-compact open set
such that $K\subseteq U$. Let $D>0$ be so that

${\mathcal{M}}_{V,C}\subseteq{\mathcal{M}}_{U,D}\,.$

Now, for all $\varphi\in C_{\mathsf{c}}(G)$ with $\sup(\varphi)\subseteq K$ we
have

$\displaystyle|\mu(\varphi)|$
$\displaystyle=\left|\lim_{i}\mu_{i}(\varphi)\right|\leq\sup_{i}\left|\mu_{i}(\varphi)\right|\leq\sup_{i}\left|\mu_{i}\right|(|\varphi|)$
$\displaystyle\leq\sup_{i}\left|\mu_{i}\right|(\|\varphi\|_{\infty}1_{K})\leq\|\varphi\|_{\infty}\sup_{i}\left|\mu_{i}\right|(U)\leq
D\|\varphi\|_{\infty}\,.$

This shows that $\mu$ is a measure. By its definition, $\mu_{i}$ converges
vaguely to $\mu$.

Finally, since $U$ is open, for all $t\in G$ we have by the inner regularity
of $|\mu|$:

$\displaystyle|\mu|(t+V)|$ $\displaystyle=\sup\\{|\mu|(\varphi):\varphi\in
C_{\mathsf{c}}(G),\varphi\leq 1_{t+V}\\}=\sup\\{|\mu(\psi)|:\psi\in
C_{\mathsf{c}}(G),|\psi|\leq 1_{t+V}\\}$
$\displaystyle=\sup\\{|\lim_{i}\mu_{i}(\psi)|:\psi\in
C_{\mathsf{c}}(G),|\psi|\leq 1_{t+V}\\}$
$\displaystyle\leq\sup\\{|\mu_{i}(\psi)|:\psi\in C_{\mathsf{c}}(G),|\psi|\leq
1_{t+V},i\in I\\}$ $\displaystyle\leq\sup\\{|\mu_{i}|(|\psi|):\psi\in
C_{\mathsf{c}}(G),|\psi|\leq 1_{t+V},i\in I\\}$
$\displaystyle\leq\sup\\{|\mu_{i}|(1_{t+V}):i\in
I\\}\leq\sup\\{\|\mu_{i}\|_{V}:i\in I\\}\leq C\,.$

Taking the supremum over all $t\in G$ we get $\mu\in{\mathcal{M}}_{C,V}(G)$. ∎

We now turn to a universal bound (compare [17, Section 9] or [27]).

######  Proposition A.3 (A universal bound).

Let $V\subset G$ be a nonempty, open relatively compact set. Then, for all
$\nu\in{\mathcal{M}}^{\infty}(G)$ and every relatively compact $B\subset G$ we
have

$|\nu|(B)\leq\frac{|B-V|}{|V|}\,\|\nu\|_{V}\,.$

###### Proof.

A direct calculation shows $1_{B}\leq\frac{1}{|V|}1_{B-V}\ast 1_{V}.$ This
gives

$\displaystyle|\nu|(B)$
$\displaystyle\leq\frac{1}{|V|}\int_{G}\int_{G}1_{V}(t-s)\,1_{B-V}(s)\,\mbox{d}s\,\mbox{d}|\nu|(t)$
$\displaystyle\stackrel{{\scriptstyle\mbox{Fubini}}}{{=\joinrel=\joinrel=\joinrel=\joinrel=\joinrel=\joinrel=}}\frac{1}{|V|}\int_{G}1_{B-V}(s)\left(\int_{G}1_{V}(t-s)\mbox{d}|\nu|(t)\right)\mbox{d}s$
$\displaystyle\leq{\|\nu\|_{V}}\frac{1}{|V|}\int_{G}1_{B-V}(s)\mbox{d}s=\frac{|B-V|}{|V|}\|\nu\|_{V}\,.$

This finishes the proof. ∎

As a consequence, we get:

######  Proposition A.4.

Let $\nu\in{\mathcal{M}}^{\infty}(G)$ be given. Let $(A_{i})$ be a van Hove
net. Then,

$\limsup_{i}\frac{|\nu|(A_{i})}{|A_{i}|}\leq\frac{{\|\nu\|_{V}}}{|V|}<\infty\,.$

###### Proof.

From the previous proposition we infer

$|\nu|(A_{i})\leq\frac{\|\nu\|_{V}}{|V|}|A_{i}-V|\,.$

Now, we have

$(A_{i}-V)\subset\overline{(A_{i}-\overline{V})}\setminus A_{i}\cup
A_{i}\subset\partial^{\overline{V}}A_{i}\cup A_{i}\,.$

This immediately gives the desired statement. ∎

######  Corollary A.5.

Let $\mu,\nu$ be translation bounded measures, $(A_{i})$ a van Hove net and
$V=-V\subseteq G$ be a fixed open, relatively compact set. Then, there exists
an index $i_{0}$ and some $\kappa$ such that, for all
$\mu,\nu\in{\mathcal{M}}^{\infty}(G)$ and all $i>i_{0}$ we have

$\|\frac{1}{|A_{i}|}\mu|_{A_{i}}\ast\widetilde{\nu|_{A_{i}}}\|_{V}\leq\kappa\|\mu\|_{V}\|\nu\|_{V}$

In particular,

$\\{\frac{1}{|A_{i}|}\mu|_{A_{i}}\ast\widetilde{\nu|_{A_{i}}}:\mu,\nu\in{\mathcal{M}}_{C,V},i>i_{0}\\}\subseteq{\mathcal{M}}_{\kappa
C,V}\,.$

###### Proof.

Note first that since $V=-V$ we have

$\|\widetilde{\nu|_{A_{i}}}\|_{V}\leq\|\nu\|_{V}\,.$

Moreover, by Proposition A.4, there exists some $i_{0}$ such that, for all
$i>i_{0}$ we have

$\left|\frac{1}{|A_{i}|}\mu_{A_{i}}\right|(G)=\frac{1}{|A_{i}|}|\mu|(A_{i})\leq\frac{2C}{|V|}\,.$

The claim follows now immediately from [30, Lemma 6.1]. ∎

##  Appendix B Universal van Hove nets

In this section we prove the existence of ”universal” van Hove nets, along
which all reflected Eberlein convolutions of functions and measures exist, as
well as all Fourier–Bohr coefficients.

To make the proofs easier to follow we do them in 3 steps.

######  Lemma B.1 (Universal van Hove net for Fourier–Bohr coefficients).

Let $G$ be a LCAG and $\\{A_{i}\\}_{i\in I}$ a van Hove net. Then, there
exists a subnet $\\{B_{j}\\}_{j\in J}$ of $A_{i}$ such that, for all $f\in
L^{\infty}(G)$ and all $\chi\in\hat{G}$ the Fourier–Bohr coefficient
$a_{\chi}^{{\mathcal{B}}}(f)$ exists.

###### Proof.

By linearity of the Fourier–Bohr coefficient, it suffices to consider $f\in
L^{\infty}(G)$ with $\|f\|_{\infty}\leq 1$. Let us consider

$X:=\\{(f,\chi):f\in L^{\infty}(G),\|f\|_{\infty}\leq
1,\chi\in\widehat{G}\\}\,.$

Then,

$\\{(\frac{1}{|A_{i}|}\int_{A_{i}}\chi(t)f(t)\mbox{d}t)_{(f,\chi)\in
X}\\}_{i\in I}$

is a net in $D^{X}$ where $D=\\{z\in{\mathbb{C}}:|z|\leq 1\\}$. This is
compact by Tychonoff’s theorem. Therefore, this net has a convergent subnet
$(y_{j})_{j\in J}$. This means that there exists a monotone final function
$h:J\to I$ such that for all $j\in J$ we have

$y_{j}=\bigl{(}\frac{1}{|A_{h(j)}|}\int_{A_{h(j)}}\chi(t)f(t)\mbox{d}t\bigr{)}_{(f,\chi)\in
X}$

Defining $B_{j}=A_{h(j)}$ for all $j\in J$ gives the claim for all $f\in
L^{\infty}(G)$ with $\|f\|_{\infty}\leq 1$. ∎

######  Lemma B.2 (Universal van Hove net for reflected Eberlein convolution
of functions).

Let $G$ be a LCAG and $\\{A_{i}\\}_{i\in I}$ a van Hove net. Then, there
exists a subnet $\\{B_{j}\\}_{j\in J}$ of $A_{i}$ such that, for all $f,g\in
L^{\infty}(G)$ reflected Eberlein convolution
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{B}}}$ exists.

###### Proof.

By linearity of the reflected Eberlein convolution in both arguments, it
suffices to consider $f,g\in L^{\infty}(G)$ with
$\|f\|_{\infty},\|g\|_{\infty}\leq 1$. Let

$X:=\\{(f,g,t):f,g\in L^{\infty}(G),t\in G,\|f\|_{\infty}\leq
1,\|g\|_{\infty}\leq 1\\}\,.$

Then,

$\\{(\frac{1}{|A_{i}|}\int_{A_{i}}f(s)\overline{g(s-t)}\mbox{d}s)_{(f,g,t)\in
X}\\}_{i\in I}$

is a net in $D^{X}$ where $D=\\{z\in{\mathbb{C}}:|z|\leq 1\\}$. This is
compact by Tychonoff’s theorem. Therefore, this net has a convergent subnet
$(y_{j})_{j\in J}$.

Similarly to the above, this yields a subnet ${\mathcal{B}}$ of
${\mathcal{A}}$ such that for all $f,g\in L^{\infty}(G)$ with
$\|f\|_{\infty}\leq 1,\|g\|_{\infty}\leq 1$ the reflected Eberlein convolution
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{B}}}$ exists. ∎

######  Lemma B.3 (Universal van Hove net for reflected Eberlein convolution
of measures).

Let $G$ be a LCAG and $\\{A_{i}\\}_{i\in I}$ a van Hove net. Then, there
exists a subnet $\\{B_{j}\\}_{j\in J}$ of $A_{i}$ such that, for all
$\mu,\nu\in{\mathcal{M}}^{\infty}(G)$ the reflected Eberlein convolution
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{B}}}$ exists.

###### Proof.

This could be shown similarly to the arguments given in the proofs of the
proceeding two lemmas. However, it is also a direct consequence of the
preceding lemma and our discussion of the Eberlein convolution in Lemma 2.8. ∎

Applying the three results in succession, we get:

######  Theorem B.4 (Existence of universal van Hove net for Fourier–Bohr
coefficients).

Every LCAG admits a van Hove net with the following properties:

  * (a)

For all $f\in L^{\infty}(G)$ the Fourier–Bohr coefficient
$a_{\chi}^{\mathcal{A}}(f)$ exists.

  * (b)

For all $f,g\in L^{\infty}(G)$,
$\text{\textlquill}f,g\text{\textrquill}_{{\mathcal{A}}}$ exists.

  * (c)

For all $\mu,\nu\in{\mathcal{M}}^{\infty}(G)$,
$\text{\textlquill}\mu,\nu\text{\textrquill}_{{\mathcal{A}}}$ exists.

  * (d)

For all $\mu\in{\mathcal{M}}^{\infty}(G)$, the autocorrelation
$\gamma=\text{\textlquill}\mu,\mu\text{\textrquill}_{{\mathcal{A}}}$ exists
with respect to ${\mathcal{A}}$.

Furthermore, any van Hove net has a subnet with these properties.

∎

######  Definition B.5.

We will refer to any net satisfying Theorem B.4 as an _universal van Hove
net_.

For such nets, [20] gives:

######  Corollary B.6.

Let ${\mathcal{A}}$ be an universal van Hove net in $G$. Then,

  * (a)

$\text{\textlquill}\,,\,\text{\textrquill}_{{\mathcal{A}}}$ is a mapping from
$L^{\infty}(G)\times L^{\infty}(G)$ into $WAP(G)$.

  * (b)

$\text{\textlquill}\,,\,\text{\textrquill}_{{\mathcal{A}}}$ is a mapping from
${\mathcal{M}}^{\infty}(G)\times{\mathcal{M}}^{\infty}(G)$ into
$\mathcal{WAP}(G)$.

∎

Let us conclude by pointing out that, when working with an universal van Hove
net, all the assumptions on the existence of the reflected Eberlein
convolution throughout Section 4 can be dropped. While one could write the
results that way, in general one works with an explicit van Hove sequence or
net, and we expect that in general an universal van Hove net cannot be
constructed explicitly.

### Acknowledgments

This work was started when NS visited Jena University, and he would like to
thank the Mathematics Department for hospitality. NS was supported by the
Natural Sciences and Engineering Council of Canada via grant 2020-00038, and
he is grateful for the support.

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