# A classification of Fourier summation formulas and crystalline measures

Felipe Gonçalves  IMPA - Estrada Dona Castorina 110, Rio de Janeiro, RJ -
Brasil, 22460-320<EMAIL_ADDRESS>
###### Abstract.

We completely classify Fourier summation formulas, and in particular, all
crystalline measures with quadratic decay. Our classification employs
techniques from almost periodic functions, Hermite-Biehler functions, de
Branges spaces and Poisson representation. We show how our classification
generalizes recent results of Kurasov & Sarnak and Olevskii & Ulanovskii. As
an application, we give a new classification result for nonnegative measures
with uniformly discrete support that are bounded away from zero on their
support. In the Appendix we attempt to give a systematic account of the
several types of Fourier summation formulas, including a new construction
using eta-products, generalizing an old example of Guinand.

###### Key words and phrases:

##  1\. Introduction

To understand the atomic structure of a crystal (of some material) one can
fire a beam of electrons into the object and observe an electron diffraction
pattern on a screen111This technique is called high resolution electron
microscopy (HRTEM), but another common technique is X-ray microscopy.. In most
materials one observes a periodic diffraction pattern, such as a two-
dimensional lattice with $6$-fold symmetry, as in Tantalum pentoxide ${\rm
Ta}_{2}{\rm O}_{5}$. However, this is not the case with Holmium-magnesium-zinc
Ho-Mg-Zn, where one observes some $5$-fold and $10$-fold symmetry, hence a
non-periodic diffraction pattern Fig. 1. The first to record such phenomena
experimentally was materials scientist Dan Shechtman, in 1982, and today these
materials are called quasicrystals. He was awarded the Nobel Prize in
Chemistry in 2011 for his breakthrough. Mathematically, if $\mu$ represents
the crystal (so $\mu\geq 0$ and $\operatorname{\mathrm{supp}}(\mu)$ is
uniformly discrete) then what we see in the diffraction pattern is effectively
$|\widehat{\mu}|^{2}$ (where $\widehat{\mu}$ is the Fourier transform of
$\mu$). A famous inverse problem is that of reconstructing atomic structure
from diffraction data, overcoming the issue that diffraction only determines
intensity of spots but loses phase information. Real-world quasicrystals (such
as Decagonite $\text{Al}_{71}\text{Ni}_{24}\text{Fe}_{5}$), although uniformly
discrete in physical space, typically have everywhere dense diffraction
spectrum222We thank Michael Baake for pointed this out, among several other
remarks.. The diffraction pattern is a super-position of Dirac combs with
several intensities (Bragg peaks) however, due to instrument precision and
tuning, we only see peaks that have, say, an intensity above $10^{-4}$ of the
brightest peak, hence it visually seems to be locally finite, such as in Fig.
1, nevertheless the real measure $\widehat{\mu}$ has a dense support. On the
other hand, stable $1$D quasicrystals with locally finite spectra do exist in
nature, and can be found as projections along pseudo axis’s, see [43, Figure
1] and the references in [18, Ch. 7]. Nowadays there is unified framework to
construct measures $\mu$ whose spectrum fits the data observed in the
diffraction patterns of physical quasicrystals, these are called model sets
(or Meyer sets) [8, 18, 27, 35, 36, 38].

Figure 1. Diffraction pattern of Holmium-magnesium-zinc Ho-Mg-Zn.

From the mathematical standpoint, and in the last few decades, many
mathematicians have wondered about what different types of measures $\mu$ with
pure point support and spectrum are there, and what kind of geometrical
properties of the spectrum are allowed under various conditions on $\mu$? The
big classification question from the mathematical side is:

Can on one classify all measures $\mu$ with pure point support and spectrum?

Freeman Dyson addressed this question in 2009 in his famous paper [17], using
the word quasicrystals for such measure, suggesting that such classification
mighty be a very difficult mathematical problem, but made the “outrageous
suggestion” it could potentially give a proof of the Riemann Hypothesis333We
do not share the same belief.. He said: “My suggestion is the following. Let
us pretend that we do not know that the Riemann Hypothesis is true. Let us
tackle the problem from the other end. Let us try to obtain a complete
enumeration and classification of one-dimensional quasicrystals. That is to
say, we enumerate and classify all point distributions that have a discrete
point spectrum.” Dyson stated this because of Guinand’s prime summation
formula (also known as Weyl’s summation formula), which we explain in the
Appendix. The notorious paper of Guinand [21], which inspired Dyson, grasps
such classification, but it was forgotten for sometime. Meyer resurrected
Guinand’s paper in [36].

We say a measure $\mu$ is crystalline if it has locally finite support and its
Fourier transform $\widehat{\mu}$ also has locally finite support444Such
measures are sometimes also called quasicrystals, Fourier quasicrystals, or
doubly sparse, depending on the author, area, context, decay conditions, etc.
We will stick with Meyer’s denomination of crystalline measure.. In this
scenario, the first geometric classification breakthrough occurred in the work
of Lev & Olevskii [29, 30], where they classify classical Poisson summation as
the only crystalline measures that have uniformly discrete support and
spectrum. Guinand’s example [21, p. 265] shows the theorem is tight (we
generalize his construction in the Appendix). Recently, the field is very
active, and the amazing works of Radchenko & Viazovska [40], Kurasov & Sarnak
[25], Olevskii & Ulanovskii [39] and Alon et. al. [2, 1], but also, the not so
recent works of Bohr [15], Besicovitch [10], Guinand [21], de Branges [12] and
Meyer [34], deeply inspired this paper.

In this paper we completely classify (what we call) Fourier summation
formulas, and in particular we classify crystalline measures that have
quadratic decay, although our results are more general and can be applied to
dense spectra. As far as we know, it is the first time classification results
are proven in such generality. Our classification involves heavily the theory
of almost periodic analytic functions [10] and de Branges spaces techniques
[12]. Almost periodic functions have permeated this are for some time now, for
example in the works of Guinand [21], Meyer [37] and Baake et. al. [4, 8, 9],
among others. On the other hand, the connection with de Branges spaces that we
make is totally new.

A Fourier summation formula should be one where the identity

$\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\mu(t)=\sum_{n\in\mathbb{N}}a(\lambda_{n})\varphi(\lambda_{n})$

holds, at the bear minimum, for all functions $\varphi\in
C^{\infty}_{c}(\mathbb{R})$, where
$\\{\lambda_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}$ is some sequence of real
nodes, $\\{a(\lambda_{n})\\}_{n\in\mathbb{N}}\subset\mathbb{C}$ are some
complex coefficients and $\mu$ is some complex-valued measure. There is a zoo
of formulas of such type which we describe in the Appendix, from ancient and
modern times. We choose the above form of because: (1) We want a summation
formula, so there must be a sum somewhere in the definition and such summation
must be singular with respect to Lebesgue measure; (2) Allowing the right hand
side above to have a vanishing singular part is boring, as otherwise any
$L^{1}$-function would give rise to a Fourier summation formula.

###  1.1. Overview

Theorems 1 and 2 are our main general classification results and we later
specialize them for measures with locally finite support (in particular, also
crystalline measures) in Theorems 3 and 4. Finally, we derive a new
characterization Theorem 5 for measures with uniformly discrete support such
that $\mu|_{\operatorname{\mathrm{supp}}(\mu)}\geq\varepsilon$, but with no
geometric assumption on $\widehat{\mu}$, which perhaps makes this result most
useful to physics.

##  2\. Main Results

We will have to introduce several things before we can state our first main
result. We say that a complex-valued measure $\mu$ on $\mathbb{R}$ is locally
finite if its total variation

$|\mu|(K)=\sup_{\sqcup_{j\in\mathbb{N}}P_{j}=K}\sum_{j\in\mathbb{N}}|\mu(P_{j})|<\infty$

is finite on every compact set $K\subset\mathbb{R}$, or equivalently, if there
are nonnegative locally finite Borel measures $\mu_{j}$ such that
$\mu=\mu_{1}-\mu_{2}+i(\mu_{3}-\mu_{4})$. We say that $\mu$ is tempered if it
is a complex-valued locally finite measure and there exists a tempered
distribution $u\in\mathcal{S}^{\prime}(\mathbb{R})$ such that
$u(\varphi)=\int_{\mathbb{R}}\varphi\mathrm{d}\mu$ for all $\varphi\in
C^{\infty}_{c}(\mathbb{R})$. Some authors might not concord with this
definition, but we are following the setup of [4]. We say $\mu$ is strongly
tempered if $\varphi\in L^{1}(|\mu|)$ for all
$\varphi\in\mathcal{S}(\mathbb{R})$. By [4, Prop. 2.5], this is equivalent to
the degree of $\mu$

$\displaystyle\deg(\mu):=\min\bigg{\\{}n\in\mathbb{Z}:\int_{\mathbb{R}}\frac{\mathrm{d}|\mu|(t)}{(1+t^{2})^{n/2}}<\infty\bigg{\\}}$
(1)

being $<\infty$. In particular, the functional
$\varphi\mapsto\int_{\mathbb{R}}\varphi\mathrm{d}\mu$ defines a tempered
distribution, hence $\mu$ is tempered. The implication [tempered]
$\Rightarrow$ [strongly tempered] is false. Indeed, the measure
$\mu=\sum_{n\geq
0}3^{n^{2}}({\boldsymbol{\delta}}_{n}-{\boldsymbol{\delta}}_{n+4^{-n^{2}}})$
is not strongly tempered but it is tempered, since $\int
2^{-t^{2}}\mathrm{d}|\mu|(t)\geq\sum_{n\geq 0}3^{n^{2}}2^{-n^{2}}=\infty$
but555Whenever it is convenient, we write $A\ll B$ to mean that there is $C>0$
such that $|A|\leq C|B|$. $|\sum_{n\geq
0}3^{n^{2}}(\varphi({n})-\varphi({n+4^{-n^{2}}}))|\ll\|\varphi^{\prime}\|_{\infty}$.

We say that a function $a:\mathbb{R}\to\mathbb{C}$ is locally summable if the
support set
$\operatorname{\mathrm{supp}}(a):=\\{\lambda\in\mathbb{R}:a(\lambda)\neq 0\\}$
is at most countable and for some enumeration
$\operatorname{\mathrm{supp}}(a)=\\{\lambda_{n}\\}_{n\geq 1}$ we have

$\sum_{\lambda_{n}\in[-T,T]}|a(\lambda_{n})|<\infty,$

for all $T>0$. This is equivalent to say that $\nu=\sum_{n\geq
1}a(\lambda_{n}){\boldsymbol{\delta}}_{\lambda_{n}}$ is locally finite. In
particular, local sums $\sum_{\lambda\in K}a(\lambda)$ are always well-defined
for any bounded set $K\subset\mathbb{R}$.

######  Definition 1 (Fourier Summation Pairs).

We say $(\mu,a)$ is a Fourier summation pair $(\mathrm{FS}$-pair$)$ if $\mu$
is a strongly tempered measure, $a:\mathbb{R}\to\mathbb{C}$ is locally
summable and

$\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\mu(t)=\sum_{\lambda\in\mathbb{R}}a(\lambda)\varphi(\lambda)$

for every $\varphi\in C^{\infty}_{c}(\mathbb{R})$. Equivalently, $\mu$ is a
strongly tempered measure and for some locally summable function
$a:\mathbb{R}\to\mathbb{C}$ we have
$\widehat{\mu}|_{C^{\infty}_{c}}=\sum_{\lambda\in\mathbb{R}}a(\lambda){\boldsymbol{\delta}}_{\lambda}$.

We use the following definition for Fourier transform:

$\widehat{\varphi}(\xi)=\int_{\mathbb{R}}\varphi(x)e^{-2\pi
ix\xi}\mathrm{d}x.$

Perhaps the two most simple examples of $\mathrm{FS}$-pairs are

$\displaystyle\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}t=\varphi(0)\ \
(\text{Fourier Inversion})\ \ \text{ and }\ \
\sum_{n\in\mathbb{Z}}\widehat{\varphi}(n)=\sum_{n\in\mathbb{Z}}\varphi(n)\ \
(\text{Poisson Summation}).$

######  Remark 1 (Real-Antipodal splitting).

In the following we show that any $\mathrm{FS}$-pair can be split into two
real-antipodal $\mathrm{FS}$-pairs. We say that a function
$a:\mathbb{R}\to\mathbb{C}$ is antipodal if
$a(-\lambda)=\overline{a(\lambda)}$ for every $\lambda\in\mathbb{R}$. We say
that an $\mathrm{FS}$-pair $(\mu,a)$ if real-antipodal is $\mu$ is real-valued
and $a(\cdot)$ is (and must be) antipodal. Indeed, if we are given an
arbitrary $\mathrm{FS}$-pair $(\mu,a)$ then we can write
$a_{1}(\lambda)=\frac{a(\lambda)+\overline{a(-\lambda)}}{2},\ \
a_{2}(\lambda)=\frac{\overline{a(-\lambda)}-a(\lambda)}{2i},$ and so
$a=a_{1}-ia_{2}$, with each $a_{j}$ antipodal, and we can write
$\mu=\mu_{1}-i\mu_{2}$, where $\mu_{1}={\rm Re}\,\mu$ and $\mu_{2}={\rm
Im}\,\mu$, and so each $\mu_{j}$ is real-valued. If $\varphi\in
C^{\infty}_{c}(\mathbb{R})$ is antipodal then $\widehat{\varphi}$ is real, and
if we further assume that either $\varphi$ is real-valued or imaginary-valued,
then from the identity

$\sum_{\lambda\in\mathbb{R}}(a_{1}(\lambda)-ia_{2}(\lambda))\varphi(\lambda)=\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\mu_{1}(t)-i\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\mu_{2}(t)$

and the fact that $\sum_{\lambda\in\mathbb{R}}a_{j}(\lambda)\varphi(\lambda)$
is real, one deduces that

$\sum_{\lambda\in\mathbb{R}}a_{j}(\lambda)\varphi(\lambda)=\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\mu_{j}(t)$

for $j=1,2$. Since for any $\varphi\in C^{\infty}_{c}(\mathbb{R})$ we have
$\varphi=\varphi_{1}-i\varphi_{2}$ where each $\varphi_{j}$ is antipodal, and
each $\varphi_{j}={\rm Re}\,\varphi_{j}-i{\rm Im}\,\varphi_{j}$, where ${\rm
Re}\,\varphi_{j}$ is real and even and $i{\rm Im}\,\varphi_{j}$ is imaginary
and odd (hence both are antipodal as well), we deduce by linearity that
$(\mu_{j},a_{j})$ is a real-antipodal $\mathrm{FS}$-pair for $j=1,2$. As an
example, note that
$\sum_{n\in\mathbb{Z}}i^{n}\varphi(n)=\sum_{n\in\mathbb{Z}}\widehat{\varphi}(n-1/4)$
has the real-antipodal splitting

$\displaystyle\sum_{n\in\mathbb{Z}}(-1)^{n}\varphi(2n)$
$\displaystyle=\frac{1}{2}\sum_{n\in\mathbb{Z}}\left(\widehat{\varphi}(n-1/4)+\widehat{\varphi}(n+1/4)\right)$
$\displaystyle\sum_{n\in\mathbb{Z}}(-1)^{n}\varphi(2n-1)$
$\displaystyle=\frac{1}{2i}\sum_{n\in\mathbb{Z}}\left(\widehat{\varphi}(n+1/4)-\widehat{\varphi}(n-1/4)\right).$

We say that a locally summable function $a:\mathbb{R}\to\mathbb{C}$ has
exponential growth if

$\sum_{\lambda\in\mathbb{R}}|a(\lambda)|e^{-c|\lambda|}<\infty$

for some $c>0$. We let $\mathbb{C}^{+}=\\{z=x+iy:y>0\\}$. Following
Besicovitch666Besicovitch considers the half-plane ${\rm Re}\,z>0$ instead of
${\rm Im}\,z>0$, so one has to mentally perform a rotation in order to use his
results. [10, Ch. 3], we say a holomorphic function
$f:\mathbb{C}^{+}\to\mathbb{C}$ is almost periodic if for every
$0<\alpha<\beta<\infty$ and $\varepsilon>0$ there is a relatively dense777A
set $\tau\subset\mathbb{R}$ is relatively dense if there exists $l>0$ such
that $(x,x+l)\cap\tau\neq\emptyset$ for every $x\in\mathbb{R}$. set of
$\varepsilon$-translations $\tau\subset\mathbb{R}$ such that

$\sup_{\alpha<{\rm Im}\,z<\beta}|f(z)-f(z+t)|\leq\varepsilon$

for every $t\in\tau$. We write $f\in\mathrm{AP}(\mathbb{C}^{+})$. If
$f\in\mathrm{AP}(\mathbb{C}^{+})$ it can be shown that

$\mathbb{E}f(\lambda):=\lim_{T\to\infty}\frac{1}{2T}\int_{-T+iy}^{T+iy}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z$

exists for all $\lambda\in\mathbb{R}$ and is independent of $y>0$ (see Section
4). It is straightforward to see that if instead
$f(\cdot+ic)\in\mathrm{AP}(\mathbb{C}^{+})$ for some $c>0$, then the above
limit exists and is independent of $y>c$. If $f\in\mathrm{AP}(\mathbb{C}^{+})$
we define the spectrum of $f$ by

${\rm spec}(f):=\\{\lambda\in\mathbb{R}:\mathbb{E}f(\lambda)\neq 0\\}.$

We say a holomorphic function $g:\mathbb{C}^{+}\to\mathbb{C}$ is of bounded
type888This is the same to say that $\log|g(z)|$ has a nonnegative harmonic
majorant. if there are two bounded holomorphic functions
$P,Q:\mathbb{C}^{+}\to\mathbb{C}$ such that $g=P/Q$.

The following is the first main result of this paper.

######  Theorem 1.

Let $(\mu,a)$ be a real-antipodal $\mathrm{FS}$-pair. Assume that
$\deg(\mu)\leq 2$ and that $a(\cdot)$ has exponential growth. Then we have:

  1. (I)

The limit

$\displaystyle
f(z)=\frac{1}{2}a(0)+\lim_{T\to\infty}\sum_{0<\lambda<T}a(\lambda)(1-\lambda/T)e^{2\pi
i\lambda z}$ (2)

converges uniformly in compact sets of $\mathbb{C}^{+}$;

  2. (II)

There is $c_{1}>0$ such that $f(\cdot+ic_{1})\in\mathrm{AP}(\mathbb{C}^{+})$;

  3. (III)

There are $c_{2},B>0$, with $c_{2}\geq c_{1}$, such that for any trigonometric
polynomial $p(x)=\sum_{n=1}^{N}p_{n}e^{2\pi i\theta_{n}x}$, with
$\\{\theta_{n}\\}\subset[0,\infty)$, we have

$\limsup_{T\to\infty}\bigg{|}\frac{1}{2T}\int_{-T}^{T}f(x+ic_{2})\overline{p(x)}\mathrm{d}x\bigg{|}\leq
B\max|p_{n}|;$

  4. (IV)

$\exp(f)$ is of bounded type and $\limsup_{y\to\infty}\frac{{\rm
Re}\,f(iy)}{y}=0$.

Conversely, suppose $f:\mathbb{C}^{+}\to\mathbb{C}$ is a given holomorphic
function such that items $({\rm II}),({\rm III})$ and $({\rm IV})$ hold true.
Then there is $c\geq 0$ such that the limit

$\mathbb{E}f(\lambda):=\lim_{T\to\infty}\frac{1}{2T}\int_{-T+iy}^{T+iy}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z$

exists for every $\lambda\in\mathbb{R}$ and $y>c$, is independent of $y$,
vanishes identically for $\lambda<0$ and defines a locally summable function
with exponential growth such that

$\sum_{\lambda\in\mathbb{R}}|\mathbb{E}f(\lambda)|e^{-2\pi y|\lambda|}<\infty$

for every $y>c$. Moreover, there is a unique real-valued locally finite
measure $\mu$ of degree at most $2$ satisfying

$\displaystyle\frac{f(z)+\overline{f(w)}}{z-\overline{w}}=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{\mathrm{d}\mu(t)}{(z-t)(\overline{w}-t)}\quad$ (3)

for all $w,z\in\mathbb{C}^{+}$. Furthermore, if we let

$\displaystyle a(\lambda)=\begin{cases}\mathbb{E}f(\lambda)&\mbox{if
}\lambda>0\\\ 2{\rm Re}\,\mathbb{E}f(0)&\mbox{if }\lambda=0\\\
\overline{\mathbb{E}f(-\lambda)}&\mbox{if }\lambda<0\end{cases}$ (4)

then $(\mu,a)$ is an real-antipodal $\mathrm{FS}$-pair and identity (2) holds.

The proof we give for Theorem 1 in the converse part actually shows a stronger
result.

######  Theorem 2.

Let $f:\mathbb{C}^{+}\to\mathbb{C}$ be a holomorphic function such that
properties $({\rm II})$ and $({\rm IV})$ in Theorem 1 hold true, but also the
following weaker version of property $({\rm III})$:

  1. (III∗)

There is $c_{2}\geq c_{1}$, and for every $M>0$ there is $B_{M}>0$, such that
for any trigonometric polynomial $p(x)=\sum_{n=1}^{N}p_{n}e^{2\pi
i\theta_{n}x}$, with $\\{\theta_{n}\\}\subset[0,M]$, we have

$\limsup_{T\to\infty}\bigg{|}\frac{1}{2T}\int_{-T}^{T}f(x+ic_{2})\overline{p(x)}\mathrm{d}x\bigg{|}\leq
B_{M}\max|p_{n}|;$

Then there is $c\geq 0$ such that the limit

$\mathbb{E}f(\lambda):=\lim_{T\to\infty}\frac{1}{2T}\int_{-T+iy}^{T+iy}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z$

exists for every $\lambda\in\mathbb{R}$ and $y>c$, is independent of $y$,
vanishes identically for $\lambda<0$ and defines a locally summable function.
Moreover, there is a unique real-valued locally finite measure $\mu$ of degree
at most $2$ satisfying (3), and if we define $a(\cdot)$ as in (4) then
$(\mu,a)$ is an real-antipodal $\mathrm{FS}$-pair and identity (2) holds.

Note that condition (III∗) is equivalent to $\mathbb{E}f(\lambda)$ being
locally summable.

######  Remark 2.

We claim that if $(\mu,a)$ is an $\mathrm{FS}$-pair such such that $\mu\geq 0$
and $a(\cdot)$ has exponential growth then $\deg(\mu)\leq 2$. Indeed, let
$\varphi\in C^{\infty}_{c}(\mathbb{R})$ be even, $0\leq\varphi\leq 1$,
$\int\varphi=\varphi(0)=1$ and
$\operatorname{\mathrm{supp}}(\varphi)\subset(-1,1)$. Let
$\varphi_{\varepsilon}(x)=\varphi(x/\varepsilon)/\varepsilon$ and
$\psi_{\varepsilon}(x)=\widehat{\varphi}(x/\varepsilon)/\varepsilon$ for
$0<\varepsilon<1$. Let $g(x)=e^{-2\pi C|x|}$, with $C>0$ sufficiently large,
and note $\widehat{g}(\xi)=\frac{C}{\pi(C^{2}+\xi^{2})}$. We have

$\int_{\mathbb{R}}(\widehat{g}\widehat{\varphi}_{\varepsilon})*\psi_{\varepsilon}(t)\mathrm{d}\mu(t)=\sum_{\lambda\in\mathbb{R}}a(\lambda)g*\varphi_{\varepsilon}(\lambda)\widehat{\psi}_{\varepsilon}(\lambda).$

It is not hard to show that
$|g*\varphi_{\varepsilon}(\lambda)\widehat{\psi}_{\varepsilon}(\lambda)|\leq
e^{-2\pi C(|\lambda|-1)}$. Thus, we conclude the right hand side above is
uniformly bounded for $0<\varepsilon<1$. Since $\mu\geq 0$ and
$(\widehat{g}\widehat{\varphi}_{\varepsilon})*\psi_{\varepsilon}\to\widehat{g}$
pointwise, as $\varepsilon\to 0$, we can apply Fatou’s Lemma to deduce
$\int\frac{\mathrm{d}\mu(t)}{C^{2}+t^{2}}<\infty$.

######  Remark 3.

Let $(\mu,a)$ be an $\mathrm{FS}$-pair such that $\mu\geq 0$ and $a(\cdot)$
satisfies

$\displaystyle\sum_{\lambda\in\mathbb{R}}|a(\lambda)|e^{-c|\lambda|/\sigma(\lambda)}<\infty,$
(5)

for some $c>0$ and a function $\sigma:\mathbb{R}\to[1,\infty)$ such that
$\int_{|x|\geq 1}\frac{1}{x\sigma(x)}\mathrm{d}x<\infty$. We claim there is
$C>0$ such that $\mu([x,x+1])\leq C$ for all $x\in\mathbb{R}$, hence
$\deg(\mu)\leq 1$. For instance, one could take
$\sigma(\lambda)=\log(10+|\lambda|)(\log\log(10+|\lambda|))^{2}$. Indeed, we
only need to consider Ingham’s construction [22]. This is an auxiliary even
function $\varphi\in\mathcal{S}(\mathbb{R})$ such that $\varphi\geq 0$,
$\operatorname{\mathrm{supp}}(\widehat{\varphi})\subset[-1,1]$,
$\widehat{\varphi}\geq 0$, $\widehat{\varphi}(0)=1$ and

$\varphi(x)\leq Ce^{-c|x|/\sigma(x)}\quad(\text{for all }x\in\mathbb{R})$

for some $C>0$. Let $c>0$ be sufficiently small such that
$\widehat{\varphi}(x)\geq 1/2$ for $x\in[-c,c]$. The decay condition on
$a(\cdot)$ and a routine approximation argument shows the identity

$\int_{\mathbb{R}}\widehat{\varphi}(t-s)\mathrm{d}\mu(t)=\sum_{\lambda\in\mathbb{R}}a(\lambda)\varphi(\lambda)e^{2\pi
i\lambda s}$

holds for all $s\in\mathbb{R}$. In particular, there is $C>0$ such that
$\mu([s-c,s+c])$ for all $s\in\mathbb{R}$.

##  3\. Applications

We now apply Theorems 1 and 2 to classify crystalline pairs. Some of the
results and proofs presented here would be better understood after reading
Sections 4, 5 and 6. We say an entire function $E$ is of Hermite-Biehler class
if

$\displaystyle|E^{*}(z)|<|E(z)|\quad\text{for all }\ z\in\mathbb{C}^{+},$ (6)

where $E^{*}(z)=\overline{E(\overline{z})}$. In this case we always write
$E=A-iB$, where $A$ and $B$ are real entire functions999Entire functions which
attain only real values on the real line. defined by $A=(E^{*}+E)/2$ and
$B=(E^{*}-E)/(2i)$, and we always write $\Theta=E^{*}/E$. In particular we
have the following identity

$\displaystyle i\frac{A}{B}=\frac{1+\Theta}{1-\Theta}.$ (7)

Note that both $A$ and $B$ have only real zeros. We denote by
$\varphi=\varphi_{E}$ a phase function associated to $E$. This is
characterized by the condition $e^{i\varphi(x)}E(x)\in\mathbb{R}$ for all real
$x$ ($\varphi$ is unique modulo an integer multiple of $\pi$). For instance,
one could take $\varphi=\frac{1}{2i}\log\Theta$ after branch cutting all zeros
and poles of $\Theta$ by vertical semi-lines. It is not hard to show that

$\varphi^{\prime}(x)={\rm
Re}\,i\frac{E^{\prime}(x)}{E(x)}=\partial_{y}\log\Theta(x+iy)|_{y=0}>0,$

for all real $x$, and so $\varphi(x)$ is an increasing function of real $x$.
We also have that

$e^{2i\varphi(x)}=\frac{A(x)^{2}}{|E(x)|^{2}}-\frac{B(x)^{2}}{|E(x)|^{2}}+2i\frac{A(x)B(x)}{|E(x)|^{2}},$

for all real $x$. As a consequence, the points $\gamma\in\mathbb{R}$ such that
$\varphi(\gamma)\equiv 0\,(\text{mod }\pi)$ coincide with the real zeros of
$B/E$ and the points $s\in\mathbb{R}$ such that
$\varphi(s)\equiv\pi/2\,(\text{mod }\pi)$ coincide with the real zeros of
$A/E$ and, because $\varphi^{\prime}>0$, these zeros are simple. In
particular, $A/B$ has only simple real zeros and simple real poles, which
interlace, and

$\\{\gamma\in\mathbb{R}:\,\varphi_{j}(\gamma)\equiv 0\,({\rm
mod}\,\pi)\\}={\rm
Zeros}(B_{j}/A_{j})\quad\text{and}\quad\frac{1}{\varphi_{j}^{\prime}(\gamma)}=\mathop{\mathrm{Res}}_{z=\gamma}\,(A_{j}/B_{j})>0.$

Moreover, if $E$ has no real zeros then $A$ and $B$ have only real simple
zeros which interlace101010For more on this we recommend [12] and the
introduction of [20].. The next result characterizes $\mathrm{FS}$-pairs
$(\mu,a)$ for which $\mu\geq 0$ has locally finite support.

######  Theorem 3.

Let $E=A-iB$ be of Hermite-Biehler class such that
$A/B\in\mathrm{AP}(\mathbb{C}^{+})$ and
$\lambda\in\mathbb{R}\mapsto\mathbb{E}(A/B)(\lambda)$ is locally summable. Let

$\displaystyle\begin{split}a(\lambda)&=\overline{a(-\lambda)}=\mathbb{E}(iA/B)(\lambda)=\lim_{T\to\infty}\frac{1}{T}\int_{-T+i}^{T+i}i\frac{A(z)}{B(z)}e^{-2\pi
i\lambda z}\mathrm{d}z\quad(\text{for }\lambda>0),\\\ \frac{1}{2}a(0)&={\rm
Re}\,\mathbb{E}(iA/B)(0)={\rm
Re}\,\lim_{T\to\infty}\frac{1}{T}\int_{-T+i}^{T+i}i\frac{A(z)}{B(z)}\mathrm{d}z,\\\
\mu&=2\pi\sum_{\,\varphi_{E}(\gamma)\equiv 0\,({\rm
mod}\,\pi)}\frac{1}{\varphi^{\prime}_{E}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}.\end{split}$
(8)

Then $(\mu,a)$ is a real-antipodal $\mathrm{FS}$-pair such that $\mu\geq 0$
and $\mu$ has locally finite support111111A set $S\subset\mathbb{R}$ is
locally finite if $\\#(S\cap(a,b))<\infty$ for any $a<b$.. Conversely, if
$(\mu,a)$ is a real-antipodal $\mathrm{FS}$-pair such that $\mu\geq 0$,
$\operatorname{\mathrm{supp}}(\mu)$ is locally finite and $a(\cdot)$ has
exponential growth, then $(\mu,a)$ has to be built from the construction
above.

######  Remark 4.

Observe that by Lemma 12(iii),(iv),(v) and identity (7),
$A/B\in\mathrm{AP}(\mathbb{C}^{+})$ if and only if
$\Theta=E^{*}/E\in\mathrm{AP}(\mathbb{C}^{+})$ and $A/B$ has locally finite
spectrum if and only if $\Theta$ has. Note also that if
$E,E^{*}\in\mathrm{AP}(\mathbb{C}^{+})$ then
$\Theta\in\mathrm{AP}(\mathbb{C}^{+})$. In particular, if $E$ is a
trigonometric polynomial of Hermite-Biehler class, then $iA/B$ belongs to
$\mathrm{AP}(\mathbb{C}^{+})$ and has locally finite spectrum, hence Theorem 3
applies and the pair $(\mu,a)$ is crystalline ($(\mu,a)\in\rm CM$, see
Appendix).

######  Remark 5 (The Kurasov & Sarnak construction).

Theorem 3 generalizes a construction of Kurasov & Sarnak [25]. They construct
a crystalline $\mathrm{FS}$-pair $(\mu,a)\in\rm CM$ by letting

$\mu=\sum_{Q(\gamma)=0}m(\gamma){\boldsymbol{\delta}}_{\gamma},$

where $Q$ is any given trigonometric polynomial with only real zeros $\gamma$
and multiplicities $m(\gamma)$. Theorem 3 majorates this construction. Indeed,
since $Q$ has finite exponential type, Hadamard’s factorization implies that

$Q(z)=z^{n}e^{c_{1}z+c_{2}}\prod_{\gamma\neq 0}(1-z/\gamma)e^{z/\gamma}$

for some $c_{1},c_{2}\in\mathbb{C}$ and $n\geq 0$, where the
$\gamma^{\prime}s$ are the real zeros of $Q$. In particular $e^{-c_{2}-iz{\rm
Im}\,c_{1}}Q$ is a real entire trigonometric polynomial, so we can assume that
$Q$ is real on the real line. If that is the case then $c_{1}\in\mathbb{R}$
and

${\rm
Re}\,i\frac{Q^{\prime}(z)}{Q(z)}=\sum_{\gamma}\frac{y}{(x-\gamma)^{2}+y^{2}}>0.$

We obtain that $E=Q^{\prime}-iQ$ is a trigonometric polynomial of Hermite-
Biehler class by a routine calculation. By the previous remark, Theorem 3
applies. It is also easy to see that the zeros of $Q$ coincide with the points
$\gamma\in\mathbb{R}$ such that $\varphi_{E}(\gamma)\equiv 0\,({\rm
mod}\,\pi)$ and
$\varphi^{\prime}_{E}(\gamma)^{-1}=\mathop{\mathrm{Res}}_{z=\gamma}\,(A/B)=m(\gamma)$.
Thus, by Theorem 3 (with $a(\cdot)$ as in (8)), both $\mu$ and $a(\cdot)$ have
locally finite support, and so $(\mu,a)$ is crystalline
($(\mu,a)\in\mathrm{RRTP}$, see Appendix).

######  Remark 6 (The Ulanovskii & Oleveskii result).

We now explain how one can prove the main result of Ulanovskii & Oleveskii
[39] using Theorem 3, but under milder decay conditions. We claim that if
$(\mu,a)$ is an $\mathrm{FS}$-pair such $\mu\geq 0$ is $\mathbb{N}$-valued,
$a(\cdot)$ has locally finite support and exponential growth then the Hermite-
Biehler function $E=A-iB$ given by Theorem 3 is a trigonometric polynomial,
and so the construction from Remark 5 applies. Indeed, since $\deg(\mu)\leq 2$
one can take $\mathcal{B}$ to have the same zeros of $B$, but with
multiplicities given by the weights of $\mu$. We can take such $\mathcal{B}$
with $\text{order}(\mathcal{B})\leq 1$ by [31, Thm. 6, p. 16]. Using (3) we
conclude that ${\rm Re}\,i\mathcal{B}^{\prime}/\mathcal{B}={\rm Re}\,A/B$ in
$\mathbb{C}^{+}$, and so Poisson representation shows that
$A/B=\mathcal{B}^{\prime}/\mathcal{B}+h$ for some $h\in\mathbb{R}$, hence we
can assume that $A=B^{\prime}$. Since $a(\cdot)$ has locally finite support,
then $B^{\prime}/B$ has locally finite spectrum contained in $[0,\infty)$.
Lemma 13 shows that $B\in\mathrm{AP}(\mathbb{C}^{+})$ and it has locally
finite spectrum bounded from below. The class $\mathrm{AP}(\mathbb{C}^{+})$ is
closed under differentiation, and so
$B^{\prime}\in\mathrm{AP}(\mathbb{C}^{+})$, and therefore,
$E\in\mathrm{AP}(\mathbb{C}^{+})$, both with locally finite spectrum bounded
from below. Note that $B^{\prime}$ also has order $\leq 1$, and so $E$ has
order $\leq 1$. We can then apply Lemma 17 to conclude that $E$ is a
trigonometric polynomial.

###### Proof of Theorem 3.

Let $f=iA/B$. Since $E=A-iB$ is Hermite-Biehler, we obtain that ${\rm
Re}\,f>0$ in $\mathbb{C}^{+}$, and so $|e^{-f}|<1$ in $\mathbb{C}^{+}$, hence
$e^{f}$ is of bounded type. Lemma 12(iv) shows that
$\Theta=\frac{1-f}{1+f}\in\mathrm{AP}(\mathbb{C}^{+})$ and that
$|\Theta|<1-c_{\varepsilon}$ for ${\rm Im}\,z>\varepsilon$, so
$f=\frac{1+\Theta}{1-\Theta}$ is bounded for ${\rm Im}\,z>\varepsilon$, for
any $\varepsilon>0$. In particular $\lim_{y\to\infty}{\rm Re}\,f(iy)/y=0$.
Moreover, $\mathbb{E}f(\lambda)$ is locally summable by hypothesis. We
conclude that Theorem 2 applies. Poisson representation [12, Thm. 4 & Prob.
89] for $f$ implies that $\mu$ has the form given by (8). Conversely, if
$(\mu,a)$ is a real-antipodal $\mathrm{FS}$-pair such that $\mu\geq 0$,
$\operatorname{\mathrm{supp}}(\mu)$ is locally finite and $a(\cdot)$ has
exponential growth, then we can apply Theorem 4 to obtain that (8) holds for
some Hermite-Biehler function $E=A-iB$ such that $A(z+ic)/B(z+ic)$ belongs to
$\mathrm{AP}(\mathbb{C}^{+})$ for some $c>0$. However, Lemma 12 also shows
that $\Theta(z+ic)=E^{*}(z+ic)/E(z+ic)$ belongs to
$\mathrm{AP}(\mathbb{C}^{+})$, and since $|\Theta|<1$ in $\mathbb{C}^{+}$,
Lemma 11 shows that $\Theta\in\mathrm{AP}(\mathbb{C}^{+})$ and, by Lemma 12
again we obtain that $A/B\in\mathrm{AP}(\mathbb{C}^{+})$. Since $a(\lambda)$
is locally summable and $a(\lambda)=\mathbb{E}f(\lambda)$ for $\lambda>0$,
then $\mathbb{E}f(\lambda)$ is locally summable. ∎

The next result classifies $\mathrm{FS}$-pairs $(\mu,a)$ for which $\mu$ has
locally finite support.

######  Theorem 4.

Let $(\mu,a)$ be an $\mathrm{FS}$-pair such that $\mu$ has locally finite
support, $\deg(\mu)\leq 2$ and $a(\cdot)$ has exponential growth. Then there
exists four Hermite-Biehler functions $E_{j}=A_{j}-iB_{j}$ and numbers
$p_{j}\in\\{0,1\\}$, for $j=1,2,3,4$, such that if we let
$f=ip_{1}A_{1}/B_{1}-ip_{2}A_{2}/B_{2}$ and
$g=ip_{3}A_{3}/B_{3}-ip_{4}A_{4}/B_{4}$ then $f(\cdot+ic)$ and $g(\cdot+ic)$
belong to $\mathrm{AP}(\mathbb{C}^{+})$ for some $c>0$, $\mathbb{E}f(\lambda)$
and $\mathbb{E}g(\lambda)$ are locally summable, and:

$\displaystyle\frac{1}{2\pi}{\mu}$
$\displaystyle=p_{1}\sum_{\,\varphi_{1}(\gamma)\equiv 0\,({\rm
mod}\,\pi)}\frac{1}{\varphi_{1}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}-p_{2}\sum_{\,\varphi_{2}(\gamma)\equiv
0\,({\rm
mod}\,\pi)}\frac{1}{\varphi_{2}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}$
$\displaystyle\ \ -ip_{3}\sum_{\,\varphi_{3}(\gamma)\equiv 0\,({\rm
mod}\,\pi)}\frac{1}{\varphi_{3}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}+ip_{4}\sum_{\,\varphi_{4}(\gamma)\equiv
0\,({\rm
mod}\,\pi)}\frac{1}{\varphi_{4}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma};$
$\displaystyle a(\lambda)$
$\displaystyle=\mathbb{E}f(\lambda)-i\mathbb{E}g(\lambda)\quad\text{and}\quad
a(-\lambda)=\overline{\mathbb{E}f(\lambda)+i\mathbb{E}g(\lambda)}\quad(\text{for
}\lambda>0);$ $\displaystyle\tfrac{1}{2}a(0)$ $\displaystyle={\rm
Re}\,\mathbb{E}f(0)-i{\rm Re}\,\mathbb{E}g(0),$

where $\varphi_{j}$ is the phase function associated with $E_{j}$. Moreover:
(i) $\mu\geq 0$ if and only if $p_{1}=1$ and $p_{2}=p_{3}=p_{4}=0$; (ii)
$a(\cdot)$ has locally finite support if and only if $f$ and $g$ have locally
finite spectrum. Furthermore, if $\mu$ is $\mathbb{N}$-valued then we can take
$A_{1}=B_{1}^{\prime}$.

###### Proof of Theorem 4.

By real-antipodal splitting we can assume that $\mu$ is real-valued (so
$p_{3}=p_{4}=0$). We can then apply Theorem 1 and obtain that

$f(z)=\frac{a(0)}{2}+\lim_{T\to\infty}\sum_{0<\lambda<T}(1-\lambda/T)a(\lambda)e^{2\pi
i\lambda z}=ih+\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{1+tz}{t-z}\frac{\mathrm{d}\mu(t)}{1+t^{2}},$

for some $h\in\mathbb{R}$, where $f$ is holomorphic in $\mathbb{C}^{+}$ and
$f(\cdot+ic)\in\mathrm{AP}(\mathbb{C}^{+})$ for some $c>0$. Let
$\mu=\sum_{\gamma\in\Lambda}r(\gamma){\boldsymbol{\delta}}_{\gamma}$ where
$\Lambda\subset\mathbb{R}$ is locally finite and
$r(\gamma)=r_{1}(\gamma)-r_{2}(\gamma)$, with $r_{j}\geq 0$. Let

$f_{j}(z)=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{1+tz}{t-z}\frac{\mathrm{d}\mu_{j}(t)}{1+t^{2}}$

for $j=1,2$, where
$\mu_{j}=\sum_{\gamma\in\Lambda}r_{j}(\gamma){\boldsymbol{\delta}}_{\gamma}$.
Observe that each $f_{j}$ is meromorphic in $\mathbb{C}$ with only simple
poles (possibly) at $\Lambda$. Since $\mu_{j}\geq 0$ we have
$if_{j}(\mathbb{C}^{+})\subset\mathbb{C}^{+}\cup\\{0\\}$, and we can then
apply a classical result [31, p. 308, Thm. 1] to deduce this happens exactly
when there are two real entire functions $A_{j}$ and $B_{j}$ with only real
zeros that interlace and such that $if_{j}=-p_{j}A_{j}/B_{j}$, with
$p_{j}\in\\{0,1\\}$ to account for the case when $\mu_{j}=0$ for some $j=0,1$.
Since ${\rm Re}\,iA_{j}/B_{j}>0$, we deduce that $E_{j}=A_{j}-iB_{j}$ is of
Hermite-Biehler class, and so, Poisson representation shows that

$\mu=2\pi p_{1}\sum_{\,\varphi_{1}(\gamma)\equiv 0\,({\rm
mod}\,\pi)}\frac{1}{\varphi_{1}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}-2\pi
p_{2}\sum_{\,\varphi_{2}(\gamma)\equiv 0\,({\rm
mod}\,\pi)}\frac{1}{\varphi_{2}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}.$

Noting that $f=f_{1}-f_{2}+ih$, we obtain
$f_{1}(\cdot+ic)-f_{2}(\cdot+ic)\in\mathrm{AP}(\mathbb{C}^{+})$, and since
$a(\cdot)$ has exponential growth we conclude that
$a(\lambda)=\mathbb{E}(f_{1}-f_{2})(\lambda)$ for $\lambda>0$ and
$\frac{1}{2}a(0)={\rm Re}\,\mathbb{E}(f_{1}-f_{2})(0)$. If $\mu\geq 0$ then
$(p_{1},p_{2})=(1,0)$. Also, $a(\cdot)$ has locally finite support if and only
if $f_{1}-f_{2}$ has locally finite spectrum. Finally, if $\mu$ is
$\mathbb{N}$ valued then, since $\deg(\mu)\leq 2$, one might take
$\mathcal{B}$ to have zeros exactly at support of $\mu$ with multiplicity
determined by the weights of $\mu$ and order at most $1$. In particular,
Hadamard factorization and log-differentiation show that ${\rm
Re}\,i\mathcal{B}^{\prime}/\mathcal{B}={\rm Re}\,iA_{1}/B_{1}$ and so
$f_{1}=ih+i\mathcal{B}^{\prime}/\mathcal{B}$, and
$\mathcal{E}=\mathcal{B}^{\prime}-i\mathcal{B}$ is now Hermite-Biehler. ∎

In what follows we say that an entire function $E$ is of Pólya class if $E$
has no zeros in $\mathbb{C}^{+}$, $|E^{*}/E|\leq 1$ in $\mathbb{C}^{+}$ and
$y\mapsto|E(x+iy)|$ is nondecreasing for $y>0$. A classical theorem [12, Thm.
7] shows that a function $E$ is of Pólya class if and only if

$\displaystyle
E(z)=E^{(r)}(0)(z^{r}/r!)e^{-dz^{2}-ibz}\prod_{n}(1-z/\overline{z_{n}})e^{z{\rm
Re}\,1/z_{n}},$ (9)

where $d\geq 0$, ${\rm Re}\,b\geq 0$, $\\{\overline{z}_{n}=x_{n}-iy_{n}\\}$
are the nonzero zeros of $E$ (with $y_{n}\geq 0$) and

$\sum_{n}\frac{1+y_{n}}{x_{n}^{2}+y_{n}^{2}}<\infty.$

The Pólya class can be also defined as functions that are uniform limits in
compact sets of polynomials with no zeros in $\mathbb{C}^{+}$ (see [12,
Problem 12]). Moreover, any function of Pólya class has order at most $2$
([12, Problem 10]).

The next result is a brand new classification theorem121212In a way, this is
the most technically hard result of the paper. for nonnegative crystalline
measures. It generalizes a result of Olevekii & Ulanovskii [39] in the
uniformly discrete case, dropping the $\mathbb{N}$-valuedness assumption.

######  Theorem 5.

Let $(\mu,a)$ be an $\mathrm{FS}$-pair. Assume that:

  1. (1)

$\mu$ has uniformly discrete support and there is $\delta>0$ such that
$\mu(\\{x\\})\geq\delta$ for all $x\in\operatorname{\mathrm{supp}}(\mu)$;

  2. (2)

There is $b>0$ such that $\operatorname{\mathrm{supp}}(a)\cap(0,b)=\emptyset$
and $a(\cdot)$ has the decay (5).

Then there exists an entire function $E=A-iB$ of Hermite-Biehler class and of
finite exponential type such that $E,E^{*}\in\mathrm{AP}(\mathbb{C}^{+})$ (and
so $A/B\in\mathrm{AP}(\mathbb{C}^{+})$), ${\rm spec}(E)$ is bounded and (8)
holds. If in addition $\operatorname{\mathrm{supp}}(a)$ is locally finite,
then $E$ is a trigonometric polynomial and Remark 5 applies, and so
$(\mu,a)\in\mathrm{RRTP}$ (see Appendix).

###### Proof.

Step 1. We first apply Theorem 3 to conclude there is an Hermite-Biehler
function $E=A-iB$ such that $A/B\in\mathrm{AP}(\mathbb{C}^{+})$ and (8) holds.
We can assume that $E$ has no real zeros as these can be removed (with a
Weierstrass product) without altering $A/B$ or the fact that $E$ is Hermite-
Biehler. In particular,
$\operatorname{\mathrm{supp}}(\mu)=\\{\gamma\in\mathbb{R}:\varphi(\gamma)\equiv
0\,({\rm mod}\,\pi)\\}=\text{Zeros}(B)$, where $\varphi$ is the phase function
associated with $E$, and these zeros are all simple. The zeros of $A$ are also
simple and interlace those of $B$. Since $\mu$ is bounded from below on its
support and $a(\cdot)$ has the decay (5), we can apply Remark 3 to obtain

$c<\varphi^{\prime}(\gamma)<1/c,$

for some $0<c<1$, whenever $B(\gamma)=0$. In particular, we have
$\sum_{B(\gamma)=0}\frac{1}{1+\gamma^{2}}<\infty$ and, since the zeros of $A$
interlace those of $B$, we also have $\sum_{A(s)=0}\frac{1}{1+s^{2}}<\infty$.
We claim that we can assume $E$ is has order at most $1$. Indeed, define the
canonical products

$\mathcal{B}(z)=bz^{p}\prod_{B(\gamma)=0,\ \gamma\neq
0}(1-z/\gamma)e^{z/\gamma}\quad\text{and}\quad\mathcal{A}(z)=z^{q}\prod_{A(s)=0,\
s\neq 0}(1-z/s)e^{z/s},$

for some $p,q\in\\{0,1\\}$, where $b\in\mathbb{R}$ is chosen so that
$(A(z)/B(z))/(\mathcal{A}(z)/\mathcal{B}(z))=1+O(z)$ for $z\to 0$. Hence both
$\mathcal{A}$ and $\mathcal{B}$ have order at most $1$ by a classical result
of Borel [31, Thm. 6, p. 16]. Since ${\rm Re}\,iA/B\geq 0$ in $\mathbb{C}^{+}$
and, a straitfoward computation, shows that ${\rm
Re}\,i\mathcal{A}/\mathcal{B}\geq 0$ in $\mathbb{C}^{+}$, we conclude that
both $A/B$ and $\mathcal{A}/\mathcal{B}$ are of bounded type and so
$F=(A/B)/(\mathcal{A}/\mathcal{B})$ also is and has no zeros. Since $F=F^{*}$,
a classical result of Krein [24] (see also [32, Thm. 1, p. 115]) shows that
$F$ is of exponential type, and so $F(z)=e^{hz}$, for some $h\in\mathbb{R}$.
Since $F$ is of bounded type we have $h=0$. We can then define
$\widetilde{E}=\mathcal{A}-i\mathcal{B}$ to finish the claim.

Step 2. Let $\\{\overline{z}_{n}=x_{n}-iy_{n}\\}$ be the zeros of $E$ with
$y_{n}>0$. We claim $\sup_{n}y_{n}<\infty$ and that $E$ is of Póyla class.
Indeed, since $f=iA/B\in\mathrm{AP}(\mathbb{C}^{+})$, by Remark 4 we have
$\Theta=E^{*}/E=(1-f)/(1+f)\in\mathrm{AP}(\mathbb{C}^{+})$. Since ${\rm
spec}(f)\subset[0,\infty)$, a simple computation using Lemma 12(i) shows
$\mathbb{E}\Theta(0)=\frac{1-\mathbb{E}f(0)}{1+\mathbb{E}f(0)}$. However,
$2{\rm Re}\,\mathbb{E}f(0)=a(0)>0$ (because $\mu\geq 0$), thus
$|\mathbb{E}\Theta(0)|<1$. Assume now $\mathbb{E}\Theta(0)\neq 0$. Then Lemma
12(i) shows straightforwardly that $\Theta(x+iy)$ has no zeros for large
$y>0$. Assume otherwise that $\mathbb{E}\Theta(0)=0$, that is,
$\mathbb{E}f(0)=1$. Since $a(\lambda)=\mathbb{E}f(\lambda)$ for $\lambda>0$ we
deduce that ${\rm spec}(f)\subset\\{0\\}\cup[b,\infty)$ where
$b=\inf\\{\lambda>0:a(\lambda)\neq 0\\}>0$ by assumption. Since $f$ has no
zeros in $\mathbb{C}^{+}$, we can then apply Lemma 12(ii) to conclude that
$b\in{\rm spec}(f)$. Writing $f=1-2g$, with ${\rm spec}(g)\subset[b,\infty)$
and $\mathbb{E}g(b)\neq 0$, we can apply Lemma 12(i) to conclude that
$|g(x+iy)|<1$ for large $y>0$, and so

$\Theta(z)=\frac{g(z)}{1-g(z)}=\sum_{n\geq 1}g(z)^{n},$

hence ${\rm spec}(\Theta)\subset[b,\infty)$ and $\mathbb{E}\Theta(b)\neq 0$.
We can then apply again Lemma 12(i) to conclude that
$\lim_{y\to\infty}\sup_{x}|e^{-2\pi
ib(x+iy)}\Theta(x+iy)-\mathbb{E}\Theta(b)|=0$, hence $\Theta(z)$ has no zeros
for large ${\rm Im}\,z$. Finally, since $E$ has order at most $1$ the
following sum is now finite

$\sum_{n}\frac{1+y_{n}}{x_{n}^{2}+y_{n}^{2}}<\infty.$

and $E$ has the following factorization

$E(z)=E(0)e^{-ihz}\prod_{n}(1-z/\overline{z_{n}})e^{z{\rm Re}\,1/z_{n}}$

for some $h\geq 0$. Hence $E$ is of Póyla class.

Step 3. We claim now that $\sup_{x\in\mathbb{R}}\varphi^{\prime}(x)<\infty$.
Since (3) holds true for $f=iA/B$, we obtain that

$K(w,z):=\frac{B(z)\overline{A(w)}-\overline{B(w)}{A(z)}}{\pi(z-\overline{w})}=\sum_{B(\gamma)=0}\frac{1}{\pi\varphi^{\prime}(\gamma)}\frac{B(z)\overline{B(w)}}{(\gamma-z)(\gamma-\overline{w})}$

for all $z,w\in\mathbb{C}$. A routine computation shows that

$\varphi^{\prime}(x)={\rm Re}\,i\frac{E^{\prime}(x)}{E(x)}=\frac{\pi
K(x,x)}{|E(x)|^{2}}=\sum_{B(\gamma)=0}\frac{\sin^{2}\varphi(x)}{\varphi^{\prime}(\gamma)(\gamma-x)^{2}}.$

Enumerate the $\text{Zeros}(B)=\\{\gamma_{n}\\}$ and let $c>0$ be so small
that $\gamma_{n+1}-\gamma_{n}\geq c$ for all $n$. Let $0<\varepsilon<c/10$ to
be chosen later. Observe first that if ${\rm
dist}(x,\\{\gamma_{n}\\})=|x-\gamma|\geq\varepsilon$ (with
$\gamma=\gamma_{n_{0}}$) then $|x-\gamma_{n_{0}+k}|\geq\varepsilon+|k|c/2$,
and so

$\varphi^{\prime}(x)\leq 2\sum_{k\geq
0}\frac{1}{c(\varepsilon+kc/2)^{2}}=O(\varepsilon^{-2}).$

On the other hand, if ${\rm
dist}(x,\\{\gamma_{n}\\})=|x-\gamma|\leq\varepsilon$, then
$|\gamma_{n_{0}+k}-x|\geq\frac{c}{2}|k|$, for $k\neq 0$, and

$\varphi^{\prime}(\gamma)\varphi^{\prime}(x)\leq\frac{\sin^{2}\varphi(x)}{(\gamma-x)^{2}}+C\sin^{2}\varphi(x)$

for some $C>0$, independent of $\varepsilon$ and $\gamma$. Using the
inequality $x^{2}\geq\sin^{2}x$ for all real $x$, and factoring
$\sin^{2}\varphi(x)$, we obtain

$\frac{\varphi^{\prime}(\gamma)\varphi^{\prime}(x)}{\sin^{2}\varphi(x)}\leq\frac{(\pi/c)^{2}}{\sin^{2}[(\gamma-x)\pi/c]}+C$

We conclude that the function

$\xi(x)=\varphi^{\prime}(\gamma)\cot\varphi(x)-\frac{\pi}{c}\cot[\pi(x-\gamma)/c]+C(x-\gamma),$

which is analytic in a complex neighbourhood of the segment
$I=[\gamma-\varepsilon,\gamma+\varepsilon]$, is nondecreasing for $x\in I$ and
$\xi(\gamma)=0$. We conclude that

$(x-\gamma)\varphi^{\prime}(\gamma)\cot\varphi(x)\geq\frac{\pi(x-\gamma)}{c}\cot[\pi(x-\gamma)/c]-C(x-\gamma)^{2}$

for $x\in I$. Since the function
$\frac{\pi(x-\gamma)}{c}\cot[\pi(x-\gamma)/c]$ has value $1$ for $x=\gamma$,
there is $\varepsilon_{0}=\varepsilon_{0}(c,C)>0$, with
$0<\varepsilon_{0}<c/10$, such that the right hand side above is bounded from
below by $\frac{\pi(x-\gamma)}{2c}\cot[\pi(x-\gamma)/c]$ for $x\in
I_{0}:=(\gamma-\varepsilon_{0},\gamma+\varepsilon_{0})$, and so

$(x-\gamma)\varphi^{\prime}(\gamma)\cot\varphi(x)\geq\frac{\pi(x-\gamma)}{2c}\cot[\pi(x-\gamma)/c]$

for $x\in I_{0}$. We now select $\varepsilon=\varepsilon_{0}$. Observing the
right hand side above is positive for $x\in I_{0}$, we can square both sides
above, factor out $(x-\gamma)^{2}$, lower bound $\varphi^{\prime}(\gamma)\geq
c$ and isolate $\sin^{2}\varphi$ to obtain

$\sin^{2}\varphi(x)\leq\frac{1}{1+C\cot^{2}[\pi(x-\gamma)/c]}$

for $C=\pi^{2}/(4c^{4})$. Note also that for $x\in I_{0}$ we have
$|x-\gamma_{n_{0}+k}|\geq|x-\gamma_{n_{0}}-kc|$ for any $k\in\mathbb{Z}$,
hence

$\displaystyle\varphi^{\prime}(x)\leq\frac{1}{c}\sum_{k\in\mathbb{Z}}\frac{\sin^{2}\varphi(x)}{(x-\gamma+kc)^{2}}$
$\displaystyle=\frac{\pi^{2}}{c^{3}}\frac{\sin^{2}\varphi(x)}{\sin^{2}[\pi(x-\gamma)/c]}$
$\displaystyle\leq\frac{\pi^{2}}{c^{3}}\frac{1}{\sin^{2}[\pi(x-\gamma)/c]+C\cos^{2}[\pi(x-\gamma)/c]}\leq\frac{\pi^{2}}{c^{3}}.$

for $x\in I_{0}$. This proves the claim.

Step 4. We claim the zeros of $E$ are separated from the real axis, that is,
there is $c>0$ such that $y_{n}\geq c$ for all $n$. Indeed, observe that since
$E(z)e^{i\varphi(z)}=E^{*}(z)e^{-i\varphi(z)}$ whenever $\varphi(z)$ is
defined (any neighborhood of $\mathbb{R}$ not containing
$\\{z_{n}\\}_{n\in\mathbb{Z}}\cup\\{\overline{z_{n}}\\}_{n\in\mathbb{Z}}$) we
obtain that

$\varphi^{\prime}(z)=\partial_{z}\frac{1}{2i}\log\Theta(z)=\frac{1}{2i}\frac{\Theta^{\prime}(z)}{\Theta(z)},$

with $\Theta=E^{*}/E$. Using the factorization of $E$ we obtain

$\Theta(z)=e^{2ihz}\prod_{n\in\mathbb{Z}}\frac{1-z/z_{n}}{1-z/\overline{z_{n}}},$

and so

$\varphi^{\prime}(x)=h+\sum_{n\in\mathbb{Z}}\frac{y_{n}}{|x-z_{n}|^{2}}$

for real $x$. Hence $\varphi^{\prime}(x_{n})\geq 1/y_{n}$, and since
$\varphi^{\prime}(x)$ is bounded, the claim follows.

Step 5. We claim that $\varphi^{\prime}\in{\rm AP}(\mathbb{R})$ and ${\rm
spec}(\varphi^{\prime})\subset[b,\infty)$. Let $c>0$ be small enough such that
$\Theta$ has no zeros or poles in the strip $S=\\{-c<{\rm Im}\,z<c\\}$. First
we show that $\Theta$ is bounded in every horizontal strip contained in $S$.
We only need to show this for $z\in S$ with ${\rm Im}\,z\leq 0$. Indeed, since

$\frac{1}{2}\partial_{y}\log|\Theta^{*}(x+iy)|=h+\sum_{n\in\mathbb{Z}}\frac{y_{n}[(x-x_{n})^{2}+y_{n}^{2}-y^{2}]}{|z-z_{n}|^{2}|z-\overline{z_{n}}|^{2}}$

and $y_{n}^{2}-y^{2}\leq 3(y_{n}-y)^{2}$ for $0\leq y\leq c$, we deduce that

$\frac{1}{2}\partial_{y}\log|\Theta^{*}(x+iy)|\leq
h+3\sum_{n\in\mathbb{Z}}\frac{y_{n}}{|x-z_{n}|^{2}}\leq
3\varphi^{\prime}(x)\leq C$

for some $C>0$. Since $\log|\Theta^{*}(x)|=0$, integration shows that
$\Theta^{*}(x+iy)\leq e^{2Cy}$ for $0\leq y\leq c$. Since
$\Theta\in\mathrm{AP}(\mathbb{C}^{+})$, we can now apply Lemma 11 to deduce
that $\Theta(\cdot-ic)\in\mathrm{AP}(\mathbb{C}^{+})$ and
$\Theta^{\prime}(\cdot-ic)\in\mathrm{AP}(\mathbb{C}^{+})$. By Lemma 12(iv), we
get that $|\Theta|$ is bounded away from zero in every horizontal strip
contained in $S$. We deduce that $\Theta^{\prime}/\Theta\in{\rm
AP}(\mathbb{R})$, and so131313If we let ${\rm AP}(S)$ be defined in the same
way as in $\mathrm{AP}(\mathbb{C}^{+})$, but considering only horizontal
strips strictly contained in $S$, we have showed that $\varphi^{\prime}\in{\rm
AP}(S)$. $\varphi^{\prime}\in{\rm AP}(\mathbb{R})$. Step 1 shows that ${\rm
spec}(\Theta)\subset\\{0\\}\cup[b,\infty)$ and $\mathbb{E}\Theta(b)\neq 0$,
hence ${\rm spec}(\Theta^{\prime}/\Theta)\subset[b,\infty)$ and ${\rm
spec}(\varphi^{\prime})\subset[b,\infty)$.

Step 6. Since $E$ is of Pólya class we have ${\rm Re}\,iE^{\prime}/E\geq 0$ in
$\mathbb{C}^{+}$, and Poisson representation guarantees that

$\frac{iE^{\prime}(z)}{E(z)}=id-ipz+\frac{1}{\pi
i}\int_{\mathbb{R}}\frac{1+tz}{t-z}\frac{\varphi^{\prime}(t)\mathrm{d}t}{1+t^{2}},$

for some $d\in\mathbb{R}$ and $p\geq 0$, where
$\deg(\varphi^{\prime}(t)\mathrm{d}t)\leq 2$. This fact, in conjunction with
the factorization of $E$ shows that

${\rm
Re}\,\frac{iE^{\prime}(z)}{E(z)}=py+\frac{y}{\pi}\int_{\mathbb{R}}\frac{\varphi^{\prime}(t)\mathrm{d}t}{(x-t)^{2}+y^{2}}=h+\sum_{n}\frac{y+y_{n}}{(x-x_{n})^{2}+(y+y_{n})^{2}}.$

Since $\sum_{n}\frac{1+y_{n}}{x_{n}^{2}+y_{n}^{2}}<\infty$, we conclude that
$p=\lim_{y\to\infty}{\rm Re}\,\frac{iE^{\prime}(iy)}{yE(iy)}=0$. We can apply
Lemma 16 so deduce that $iE^{\prime}/E\in\mathrm{AP}(\mathbb{C}^{+})$ and
${\rm spec}(iE^{\prime}/E)\subset\\{0\\}\cup[b,\infty)$. We can apply Lemma 13
to conclude that $E\in\mathrm{AP}(\mathbb{C}^{+})$ and that ${\rm spec}(E)$ is
bounded from below. Since $E^{*}=\Theta E$, we conclude that
$E^{*}\in\mathrm{AP}(\mathbb{C}^{+})$ also and that ${\rm spec}(E^{*})$ is
bounded from below. Finally, we can use Lemma 17 to deduce the spectrum of $E$
(and of $E^{*}$) is bounded and $E$ is of exponential type. To finish, if in
addition $\operatorname{\mathrm{supp}}(a)$ is locally finite, then so is ${\rm
spec}(A/B)$. Since $a(\cdot)$ has exponential growth (due to (5)), we obtain
that $\Theta$ has locally finite spectrum, thus so $\varphi^{\prime}$ has
locally finite spectrum. The previous applications of Lemmas 16 and 13 (and
their content) show that $E$ has locally finite spectrum as well, and so Lemma
17 shows that $E$ is a trigonometric polynomial. ∎

We believe that Theorem 5 should still hold only assuming that
$\operatorname{\mathrm{supp}}(\mu)$ is locally finite. Note that by the decay
of $a(\cdot)$ and Remark 3, $\operatorname{\mathrm{supp}}(\mu)$ is contained
in a finite union of uniformly discrete sets. Also, the uniformly discreteness
of $\operatorname{\mathrm{supp}}(\mu)$ plays a role only in Step 3 above,
where we show $\varphi^{\prime}$ is bounded as a stepping stone to show the
zeros of $E$ cannot get close to the real axis. Note that if
$\varphi^{\prime}$ is bounded then $\operatorname{\mathrm{supp}}(\mu)$ is
uniformly discrete. However, we still think there must be a way to circumvent
this issue, and nevertheless conclude that the zeros of $E$ are separated from
the real axis. Unfortunately we were not able to come up with such maneuver,
despite many efforts.

###  3.1. de Branges spaces - a Hilbert space interpretation of
$\mathrm{FS}$-pairs

A de Branges space [12] (see also the introduction of [20]) is a Hilbert space
$(\mathfrak{H},\|\cdot\|)$ of entire functions $F:\mathbb{C}\to\mathbb{C}$
satisfying:

  1. (H1)

If $F\in\mathfrak{H}$ and $F(w)=0$ for some $w\in\mathbb{C}$, then
$G(z)=F(z)\frac{z-\overline{w}}{z-w}$ belongs to $\mathfrak{H}$ and
$\|G\|=\|F\|$;

  2. (H2)

The functional $F\in\mathfrak{H}\mapsto F(w)$ is continuous for every
$w\in\mathbb{C}$;

  3. (H3)

If $F\in\mathfrak{H}$ then $F^{*}\in\mathfrak{H}$ and $\|F^{*}\|=\|F\|$.

Because of (H2), the space $\mathfrak{H}$ comes equipped with an unique
reproducing kernel $K(w,z)$, that is, $F(w)=\langle F,K(w,\cdot)\rangle$ for
any $w\in\mathbb{C}$ and $F\in\mathfrak{H}$. De Branges proved [12, Thm. 23]
that for any such Hilbert space $\mathfrak{H}$ there exists an Hermite-Biehler
function $E$ such that $\mathfrak{H}=\mathfrak{H}(E)$ isometrically, where
$\mathfrak{H}(E)$ is the space of entire functions $F$ such that141414The
original definition involves bounded type theory, but this is an equivalent
definition, see [20].

$\|F\|^{2}:=\sup_{y\in\mathbb{R}}\int_{\mathbb{R}}\left|\frac{F(x+iy)}{E(x+i|y|)}\right|^{2}\mathrm{d}x<\infty,$

in which case the sup above is attained at $y=0$. De Branges also shows [12,
Thm. 19] that any such space $\mathfrak{H}(E)$ is a Hilbert space that
satisfies the above axioms. When we identify $\mathfrak{H}=\mathfrak{H}(E)$
for some Hermite-Biehler function $E=A-iB$ we have

$\displaystyle
K(w,z)=\frac{B(z)\overline{A(w)}-\overline{B(w)}{A(z)}}{\pi(z-\overline{w})}=\frac{E(z)\overline{E(w)}-E^{*}(z)E(\overline{w})}{2\pi
i(\overline{w}-z)}$ (10)

and

$F(w)=\int_{\mathbb{R}}\frac{F(t)\overline{K(w,t)}}{|E(t)|^{2}}\mathrm{d}t$

for every $w\in\mathbb{C}$ and $F\in\mathfrak{H}$. The function realizing
$\mathfrak{H}=\mathfrak{H}(E)$ is not unique. Indeed, if
$\mathfrak{H}(E)=\mathfrak{H}(E_{1})$ isometrically, one can then apply [12,
Problem 69] (for $S=A_{1}$ and $S=B_{1}$) to obtain

$E_{1}(z)=\frac{e^{i\beta}}{\sqrt{1-|p|^{2}}}(E(z)-\overline{p}E^{*}(z))$

for some $\beta\in\mathbb{R}$ and $p\in\mathbb{C}$, with $|p|<1$. Conversely,
a routine computation shows that any $E_{1}$ defined in the above way
satisfies (10), and since $K(z,z)>0$ for $z\in\mathbb{C}^{+}$, we conclude
that $E_{1}$ is Hermite-Biehler, and so $\mathfrak{H}(E)=\mathfrak{H}(E_{1})$
isometrically. Moreover, their theta functions are related by the Möbius
transformation

$\Theta_{1}(z)=e^{2i\beta}\frac{\Theta(z)-p}{1-\Theta(z)\overline{p}}.$

A major result [12, Thm. 22] is that, for any $\alpha\in[0,\pi)$, the set
$\\{K(\gamma,z)\\}_{\varphi(\gamma)\equiv\alpha\,(\text{mod }\pi)}$ is
orthogonal in $\mathfrak{H}(E)$, and its orthogonal complement is one-
dimensional and spanned by $e^{i\alpha}E-e^{-i\alpha}E^{*}$. In particular,
when $\alpha=0$, $E$ has no real zeros and $B\notin\mathfrak{H}(E)$, then
representation of functions in the basis $\\{K(\gamma,z)\\}_{B(\gamma)=0}$
implies the identities

$\pi
K(w,z):=\frac{B(z)\overline{A(w)}-\overline{B(w)}{A(z)}}{z-\overline{w}}=\sum_{B(\gamma)=0}\frac{1}{\varphi^{\prime}(\gamma)}\frac{B(z)\overline{B(w)}}{(\gamma-z)(\gamma-\overline{w})}$

and

$F(z)=\sum_{B(\gamma)=0}\frac{F(\gamma)B(z)}{B^{\prime}(\gamma)(z-\gamma)},\quad\text{for
any }F\in\mathfrak{H}(E),$

with convergence in $\mathfrak{H}(E)$ and uniformly (and absolutely) in
compact sets of $\mathbb{C}$. Letting $f=iA/B$, if one divides the expansion
of $K$ above by $iB(z)\overline{B(w)}$ we get

$\frac{f(z)+\overline{f(w)}}{z-\overline{w}}=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{\mathrm{d}\mu(t)}{(t-z)(t-\overline{w})}$

with
$\mu=\sum_{B(\gamma)=0}\frac{2\pi}{\varphi^{\prime}(\gamma)}{\boldsymbol{\delta}}_{\gamma}$.
Assume now that $f=iA/B\in\mathrm{AP}(\mathbb{C}^{+})$ and that
$\mathbb{E}f(\cdot)$ is locally summable. Thus, Theorem 3 applies and
$(\mu,a)$ is a $\mathrm{FS}$-pair (with $a(\cdot)$ as in (8)). There is no
particular reason (other than convenience) to take $\alpha=0$ here. One could
select as well any $\alpha\in[0,\pi)$, but now the $\mathrm{FS}$-pair would
be151515The proof of Theorem 1 implies that
$(e^{i\alpha}E-e^{-i\alpha}E^{*})\notin\mathfrak{H}(E)$

$\displaystyle\begin{split}\mu&=\sum_{\varphi(\gamma)\equiv\alpha\,(\text{mod
}\pi)}\frac{2\pi}{\varphi^{\prime}(\gamma)}{\boldsymbol{\delta}}_{\gamma},\\\
a(\lambda)&=\overline{a(-\lambda)}=\mathbb{E}f(\lambda)\ (\lambda>0)\\\
a(0)&=2{\rm Re}\,\mathbb{E}f(0),\end{split}$ (11)

assuming
$f=(1+e^{-2i\alpha}\Theta)/(1-e^{-2i\alpha}\Theta)\in\mathrm{AP}(\mathbb{C}^{+})$
(note if $E_{\alpha}=e^{i\alpha}E=A_{\alpha}-iB_{\alpha}$ then
$f=iA_{\alpha}/B_{\alpha}$).

One could ask the question whether there is another natural axiom, as the ones
above, that would force the space $\mathfrak{H}$ to “produce” an
$\mathrm{FS}$-pair. Such axiom indeed exists.

######  Proposition 6.

Let $(\mathfrak{H},\|\cdot\|)$ be a Hilbert space of entire functions
satisfying (H1), (H2) and (H3), with reproducing kernel $K(w,z)$. Assume also
that:

  1. (H4)

There is $\alpha\in\mathbb{C}^{+}$ such that the function

$f(z)=\frac{{(\overline{\alpha}-z)K(\alpha,z)}-(\alpha-z)K(\overline{\alpha},z)}{{(\overline{\alpha}-z)K(\alpha,z)}+(\alpha-z)K(\overline{\alpha},z)}$

belongs to $\mathrm{AP}(\mathbb{C}^{+})$ and $\mathbb{E}f(\cdot)$ is locally
summable.

Then there is an Hermite-Biehler function $E=A-iB$ such that
$\mathfrak{H}=\mathfrak{H}(E)$ isometrically, and, if we define $\mu$ and
$a(\cdot)$ as in (8), we have that $(\mu,a)$ is a $\mathrm{FS}$-pair.

###### Proof.

We can then apply [12, Thm. 23, p. 58] to obtain that the identity

$\displaystyle\frac{(K(w,z)-K(\beta,z)K(\beta,\beta)^{-1}K(w,\beta))(z-\overline{\beta})}{z-\beta}=\frac{(K(w,z)-K(\overline{\beta},z)K(\overline{\beta},\overline{\beta})^{-1}K(w,\overline{\beta}))(\overline{w}-\overline{\beta})}{\overline{w}-\beta}$

holds for all $w,z\in\mathbb{C}$ and $\beta\in\mathbb{C}^{+}$. Let
$L(w,z)=2\pi i(\overline{w}-z)K(w,z)$ and

$E_{1}(z)=L(\alpha,\alpha)^{-1/2}L(\alpha,z).$

A routine computation, using the above identity, now shows that

$K(w,z)=\frac{E_{1}(z)\overline{E_{1}(w)}-E_{1}^{*}(z)E_{1}(\overline{w})}{2\pi
i(\overline{w}-z)}.$

Since $0<K(z,z)$ for $z\in\mathbb{C}^{+}$, we conclude that $E_{1}$ is of
Hermite-Biehler class, thus $\mathfrak{H}=\mathfrak{H}(E_{1})$ isometrically.
Letting $E_{1}=A_{1}-iB_{1}$ and noting that
$L(\alpha,z)^{*}=-L(\overline{\alpha},z)$, we deduce that

$f(z)=\frac{1+\frac{L(\alpha,z)^{*}}{L(\alpha,z)}}{1-\frac{L(\alpha,z)^{*}}{L(\alpha,z)}}=\frac{1+\Theta_{1}(z)}{1-\Theta_{1}(z)}=i\frac{A_{1}(z)}{B_{1}(z)}.$

We can then apply Theorem 3 to finish the proof. ∎

##  4\. Almost Periodic Functions in $\mathbb{R}$

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ in almost
periodic (in the sense of Bohr [15]) if for every $\varepsilon>0$ the
following set is relatively dense

$\tau_{\varepsilon}(f):=\\{t\in\mathbb{R}:\sup_{x\in\mathbb{R}}|f(x)-f(x+t)|\leq\varepsilon\\}.$

This set is called the set of $\varepsilon$-translations for $f$. We denote by
${\rm AP}(\mathbb{R})$ the set of continuous almost period functions
$f:\mathbb{R}\to\mathbb{C}$. It is not hard to show that any $f\in{\rm
AP}(\mathbb{R})$ is bounded and uniformly continuous, and that
$\tau_{\varepsilon}(f)$ is also closed with non-empty interior. The following
is a very useful criteria for almost periodicity (see [3, p. 7]). In what
follows $C(\mathbb{R})$ denotes the usual Banach algebra of bounded continuous
functions with the topology induced by the sup-norm
$\|f\|_{\infty}=\sup_{x\in\mathbb{R}}|f(x)|$.

In the remaining part of this section we compile necessary facts about almost
periodic functions and we provide some proofs for completeness.

######  Theorem 7 (Bochner’s criterion).

Let $f:\mathbb{R}\to\mathbb{C}$ be continuous. Then $f\in{\rm AP}(\mathbb{R})$
if and only if the set of functions $\\{f(\cdot+h)\\}_{h\in\mathbb{R}}$ is
pre-compact in $C(\mathbb{R})$.

######  Remark 7.

Bochner’s criterion works also for functions $f:\mathbb{R}\to\mathbb{C}^{n}$.
In particular, if $f_{j}\in{\rm AP}(\mathbb{R})$ for $j=1,...,N$, then
$\cap_{j=1}^{N}\tau_{\varepsilon}(f_{j})$ is nonempty and relatively dense.

It is now straightforward to show that ${\rm AP}(\mathbb{R})$ closed under
multiplication, addition and uniform convergence, hence ${\rm AP}(\mathbb{R})$
is a closed subalgebra of $C(\mathbb{R})$. Moreover, it also follows that for
every continuous $g:\mathbb{C}\to\mathbb{C}$ we have $g\circ f\in{\rm
AP}(\mathbb{R})$ whenever $f\in{\rm AP}(\mathbb{R})$. Since exponentials
$e^{2\pi ix\lambda}$, with $\lambda\in\mathbb{R}$, are periodic, we conclude
by Bochner’s criterion that any trigonometric polynomial

$p(x)=\sum_{n=1}^{N}a_{n}e^{2\pi
i\lambda_{n}x}\quad\quad(\lambda_{n}\in\mathbb{R})$

belongs to ${\rm AP}(\mathbb{R})$.

######  Lemma 8.

Given any $f\in{\rm AP}(\mathbb{R})$ and $\varepsilon>0$, there is a
trigonometric polynomial $p$ such that $\|f-p\|_{\infty}<\varepsilon$. In
particular, ${\rm AP}(\mathbb{R})$ is the closure in $C(\mathbb{R})$ of the
algebra of trigonometric polynomials.

###### Proof.

We can assume $\|f\|_{\infty}\leq 1$. For a given $\varepsilon>0$ let
$\tau_{\varepsilon}$ be the set of $\varepsilon$-translations for $f$. First
we will need some auxiliary functions. We claim that for every sufficiently
large $M\geq 1$ there exists $\varphi_{M}\in
C^{\infty}_{c}(\tau_{\varepsilon}\cap(-M,M))$ such that $0\leq\varphi_{M}\leq
1$, $\int_{\mathbb{R}}\varphi_{M}=1$,
$\int_{\mathbb{R}}|\varphi_{M}^{\prime}|=O_{\varepsilon}(1)$ and

$\sum_{n\in\mathbb{Z}}|\widehat{\varphi}(n/(4M))|^{1/2}=O_{\varepsilon}(1).$

Assuming this claim is true, we now finish the proof. Let

$f_{M}(x)=\int_{\mathbb{R}}f(x+t)\varphi_{M}(t)\mathrm{d}t$

and note that $\|f-f_{M}\|_{\infty}\leq\varepsilon$ for any $M$. Also note
that if $|x|\leq M$ then

$f_{M}(x)=\int_{-2M}^{2M}f(t)\varphi_{M}(t-x)\mathrm{d}t.$

The function $\varphi_{M}(x)$ can be identified with a $4M$-periodic
$C^{\infty}$-function and so we have

$\varphi_{m}(x)=\sum_{n\in\mathbb{Z}}\theta_{n}e^{2\pi inx/(4M)}$

where
$\sum_{n\in\mathbb{Z}}|\theta_{n}|^{2}=(4M)^{-1}\int_{-2M}^{2M}|\varphi_{M}(x)|^{2}\mathrm{d}x$
and

$\theta_{n}=(4M)^{-1}\int_{-2M}^{2M}\varphi_{M}(x)e^{-2\pi
inx/(4M)}\mathrm{d}x=\widehat{\varphi}_{M}(n/(4M))/(4M).$

We obtain

$f_{M}(x)=\sum_{n\in\mathbb{Z}}\widetilde{\theta}_{n}e^{-2\pi
inx/(4M)}\quad\text{ for }|x|\leq M,$

where $\widetilde{\theta}_{n}=\theta_{n}\int_{-2M}^{2M}f(t)e^{2\pi
int/(4M)}\mathrm{d}t$. Note this representation converges absolutely since
$|\widetilde{\theta}_{n}|\leq
4M|\theta_{n}|=|\widehat{\varphi}_{M}(n/(4M))|\leq 1$,
$\sum_{n\in\mathbb{Z}}|\widetilde{\theta}_{n}|^{1/2}=O_{\varepsilon}(1)$, and
so, $\sum_{n\in\mathbb{Z}}|\widetilde{\theta}_{n}|=O_{\varepsilon}(1)$. Now
enumerate $(\widetilde{\theta}_{n})_{n\in\mathbb{Z}}$ in decreasing order of
magnitude and call this new sequence $(\alpha_{M,n})_{n\geq 1}$. We obtain

$f_{M}(x)=\sum_{n\geq 1}\alpha_{M,n}e^{2\pi i\lambda_{M,n}x}\quad\text{ for
}|x|\leq M,$

for some $\lambda_{M,n}\in\frac{1}{4M}\mathbb{Z}$. Noticing that

$|\alpha_{M,n}|^{1/2}n\leq\sum_{j=1}^{n}|\alpha_{M,j}|^{1/2}=O_{\varepsilon}(1),$

we obtain $|\alpha_{M,n}|\leq n^{-2}$. A standard Cantor’s diagonal procedure
guarantees the existence of $(\alpha_{n})_{n\geq 1}$ such that
$\sum_{n\in\mathbb{Z}}|\alpha_{n}|=O_{\varepsilon}(1)$ and a subsequence
$M_{k}\to\infty$ such that $\lim_{k}\alpha_{M_{k},n}=\alpha_{n}$ for all
$n\geq 1$. Let now $I\subset\mathbb{N}$ be the $n$’s such that
$\sup_{k}|\lambda_{M_{k},n}|<\infty$. By further taking a subsequence of the
$M_{k}$’s we can assume that $\lim_{k}\lambda_{M_{k},n}\to\lambda_{n}$ for
$n\in I$. Note now that

$|\alpha_{M_{k},n}|\leq|\widehat{\varphi}_{M}(-\lambda_{M_{k},n})|\leq\frac{1}{2\pi|\lambda_{M_{k},n}|}\int_{\mathbb{R}}|\varphi_{M}^{\prime}(x)|\mathrm{d}x=O_{\delta}(|\lambda_{M_{k},n}|^{-1}).$

In particular, $\lim_{k}\alpha_{M_{k},n}=\alpha_{n}=0$ if $n\notin I$.
Finally, the uniform bound $|\alpha_{M,n}|\leq n^{-2}$ forces $f_{M_{k}}$ to
converge uniformly in compact sets to

$g(x)=\sum_{n\geq 1}\alpha_{n}e^{2\pi i\lambda_{n}x},$

which in particular implies that

$\|f-g\|_{\infty}\leq\varepsilon.$

However, we can now simply truncated $g$ to find a trigonometric polynomial
$p$ such that $\|f-p\|_{\infty}\leq 2\varepsilon$.

It remains to construct the auxiliary functions $\varphi_{M}$. Since $f$ is
also uniformly continuous, it is easy to show that
$\\{t_{n}\\}_{n\in\mathbb{Z}}+(-\delta,\delta)\subset\tau_{\varepsilon/4}$ for
some small $\delta=\delta_{\varepsilon}>0$ and some sequence
$\\{t_{n}\\}_{n\in\mathbb{Z}}$ satisfying $1/\delta\leq t_{n+1}-t_{n}\leq
3/\delta$ for all $n$. Take $h\in C^{\infty}_{c}(-1,1)$ even with $0\leq h\leq
1$ and $\int_{\mathbb{R}}h=1$. Let $h_{\delta}(x)=h(x/\delta)/\delta$,
$\psi(x)=(2N+1)^{-1}\sum_{n=-N}^{N}h_{\delta}(x-t_{n})$ and
$\varphi_{M}(x)=\psi*\psi*\psi*\psi$, where $N\geq 1$ is selected to be the
largest such that $\\{t_{n}\\}_{|n|\leq
N}+(-\delta,\delta)\subset\tau_{\varepsilon/4}\cap(-M/4,M/4)$. Note we must
have $N\geq\frac{\delta M}{10}$ for sufficiently large $M$. Since
$\tau_{\varepsilon/4}+\tau_{\varepsilon/4}+\tau_{\varepsilon/4}+\tau_{\varepsilon/4}\subset\tau_{\varepsilon}$
we conclude that $\varphi_{M}\in
C^{\infty}_{c}(\tau_{\varepsilon}\cap(-M,M))$. Also note that
$\int_{\mathbb{R}}\varphi_{M}=(\int_{\mathbb{R}}\psi)^{4}=1$ and, since
$\varphi_{M}^{\prime}=\psi^{\prime}*\psi*\psi*\psi$, that

$\int_{\mathbb{R}}|\varphi^{\prime}_{M}|\leq\int_{\mathbb{R}}|\psi^{\prime}|=O(1/\delta).$

Finally note that

$\displaystyle\sum_{n\in\mathbb{Z}}|\widehat{\varphi}_{M}(n/(4M))|^{1/2}=\sum_{n\in\mathbb{Z}}|\widehat{\psi}(n/(4M))|^{2}$
$\displaystyle=4M\int_{-2M}^{2M}|\psi(x)|^{2}\mathrm{d}x$
$\displaystyle=\frac{4M}{\delta(2N+1)}\int_{-1}^{1}|h(x)|^{2}\mathrm{d}x\leq\frac{4M}{\delta(2N+1)}\ll\delta^{-2},$

for $M$ large enough. This finishes the proof. ∎

We define

$\mathbb{E}f:=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(x)\mathrm{d}x\quad\text{and}\quad\mathbb{E}f(\lambda):=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(x)e^{-2\pi
i\lambda x}\mathrm{d}x,$

whenever these limits exist (so $\mathbb{E}f=\mathbb{E}f(0)$).

######  Lemma 9.

For any $f\in{\rm AP}(\mathbb{R})$ we have:

  1. (1)

The average $\mathbb{E}f(\lambda)$ exists;

  2. (2)

The set ${\rm spec}(f):=\\{\lambda\in\mathbb{R}:\mathbb{E}f(\lambda)\neq 0\\}$
it at most countable;

  3. (3)

If ${\rm spec}(f)=\emptyset$ then $f=0$;

  4. (4)

If $\sum_{\lambda\in\mathbb{R}}|\mathbb{E}f(\lambda)|<\infty$, then the
following series converges absolutely and uniformly

$f(x)=\sum_{\lambda\in\mathbb{R}}\mathbb{E}f(\lambda)e^{2\pi i\lambda x};$

  5. (5)

We have $\mathbb{E}|f|^{2}=\sum_{\lambda}|\mathbb{E}f(\lambda)|^{2}$ and for
every $\varepsilon>0$ there is $S\subset{\rm spec}(f)$ finite such that
$\mathbb{E}_{x}|f(x)-\sum_{\lambda\in S}\mathbb{E}f(\lambda)e^{2\pi i\lambda
x}|^{2}<\varepsilon$.

###### Proof.

For item (1), we can assume $\lambda=0$. Let $\varepsilon>0$ be given and take
a trigonometric polynomial $p$ such that $\|f-p\|_{\infty}\leq\varepsilon$.
Let $A=\mathbb{E}p$, which exists by direct computation. We have

$\displaystyle\frac{1}{2T}\int_{-T}^{T}f(x)\mathrm{d}x-\frac{1}{2T^{\prime}}\int_{-T^{\prime}}^{T^{\prime}}f(x)\mathrm{d}x$
$\displaystyle=O(2\varepsilon)+\frac{1}{2T}\int_{-T}^{T}p(x)\mathrm{d}x-\frac{1}{2T^{\prime}}\int_{-T^{\prime}}^{T^{\prime}}p(x)\mathrm{d}x$
$\displaystyle=O(2\varepsilon)+A-A+o_{T,T^{\prime}}(1).$

For item (2), note first that $\mathbb{E}f(\lambda)$ exists since
$f(x)e^{-2\pi i\lambda x}$ is also almost periodic (by Bochner’s criterion).
Let $p_{n}$ be a trigonometric polynomial such that
$\|f-p_{n}\|_{\infty}<\frac{1}{n}.$ Assume that $|\mathbb{E}f(\lambda)|>0$ and
let $1/n<|\mathbb{E}f(\lambda)|$. We obtain that
$|\mathbb{E}p_{n}(\lambda)|\geq|\mathbb{E}f(\lambda)|-\frac{1}{n}>0$, hence
$\mathbb{E}p_{n}(\lambda)\neq 0$. We conclude that ${\rm
spec}(f)\subset\bigcup_{n\geq 1}{\rm spec}(p_{n})$. For item (3), let
$\varepsilon>0$ be given and take a trigonometric polynomial $p$ such that
$\|\bar{f}-p\|_{\infty}\leq\varepsilon$. Now, observing that

$\frac{1}{2T}\int_{-T}^{T}|f(x)|^{2}\mathrm{d}x=\frac{1}{2T}\int_{-T}^{T}f(x)p(x)\mathrm{d}x+\frac{1}{2T}\int_{-T}^{T}f(x)(\bar{f}(x)-p(x))\mathrm{d}x=o_{T}(1)+O(\varepsilon\|f\|_{\infty}),$

we conclude that $\lim_{T}\frac{1}{2T}\int_{-T}^{T}|f(x)|^{2}\mathrm{d}x=0$.
This is impossible if $f$ is nonzero, because $|f|^{2}$ is also almost
periodic and uniformly continuous, and so if $|f(x)|^{2}\geq c$ for $x$ in
some interval $I$ then $|f(x)|^{2}\geq c/2$ for $x\in\tau_{c/2}(|f|^{2})\ +I$.
Since $\tau_{c/2}(|f|^{2})$ contains some increasing sequence
$\\{t_{n}\\}_{n\in\mathbb{Z}}$ satisfying $t_{n+1}-t_{n}=O(1)$, it implies
that $\mathbb{E}|f|^{2}\geq\mathbb{E}({\bf 1}_{\tau_{c/2}(|f|^{2})+I})>0$,
which is absurd. We conclude that $f=0$. For item (4), note that

$g(x)=\sum_{\lambda\in\mathbb{R}}\mathbb{E}f(\lambda)e^{2\pi i\lambda x},$

is well-defined and converges absolutely and uniformly on $\mathbb{R}$. Also
note that dominated convergence implies that
$\mathbb{E}g(\lambda)=\mathbb{E}f(\lambda)$ for all $\lambda\in\mathbb{R}$,
and so by item (3) $f=g$. For item (5), let $\Lambda^{\prime}:=\cup_{m\geq
1}{\rm spec}(p_{m})$ for some trigonometric polynomials $p_{m}$ such that
$\|f-p_{m}\|_{\infty}<1/m$ and enumerate
$\Lambda^{\prime}=\\{\lambda_{n}\\}_{n\geq 1}$. It is enough to show that
$\mathbb{E}|f|^{2}=\sum_{n\geq 1}|\mathbb{E}f(\lambda_{n})|^{2}$. Let
$f_{N}(x)=\sum_{n=1}^{N}\mathbb{E}f(\lambda_{n})e^{2\pi i\lambda_{n}x}$. A
straightforward computation shows that
$\mathbb{E}|f|^{2}-\mathbb{E}|f_{N}|^{2}=\mathbb{E}|f-f_{N}|^{2}$ and so
$\sum_{n=1}^{N}|\mathbb{E}f(\lambda_{n})|^{2}\leq\mathbb{E}|f|^{2}$ for all
$N$. Another routine computation shows

$\mathbb{E}|f-p|^{2}=\mathbb{E}|f|^{2}-\mathbb{E}|f_{N}|^{2}+\mathbb{E}|f_{N}-p|^{2}\geq\mathbb{E}|f|^{2}-\mathbb{E}|f_{N}|^{2},$

for any trigonometric polynomial of the form $p(x)=\sum_{n=1}^{N}b_{n}e^{2\pi
i\lambda_{n}x}$, for some $b_{n}\in\mathbb{C}$. Since $p_{m}$ is of such form
for some $N_{m}$, we conclude that $\mathbb{E}|f-f_{N_{m}}|^{2}<1/m$ and that

$\mathbb{E}|f_{N_{m}}|^{2}\leq\mathbb{E}|f|^{2}\leq\mathbb{E}|f_{N_{m}}|^{2}+2/m.$

∎

The following proposition (see [10, p. 46]) justifies the notation

$f(x)\sim\sum_{\lambda\in\mathbb{R}}\mathbb{E}f(\lambda)e^{2\pi i\lambda x}.$

######  Proposition 10 (Bochner’s approximation).

For any $f\in{\rm AP}(\mathbb{R})$ there is an sequence of functions
$c_{n}:\mathbb{R}\to[0,1]$, $c_{n}\leq c_{n+1}$, each $c_{n}$ with finite
support, satisfying

$\lim_{n\to\infty}c_{n}(\lambda)=\begin{cases}1&\mbox{if }\
\mathbb{E}f(\lambda)\neq 0\\\ 0&\mbox{if }\
\mathbb{E}f(\lambda)=0,\end{cases}$

and such that

$\lim_{n\to\infty}\sup_{x\in\mathbb{R}}|f(x)-\sum_{\lambda\in\mathbb{R}}c_{n}(\lambda)\mathbb{E}f(\lambda)e^{2\pi
i\lambda x}|=0.$

##  5\. Almost Periodic Functions in $\mathbb{C}^{+}$

Recall that we have defined in the introduction the space
$\mathrm{AP}(\mathbb{C}^{+})$ of holomorphic almost periodic functions
$f:\mathbb{C}^{+}\to\mathbb{C}$ such that for every $\varepsilon>0$ there is a
relatively dense set of $\varepsilon$-translations
$\tau_{\varepsilon}(f)\subset\mathbb{R}$ satisfying

$\sup_{\varepsilon<{\rm Im}\,z<1/\varepsilon}|f(z)-f(z+t)|<\varepsilon$

for every $t\in\tau_{\varepsilon}(f)$. We say a function
$f:\mathbb{C}^{+}\to\mathbb{C}$ is bounded on strips if

$\sup_{\varepsilon<{\rm Im}\,z<1/\varepsilon}|f(z)|<\infty$

for every $\varepsilon>0$. We now give an alternative characterization of
almost periodicity.

######  Lemma 11.

Let $f:\mathbb{C}^{+}\to\mathbb{C}$ be holomorphic. Then following are
equivalent:

  1. (1)

$f$ is bounded on strips and for every $h>0$ we have $f(\cdot+ih)\in{\rm
AP}(\mathbb{R})$;

  2. (2)

$f$ is bounded on strips and there is $h>0$ such that $f(\cdot+ih)\in{\rm
AP}(\mathbb{R})$;

  3. (3)

$f\in\mathrm{AP}(\mathbb{C}^{+})$;

In this case the quantity

$\mathbb{E}f(\lambda):=\lim_{T\to\infty}\frac{1}{2T}\int_{-T+iy}^{T+iy}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z$

exists for every $\lambda\in\mathbb{R}$, is independent of $y>0$, it is
nonzero in at most countably many $\lambda$’s and

$\sum_{\lambda\in\mathbb{R}}|\mathbb{E}f(\lambda)|^{2}e^{-4\pi
y\lambda}=\mathbb{E}[|f(\cdot+iy)|^{2}]$

for every $y>0$.

###### Proof.

The implications $(1)\Rightarrow(2)$ and $(3)\Rightarrow(1)$ are obvious. We
now show $(2)\Rightarrow(3)$. Let $\tau_{\varepsilon}$ be the set of
$\varepsilon$-translations for $f(\cdot+ih)$. We claim that
$\tau_{\varepsilon}$ also works in any horizontal strip containing the line
${\rm Im}\,z=h$. Indeed, let $t\in\tau_{\varepsilon}$ and
$0<y_{1}=y_{2}/2<y_{2}<h$. For all $y$ with $y_{2}<y<h$, we can then apply
Hadamard’s there lines lemma (for ${\rm Im}\,z\in\\{y_{1},y,h\\}$) to conclude
that

$\displaystyle\sup_{x}\log|f(x+iy)-f(x+iy+t)|$
$\displaystyle\leq\frac{y-y_{1}}{h-y_{1}}\sup_{x}\log|f(x+ih)-f(x+ih+t)|+\frac{h-y}{h-y_{1}}\sup_{x}\log|f(x+iy_{1})-f(x+iy_{1}+t)|$
$\displaystyle\leq\frac{y_{2}-y_{1}}{h-y_{1}}\log\varepsilon+\frac{h-y_{2}}{h-y_{1}}B_{1}$

for some $B_{1}>0$. Hence, there is $\alpha=\alpha(y_{2},h)>0$ such that

$\sup_{y_{2}<{\rm
Im}\,z<h}|f(z)-f(z+t)|\leq\frac{1}{\alpha}\varepsilon^{\alpha}.$

We can apply the same procedure for $h<{\rm Im}\,z<y_{3}$ for any $y_{3}>h$ to
prove the claim. This shows that $f\in\mathrm{AP}(\mathbb{C}^{+})$. Now notice
that by Lemma 9 the limit $\mathbb{E}f(\lambda)$ exists for every $y>0$. To
show is independent of $y$, we can use Cauchy’s formula to deduce that if
$0<y_{1}<y_{2}$ then

$\displaystyle\bigg{[}\frac{1}{2T}\int_{-T+iy_{1}}^{T+iy_{1}}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z-\frac{1}{2T}\int_{-T+iy_{2}}^{T+iy_{2}}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z\bigg{]}$
$\displaystyle=\bigg{[}\frac{1}{2T}\int_{-T+iy_{1}}^{-T+iy_{2}}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z-\frac{1}{2T}\int_{T+iy_{1}}^{T+iy_{2}}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z\bigg{]}=O((y_{2}-y_{1})e^{2\pi|\lambda|y_{2}}/T).$

Taking $T\to\infty$, we conclude that $\mathbb{E}f(\lambda)$ is independent of
$y$. Since $e^{-2\pi\lambda
y}\mathbb{E}f(\lambda)=\mathbb{E}[f(\cdot+iy)](\lambda)$ we deduce that
$\lambda\in\mathbb{R}\mapsto\mathbb{E}f(\lambda)$ has countable support and
$\sum_{\lambda\in\mathbb{R}}|e^{-2\pi\lambda
y}\mathbb{E}f(\lambda)|^{2}=\mathbb{E}|f(\cdot+iy)|^{2}$. This finishes the
proof. ∎

For a function $f\in\mathrm{AP}(\mathbb{C}^{+})$ (or $f\in{\rm
AP}(\mathbb{R})$) we define the spectrum of $f$ to be

${\rm spec}(f):=\\{\lambda\in\mathbb{R}:\mathbb{E}f(\lambda)\neq 0\\}.$

######  Lemma 12.

The following hold:

  1. (i)

Let $f\in\mathrm{AP}(\mathbb{C}^{+})$. Then $f$ is bounded in
$\mathbb{C}^{+}+ic$, for some $c>0$, if and only if ${\rm
spec}(f)\subset[0,\infty)$. In this case $f$ is bounded in $\mathbb{C}^{+}+ic$
for any $c>0$ and $\lim_{y\to\infty}\sup_{x}|f(x+iy)-\mathbb{E}f(0)|=0$;

  2. (ii)

If $f\in\mathrm{AP}(\mathbb{C}^{+})$, ${\rm spec}(f)$ is bounded from below
and $f(z)\neq 0$ for all ${\rm Im}\,z>c$, for some $c>0$, then $\inf{\rm
spec}(f)\in{\rm spec}(f)$;

  3. (iii)

If $f,g\in\mathrm{AP}(\mathbb{C}^{+})$ have spectrum bounded from below then
so has $fg\in\mathrm{AP}(\mathbb{C}^{+})$. If in addition $f,g$ have locally
finite spectrum, then so has $fg$.

  4. (iv)

If $f\in\mathrm{AP}(\mathbb{C}^{+})$ and $f\neq 0$ in $\mathbb{C}^{+}$, then
$\inf_{\varepsilon<{\rm Im}\,z<1/\varepsilon}|f(z)|>0$ for all $\varepsilon>0$
and $1/f\in\mathrm{AP}(\mathbb{C}^{+})$. Moreover, if $f$ has spectrum bounded
from below then so has $1/f$, and if in addition $f$ has locally finite
spectrum then so has $1/f$.

  5. (v)

If $f\in\mathrm{AP}(\mathbb{C}^{+})$ and $|f(z)|<1$ for all
$z\in\mathbb{C}^{+}$ then for every $\varepsilon>0$ there is $c>0$ such that
$|f(z)|<1-c$ if ${\rm Im}\,z>\varepsilon$.

###### Proof.

Items (i) and (ii): These are direct applications of [10, Thm. p. 162 & p.
152]. Item (iii): Note that $f_{1}(z)=e^{-2\pi iMz}f(z)$ and
$g_{1}(z)=e^{-2\pi iMz}g(z)$ have spectrum contained in $[0,\infty)$ for some
$M>0$, thus both are bounded for ${\rm Im}\,z>1$ and so is $h=f_{1}g_{1}$.
Thus, ${\rm spec}(fg)={\rm spec}(h)+2M\subset[0,\infty)$, hence $fg$ have
spectrum bounded from below. We also have ${\rm spec}(fg)\subset{\rm
spec}(f)+{\rm spec}(g)={\rm spec}(f_{1})+{\rm spec}(g_{1})-2M$, and since sums
of locally finite sets contained in $[0,\infty)$ is also locally finite and
contained in $[0,\infty)$, we conclude that the spectrum of $fg$ is locally
finite if both $f$ and $g$ have locally finite spectrum. Item (iv): By a
clever application of Hadamard’s Three-Lines theorem [10, Thm. 11, p. 139 &
Thm. 9, p. 144], one can show that if $w$ is an accumulation point of $f$ in
an horizontal line, then $f(z)=w$ has a solution in any horizontal strip
containing this line. In particular, if $f$ never vanishes in $\mathbb{C}^{+}$
then $|f|$ is bounded away from zero in any horizontal strip, thus
$1/f\in\mathrm{AP}(\mathbb{C}^{+})$. If $f\sim\sum_{\lambda\geq
M}\mathbb{E}f(\lambda)e^{2\pi i\lambda z}$ where $M=\inf{\rm spec}(f)$, since
$f$ has no zeros, $M\in{\rm spec}(f)$, $p=\mathbb{E}f(M)\neq 0$ and
$\lim_{y\to\infty}\sup_{x}|f(x+iy)e^{-2\pi iM(x+it)}-p|=0$. In particular
$g(z)=1-f(z)e^{-2\pi iMz}/p$ is bounded in absolute value by $1/2$ for ${\rm
Im}\,z>c$, for some large $c>0$ and ${\rm spec}(g)\subset(0,\infty)$. We
obtain

$\frac{1}{f(z)}=\frac{e^{-2\pi iMz}}{p}\frac{1}{1-g(z)}=\frac{e^{-2\pi
iMz}}{p}\sum_{n\geq 0}g(z)^{n},$

where the sum converges absolutely and uniformly for ${\rm Im}\,z>c$. We
conclude that $\mathbb{E}(1/f)(\lambda)=\sum_{n\geq
0}\mathbb{E}(g^{n})(\lambda+M)$ (with absolute convergence), and so
$\mathbb{E}(1/f)(\lambda)=0$ if $\lambda<-M$. If in addition ${\rm spec}(f)$
is locally finite, then ${\rm spec}(g)$ is locally finite and contained in
$[\delta,\infty)$, for some $\delta>0$. Since ${\rm
spec}(1/f)+\\{M\\}\subset\cup_{n\geq 0}\big{(}\oplus^{n}{\rm spec}(g)\big{)}$,
we conclude that $({\rm spec}(1/f)+\\{M\\})\cap[0,T]\subset\cup_{0\leq n\leq
T/\delta}\big{(}\oplus^{n}{\rm spec}(g)\cap[0,T]\big{)}$, and so ${\rm
spec}(1/f)$ is locally finite. Item (v): Since $f$ is bounded we deduce that
$\mathbb{E}f(\lambda)=0$ for $\lambda<0$ and, by [10, Thm. 9, p. 144], that
$\sup_{\varepsilon<{\rm Im}\,z<1/\varepsilon}|f(z)|<1$ for every
$\varepsilon>0$. Hence $|\mathbb{E}f(0)|<1$. On the other hand
$\lim_{y\to\infty}\sup_{x}|f(x+iy)-\mathbb{E}f(0)|=0$. ∎

######  Lemma 13.

Let $f\in\mathrm{AP}(\mathbb{C}^{+})$ and assume ${\rm
spec}(f)\subset\\{0\\}\cup[b,\infty)$ for some $b>0$. If
$F:\mathbb{C}^{+}\to\mathbb{C}$ is holomorphic and $iF^{\prime}/F=f$, then
$F\in\mathrm{AP}(\mathbb{C}^{+})$ and ${\rm
spec}(F)\subset\\{-\mathbb{E}f(0)/(2\pi)\\}\cup[-\mathbb{E}f(0)/(2\pi)+b,\infty)$.
Moreover, if in addition ${\rm spec}(f)$ is locally finite, then so it is
${\rm spec}(F)$.

###### Proof.

Let $p=\mathbb{E}f(0)$. Since $F$ has no zeros in $\mathbb{C}^{+}$ we can
write $F(z)=e^{-ig(z)-ipz}$ for some $g:\mathbb{C}^{+}\to\mathbb{C}$
holomorphic. Since $g^{\prime}(z)+p=iF^{\prime}(z)/F(z)=f(z)$ we conclude that
$g^{\prime}\in\mathrm{AP}(\mathbb{C}^{+})$, ${\rm spec}(g)\subset[b,\infty)$
and $\mathbb{E}(g^{\prime})(\lambda)={\boldsymbol{1}}_{\lambda\geq
b}\mathbb{E}f(\lambda)$. We can now apply [10, Thm. 9, p. 152] to conclude
that $g\in\mathrm{AP}(\mathbb{C}^{+})$ and that
$\mathbb{E}g(\lambda)={\boldsymbol{1}}_{\lambda\geq
b}\mathbb{E}f(\lambda)/(2\pi i\lambda)$ for $\lambda\neq 0$. If
$\Phi:\mathbb{C}\to\mathbb{C}$ is entire and $h\in\mathrm{AP}(\mathbb{C}^{+})$
then it is easy to conclude that $\Phi\circ h\in\mathrm{AP}(\mathbb{C}^{+})$.
Indeed, since $h$ is bounded in strips and since $\Phi^{\prime}$ is bounded in
bounded sets, we conclude that for any horizontal strip contained in
$\mathbb{C}^{+}$ there is $C>0$ such that $|\Phi(h(z))-\Phi(h(w))|\leq
C|h(z)-h(w)|$ for all $z$ and $w$ in that strip. This shows that $\Phi\circ
h\in\mathrm{AP}(\mathbb{C}^{+})$. This implies that the power series expansion

$\Phi(h(z))=\sum_{n\geq 0}\frac{\Phi^{(n)}(0)}{n!}h(z)^{n}.$

converges absolutely and uniformly on any horizontal strip. Now assume that
${\rm spec}(h)\subset[b,\infty)$ for some $b>0$. Since ${\rm
spec}(h^{n})\subset[nb,\infty)$ and

$\mathbb{E}[\Phi\circ h](\lambda)=\sum_{n\geq
0}\frac{\Phi^{(n)}(0)}{n!}\mathbb{E}[h^{n}](\lambda),$

we conclude that ${\rm spec}(\Phi\circ h)\subset\\{0\\}\cup[b,\infty)$. If in
addition ${\rm spec}(h)$ is locally finite, since

${\rm spec}(\Phi\circ h)\cap[0,T]\subset\cup_{0\leq n\leq
T/b}\big{(}\oplus^{n}{\rm spec}(h)\cap[0,T]\big{)},$

we conclude that ${\rm spec}(\Phi\circ h)$ is locally finite. Considering
$\Phi(z)=e^{-iz}$ and $h(z)=g(z)-\mathbb{E}g(0)$, we deduce that
$F(z)=\Phi(h(z))e^{-i\mathbb{E}g(0)-ipz}$ belongs to
$\mathrm{AP}(\mathbb{C}^{+})$ and that ${\rm
spec}(F)\subset\\{-p/(2\pi)\\}\cup[-p/(2\pi)+b,\infty)$. Moreover, ${\rm
spec}(F)$ is locally finite if ${\rm spec}(f)$ also is. ∎

We also have approximation by trigonometric polynomials (see [10, p. 148])

######  Proposition 14 (Bochner’s approximation).

Let $f\in\mathrm{AP}(\mathbb{C}^{+})$. Then for every $b>1$ there is a
sequence of functions $c_{n}:\mathbb{R}\to[0,1]$ as in Proposition 10 such
that

$\lim_{n\to\infty}\sup_{1/b<{\rm
Im}\,z<b}|f(z)-\sum_{\lambda\in\mathbb{R}}c_{n}(\lambda)\mathbb{E}f(\lambda)e^{2\pi
i\lambda z}|=0.$

##  6\. Representations of Analytic Functions

We say a holomorphic function $g:\mathbb{C}^{+}\to\mathbb{C}$ if of bounded
type if there are two bounded holomorphic functions
$P,Q:\mathbb{C}^{+}\to\mathbb{C}$ such that $g=P/Q$. We write $g\in{\rm
BT}(\mathbb{C}^{+})$. If $g$ is of bounded type then the mean type

$\vartheta(g):=\limsup_{y\to\infty}\frac{\log|g(iy)|}{y}$

exists and is finite. For more information about these definitions we
recommend [12, 19].

######  Lemma 15.

Let $f:\mathbb{C}^{+}\to\mathbb{C}$ be holomorphic and write $g=\exp(f)$. The
following are equivalent:

  1. (1)

There is a real-valued locally finite measure $\mu$ with $\deg(\mu)\leq 2$
such that

$\displaystyle\frac{f(z)+\overline{f(w)}}{z-\overline{w}}=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{\mathrm{d}\mu(t)}{(t-z)(t-\overline{w})}$ (12)

holds for every $z,w\in\mathbb{C}^{+}$;

  2. (2)

$g\in{\rm BT}(\mathbb{C}^{+})$ and $\vartheta(g)=0$.

###### Proof.

First we show that $(1)\Rightarrow(2)$. Indeed, note first that

${\rm
Re}\,f(x+iy)=\frac{y}{\pi}\int_{\mathbb{R}}\frac{\mathrm{d}\mu(t)}{(t-x)^{2}+y^{2}}.$

Writing $\mu=\mu_{1}-\mu_{2}$ where each $\mu_{j}$ is nonnegative and defining

$f_{j}(z)=\frac{1}{\pi
i}\int_{\mathbb{R}}\frac{1+tz}{t-z}\frac{\mathrm{d}\mu_{j}(t)}{1+t^{2}}$

we conclude that ${\rm Re}\,f_{j}\geq 0$ in $\mathbb{C}^{+}$ and that ${\rm
Re}\,f={\rm Re}\,f_{1}-{\rm Re}\,f_{2}$, hence $f=f_{1}-f_{2}+ih$ for some
constant $h\in\mathbb{R}$. However, since each $g_{j}=\exp(-f_{j})$ is bounded
in absolute value by $1$ in $\mathbb{C}^{+}$ and $g=e^{ih}g_{2}/g_{1}$, we
deduce that $g\in{\rm BT}(\mathbb{C}^{+})$. Note also that

$\vartheta(g)=\limsup_{y\to\infty}\frac{{\rm
Re}\,f(iy)}{y}=\limsup_{y\to\infty}\frac{1}{\pi}\int_{\mathbb{R}}\frac{\mathrm{d}\mu(t)}{t^{2}+y^{2}}=0.$

Now we show $(2)\Rightarrow(3)$. Since $g$ has no zeros, Nevalinnas’s
factorization for functions of bounded type [12, Theprem 9] implies that there
exists a unique real-valued locally finite measure $\mu$, with $\deg(\mu)\leq
2$, and some $c,h\in\mathbb{R}$ such that

$e^{f(z)}=g(z)=e^{-ihz}\exp\bigg{(}ic+\frac{1}{\pi
i}\int_{\mathbb{R}}\frac{1+tz}{t-z}\frac{\mathrm{d}\mu(t)}{1+t^{2}}\bigg{)}.$

Since $h=\vartheta(g)=0$, a simple computation shows that (12) holds. ∎

######  Lemma 16.

Let $f\in{\rm AP}(\mathbb{R})$ and assume that ${\rm
spec}(f)\cap(0,b)=\emptyset$ for some $b>0$. Then

$F(z)=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{1+tz}{t-z}\frac{f(t)\mathrm{d}t}{1+t^{2}}$

belongs to $\mathrm{AP}(\mathbb{C}^{+})$ and
$\mathbb{E}F(\lambda)={\boldsymbol{1}}_{\lambda\geq b}\mathbb{E}f(\lambda)$
for $\lambda\neq 0$.

###### Proof.

Since any $f\in{\rm AP}(\mathbb{R})$ is bounded, the integral defining $F$
converges absolutely and defines an holomorphic function $F$. Noticing that
$\frac{1+tz}{(t-z)(1+t^{2})}=\frac{1}{t-z}-\frac{t}{1+t^{2}}$ we obtain that

$F^{\prime}(z)=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{f(t)\mathrm{d}t}{(t-z)^{2}},$

and so $|F^{\prime}(x+iy)|\leq\frac{1}{2y}\max_{t\in\mathbb{R}}|f(t)|$ for
$y>0$, hence $F^{\prime}$ is bounded on strips. Since

$F^{\prime}(x+i)=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{f(t+x)\mathrm{d}t}{(t-i)^{2}}$

it is clear that any set of $\varepsilon$-translations for $f$ is also one for
$F^{\prime}(\cdot+i)$, and so $F^{\prime}(\cdot+i)\in{\rm AP}(\mathbb{R})$,
and Lemma 11 shows that $F^{\prime}\in\mathrm{AP}(\mathbb{C}^{+})$. Dominated
Convergence implies that

$\mathbb{E}F^{\prime}(\lambda)=e^{2\pi\lambda}\mathbb{E}f(\lambda)\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{e^{2\pi i\lambda
t}}{(t-i)^{2}}=e^{2\pi\lambda}\mathbb{E}f(\lambda)2\pi
i|\lambda|e^{-2\pi|\lambda|}{\boldsymbol{1}}_{\lambda>0}=2\pi
i\max\\{\lambda,0\\}\mathbb{E}f(\lambda),$

and, since ${\rm spec}(f)\cap(0,b)=\emptyset$ for some $b>0$, we obtain ${\rm
spec}(F^{\prime})\subset[b,\infty)$. We can now apply [10, Thm. 9, p. 152] to
conclude that $F\in\mathrm{AP}(\mathbb{C}^{+})$. ∎

######  Lemma 17.

Let $f:\mathbb{C}\to\mathbb{C}$ be entire such that
$f,f^{*}\in\mathrm{AP}(\mathbb{C}^{+})$ and both have spectrum bounded from
below. If $f$ has finite order, that is,

${\rm
order}(f):=\inf\\{\rho>0:\limsup_{|z|\to\infty}\frac{\log^{+}|f(z)|}{|z|^{\rho}}<\infty\\}<\infty,$

then $f$ has finite exponential type, that is,

$\tau=\limsup_{|z|\to\infty}\frac{\log|f(z)|}{|z|}<\infty.$

In this case $f$ and $f^{*}$ are of bounded type, $f$ is bounded on the real
line and $|f(z)|\leq Me^{\tau|y|}$ for all $z=x+iy$, where
$M=\sup_{x\in\mathbb{R}}|f(x)|$ and
$\tau=\max\\{\vartheta(f^{*}),\vartheta(f)\\}$. Moreover,
$\mathbb{E}f(\lambda)=\overline{\mathbb{E}(f^{*})(-\lambda)}$ for all
$\lambda\in\mathbb{R}$ and

${\rm spec}(f)=-{\rm spec}(f^{*})\subset[-\tau/(2\pi),\tau/(2\pi)].$

If in addition ${\rm spec}(f)$ is locally finite, then $f$ is trigonometric
polynomial.

###### Proof.

Observe that since $f$ has spectrum bounded from below, then Lemma 12(i) shows
that for $m=\min\\{\inf{\rm spec}(f),\inf{\rm spec}(f^{*})\\}$ we have that
$|f(z)e^{-2\pi imz}|+|f^{*}(z)e^{-2\pi imz}|$ is bounded for ${\rm Im}\,z>c$,
for any $c>0$. This in particular shows that $f(z)e^{-2\pi|m|z}$ is bounded on
the sides of any sector $S_{c}=\\{z=x+iy:x>|y|/c\\}$. Since $f$ has finite
order, one can apply Phranmën-Lindelöf for sectors (with small $c>0$) [32,
Thm. 1, p. 37] to conclude that $f(z)e^{-2\pi|m|z}$ is bounded in $S_{c}$. We
then apply the same argument replacing $f(z)$ by $f^{*}(-z)$ to conclude that
$f^{*}(-z)e^{-2\pi|m|z}$ is bounded in $S_{c}$. All these imply that $f$ has
finite exponential type, that is,

$|f(z)|\ll e^{C|z|}$

for some $C>0$. However, since $f(z+i)e^{-2\pi im(z+i)}$ is bounded on the
real line, another application of Phranmën-Lindelöf [32, Thm. 3, p. 38] shows
that $|f(z+i)e^{-2\pi im(z+i)}|\ll e^{C^{\prime}|y|}$ for some $C^{\prime}>0$,
and so $|f(z)|\ll e^{C^{\prime\prime}|y|}$ for some $C^{\prime\prime}>0$. In
particular $f$ is bounded on every horizontal strip and, by the same result,
we can take $C^{\prime}=\tau$. A straightforward calculation using Cauchy’s
integral formula shows that

$\mathbb{E}f(\lambda)-\overline{\mathbb{E}(f^{*})(-\lambda)}=\lim_{T\to\infty}\frac{1}{2T}\left(\int_{T-i}^{T+i}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z-\int_{-T-i}^{-T+i}f(z)e^{-2\pi i\lambda
z}\mathrm{d}z\right)=0,$

since $f$ is now bounded on horizontal strips. In particular, ${\rm
spec}(f)=-{\rm spec}(f^{*})$. A well-known result of Krein [24] (see also [32,
Thm. 1, p. 115]) shows that $f,f^{*}\in{\rm BT}(\mathbb{C}^{+})$ and that
$\tau=\max\\{\vartheta(f^{*}),\vartheta(f)\\}$. Since $g(z)=f(z)e^{i\tau z}$
is bounded in $\mathbb{C}^{+}$, Lemma 12(i) shows that ${\rm
spec}(f)+\tau/(2\pi)={\rm spec}(g)\subset[0,\infty)$, hence ${\rm
spec}(f)\subset[-\tau/(2\pi),\infty)$. The same argument shows that ${\rm
spec}(f^{*})\subset[-\tau/(2\pi),\infty)$, but since
$\mathbb{E}f(\lambda)=\overline{\mathbb{E}(f^{*})(-\lambda)}$, we obtain that
${\rm spec}(f)\subset[-\tau/(2\pi),\tau/(2\pi)]$. Finally, if ${\rm spec}(f)$
is locally finite, Proposition 14 proves straightforwardly that $f$ is a
trigonometric polynomial. ∎

##  7\. Proof of Theorem 1

We start we some lemmas.

######  Lemma 18.

For $w,z\in\mathbb{C}^{+}$ let

$\displaystyle g(w,z,x)=\frac{e^{-2\pi
i\overline{w}|x|}{\boldsymbol{1}}_{x<0}+e^{2\pi iz|x|}{\boldsymbol{1}}_{x\geq
0}}{z-\overline{w}}.$ (13)

Then

$\widehat{g}(w,z,\xi)=\frac{1}{2\pi i(\xi-z)(\xi-\overline{w})}.$

###### Proof.

Letting $f_{z}(x)=e^{2\pi
iz|x|}{\boldsymbol{1}}_{x>0}+\frac{1}{2}{\boldsymbol{1}}_{x=0}$ we have

$\widehat{f}_{z}(\xi)=\int_{0}^{\infty}e^{2\pi
ix(z-\xi)}\mathrm{d}x=\frac{1}{2\pi i(\xi-z)}.$

The lemma follows since
$g(w,z,x)=(z-\overline{w})^{-1}(f_{-\overline{w}}(-x)+f_{z}(x))$. ∎

######  Lemma 19.

Let $(\mu,a)$ be a $\mathrm{FS}$-pair with $\deg(\mu)\leq 2$. Then

$\displaystyle\lim_{T\to\infty}\sum_{|\lambda|<T}a(\lambda)g(w,z,\lambda)(1-|\lambda|/T)=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{\mathrm{d}\mu(t)}{(t-z)(t-\overline{w})}$ (14)

uniformly for $w$ and $z$ in the region

$R_{c}:=\\{z\in\mathbb{C}^{+}:|{\rm Re}\,z|\leq 1/c\text{ and }{\rm Im}\,z\geq
c\\},\quad(\text{for any }c>0).$

###### Proof.

Let $\varphi\in C_{c}^{\infty}(\mathbb{R})$, $\int\varphi=1$, $\varphi\geq 0$,
$\operatorname{\mathrm{supp}}\varphi\subset(-1,1)$ and define
$\varphi_{\varepsilon}(x)=\varphi(x/\varepsilon)/\varepsilon$ for
$0<\varepsilon<1$. Let $w,z\in R_{c}$, $T>2/c$ and

$g_{\varepsilon,T}(x):=(g(w,z,\cdot)(1-\tfrac{1}{T}|\cdot|)_{+})*\varphi_{\varepsilon}(x),$

where $s_{+}=\max\\{0,s\\}$. Note that $g_{\varepsilon,T}\in
C_{c}^{\infty}(-\varepsilon-T,T+\varepsilon)$ and

$\widehat{g}_{\varepsilon,T}(\xi)=(\widehat{g}(w,z,\cdot)*{S_{T}}(\xi))\widehat{\varphi}(\varepsilon\xi),$

where $S_{T}(x)=\frac{\sin^{2}(T\pi x)}{T(\pi x)^{2}}$. Note that
$|\widehat{\varphi}(\varepsilon\xi)|\leq 1$. We claim that161616We write
$A\ll_{p}B$ if there is a numerical constant $K>0$, depending only in the
parameter $p$, such that $A\leq KB$.

$\displaystyle|\widehat{g}(w,z,\cdot)*{S_{T}}(\xi)|\ll_{c}\frac{1}{\xi^{2}+c^{2}}.$
(15)

We shall prove this claim in the end. Since $(\mu,a)$ is a $\mathrm{FS}$-pair,
in particular we obtain

$\sum_{|\lambda|<{T+\varepsilon}}a(\lambda)g_{\varepsilon,T}(\lambda)=\int_{\mathbb{R}}\widehat{g}_{\varepsilon,T}(t)\mathrm{d}\mu(t).$

Since $a(\cdot)$ is locally summable, and as $\varepsilon\to 0$,
$g_{\varepsilon,T}(x)\to g(w,z,x)(1-|x|/T)_{+}$ uniformly in $x\in\mathbb{R}$
and $\widehat{g}_{\varepsilon,T}(\xi)\to\widehat{g}(w,z,\cdot)*S_{T}(\xi)$
pointwise, Dominated Convergence implies that

$\sum_{|\lambda|<T}a(\lambda)g(w,z,\lambda)(1-|\lambda|/T)=\int_{\mathbb{R}}\widehat{g}(w,z,\cdot)*{S_{T}}(t)\mathrm{d}\mu(t).$

It is now enough to show the right hand side above converges uniformly in the
region $(w,z)\in R_{c}^{2}$. This can be done by noticing that

$|\widehat{g}(w,z,\xi_{1})-\widehat{g}(w,z,\xi_{2})|=\frac{1}{2\pi|z-\overline{w}|}\bigg{|}\frac{1}{\xi_{1}-z}-\frac{1}{\xi_{1}-\overline{w}}-\frac{1}{\xi_{2}-z}+\frac{1}{\xi_{2}-\overline{w}}\bigg{|}\leq\frac{|\xi_{1}-\xi_{2}|}{4\pi
c^{3}},$

and so $\widehat{g}(w,z,\cdot)*{S_{T}}(\xi)\to\widehat{g}(w,z,\xi)$ uniformly
in $x\in\mathbb{R}$ and $(w,z)\in R_{c}$, as $T\to\infty$. Finally, the
uniform bound (15) guarantees the desired uniform convergence. It remains to
prove the claim (15). First note that

$|\widehat{g}(w,z,\xi)|\ll\frac{1}{(\xi-{\rm
Re}\,z)^{2}+c^{2}}+\frac{1}{(\xi-{\rm
Re}\,w)^{2}+c^{2}}\ll_{c}\frac{1}{c^{2}+\xi^{2}},$

and so we can use a contour change (and that $T>2/c$) to obtain

$\displaystyle|g(w,z,\cdot)*{S_{T}}(\xi)|$
$\displaystyle\ll_{c}\frac{1}{T}\int_{\mathbb{R}}\frac{\sin^{2}(\pi
Tt)}{((\xi-t)^{2}+c^{2})t^{2}}\mathrm{d}t=\frac{1}{T}\int_{\mathbb{R}}\frac{\sin^{2}(\pi
T(t+i/T))}{((\xi-t-i/T)^{2}+c^{2})(t+i/T)^{2}}\mathrm{d}t$
$\displaystyle\ll\frac{1}{T}\int_{\mathbb{R}}\frac{1}{((\xi-t)^{2}+c^{2})(t^{2}+1/T^{2})}\mathrm{d}t=\frac{\pi(c+1/T)/c}{\xi^{2}+(c+1/T)^{2}}\ll\frac{1}{c^{2}+\xi^{2}}.$

∎

###### Proof of Theorem 1 (and Theorem 2).

Necessity. Assume that $(\mu,a)$ is an real-antipoal $\mathrm{FS}$-pair such
that $\deg(\mu)\leq 2$ and $a(\cdot)$ has exponential growth. Since
$a(-\lambda)=\overline{a(\lambda)}$, we can apply Lemma 19 (multiplying both
sides by $(z-\overline{w})$) to obtain

$\displaystyle
a(0)+\lim_{T\to\infty}\bigg{(}\sum_{0<\lambda<T}a(\lambda)e^{2\pi i\lambda
z}(1-\lambda/T)+\sum_{0<\lambda<T}\overline{a(\lambda)}e^{-2\pi
i\lambda\overline{w}}(1-\lambda/T)\bigg{)}$ $\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{R}}\bigg{(}\frac{1}{t-z}-\frac{1}{t-\overline{w}}\bigg{)}{\mathrm{d}\mu(t)}=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{1+tz}{t-z}\frac{\mathrm{d}\mu(t)}{1+t^{2}}-\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{1+t\overline{w}}{t-\overline{w}}\frac{\mathrm{d}\mu(t)}{1+t^{2}},$

where the limit exists and is uniform for $w$ and $z$ in compact sets of
$\mathbb{C}^{+}$. In particular, we conclude there exists $h\in\mathbb{R}$
such that

$\displaystyle
f(z)=\frac{1}{2}a(0)+\lim_{T\to\infty}\sum_{0<\lambda<T}a(\lambda)e^{2\pi
i\lambda z}(1-\lambda/T)=ih+\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{1+tz}{t-z}\frac{\mathrm{d}\mu(t)}{1+t^{2}},$ (16)

where the limit above exists and is uniform in compact sets. A simple
computation shows

$\displaystyle\frac{f(z)+\overline{f(w)}}{z-\overline{w}}=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{\mathrm{d}\mu(t)}{(t-z)(t-\overline{w})}.$ (17)

This proves assertion (I). Since $a(\cdot)$ has exponential growth, we let
$c=\inf\\{b>0:\sum_{\lambda\in\mathbb{R}}|a(\lambda)|e^{-2\pi
b|\lambda|}<\infty\\}$. Then we must have

$f(z)=\frac{1}{2}a(0)+\sum_{\lambda>0}a(\lambda)e^{2\pi i\lambda z}$

for ${\rm Im}\,z>c$, as the above series converges absolutely and uniformly
for ${\rm Im}\,z>c+\varepsilon$, for any $\varepsilon>0$. Thus, after choosing
an enumeration for $\operatorname{\mathrm{supp}}(a)$, we conclude that
$f(\cdot+ih)$ is the uniform limit of trigonometric polynomials and we obtain
that $f(\cdot+ih)\in{\rm AP}(\mathbb{R})$ for every $h>c$. By Lemma 11 we
obtain $f(\cdot+ic)\in\mathrm{AP}(\mathbb{C}^{+})$. This proves assertion
(II). Assertion (III) is direct since if $p(x)=\sum_{\theta\in
S}p_{\theta}e^{2\pi i\theta x}$, for some finite set $S\subset[0,\infty)$,
then

$\displaystyle\limsup_{T\to\infty}\bigg{|}\frac{1}{2T}\int_{-T}^{T}f(x+2ic)\overline{p(x)}\mathrm{d}x\bigg{|}$
$\displaystyle=\bigg{|}\frac{1}{2}a(0){\boldsymbol{\delta}}_{S}(0)+\sum_{\lambda>0}a(\lambda)e^{-4\pi
c\lambda}{\boldsymbol{\delta}}_{S}(\lambda)p_{\lambda}\bigg{|}$
$\displaystyle\ll\max|p_{\theta}|\sum_{\lambda\in\mathbb{R}}|a(\lambda)|e^{-4\pi
c|\lambda|}\ll\max|p_{\theta}|.$

Finally, by Lemma 15, assertion (IV) is implied by representation (17).

Sufficiency. We prove sufficiency replacing property $({\rm III})$ by $({\rm
III}^{*})$ as in Theorem 2, and so also proving Theorem 2. Let
$c_{1}=\inf\\{b>0:f(\cdot+ib)\in\mathrm{AP}(\mathbb{C}^{+})\\}$ and
$f_{1}(z)=f(z+c_{1})$. By condition $({\rm II})$, we can apply Lemma 11 to
deduce the limit

$\mathbb{E}f_{1}(\lambda)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T+iy}^{T+iy}f_{1}(z)e^{-2\pi
i\lambda z}\mathrm{d}z=e^{-2\pi\lambda
c_{1}}\lim_{T\to\infty}\frac{1}{2T}\int_{-T+iy+ic_{1}}^{T+iy+ic_{1}}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z$

exists, does not depend on $y>0$ and is nonzero only for countable many real
$\lambda$’s. We also have that
$\sum_{\lambda}|\mathbb{E}f_{1}(\lambda)|^{2}e^{-4\pi h\lambda}<\infty$ for
all $h>0$. This shows that

$\mathbb{E}f(\lambda)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T+iy}^{T+iy}f(z)e^{-2\pi
i\lambda z}\mathrm{d}z$

is independent of $y>c_{1}$ and is nonzero only for countable many real
$\lambda$’s. By condition $({\rm IV})$, we can apply Lemma 15 to deduce there
is a real-valued locally finite measure $\mu$ of degree at most $2$ such that
(17) is true. In particular,

$2{\rm
Re}\,f(x+iy)=\frac{y}{\pi}\int_{\mathbb{R}}\frac{\mathrm{d}\mu(t)}{(t-x)^{2}+y^{2}}.$

As in Lemma 15 one can then write $f=f_{1}-f_{2}$ such that ${\rm
Re}\,f_{j}\geq 0$ in $\mathbb{C}^{+}$ and
$f_{j}(i)=\int_{\mathbb{R}}\frac{\mathrm{d}\mu_{j}(t)}{2\pi(1+t^{2})}$. We can
then apply Harnack’s inequality [10, p. 136], that translates to

$\bigg{|}\frac{f_{j}(z)-f_{j}(i)}{f_{j}(z)+f_{j}(i)}\bigg{|}\leq\bigg{|}\frac{z-i}{z+i}\bigg{|}.$

Since

$\frac{|z+i|+|z-i|}{|z+i|-|z-i|}\leq\frac{(1+|z|)^{2}}{4y}\quad(z=x+iy),$

we conclude that $|f_{j}(z)|\leq\frac{(1+|z|)^{2}}{4y}f_{j}(i)$ and so
$|f(z)|\leq\frac{(1+|z|)^{2}}{4y}\int_{\mathbb{R}}\frac{\mathrm{d}|\mu|(t)}{2\pi(1+t^{2})}$.
We deduce that

$\liminf_{r\to\infty}\frac{1}{r}\int_{0}^{\pi}\log^{+}|f(re^{i\theta})|\sin\theta\mathrm{d}\theta=0,$

and so one can now apply the Phranmën-Lindelöf principle (as in [12, Thm. 1])
to obtain that $f(z+ic)$ is bounded in $\mathbb{C}^{+}$ for every $c>c_{1}$.
Lemma 12(i) shows that $\mathbb{E}f(\lambda)=0$ for $\lambda<0$. By condition
$({\rm IV})$, there is $B_{M}>0$ and $c_{2}\geq c_{1}$ (with equality only if
$f(\cdot+ic_{1})\in{\rm AP}(\mathbb{R})$) such that

$B_{M}\max|p_{\theta}|\geq\limsup_{T\to\infty}\bigg{|}\frac{1}{2T}\int_{-T}^{T}f(x+ic_{2})\overline{p(x)}\mathrm{d}x\bigg{|}=\bigg{|}\sum_{0\leq\lambda\leq
M}\mathbb{E}f(\lambda)e^{-2\pi
c_{2}\lambda}{\boldsymbol{\delta}}_{S}(\lambda)p_{\lambda}\bigg{|}$

whenever $p(x)=\sum_{\theta\in S}p_{\theta}e^{2\pi i\theta x}$ for some finite
set $S\subset[0,M]$. We conclude that

$\sum_{0\leq\lambda\leq M}|\mathbb{E}f(\lambda)|\leq B_{M}e^{2\pi c_{2}M}$

for every $M>0$, so the function $a(\lambda)$ defined in (4) is locally
summable. Finally, let us prove that $(\mu,a)$ is a $\mathrm{FS}$-pair. For
$z\in\mathbb{C}^{+}$ and $t\in\mathbb{R}$ let $P_{z}(t)=\frac{z}{\pi
i(t^{2}-z^{2})}$. Since identity (17) holds true, a simple computation shows

${f(z+s)+\overline{f(-\bar{z}+s)}}=P_{z}*\mu(s)$

for all $s\in\mathbb{R}$. By Bochner’s approximation on ${\rm
AP}(\mathbb{R})$, for every $h>0$, one can find a sequence of functions
$d_{n}:\mathbb{R}\to[0,1)$, each with finite support and such that
$\lim_{n}d_{n}(\lambda)={\boldsymbol{\delta}}_{{\rm spec}(f)}$, and that

$\lim_{n}\sup_{{\rm Im}\,z=c_{1}+h}|f(z)-\sum_{\lambda\geq
0}\mathbb{E}f(\lambda)d_{n}(\lambda)e^{2\pi i\lambda z}|=0.$

Let $\varphi\in C_{c}^{\infty}((-M,M))$ be antipodal. It is now
straightforward (using Bochner’s approximation, Dominated Convergence and
local summability) to conclude that

$\int_{\mathbb{R}}f(z+s)\widehat{\varphi}(s)\mathrm{d}s=\sum_{0\leq\lambda<M}\mathbb{E}f(\lambda)\varphi(\lambda)e^{2\pi
i\lambda z}.$

if ${\rm Im}\,z>c_{1}$. Another routine computation (using that $a(0)=2{\rm
Re}\,\mathbb{E}f(0)$) shows

$\displaystyle\sum_{|\lambda|<M}a(\lambda)\varphi(\lambda)e^{2\pi
i|\lambda|z}=\int_{\mathbb{R}}[{f(z+s)+\overline{f(-\bar{z}+s)}}]\widehat{\varphi}(s)\mathrm{d}s$
$\displaystyle=\int_{\mathbb{R}}P_{z}*\mu(s)\widehat{\varphi}(s)\mathrm{d}s$
$\displaystyle=\int_{\mathbb{R}}P_{z}*\widehat{\varphi}(t)\mathrm{d}\mu(t)$

if ${\rm Im}\,z>c_{1}$. By real-antipodal splitting and linearity, the above
formula holds for all $\varphi\in C_{c}^{\infty}(\mathbb{R})$. Now observe
that both sides extend analytically to ${\rm Im}\,z>0$. Taking $z=iy$ and
noting that $P_{iy}$ is Poisson’s kernel, a routine application of Dominated
Convergence and approximate identity arguments (using that $a(\cdot)$ is
locally summable and $\deg(\mu)\leq 2$) allows us to take $y\to 0$ and
conclude that

$\sum_{|\lambda|<M}a(\lambda)\varphi(\lambda)=\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\mu(t).$

Therefore $(\mu,a)$ is an $\mathrm{FS}$-pair. This finishes the proof. ∎

## Acknowledgments

We thank Oleksiy Klurman and Bryce Kerr for all the conversations we had
around this topic during their visit to Rio. Theorems do come out of the
smoke. We thank Friedrich Littmann for the fruitful conversations and comments
which inspired Theorem 5. We also thank Michael Baake for further references
and remarks. The author acknowledges the support from FAPERJ, grant
E-26/200.209/2023, and the Serrapilheira Institute, grant Serra-2211-41824.

## Appendix: $\mathrm{FS}$-pairs, an account of various examples.

Throughout this section we define certain families of $\mathrm{FS}$-pairs
which will be closed under scaling, translation, modulation and multiplication
by scalar.

### Finite Support

Let $(\mu,a)$ be a $\mathrm{FS}$-pair such that $a(\cdot)$ has finite support,
this is, there is $\\{p_{n}\\}_{n=1}^{N}\subset\mathbb{C}$ and
$\\{\lambda_{n}\\}_{n=1}^{N}\subset\mathbb{R}$ such that

$\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\mu(t)=\sum_{n=1}^{N}p_{n}\varphi(\lambda_{n})$

for all $\varphi\in C^{\infty}_{c}(\mathbb{R})$. It is easy to see that we
must have $\mathrm{d}\mu(t)=\sum_{n=1}^{N}p_{n}e^{2\pi
i\lambda_{n}t}\mathrm{d}t$, since the difference of these two measures would
vanish identically over $C^{\infty}_{c}(\mathbb{R})$.

### Uniformly Discrete Poisson Summation (${\rm UDPS}$)

We denote ${\rm UDPS}$ the set of $\mathrm{FS}$-pairs $(\mu,a)$ such that
there is $\alpha>0$, $\\{\theta_{j}\\}_{j=1}^{N}\subset[0,\alpha)$ and trig-
polynomials $\\{Q_{j}\\}_{j=1}^{N}$ where

$\mu=\sum_{j=1}^{N}\sum_{\lambda\in\alpha\mathbb{Z}+\theta_{j}}Q_{j}(\lambda){\boldsymbol{\delta}}_{\lambda}.$

It is easy to see that if $(\mu,a)\in{\rm UDPS}$ then $a(\cdot)$ is also of
the above form. The breakthrough of Lev & Olveskii [30] shows that
$(\mu,a)\in{\rm UDPS}$ if and only if $\mu$ and $a(\cdot)$ have uniformly
discrete support (assuming both are strongly tempered). We observe that we can
always assume that all $Q_{j}^{\prime}s$ are equal since by classical
interpolation results (for instance, Lagrange interpolation via Chebyshev
polynomials) one can always find trigonometric polynomials $P_{j}$, which are
$\alpha$-periodic, and such that
$P_{j}(\theta_{k})={\boldsymbol{\delta}}_{j,k}$. Thus, if we let
$Q=\sum_{j=1}^{N}P_{j}Q_{j}$ then $Q(\alpha n+\theta_{j})=Q_{j}(\alpha
n+\theta_{j})$ for all $n\in\mathbb{Z}$, and so

$\mu=\sum_{\lambda\in\cup_{j=1}^{N}(\alpha\mathbb{Z}+\theta_{j})}Q(\lambda){\boldsymbol{\delta}}_{\lambda}.$

### Real Rooted Trigonometric Polynomial ($\mathrm{RRTP}$)

We let the class $\mathrm{RRTP}$ to be the family of $\mathrm{FS}$-pairs
$(\mu,a)$ such that there are four trigonometric polynomials $E_{j}$
($j=1,2,3,4$) of Hermite-Biehler class, that is,
$|E_{j}(z)|>|E_{j}(\overline{z})|$ for ${\rm Im}\,z>0$, and such that

$\displaystyle\mu$ $\displaystyle=\sum_{\,\varphi_{1}(\gamma)\equiv 0\,({\rm
mod}\,\pi)}\frac{Q_{1}(\gamma)}{\varphi_{1}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}-\sum_{\,\varphi_{2}(\gamma)\equiv
0\,({\rm
mod}\,\pi)}\frac{Q_{2}(\gamma)}{\varphi_{2}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}$
$\displaystyle-i\sum_{\,\varphi_{3}(\gamma)\equiv 0\,({\rm
mod}\,\pi)}\frac{Q_{3}(\gamma)}{\varphi_{3}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma}+i\sum_{\,\varphi_{4}(\gamma)\equiv
0\,({\rm
mod}\,\pi)}\frac{Q_{4}(\gamma)}{\varphi_{4}^{\prime}(\gamma)}{{\boldsymbol{\delta}}}_{\gamma},$

where $Q_{j}$ are trigonometric polynomials, $\varphi_{j}$ is the phase
function associated with $E_{j}$, defined by the condition that
$E_{j}(x)e^{i\varphi_{j}(x)}$ is real for all $x\in\mathbb{R}$ ($\varphi_{j}$
is uniquely defined modulo $\pi$, see (6)). We observe that if we write
$E_{j}=A_{j}-iB_{j}$, were $A_{j},B_{j}$ are trigonometric polynomials which
are real on the real line, then both $A,B$ have only real zeros, $A/B$ have
only simple real zeros and simple poles which interlace,

$\\{\gamma\in\mathbb{R}:\,\varphi_{j}(\gamma)\equiv 0\,({\rm
mod}\,\pi)\\}={\rm
Zeros}(B_{j}/A_{j})\quad\text{and}\quad\frac{1}{\varphi_{j}^{\prime}(\gamma)}=\mathop{\mathrm{Res}}_{z=\gamma}\,(A_{j}/B_{j})>0,$

By Theorem 3 we conclude that $(\mu,\widehat{\mu})\in\rm CM$ (see below). By
Remark 5, $\mathrm{RRTP}$ contains the pairs $(\mu,\widehat{\mu})$ where

$\mu=\sum_{P(\gamma)=0}m(\gamma)Q(\gamma){\boldsymbol{\delta}}_{\gamma}$

and $Q,P$ are trigonometric polynomials, $P$ with only real roots and
$m(\gamma)$ is the multiplicity of the zero $\gamma$. These measures were
first introduced and studied by Kurasov and Sarnak [25]. Observe that ${\rm
UDPS}\subset\mathrm{RRTP}$ since
$\cup_{j}\alpha\mathbb{Z}+\theta_{j}=\text{Zeros}(\prod_{j}\sin(\pi(z-\theta_{j})/\alpha))$.
Indeed $\mathrm{RRTP}$ is a much richer class since

$P(z)=\det(U+\text{diag}(e^{2\pi il_{1}z},...,e^{2\pi il_{n}z}))$

has only real roots if $U$ is a $n\times n$ unitary matrix and
$l_{1},....,l_{n}\in\mathbb{R}$. The zeros of such polynomials are generically
not contained in a finite union of arithmetic progressions [2]. A simple
example is $P(x)=\sin(x)+\frac{1}{10}\sin(\sqrt{2}x)$. Trigonometric
polynomials with only real zeros have been classified recently in [1] as
restrictions of Lee-Yang polynomials. A polynomial
$P:\mathbb{C}^{n}\to\mathbb{C}$ is Lee-Yang if $P(z_{1},...,z_{n})\neq 0$
whenever
$\min\\{\max\\{|z_{1}|,...,|z_{n}|\\},\max\\{|z_{1}|^{-1},...,|z_{n}|^{-1}\\}\\}<1$.
Indeed, the main result in [1] states that if $Q$ is a trigonometric
polynomial with only real zeros then there exists a Lee-Yang polynomial
$P(z_{1},...,z_{n})$ and positive reals $l_{1},...,l_{n}$, linear independent
over $\mathbb{Q}$, such that

$Q(x)=P(e^{il_{1}x},...,e^{il_{n}x}).$

### Crystalline Measures ($\rm CM$)

We let $\rm CM$ be the class of $\mathrm{FS}$-pairs $(\mu,a)$ such that both
$\mu$ and $a(\cdot)$ have locally finite support. It is not hard to see that
$\mathrm{RRTP}\subset\rm CM$. One main theme investigated in [25, 2, 1] is to
find pairs $(\mu,a)\in\rm CM$ such that $\operatorname{\mathrm{supp}}(a)$
and/or $\operatorname{\mathrm{supp}}(\mu)$ intersects any infinite arithmetic
progression at most finitely171717So very non-periodic. Guinand [21, p. 265],
in 1959, was “almost the first” to produce such pair, as we shall explain
below. He constructs a (self-dual) $\mathrm{FS}$-pair $(\mu,\mu)$ with

$\mu=\sum_{n\geq
0}c_{n}({\boldsymbol{\delta}}_{\sqrt{n+1/9}}+{\boldsymbol{\delta}}_{-\sqrt{n+1/9}}),$

where $c_{n}$ are the cofficients of the modular form181818Guinand did not
realized this nor constructed his example in the way we do here.

$\displaystyle\begin{split}\frac{\eta(z)^{2/3}\eta(4z)^{2/3}}{\eta(2z)^{1/3}}&=q^{1/9}\left(1-\frac{2}{3}\\*q-\frac{4}{9}\\*q^{2}-\frac{40}{81}\\*q^{3}-\frac{160}{243}\\*q^{4}+\frac{268}{729}\\*q^{5}+\frac{1808}{6561}\\*q^{6}+\cdots\right)\\\
&=\sum_{n\geq 0}c_{n}q^{1/9+n}.\end{split}$ (18)

Above

$\eta(z)=q^{1/24}\prod_{n\geq 1}(1-q^{n})=\sum_{n\geq
1}\chi_{12}(n)q^{n^{2}/24}$

is the Dedekind eta-function, $q=e^{2\pi iz}$ and $\chi_{12}$ is the Dirichlet
character of modulus $12$ with $\chi(1)=\chi(11)=-\chi(5)=-\chi(7)=1$ and zero
otherwise. Hence, if $n=9m^{2}+2m$ then $\sqrt{n+1/9}=3m+1/3$, so the support
of $\mu$ contains an arithmetic progression. We will show below that, with the
right point of view, one can embed Guinand’s example and also classical
Poisson summation in a real one parameter family, for which almost all members
have the property that their support intersect any infinite arithmetic
progression at most finitely. For that we need first the following lemma.

######  Lemma 20.

Let $F\in\mathrm{AP}(\mathbb{C}^{+})$, with locally finite spectrum contained
in $[0,\infty)$ and assume that

$\sum_{n\geq 0}|\mathbb{E}F(\gamma_{n})|(1+\gamma_{n})^{-k}<\infty$

for some $k>0$, where ${\rm spec}(F)=\\{\gamma_{n}\\}_{n\geq 0}$, Assume that
$G(z)=F(-1/z)\sqrt{i/z}$ also belongs to $\mathrm{AP}(\mathbb{C}^{+})$, has
locally finite spectrum contained in $[0,\infty)$ and $\mathbb{E}G$ decays
similarly as $\mathbb{E}F$. Let

$\mu=\sum_{n\geq
0}\mathbb{E}F(\gamma_{n})({\boldsymbol{\delta}}_{-\sqrt{2\gamma_{n}}}+{\boldsymbol{\delta}}_{-\sqrt{2\gamma_{n}}}).$

Then $\mu$ is a strongly tempered measure and

$\widehat{\mu}=\sum_{n\geq
0}\mathbb{E}G(\lambda_{n})({\boldsymbol{\delta}}_{\sqrt{2\lambda_{n}}}+{\boldsymbol{\delta}}_{-\sqrt{2\lambda_{n}}})$

where ${\rm spec}(G)=\\{\lambda_{n}\\}_{n\geq 0}$. Hence
$(\mu,\widehat{\mu})\in\rm CM$.

###### Proof.

The proof is inspired by [16, Lemma 2.1]. Consider the gaussian
$g_{z}(t)=e^{\pi izt^{2}}$ for $z\in\mathbb{C}^{+}$. Observe that
$\widehat{g}_{z}(t)=\sqrt{i/z}g_{-1/z}(t)$ and that $g_{z}(\sqrt{2s})=e^{2\pi
izs}$ for $s\geq 0$. Assume first that $\mathbb{E}G$ also has polynomial
growth. Then the identity

$\displaystyle G(z)=\sum_{n\geq
0}\mathbb{E}G(\lambda_{n})g_{z}(\sqrt{2\lambda_{n}})=F(-1/z)\sqrt{i/z}=\sum_{n\geq
0}\mathbb{E}F(\gamma_{n})\widehat{g}_{z}(\sqrt{2\lambda_{n}})$

shows that

$\sum_{n\geq
0}\mathbb{E}F(\gamma_{n})(\widehat{\varphi}(\sqrt{2\gamma_{n}})+\widehat{\varphi}(\sqrt{-2\gamma_{n}}))=\sum_{n\geq
0}\mathbb{E}G(\lambda_{n})(\varphi(\sqrt{2\lambda_{n}})+\varphi(-\sqrt{2\lambda_{n}})),$

for any $\varphi\in{\rm span}_{\mathbb{C}}\\{g_{z}:z\in\mathbb{C}^{+}\\}$. An
approximation argument similar to the one employed in [16, Lemma 2.1] shows
that the above identity actually holds for any
$\varphi\in\mathcal{S}(\mathbb{R})$. ∎

The following construction was inspired by a similar one partially
communicated in 2018 by Danylo Radchenko in a seminar at the University of
Bonn. Let $N\geq 1$ be an integer and ${\boldsymbol{r}}=\\{r_{d}\\}_{d|N}$ be
a sequence of reals indexed by the divisors of $N$ such that

$\displaystyle r_{d}=r_{N/d},\quad\sum_{d|N}r_{d}=1\ \text{ and }\
\sum_{d|N}dr_{d}=\frac{24k}{b}$ (19)

where $b\geq 1$ and $k\geq 0$ are coprime integers with $b=1$ if $k=0$.
Consider the eta-product

$\eta({\boldsymbol{r}},z)=\prod_{d|N}\eta(dz)^{r_{d}}=\sum_{n\geq
k}\alpha_{n}q^{n/b}.$

The eta-function $\eta:\mathbb{C}^{+}\to\mathbb{C}$ is an holomorphic function
that satisfies the functional identities $\eta(z+\ell)=e^{\pi
i\ell/12}\eta(z)$, for $\ell\in\mathbb{Z}$, and
$\eta(-1/z)=\sqrt{z/i}\,\eta(z)$. We obtain that

$\displaystyle\eta({\boldsymbol{r}},z+b)=e^{\pi
ib(\sum_{d|N}dr_{d})/12}\eta({\boldsymbol{r}},z)=\eta({\boldsymbol{r}},z)$
$\displaystyle\eta({\boldsymbol{r}},-1/z)=\prod_{d|N}\eta(-d/z)^{r_{d}}=\prod_{d|N}(z/(di))^{r_{d}/2}\prod_{d|N}\eta(z/d)^{r_{d}}$
$\displaystyle=\sqrt{z/(i\sqrt{N})}\prod_{d|N}\eta(dz/N)^{r_{N/d}}$
$\displaystyle=\sqrt{z/(i\sqrt{N})}\eta({\boldsymbol{r}},z/N).$

Above we use that $\prod_{d|N}d^{-r_{d}/2}=N^{-1/4}$, which follows from the
conditions on $r_{d}$. In particular we conclude that the function

$F_{+}({\boldsymbol{r}},z)=\eta({\boldsymbol{r}},z/\sqrt{N})=\sum_{n\geq
k}\alpha_{n}q^{n/(b\sqrt{N})}.$

satisfies the functional equations

$\displaystyle
F_{+}({\boldsymbol{r}},z+b\sqrt{N})=F_{+}({\boldsymbol{r}},z)\quad\text{and}\quad\sqrt{i/z}\,F_{+}({\boldsymbol{r}},-1/z)=F_{+}({\boldsymbol{r}},z)$

for all $z\in\mathbb{C}^{+}$. It can be shown that the function
$\eta({\boldsymbol{r}},z)$ is a holomorphic modular form of level $N$, and so
the Hecke bound produces $|\alpha_{n}|\ll n^{1/4}$ (see also [11]). In
particular, $\mu_{+}$ is strongly tempered. Hence $F_{+}$ satisfies the
hypotheses of Lemma 20. Thus, the measure

$\mu_{+}=\sum_{n\geq
k}\alpha_{n}({\boldsymbol{\delta}}_{\sqrt{2n/(b\sqrt{N})}}+{\boldsymbol{\delta}}_{-\sqrt{2n/(b\sqrt{N})}}),$

satisfies

$\widehat{\mu}_{+}=\mu_{+}.$

Poisson Summation can be recovered this way by

$\frac{\eta(2z)^{5}}{\eta(z)^{2}\eta(4z)^{2}}=1+2q+2q^{4}+2q^{9}+2q^{25}+\cdots,$

with parameters $N=4$, $(r_{1},r_{2},r_{4})=(-2,5,-2)$, $b=1$ and $k=0$.
Guinand’s example (18) has parameters $N=4$,
$(r_{1},r_{2},r_{4})=(2/3,-1/3,2/3)$, with the others $r_{d}=0$ for $d|36$,
$b=3$ and $k=1$.

Assume now that $N$ is a square. Using the modular $\lambda$-invariant
(Hauptmodul for $\Gamma(2)$)

$\lambda(z)=\frac{16\eta(2z)^{16}\eta(z/2)^{8}}{\eta(z)^{24}}=16q^{1/2}-128q+704q^{3/2}+\cdots,$

which can be shown to satisfy the transformation laws

$\displaystyle\lambda(-1/z)=1-\lambda(z)\quad\text{and}\quad\lambda(z+1)=\lambda(z)/(\lambda(z)-1),$

we can define instead

$F_{-}({\boldsymbol{r}},z)=(1-2\lambda(z))\eta({\boldsymbol{r}},z/\sqrt{N})=\sum_{n\geq
2k}\beta_{n}q^{n/(2b\sqrt{N})}$

and deduce that

$\displaystyle
F_{-}({\boldsymbol{r}},z+2b\sqrt{N})=F_{-}({\boldsymbol{r}},z)\quad\text{ and
}\quad\sqrt{i/z}\,F_{-}({\boldsymbol{r}},-1/z)=-F_{-}({\boldsymbol{r}},z).$

Again we conclude that the measure

$\mu_{-}=\sum_{n\geq
2k}\beta_{n}({\boldsymbol{\delta}}_{\sqrt{n/(b\sqrt{N})}}+{\boldsymbol{\delta}}_{-\sqrt{n/(b\sqrt{N})}}),$

satisfies $\widehat{\mu}_{-}=-\mu_{-}.$ It is now obvious that one can build
infinitely many examples of such measures $\mu_{\pm}=\pm\widehat{\mu}_{\pm}$
with locally finite support.

Consider now the following family

$\displaystyle\frac{\eta(z)^{l}\eta(4z)^{l}}{\eta(2z)^{2l-1}}$
$\displaystyle=q^{(l+2)/24}\left(1-lq+\frac{(l-1)(2+l)}{2}q^{2}+\sum_{n\geq
3}\alpha_{n,l}q^{n}\right)$
$\displaystyle(1-2\lambda(2z))\frac{\eta(z)^{l}\eta(4z)^{l}}{\eta(2z)^{2l-1}}$
$\displaystyle=q^{(l+2)/24}\left(1-(32+l)q+\tfrac{1}{2}(l^{2}+65l+510)q^{2}+\sum_{n\geq
3}\beta_{n,l}q^{n}\right)$

for $N=4$ and $l\geq-2$. The examples above produce measures
$\mu_{\pm}=\pm\widehat{\mu}_{\pm}$ with support contained in

$\\{\sqrt{c+n}\\}_{n\geq 0}\cup\\{-\sqrt{c+n}\\}_{n\geq 0},$

for any real $c=(l+2)/24\geq 0$. Note that for $l=-2$ we recover classical
Poisson summation and for $l=2/3$ we recover Guinand’s example. Perhaps other
interesting constructions may be derived from [28] and the references therein.
It is not hard to show that $n!\alpha_{n,l}$ and $n!\beta_{n,l}$ are
polynomials in $l$ with integer coefficients and degree at most $n$.

Claim: We claim that if $c$ is irrational then $\\{\sqrt{c+n}\\}_{n\geq 0}$
intersects any infinite arithmetic progression at most $2$ times. Indeed,
having three hits implies that

$\frac{\sqrt{n_{2}+c}-\sqrt{n_{1}+c}}{k_{1}}=\frac{\sqrt{n_{3}+c}-\sqrt{n_{2}+c}}{k_{2}}$

is satisfied for some integers $n_{3}>n_{2}>n_{1}\geq 0$ and $k_{1},k_{2}>0$.
A boring computation to isolate $n_{2}+c$ shows that $c$ is rational, which is
absurd.

Such measures are also connected with new Fourier interpolation formulae
produced in [40], which as a byproduct produces crystalline pairs with space
and spectral supports contained in $\\{\pm\sqrt{n}\\}_{n\geq
0}\cup\\{x_{0}\\}$, for any $x_{0}$ not an integer square root. These nodes
can also be perturbed to be $\\{\pm\sqrt{n+\varepsilon_{n}}\\}_{n\geq 0}$, as
long as $|\varepsilon_{n}|\leq\delta n^{-5/4}$ for some small $\delta>0$, see
[42]. The recent papers [41, 26] also suggest that there should exist a
crystalline pair with space and spectral supports contained in $\\{\pm
n^{\alpha}\\}_{n\geq 1}$ and $\\{\pm n^{(1-\alpha)}\\}_{n\geq 1}$
respectively, whenever $0<\alpha<1$. Moreover, [26, Cor. 3] proves the
striking result that for every (supercritical) pair of increasing sequences
$\mathcal{T}=\\{t_{j}\\}_{j\in\mathbb{Z}}$ and
$\mathcal{S}=\\{s_{j}\\}_{j\in\mathbb{Z}}$ such that

$\max\left(\limsup_{j\to\infty}|t_{j}|^{p-1}(t_{j+1}-t_{j}),\limsup_{j\to\infty}|s_{j}|^{q-1}(s_{j+1}-s_{j})\right)<\frac{1}{2},$

for some $1<p,q<\infty$ with $\frac{1}{p}+\frac{1}{q}=1$, there is a
$\mathrm{FS}$-pair $(\mu,a)$ with $\mu$ supported in $\mathcal{T}$ and
$a(\cdot)$ supported in $\mathcal{S}$. An important simple example is
$\mathcal{T}=\\{\pm n^{\alpha}\\}_{n\geq 0}$ and $\mathcal{S}=\\{\pm
n^{\beta}\\}_{n\geq 0}$ for positive $\alpha,\beta$ with $\alpha+\beta<1$. In
fact, $(\mathcal{T},\mathcal{S})$ is both a Fourier uniqueness pair and an
interpolating pair.

### Almost Crystalline Measures

We say $(\mu,a)$ is an almost Fourier summation pair ($\mathrm{AFS}$-pair) if
$\mu$ is strongly tempered, $a(\cdot)$ is locally summable function and

$(a-\widehat{\mu})|_{C_{c}^{\infty}(\mathbb{R})}=(\nu_{1}-\widehat{\nu}_{2})|_{C_{c}^{\infty}(\mathbb{R})},$

where $\nu_{1}$ and $\nu_{2}$ are absolutely continuous, $\nu_{1}\neq 0$ is
tempered and $\nu_{2}$ is strongly tempered. In particular, if $\varphi\in
C_{c}^{\infty}(\mathbb{R})$ we have

$\sum_{\lambda\in\mathbb{R}}a(\lambda)\varphi(\lambda)-\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\mu(t)=\int_{\mathbb{R}}\varphi(t)\mathrm{d}\nu_{1}(t)-\int_{\mathbb{R}}\widehat{\varphi}(t)\mathrm{d}\nu_{2}(t).$

We then let the class $\rm ACM$ (Almost Crystalline Measures) denote the
$\mathrm{AFS}$-pairs $(\mu,a)$ such that both $\mu$ and $a(\cdot)$ have
locally finite support. A typical example is Guinand’s prime summation formula
(also known as Riemann–Weil explicit formula), which states that for every
$\varphi\in C_{c}^{\infty}(\mathbb{R})$ we have

$\displaystyle\frac{\log(\pi/N)}{2\pi}\varphi(0)+\frac{1}{2\pi}\sum_{n\geq
1}\frac{\Lambda(n)\chi(n)}{\sqrt{n}}\bigg{(}\varphi\left(\frac{\log
n}{2\pi}\right)+\varphi\left(\frac{-\log n}{2\pi}\right)\bigg{)}$
$\displaystyle=-\sum_{\gamma}\widehat{\varphi}(\gamma)+2\int_{\mathbb{R}}\varphi(t)\cosh(\pi
t)\mathrm{d}t+\frac{1}{2\pi}\int_{\mathbb{R}}\widehat{\varphi}(t){\rm
Re}\,\psi(\tfrac{1}{4}+i\tfrac{t}{2})\mathrm{d}t$

where $\psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$ is the digamma function,
$\Lambda(n)$ is the von Mangoldt function, $\rho=1/2+i\gamma$ are the zeros of
the Dirichlet $L$-function

$L(\chi,s)=\sum_{n\geq 1}\chi(n)n^{-s}$

on the critical strip $0<{\rm Re}\,\rho<1$ and $\chi$ is a primitive, real and
even Dirichlet character of modulus $N$. Since
$|\psi(1/4+it/2)|\ll\log(2+|t|)$ and
$\sum_{\gamma}\frac{1}{1+\gamma^{2}}<\infty$, we conclude, assuming the
Riemann Hypothesis for $L(\chi,s)$, that if we let

$\displaystyle a$
$\displaystyle=\frac{\log(\pi/N)}{2\pi}{\boldsymbol{1}}_{0}+\frac{1}{2\pi}\sum_{n\geq
1}\frac{\Lambda(n)\chi(n)}{\sqrt{n}}({\boldsymbol{1}}_{\frac{\log
n}{2\pi}}+{\boldsymbol{1}}_{-\frac{\log n}{2\pi}})$ $\displaystyle\mu$
$\displaystyle=-\sum_{\gamma}{\boldsymbol{\delta}}_{\gamma}$
$\displaystyle\nu_{1}$ $\displaystyle=2\cosh(\pi t)\mathrm{d}t$
$\displaystyle\nu_{2}$ $\displaystyle=-\frac{1}{2\pi}{\rm
Re}\,\psi(\tfrac{1}{4}+i\tfrac{t}{2})\mathrm{d}t,$

then $(\mu,a)\in\rm ACM$. The difference of any two of such pairs above for
two characters $\chi_{1}$ and $\chi_{2}$ real, even and primitive, but
different moduli $N$, produces a pair in $\rm CM$ (this was pointed out first
by Guinand [21]). Such measures are also connected with new Fourier
interpolation formulae produced in [14], and these produce, under GRH, $\rm
CM$-pairs $(\mu,a)$ with
$\operatorname{\mathrm{supp}}(\mu)=\\{\gamma\in\mathbb{R}:L(\chi,1/2+i\gamma)=0\\}\cup\\{\gamma_{0}\\}$
and
$\operatorname{\mathrm{supp}}(a)=\\{\operatorname{\mathrm{sgn}}(n)\frac{\log|n|}{4\pi}\\}_{n\geq
1}$, where $\gamma_{0}$ is any real such that $L(\chi,1/2+i\gamma_{0})\neq 0$.

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