# Phaseless sampling on square-root lattices

Philipp Grohs*,†,‡ *Faculty of Mathematics, University of Vienna, Oskar-
Morgenstern-Platz 1, 1090 Vienna, Austria  †Research Network
DataScience@UniVie, University of Vienna, Kolingasse 14-16, 1090 Vienna,
Austria  ‡Johann Radon Institute of Applied and Computational Mathematics,
Austrian Academy of Sciences, Altenbergstrasse 69, 4040 Linz, Austria
<EMAIL_ADDRESS>and  Lukas Liehr*<EMAIL_ADDRESS>
###### Abstract.

For which window functions $g\in{L^{2}({\mathbb{R}}^{d})}$ and which sampling
sets $\Lambda\subseteq{\mathbb{R}}^{2d}$ is every
$f\in{L^{2}({\mathbb{R}}^{d})}$ uniquely determined (up to a global phase
factor) by phaseless samples of the form

$|V_{g}f(\Lambda)|=\left\\{|V_{g}f(\lambda)|:\lambda\in\Lambda\right\\},$

where $V_{g}f$ denotes the short-time Fourier transform (STFT) of $f$ with
respect to $g$? This problem is of central importance in STFT phase retrieval
and differs from ordinary sampling problems in a rather fundamental and subtle
way: recent results demonstrate that uniqueness is unachievable if $\Lambda$
is a lattice, i.e
$\Lambda=A{\mathbb{Z}}^{2d},A\in\mathrm{GL}(2d,{\mathbb{R}})$. Driven by this
discretization barrier, the present article centers around the initiation of a
new sampling scheme which allows for unique recovery of any square-integrable
function via phaseless STFT-sampling. Precisely, we show that square-root
lattices, i.e. sets of the form

$\Lambda=A\left(\sqrt{{\mathbb{Z}}}\right)^{2d},\
\sqrt{{\mathbb{Z}}}=\\{\pm\sqrt{n}:n\in{\mathbb{N}}_{0}\\},$

guarantee uniqueness of the STFT phase retrieval problem. The result holds for
a large class of window functions, including Gaussians.

Keywords. phase retrieval, sampling, square-root lattice, time-frequency
analysis, short-time Fourier transform

AMS subject classifications. 32A15, 32A60, 94A12, 94A20

##  1\. Introduction

Consider a window function $g\in{L^{2}({\mathbb{R}}^{d})}$ and a set
$\Lambda\subseteq{\mathbb{R}}^{2d}$ of sampling locations. Let
$V_{g}f:{\mathbb{R}}^{2d}\to{\mathbb{C}}$ denote the short-time Fourier
transform (STFT) of $f\in{L^{2}({\mathbb{R}}^{d})}$ with respect to $g$, given
by

$V_{g}f(x,\omega)=\int_{{{\mathbb{R}}^{d}}}f(t)\overline{g(t-x)}e^{-2\pi
i\omega\cdot t}\,dt.$

The well-established theory in time-frequency analysis ensures that every
$f\in{L^{2}({\mathbb{R}}^{d})}$ is uniquely determined by the STFT-samples
$\\{V_{g}f(\lambda):\lambda\in\Lambda\\}$, provided that the window function
$g$ satisfies certain mild conditions and
$\Lambda=A{\mathbb{Z}}^{2d},A\in\mathrm{GL}(2d,{\mathbb{R}}),$ is a
sufficiently dense lattice. Consequently, arbitrary square-integrable
functions can be distinguished via sampling of their STFTs. Suppose now that
merely the absolute values of the STFT-samples are given, i.e. the sampling
set is of the form

(1)
$|V_{g}f(\Lambda)|\coloneqq\left\\{|V_{g}f(\lambda)|:\lambda\in\Lambda\right\\}.$

This leads to the phaseless sampling problem and the question is whether a
distinction of square-integrabe functions is possible by looking at the
respective phaseless STFT samples as given in (1). The investigation of this
problem is of utmost importance in a series of applications, prominent
examples include coherent diffraction imaging [29, 8], audio processing [25],
and quantum mechanics [24]. It is known as the uniqueness problem in _STFT
phase retrieval_ and has received considerable attention in recent years.
Since $|V_{g}f(\Lambda)|=|V_{g}(\tau f)(\Lambda)|$ whenever
$\tau\in{\mathbb{T}}\coloneqq\\{z\in{\mathbb{C}}:|z|=1\\}$ is a complex number
of modulus one, it is evident that a distinction from phaseless samples of the
form (1) is only achievable up to the ambiguity arising from multiplication by
a number in ${\mathbb{T}}$. To that end, we define for two functions
$f,h\in{L^{2}({\mathbb{R}}^{d})}$ the equivalence relation

$f\sim h\iff\exists\tau\in{\mathbb{T}}:f=\tau h,$

and say that $f$ and $h$ agree up to a global phase if $f\sim h$. Compared to
ordinary sampling of the STFT, the uniqueness problem from phaseless samples
differs in a fundamental way: recently established discretization barriers
show that if $\Lambda$ is a lattice, then functions in
${L^{2}({\mathbb{R}}^{d})}$ are never determined by samples of the form
$|V_{g}f(\Lambda)|$, independently of the choice of both the lattice $\Lambda$
and the window function $g$ [12, 11, 4]. New techniques and new sampling
schemes are therefore required to settle the uniqueness problem in the absence
of phase information.

The objective of the present article centers around the derivation of
phaseless sampling results from square-root lattices. Precisely, we determine
a function space $\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ (depending on a
decay parameter $a$ and a growth parameter $b$) which consists of analytic
functions on ${\mathbb{C}}^{d}$ such that the restriction to
${{\mathbb{R}}^{d}}$ of every $g\in\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ is
an element of ${L^{2}({\mathbb{R}}^{d})}$. This function space has the
following property: whenever $g$ belongs to
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$, then every
$f\in{L^{2}({\mathbb{R}}^{d})}$ is determined up to a global phase by
$|V_{g}f(\Lambda)|$, provided that $\Lambda$ is a square-root lattice,

$\Lambda=A\left(\sqrt{{\mathbb{Z}}}\right)^{2d},\quad\sqrt{{\mathbb{Z}}}\coloneqq\\{\pm\sqrt{n}:n\in{\mathbb{N}}_{0}\\},\quad
A\in\mathrm{GL}(2d,{\mathbb{R}}),$

which satisfies a generalized density condition (see Section 1.1 for further
clarifications). Thus, sampling on ordinary lattices never yields uniqueness,
while a replacement by a square-root lattice does the job. It is worth
mentioning that $\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ contains Gaussians,
the most important window functions in contemporary STFT phase retrieval
research. Moreover, a suitable choice of $a$ and $b$ implies that
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ is dense in
${L^{2}({\mathbb{R}}^{d})}$.

###  1.1. Main results

Recall that a lattice in ${{\mathbb{R}}^{d}}$ is a set of the form
$A{\mathbb{Z}}^{d}$ for some invertible matrix
$A\in\mathrm{GL}(d,{\mathbb{R}})$. We call a subset
$\Lambda\subseteq{{\mathbb{R}}^{d}}$ a _square-root lattice_ if there exists
an invertible matrix $A\in\mathrm{GL}(d,{\mathbb{R}})$ such that

$\Lambda=A\left(\sqrt{{\mathbb{Z}}}\right)^{d}=\left\\{Az:z\in\underbrace{\sqrt{{\mathbb{Z}}}\times\dots\times\sqrt{{\mathbb{Z}}}}_{d\
\mathrm{times}}\right\\},$

where $\sqrt{{\mathbb{Z}}}\coloneqq\\{\pm\sqrt{n}:n\in{\mathbb{N}}_{0}\\}$. In
analogy to an ordinary lattice, the matrix $A$ is called the generating matrix
of the square-root lattice. A square-root lattice $\Lambda$ is called
rectangular if its generating matrix is diagonal. The present article
establishes uniqueness results for the STFT phase retrieval problem via
sampling on square-root lattices. The window functions which define the STFT
and for which square-root sampling implies uniqueness belong to a space of
analytic functions. Before defining this space, we settle some notation.
Whenever $z\in{{\mathbb{C}}^{d}}$, we write $z=x+iy$, where
$x\in{{\mathbb{R}}^{d}}$ is the real part of $z$ and $y\in{{\mathbb{R}}^{d}}$
is the imaginary part of $z$. For two functions
$v,w:{{\mathbb{C}}^{d}}\to[0,\infty)$ we write $v\lesssim w$, if and only if
there exists a positive constant $c>0$ such that $v(z)\leq cw(z)$ for every
$z\in{{\mathbb{C}}^{d}}$.

The space of analytic functions we are dealing with will be denoted by
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$. For
$a,b\in{\mathbb{R}}_{>0}^{d}\coloneqq(0,\infty)^{d}$ it is defined by

$\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})\coloneqq\left\\{F\in\mathcal{O}({\mathbb{C}}^{d}):|F(x+iy)|\lesssim\prod_{j=1}^{d}e^{-a_{j}x_{j}^{2}}e^{b_{j}y_{j}^{2}}\right\\},$

where $\mathcal{O}({\mathbb{C}}^{d})$ denotes the space of entire functions of
$d$ complex variables. The function class
$\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$ is a linear subspace of
$\mathcal{O}({{\mathbb{C}}^{d}})\cap{L^{2}({\mathbb{R}}^{d})}$ (the space of
all entire functions which are square-integrable on ${{\mathbb{R}}^{d}}$) and
if $a<b$ (i.e. $a_{j}<b_{j}$ for every $j\in\\{1,\dots,d\\}$), then
$\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$ is dense in
${L^{2}({\mathbb{R}}^{d})}$. Note that we will repeatedly identify entire
functions with their restriction on ${{\mathbb{R}}^{d}}$, and vice versa,
functions on ${{\mathbb{R}}^{d}}$ with their analytic extension on
${\mathbb{C}}^{d}$ (provided the extension exists). We refer to Section 2.1
for a detailed discussion of properties of
$\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$. Window functions which belong to the
function space $\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$ lead to phaseless
sampling results from square-root lattices. This is the content of the
following theorem which constitutes the main result of the article.

######  Theorem 1.1.

Let $a,b\in{\mathbb{R}}^{d}_{>0}$ and let
$0\neq\varphi\in\mathcal{O}^{a}_{b}({\mathbb{C}}^{d})$ be a window function.
Suppose that $\Lambda=A(\sqrt{{\mathbb{Z}}})^{2d}$ is a rectangular square-
root lattice such that the generating matrix
$A=\mathrm{diag}(\tau_{1},\dots,\tau_{d},\nu_{1},\dots,\nu_{d})$ with
$\tau,\nu\in{\mathbb{R}}^{d}_{>0}$ satisfies

$\tau_{j}<\sqrt{\frac{1}{2b_{j}e}},\quad\nu_{j}<\sqrt{\frac{a_{j}}{2\pi^{2}e}},\quad
j\in\\{1,\dots,d\\}.$

Then the following statements are equivalent for every
$f,h\in{L^{2}({\mathbb{R}}^{d})}$:

  1. (1)

$|V_{\varphi}f(\lambda)|=|V_{\varphi}h(\lambda)|$ for every
$\lambda\in\Lambda$,

  2. (2)

$f\sim h$.

Notice that for $\gamma>0$ and for a suitable choice of $a$ and $b$, the
Gaussian
$\varphi(x)=e^{-\gamma\|x\|_{2}^{2}},\,\|x\|_{2}^{2}=\sum_{j=1}^{d}x_{j}^{2}$,
is an element of $\mathcal{O}^{a}_{b}({\mathbb{C}}^{d})$. In view of both
theory and applications, Gaussians represent the perhaps most significant
window functions as they allow for a connection to the theory of Fock spaces.
It is this particular relation which lead to the most extensive results in
STFT phase retrieval with Gaussian windows [18, 1, 14, 13, 3, 17, 19, 2, 15].
We therefore state two separate results on square-root sampling for Gaussian-
type windows where we also allow for multiplication by entire functions of
exponential type. The first consequence of Theorem 1.1 reads as follows.

######  Corollary 1.2.

Let $\gamma>0$, let $p:{\mathbb{C}}^{d}\to{\mathbb{C}}$ be an entire function
of exponential type and let $\varphi\in{L^{2}({\mathbb{R}}^{d})}$ be the
window function $\varphi(x)=p(x)e^{-\gamma\|x\|_{2}^{2}}.$ If $\alpha,\beta>0$
satisfy

$\alpha<\sqrt{\frac{1}{2\gamma
e}},\quad\beta<\sqrt{\frac{\gamma}{2\pi^{2}e}},$

then the following statements are equivalent for every
$f,h\in{L^{2}({\mathbb{R}}^{d})}$:

  1. (1)

$|V_{\varphi}f(\lambda)|=|V_{\varphi}h(\lambda)|$ for every
$\lambda\in\alpha(\sqrt{{\mathbb{Z}}})^{d}\times\beta(\sqrt{{\mathbb{Z}}})^{d}$,

  2. (2)

$f\sim h$.

Every polynomial is an entire function of exponential type. Hermite functions
are products of Gaussians with polynomials. Recall that the Hermite functions
on the real line are given by

$h_{n}(t)=\frac{2^{1/4}}{\sqrt{n!}}\left(-\frac{1}{\sqrt{2\pi}}\right)^{n}e^{\pi
t^{2}}\left(\frac{d}{dt}\right)^{n}e^{-2\pi t^{2}},\quad
n\in{\mathbb{N}}_{0}.$

The system of Hermite basis functions
$\\{\mathfrak{h}_{k}:k\in{\mathbb{N}}_{0}^{d}\\}\subseteq{L^{2}({\mathbb{R}}^{d})}$
is defined via tensorisation,

$\mathfrak{h}_{k}(t_{1},\dots,t_{d})\coloneqq\prod_{j=1}^{d}h_{k_{j}}(t_{j}).$

The prototypical example of a Hermite function is the standard Gaussian
$\mathfrak{h}_{0}$ which is given by
$\mathfrak{h}_{0}(t)=2^{d/4}e^{-\pi\|t\|_{2}^{2}}$. Any other Hermite function
is the product of $\mathfrak{h}_{0}$ with a polynomial. The choice of Hermite
functions as window functions implies the following concise statement on STFT
phase retrieval with Hermite window functions.

######  Corollary 1.3.

Let $\mathfrak{h}\in{L^{2}({\mathbb{R}}^{d})}$ be an arbitrary Hermite
function. If

$0<\alpha<\sqrt{\frac{1}{2\pi e}},$

then the following statements are equivalent for every
$f,h\in{L^{2}({\mathbb{R}}^{d})}$:

  1. (1)

$|V_{\mathfrak{h}}f(\lambda)|=|V_{\mathfrak{h}}h(\lambda)|$ for every
$\lambda\in\alpha(\sqrt{{\mathbb{Z}}})^{2d}$,

  2. (2)

$f\sim h$.

Figure 1. The left figure depicts an ordinary lattice of the form
$A{\mathbb{Z}}^{2}$ while the right figure visualizes a square-root lattice
$B(\sqrt{{\mathbb{Z}}})^{2}$ $(A,B\in\mathrm{GL}(2,{\mathbb{R}}))$.

###  1.2. Existing work on square-root sampling

The purpose of this section is to mention several existing results on square-
root sampling. The first one is a sampling result for the short-time Fourier
transform with Gaussian window function.

Ascensi, Lyubarskii and Seip [6]. Consider the set
$\Lambda\subseteq{\mathbb{R}}^{2}\simeq{\mathbb{C}}$ defined by

(2)
$\Lambda=\sqrt{2}\left(\\{0\\}\times\sqrt{{\mathbb{Z}}}\right)\cup\sqrt{2}\left(\sqrt{{\mathbb{Z}}}\times\\{0\\}\right)\cup\\{(1,0),(0,1)\\}.$

According to [6, Theorem 1], the set $\Lambda$ is a uniqueness set for the
Bargmann-Fock space $\mathcal{F}^{2}({\mathbb{C}})$ which is defined as the
collection of all entire functions $F\in\mathcal{O}({\mathbb{C}})$ so that

$\int_{\mathbb{C}}|F(z)|^{2}e^{-\pi|z|^{2}}\,dA(z)<\infty,$

where $dA(z)$ denotes the Euclidean area measure on ${\mathbb{C}}$. The
Bargmann-transform constitutes a unitary isomorphism between
$\mathcal{F}^{2}({\mathbb{C}})$ and ${L^{2}({\mathbb{R}})}$. In addition, it
yields an intimate relation between the short-time Fourier transform
$V_{\varphi}$ with window function $\varphi(t)=2^{1/4}e^{-\pi t^{2}}$ and
$\mathcal{F}^{2}({\mathbb{C}})$ [20, Section 3.4]. In view of this one-to-one
correspondence, the theorem of Ascensi, Lyubarskii and Seip implies that if
$\Lambda$ is defined as in (2), then the following statements are equivalent
for every $f,h\in{L^{2}({\mathbb{R}}^{d})}$:

  1. (1)

$V_{\varphi}f(\lambda)=V_{\varphi}h(\lambda)$ for every $\lambda\in\Lambda$,

  2. (2)

$f=h$.

This shows that in the presence of phase information, square-root samples
which are distributed over the time- and frequency axis give rise to unique
determination of every $f\in{L^{2}({\mathbb{R}})}$.

Viazovska and Radchenko [26]. Let $\mathcal{S}_{e}({\mathbb{R}})$ denote the
space of even Schwartz functions $f:{\mathbb{R}}\to{\mathbb{R}}$. According to
[26, Corollary 1], the following two statements are equivalent for every
$f,h\in\mathcal{S}_{e}({\mathbb{R}})$:

  1. (1)

$f(\lambda)=h(\lambda)$ and $\hat{f}(\lambda)=\hat{h}(\lambda)$ for every
$\lambda\in\sqrt{{\mathbb{Z}}_{\geq 0}}$,

  2. (2)

$f=h$.

Thus, every even Schwartz function is determined by square-root samples of
itself and its Fourier transform. The proof of this statement is based on the
theory of modular forms. We find the fact that both our results as well as the
results of [26] provide unique recoverability from square-root samples quite
intriguing.

##  2\. Preliminaries

In this section, we collect several preliminary results which are needed in
the remainder of the article, in particular, in the proofs of the assertions
presented in Section 1.1. The focus of study is the function space
$\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$ and uniqueness sets for spaces of
analytic functions.

###  2.1. The function space $\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$

Recall that if $a,b\in{\mathbb{R}}^{d}_{>0}$, then
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ denotes the collection of all entire
functions $F\in\mathcal{O}({{\mathbb{C}}^{d}})$ which satisfy the estimate

$|F(x+iy)|\lesssim\prod_{j=1}^{d}e^{-a_{j}x_{j}^{2}}e^{b_{j}y_{j}^{2}},\quad
x,y\in{{\mathbb{R}}^{d}}.$

Further, we define

$\mathcal{O}^{b}({\mathbb{C}}^{d})\coloneqq\left\\{F\in\mathcal{O}({\mathbb{C}}^{d}):|F(x+iy)|\lesssim\prod_{j=1}^{d}e^{b_{j}y_{j}^{2}}\right\\}.$

It is evident that for every $a\in{\mathbb{R}}^{d}_{>0}$ we have
$\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})\subseteq\mathcal{O}^{b}({\mathbb{C}}^{d})$.
The parameters $a$ and $b$ govern the growth and the decay of
$F\in\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$: the constant $b$ upper bounds the
growth of $F$ on ${{\mathbb{C}}^{d}}$ whilst $a$ simultaneously lower bounds
the decay rate of $F$ on ${{\mathbb{R}}^{d}}$. Every
$F\in\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$ is in particular an entire
function of order at most two in each complex variable. Recall that an entire
function $F\in\mathcal{O}({\mathbb{C}})$ is said to be of finite order if
there exists a positive number $c>0$ such that $|F(z)|\lesssim e^{|z|^{c}}.$
The smallest non-negative number $\rho\geq 0$ such that

$|F(z)|\lesssim e^{|z|^{\rho+\varepsilon}}$

for every $\varepsilon>0$ is called the order of $F$. The order of a non-
constant entire function $F$ is given by the formula

$\rho=\limsup_{r\to\infty}\frac{\log\log M(r)}{\log r},$

where $M(r)=\max_{z\in\partial B_{r}(0)}|F(z)|$ with $\partial
B_{r}(0)\coloneqq\\{z\in{\mathbb{C}}:|z|=r\\}$. An entire function
$G:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$ is said to be of exponential type if it
is of order one in each variable, that is, there exists a positive real number
$\sigma>0$ such that

$|G(z)|\lesssim e^{\sigma\|z\|_{1}}$

where
$\|z\|_{1}=\sum_{j=1}^{d}|z_{j}|,\,z=(z_{1},\dots,z_{d})^{T}\in{{\mathbb{C}}^{d}}$.
The following short Lemma makes an assertion about the product of an entire
function of exponential type with a function in
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$. It will be of frequent use
throughout the remainder of the article.

######  Lemma 2.1.

Let $a,b\in{\mathbb{R}}^{d}_{>0}$, let
$F\in\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ and let $G$ be an entire
function of exponential type. Then for every
$\varepsilon=(\varepsilon_{1},\dots,\varepsilon_{d})^{T}\in{\mathbb{R}}_{>0}^{d}$
with $\varepsilon_{j}\in(0,a_{j})$ for all $j\in\\{1,\dots,d\\}$, it holds
that $GF\in\mathcal{O}_{a-\varepsilon}^{b+\varepsilon}({{\mathbb{C}}^{d}})$.

###### Proof.

Since $G$ is of exponential type, there exists a positive real number
$\sigma>0$ such that

$|G(z)|\lesssim
e^{\sigma\|z\|_{1}}\leq\prod_{j=1}^{d}e^{\sigma|x_{j}|}e^{\sigma|y_{j}|}.$

Notice that for every $\delta>0$ there exists a constant $C=C(\delta)>0$ such
that for every $t\in{\mathbb{R}}$ it holds that $e^{\sigma|t|}\leq Ce^{\delta
t^{2}}$. Thus,

$|G(x+iy)F(x+iy)|\lesssim\prod_{j=1}^{d}e^{-(a_{j}-\varepsilon_{j})x_{j}^{2}}e^{(b_{j}+\varepsilon_{j})y_{j}^{2}},$

and therefore
$GF\in\mathcal{O}_{a-\varepsilon}^{b+\varepsilon}({{\mathbb{C}}^{d}}).$ ∎

We now summarize several important properties of the function class
$\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$.

######  Proposition 2.2.

Let $a,b\in{\mathbb{R}}^{d}_{>0}$. Then the following properties hold:

  1. (1)

The set $\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ is a linear subspace of
$\mathcal{O}({{\mathbb{C}}^{d}})\cap L^{2}({{\mathbb{R}}^{d}})$.

  2. (2)

For every $\mathfrak{a},\mathfrak{b}\in{\mathbb{R}}_{>0}^{d}$ such that
$\mathfrak{a}_{j}\in(0,a_{j})$ and $\mathfrak{b}_{j}>b_{j}$ for all
$j\in\\{1,\dots,d\\}$, it holds that

$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})\subseteq\mathcal{O}_{\mathfrak{a}}^{\mathfrak{b}}({{\mathbb{C}}^{d}}).$

  3. (3)

If $a<b$, then $\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ is infinite-
dimensional and dense in ${L^{2}({\mathbb{R}}^{d})}$.

  4. (4)

If $a=b$, then $\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})={\mathbb{C}}\varphi$
where $\varphi\in\mathcal{O}({{\mathbb{C}}^{d}})$ is the Gaussian

$\varphi(z_{1},\dots,z_{d})=\prod_{j=1}^{d}e^{-a_{j}z_{j}^{2}}.$

  5. (5)

If $a_{j}>b_{j}$ for some $j\in\\{1,\dots,d\\}$, then
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})=\\{0\\}$.

###### Proof.

The property of $\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ being a linear space
of entire functions follows directly from its definition. Since every
$F\in\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ has Gaussian decay on
${{\mathbb{R}}^{d}}$, we further deduce the inclusion
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})\subseteq L^{2}({{\mathbb{R}}^{d}})$.
The inclusion
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})\subseteq\mathcal{O}_{\mathfrak{a}}^{\mathfrak{b}}({{\mathbb{C}}^{d}})$
with $\mathfrak{a},\mathfrak{b}$ defined as above follows at once from the
definition of $\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$. This shows the first
two claims of the Proposition. To prove the third statement, suppose that
$a<b$ and define $c\coloneqq\frac{1}{2}(a+b)$. Then for every
$j\in\\{1,\dots,d\\}$ and every $z\in{{\mathbb{C}}^{d}}$ we have

(3) $c_{j}>a_{j},\quad
c_{j}<b_{j},\quad\left|e^{-c_{j}z_{j}^{2}}\right|=e^{-c_{j}x_{j}^{2}}e^{c_{j}y_{j}^{2}}.$

This shows that the Gaussian $G\in\mathcal{O}({{\mathbb{C}}^{d}})$ defined via

$G(z)=\prod_{j=1}^{d}e^{-c_{j}z_{j}^{2}}$

is an element of $\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$. Now let
$p\in\mathcal{O}({{\mathbb{C}}^{d}})$ be a polynomial. By virtue of Lemma 2.1
and the fact that $p$ is an entire function of exponential type, it holds that
$pG\in\mathcal{O}_{c-\varepsilon}^{c+\varepsilon}({{\mathbb{C}}^{d}})$ for
every
$\varepsilon=(\varepsilon_{1},\dots,\varepsilon_{d})^{T}\in{\mathbb{R}}_{>0}^{d}$
with $\varepsilon_{j}\in(0,a_{j})$ for all $j\in\\{1,\dots,d\\}$. In view of
Equation (3), it follows that a sufficiently small choice of $\varepsilon_{j}$
implies that we have both $c_{j}-\varepsilon_{j}>a_{j}$ and
$c_{j}+\varepsilon_{j}<b_{j}$ for every $j\in\\{1,\dots,d\\}$. Thus, the
previous inclusion shows that $pG\in\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$.
Since $p$ was arbitrary, every function which is a product of a polynomial
with the Gaussian $G$ is an element of
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$. In particular,
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ contains a system of scaled Hermite
functions, the span of which is known to be dense in
${L^{2}({\mathbb{R}}^{d})}$. The statement follows from that. For the fourth
assertion, we observe that if $F\in\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$,
then for every $z\in{{\mathbb{C}}^{d}}$ it holds that

(4)
$\begin{split}\left|F(z)\prod_{j=1}^{d}e^{\frac{a_{j}+b_{j}}{2}z_{j}^{2}}\right|&=|F(z)|\prod_{j=1}^{d}e^{\frac{a_{j}+b_{j}}{2}x_{j}^{2}}\prod_{j=1}^{d}e^{-\frac{a_{j}+b_{j}}{2}y_{j}^{2}}\\\
&\lesssim\prod_{j=1}^{d}e^{\frac{b_{j}-a_{j}}{2}|z_{j}|^{2}}.\end{split}$

If $a=b$, then the right-hand side of Equation (4) is bounded, whence
Liouville’s theorem applies and shows the existence of a constant
$\nu\in{\mathbb{C}}$ such that $F=\nu\varphi$, where $\varphi$ is defined as
in the statement of the Proposition. Hence,
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})\subseteq{\mathbb{C}}\varphi$. The
reverse inclusion is trivial. In order to prove the last claim, we write for
$j\in\\{1,\dots,d\\}$ and for $z\in{{\mathbb{C}}^{d}}$

$z^{\prime}\coloneqq(z_{1},\dots,z_{j-1},z_{j+1},\dots,z_{d})^{T}\in{\mathbb{C}}^{d-1}.$

If $d=1$, we skip the definition of $z^{\prime}$. We then have

$\left|F(z_{1},\dots,z_{d})e^{\frac{a_{j}+b_{j}}{2}z_{j}^{2}}\right|\lesssim
e^{-\frac{b_{j}-a_{j}}{2}|z_{j}|^{2}}H(z^{\prime}),$

where $H$ is a function of $z^{\prime}\in{\mathbb{C}}^{d-1}$. Suppose that
$a_{j}>b_{j}$ holds for some $j\in\\{1,\dots,d\\}$. Then, it follows from
Liouville’s theorem that the map $z_{j}\mapsto F(z_{1},\dots,z_{d})$ vanishes
identically. Since $z^{\prime}$ was arbitrary, we have $F=0$. ∎

###  2.2. Uniqueness sets for $\mathcal{O}^{b}({\mathbb{C}}^{d})$

Let $\mathcal{S}\subseteq\mathcal{O}({{\mathbb{C}}^{d}})$ be a linear space. A
set $\Lambda\subseteq{{\mathbb{C}}^{d}}$ is said to be a uniqueness set for
$\mathcal{S}$ if for every $F\in\mathcal{S}$ it holds that

$(F(\lambda)=0\ \ \forall\lambda\in\Lambda)\implies F=0.$

Clearly, every open set in $\Lambda\subseteq{{\mathbb{C}}^{d}}$ is a
uniqueness set for $\mathcal{O}({{\mathbb{C}}^{d}})$. If $\Lambda$ is
contained in ${{\mathbb{R}}^{d}}$, then the following can be said [11, Lemma
3.4].

######  Lemma 2.3.

Let $\Lambda\subseteq{{\mathbb{R}}^{d}}$ be a Lebesgue-measurable set such
that $\mathcal{L}^{d}(\Lambda)>0$, where $\mathcal{L}^{d}$ denotes the
$d$-dimensional Lebesgue measure. Then $\Lambda$ is a uniqueness set for
$\mathcal{O}({{\mathbb{C}}^{d}})$.

We now aim to derive uniqueness sets for the class
$\mathcal{O}^{b}({\mathbb{C}}^{d}),b\in{\mathbb{R}}^{d}_{>0}$, which are
discrete subsets of ${{\mathbb{R}}^{d}}$. To that end, we define the zero-
counting function. If $0<r<R$ and $F\in\mathcal{O}(B_{R}(0))$ is an analytic
function on the open unit ball $B_{R}(0)=\\{z\in{\mathbb{C}}:|z|<R\\}$, then
we denote by $n(r)$ the number of zeros $z$ of $F$ for which $|z|\leq r$
(counting multiplicities). This defines a map $n:[0,R)\to{\mathbb{N}}_{0}$.
Jensen’s formula relates the distribution of zeros of an analytic function to
its growth [28, p. 50].

######  Theorem 2.4 (Jensen).

Let $0<r<R$ and suppose that $F\in\mathcal{O}(B_{R}(0))$ satisfies $F(0)\neq
0$. If $z_{1},\dots,z_{n}$ are the zeros of $F$ for which $|z_{j}|\leq r$
(counting multiplicities), then

$\begin{split}\frac{1}{2\pi}\int_{0}^{2\pi}\log|F(re^{i\theta})|\,d\theta&=\log|F(0)|+\sum_{j=1}^{n}\log\left(\frac{r}{|z_{j}|}\right)\\\
&=\log|F(0)|+\int_{0}^{r}\frac{n(t)}{t}\,dt.\end{split}$

With the aid of Jensen’s formula we are well-equipped to prove the following
uniqueness statement.

######  Proposition 2.5.

Let $b=(b_{1},\dots,b_{d})\in{\mathbb{R}}^{d}_{>0}$. For each
$j\in\\{1,\dots,d\\}$ let

$\Lambda_{j}\coloneqq\\{\pm\lambda_{j}(k):k\in{\mathbb{N}}_{0}\\}\subseteq{\mathbb{R}}$

where $\lambda_{j}:{\mathbb{N}}_{0}\to{\mathbb{R}}_{>0}$ is an increasing
function. If

$\liminf_{k\to\infty}\frac{\lambda_{j}(k)}{\sqrt{k}}<\frac{1}{\sqrt{b_{j}e}}$

for every $j\in\\{1,\dots,d\\}$, then
$\Lambda\coloneqq\Lambda_{1}\times\dots\times\Lambda_{d}$ is a uniqueness set
for $\mathcal{O}^{b}({{\mathbb{C}}^{d}})$.

###### Proof.

We prove the statement by induction over the dimension $d\in{\mathbb{N}}$.

Step 1: $d=1$. Let $\Lambda\coloneqq\Lambda_{1}$ and suppose that
$F:{\mathbb{C}}\to{\mathbb{C}}$ is an entire function which satisfies the
bound $|F(z)|\lesssim e^{by^{2}}\leq e^{b|z|^{2}}$ for $b\coloneqq b_{1}$.
Without loss of generality we may suppose that $F(0)=1$. For if
$F(0)\notin\\{0,1\\}$ then one may use a scaling argument, and if $F$ has a
zero of order $m$ at zero then one may consider $F(z)/z^{m}$. Let $n(r)$
denote the number of zeros of $F$ in the closed ball
$\overline{B_{r}(0)}\coloneqq\\{z\in{\mathbb{C}}:|z|\leq
r\\}\subseteq{\mathbb{C}}$ of radius $r>0$. Combining the assumption that
$F(0)=1$ with Jensen’s formula implies the identity

$\int_{0}^{r}\frac{n(t)}{t}\,dt=\frac{1}{2\pi}\int_{0}^{2\pi}\log|F(re^{i\theta})|\,d\theta.$

Since $|F(z)|\leq Ce^{b|z|^{2}}$ for some constant $C>0$, the previous
equality yields the estimate

$\int_{0}^{r}\frac{n(t)}{t}\,dt\leq\log C+br^{2}.$

Now let $s>1$. The property of $t\mapsto n(t)$ being non-decreasing shows that

$n(r)\log
s=n(r)\int_{r}^{sr}\frac{1}{t}\,dt\leq\int_{r}^{sr}\frac{n(t)}{t}\,dt\leq\log
C+bs^{2}r^{2}.$

Assume that $F$ vanishes at $\Lambda\setminus\\{0\\}$ (excluding zero because
of the assumption that $F(0)$ is, without loss of generality, equal to one),
but does not vanish identically. Choosing $r=\lambda(k)$ and $s>1$ yields
$n(r)\geq 2k$ and

$2k\log s\leq\log C+bs^{2}\lambda(k)^{2}.$

This in turn shows the validity of the inequality

(5) $\sqrt{\frac{2\log s}{bs^{2}}}\leq\sqrt{\frac{\log
C}{bs^{2}k}}+\frac{\lambda(k)}{\sqrt{k}},$

where we used the elementary estimate $\sqrt{p+q}\leq\sqrt{p}+\sqrt{q}$ if
$p,q\geq 0$. Maximizing the left-hand side of Equation (5) with respect to
$s>1$ gives

$\sup_{s>1}\sqrt{\frac{2\log s}{bs^{2}}}=\frac{1}{\sqrt{be}},$

resulting in the lower $\liminf$-bound

$\frac{1}{\sqrt{be}}\leq\liminf_{k\to\infty}\frac{\lambda(k)}{\sqrt{k}}.$

Hence, if

$\liminf_{k\to\infty}\frac{\lambda(k)}{\sqrt{k}}<\frac{1}{\sqrt{be}},$

then this gives a contradiction and $F$ must vanish identically.

Step 2: $d>1$. Suppose that the assertion holds for $d-1$. For a fixed
$p\in\Lambda_{1}$, the map

$(z_{2},\dots,z_{d})\mapsto F(p,z_{2},\dots,z_{d})$

is an entire function of $d-1$ complex variables which belongs to
$\mathcal{O}^{b^{\prime}}({\mathbb{C}}^{d-1})$, where
$b^{\prime}=(b_{2},\dots,b_{d})^{T}\in{\mathbb{R}}_{>0}^{d-1}$. This function
vanishes at $\Lambda_{2}\times\dots\times\Lambda_{d}$. By the induction
hypothesis, it must vanish identically. Now fix
$(p_{2},\dots,p_{d})\in{\mathbb{C}}^{d-1}$ and consider the map

$S(z)\coloneqq F(z,p_{2},\dots,p_{d}).$

Then $S$ is an entire function which vanishes at $\Lambda_{1}$ and satisfies
the bound

$|S(z)|\lesssim\left(\prod_{j=2}^{d}e^{b_{j}|p_{j}|^{2}}\right)e^{b_{1}y_{1}^{2}},\quad
z=x_{1}+iy_{1}.$

Therefore, it holds that $S\in\mathcal{O}^{b_{1}}({\mathbb{C}})$. The case
$d=1$ shows that $S$ vanishes identically and the arbitrariness of
$(p_{2},\dots,p_{d})\in{\mathbb{C}}^{d-1}$ concludes the proof of the
statement. ∎

##  3\. Proof of the main results

This section is devoted to the proofs of Theorem 1.1, Corollary 1.2 as well as
Corollary 1.3. After proving the statements, we provide a list of remarks
concerning extensions and comparisons to previous and future work. We start
with the proof of Theorem 1.1. To that end, we introduce the following
notation: if $p,q\in[1,\infty]$ are conjugate to each other, i.e.
$\frac{1}{p}+\frac{1}{q}=1$, and if $v\in L^{p}({{\mathbb{R}}^{d}}),w\in
L^{q}({{\mathbb{R}}^{d}})$, we define

$\langle v,w\rangle\coloneqq\langle
v,w\rangle_{L^{p}({{\mathbb{R}}^{d}})\times
L^{q}({{\mathbb{R}}^{d}})}\coloneqq\int_{{\mathbb{R}}^{d}}v(t)\overline{w(t)}\,dt.$

For a function $f:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$, the shift of $f$ by
$\nu\in{{\mathbb{C}}^{d}}$ is defined via

$T_{\nu}f(t)\coloneqq f(t-\nu).$

The Fourier transform of a function $f\in L^{1}({{\mathbb{R}}^{d}})$ is
defined by

${\mathcal{F}}f(\omega)=\int_{{\mathbb{R}}^{d}}f(t)e^{-2\pi i\omega\cdot
t}\,dt$

where $\omega\cdot t=\sum_{j=1}^{d}\omega_{j}t_{j}$. The extension of
${\mathcal{F}}$ from $L^{1}({{\mathbb{R}}^{d}})\cap{L^{2}({\mathbb{R}}^{d})}$
to a unitary operator on ${L^{2}({\mathbb{R}}^{d})}$ is carried out in the
usual way. Notice that the short-time Fourier transform satisfies
$V_{g}f(x,\omega)={\mathcal{F}}(f\overline{T_{x}g})(\omega)$. The reflection
of a function $f:{{\mathbb{R}}^{d}}\to{\mathbb{C}}$ is defined via
$\mathcal{R}f(t)=f(-t)$. Finally, we define for a vector
$\omega\in{{\mathbb{C}}^{d}}$ and $f:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$ the
map $f_{\omega}:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$ via

$f_{\omega}(t)\coloneqq(T_{\omega}f(t))\overline{f(t)}.$

We call $f_{\omega}$ the tensor product of $f$. For the proof of Theorem 1.1
we require two Lemmas. The first one is a well-known statement which relates
STFTs of the form $V_{f}f$ to the equivalence relation $\sim$ [20, 7].

######  Lemma 3.1.

Let $f,h\in{L^{2}({\mathbb{R}}^{d})}$ such that
$V_{f}f(x,\omega)=V_{h}h(x,\omega)$ for every
$(x,\omega)\in{\mathbb{R}}^{2d}$. Then $f\sim h$.

The second Lemma concerns parameter integrals with respect to a complex
variable [27, 22].

######  Lemma 3.2.

Let $F:{{\mathbb{R}}^{d}}\times{\mathbb{C}}^{d}\to{\mathbb{C}}$ be a function
subjected to the following assumptions:

  1. (1)

$F(\cdot,z)$ is Lebesgue-measurable for every $z\in{\mathbb{C}}^{d}$,

  2. (2)

$F(t,\cdot)$ is an entire function for every $t\in{{\mathbb{R}}^{d}}$,

  3. (3)

The function $z\mapsto\int_{{\mathbb{R}}^{d}}|F(t,z)|\,dt$ is locally bounded,
that is, for every $z_{0}\in{\mathbb{C}}^{d}$ there exists a positive constant
$\delta>0$ such that

$\sup_{\begin{subarray}{c}z\in{\mathbb{C}}^{d}\\\
|z-z_{0}|\leq\delta\end{subarray}}\int_{{\mathbb{R}}^{d}}|F(t,z)|\,dt<\infty.$

Then the function $z\mapsto\int_{{\mathbb{R}}^{d}}F(t,z)\,dt$ defines an
entire function of $d$ complex variables.

###### Proof of Theorem 1.1.

The fact that (2) implies (1) is trivial. The proof of the converse
implication is divided into four steps.

Step 1: An estimate. For $t\in{\mathbb{R}}^{d}$ denote by
$\varphi_{t}(y)=\varphi(y-t)\overline{\varphi(y)}$ the tensor product of
$\varphi$. Since $\varphi$ is an entire function, both maps
$t\mapsto\varphi(y-t)$ and $y\mapsto\varphi(y-t)$ define entire functions.
Moreover, for $y\in{{\mathbb{R}}^{d}}$ we have

$\overline{\varphi(y)}=\overline{\varphi(\overline{y})}$

and the right-hand side extends from ${{\mathbb{R}}^{d}}$ to an entire
function on ${\mathbb{C}}^{d}$. Hence, $\varphi_{t}(y)$ extends in both
variables $t$ and $y$ to an entire function. Let $x+i\rho\in{\mathbb{C}}^{d}$
with $x,\rho\in{\mathbb{R}}^{d}$ and consider the modulus of a complex shift
of $\varphi_{t}$ by $x+i\rho$, i.e. the map $|T_{x+i\rho}\varphi_{t}|$. Using
the definition of the class $\mathcal{O}_{a}^{b}({\mathbb{C}}^{d})$ results in
the estimate

$\begin{split}|T_{x+i\rho}\varphi_{t}(y)|&=|\varphi(y-(x+i\rho)-t)||\varphi(y-(x+i\rho))|\\\
&\lesssim\prod_{j=1}^{d}e^{-a_{j}(y_{j}-x_{j}-t_{j})^{2}}\prod_{j=1}^{d}e^{b_{j}\rho_{j}^{2}}\prod_{j=1}^{d}e^{-a_{j}(y_{j}-x_{j})^{2}}\prod_{j=1}^{d}e^{b_{j}\rho_{j}^{2}}\\\
&=\prod_{j=1}^{d}e^{-2a_{j}(y_{j}-x_{j}-\frac{1}{2}t_{j})^{2}}e^{-\frac{a_{j}}{2}t_{j}^{2}}e^{2b_{j}\rho_{j}^{2}}.\end{split}$

The modulus of the inner product of a function $u\in
L^{1}({{\mathbb{R}}^{d}})$ with a complex shift of $\varphi_{t}$ is therefore
upper bounded by

(6) $\begin{split}|\langle
u,T_{x+i\rho}\varphi_{t}\rangle|&\lesssim\int_{{\mathbb{R}}^{d}}|u(y)|\prod_{j=1}^{d}e^{-2a_{j}(y_{j}-x_{j}-\frac{1}{2}t_{j})^{2}}e^{-\frac{a_{j}}{2}t_{j}^{2}}e^{2b_{j}\rho_{j}^{2}}\,dy\\\
&\leq\prod_{j=1}^{d}e^{-\frac{a_{j}}{2}t_{j}^{2}}e^{2b_{j}\rho_{j}^{2}}\|u\|_{L^{1}({{\mathbb{R}}^{d}})}\end{split}$

Step 2: Holomorphy properties. Now replace in the previous estimate the
function $u$ by the tensor product $f_{t}$ where
$f\in{L^{2}({\mathbb{R}}^{d})}$. The Cauchy-Schwarz inequality shows that

(7)
$\|f_{t}\|_{L^{1}({{\mathbb{R}}^{d}})}\leq\|f\|_{L^{2}({{\mathbb{R}}^{d}})}^{2}.$

Consider the Fourier integral

(8) $F(z)\coloneqq\int_{{\mathbb{R}}^{d}}\langle f_{t},T_{x}\varphi_{t}\rangle
e^{2\pi iz\cdot t}\,dt.$

For $z=v+i\delta\in{\mathbb{C}}^{d},v,\delta\in{\mathbb{R}}^{d},$ the Fourier
integral $F$ satisfies the estimate

$\displaystyle|F(z)|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{}}}{\leq}}\int_{{\mathbb{R}}^{d}}|\langle
f_{t},T_{x}\varphi_{t}\rangle|\prod_{j=1}^{d}e^{-2\pi\delta_{j}t_{j}}\,dt$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Eq.\eqref{eq:3},\eqref{eq:replacement}}}}{\leq}}\|f\|_{L^{2}({{\mathbb{R}}^{d}})}^{2}\int_{{\mathbb{R}}^{d}}\prod_{j=1}^{d}e^{-\frac{a_{j}}{2}t_{j}^{2}}e^{-2\pi\delta_{j}t_{j}}\,dt$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{}}}{=}}\prod_{j=1}^{d}e^{\frac{2\pi^{2}}{a_{j}}\delta_{j}^{2}}\|f\|_{L^{2}({{\mathbb{R}}^{d}})}^{2}\int_{{\mathbb{R}}^{d}}\prod_{j=1}^{d}e^{-\frac{a_{j}}{2}(t_{j}+\frac{2\pi\delta_{j}}{a_{j}})^{2}}\,dt$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{}}}{=}}\prod_{j=1}^{d}e^{\frac{2\pi^{2}}{a_{j}}\delta_{j}^{2}}\|f\|_{L^{2}({{\mathbb{R}}^{d}})}^{2}\int_{{\mathbb{R}}^{d}}\prod_{j=1}^{d}e^{-\frac{a_{j}}{2}t_{j}^{2}}\,dt$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{}}}{\leq}}C(a)\|f\|_{L^{2}({{\mathbb{R}}^{d}})}^{2}\prod_{j=1}^{d}e^{\frac{2\pi^{2}}{a_{j}}\delta_{j}^{2}}$

for some constant $C(a)$ depending only on $a$. The fact that the integrand of
the Fourier integral $F$ is holomorphic in $z$ and measurable in $t$, combined
with the previous estimate, shows that $F$ defines an entire function of $d$
complex variables (cf. Lemma 3.2).

Step 3: Spectrogram properties. Consider the absolute value of the STFT of a
function $f\in{L^{2}({\mathbb{R}}^{d})}$ with respect to the window function
$\varphi$ in the second argument, i.e. the map
$\omega\mapsto|V_{\varphi}f(x,\omega)|^{2}$ for some fixed
$x\in{{\mathbb{R}}^{d}}$. Using the definition of the STFT as the Fourier
transform of the product of $f$ with a shift of $\varphi$ yields the identity

(9)
$\begin{split}|V_{\varphi}f(x,\cdot)|^{2}&={\mathcal{F}}(f\overline{T_{x}\varphi})\overline{{\mathcal{F}}(f\overline{T_{x}\varphi})}={\mathcal{F}^{-1}}\mathcal{R}(f\overline{T_{x}\varphi}){\mathcal{F}^{-1}}(\overline{f}T_{x}\varphi)\\\
&={\mathcal{F}^{-1}}(\mathcal{R}(f\overline{T_{x}\varphi})*(\overline{f}T_{x}\varphi)).\end{split}$

The convolution
$\mathcal{R}(f\overline{T_{x}\varphi})*(\overline{f}T_{x}\varphi)$ evaluated
at $s\in{{\mathbb{R}}^{d}}$ is given by

(10)
$\begin{split}\mathcal{R}(f\overline{T_{x}\varphi})*(\overline{f}T_{x}\varphi)(s)&=\int_{{\mathbb{R}}^{d}}f(-(s-t))\overline{\varphi(-(s-t)-x)}\overline{f(t)}\varphi(t-x)\,dt\\\
&=\int_{{\mathbb{R}}^{d}}f_{s}(t)\overline{T_{x}\varphi_{s}(t)}\,dt\\\
&=\langle f_{s},T_{x}\varphi_{s}\rangle_{L^{1}({{\mathbb{R}}^{d}})\times
L^{\infty}({{\mathbb{R}}^{d}})}\\\ &=\langle
f_{s},T_{x}\varphi_{s}\rangle.\end{split}$

Combining Equation (9) with Equation (10) shows that the spectrogram satisfies
the identity

(11) $|V_{\varphi}f(x,\omega)|^{2}={\mathcal{F}^{-1}}(t\mapsto\langle
f_{t},T_{x}\varphi_{t}\rangle)(\omega)=\int_{{\mathbb{R}}^{d}}\langle
f_{t},T_{x}\varphi_{t}\rangle e^{2\pi i\omega\cdot t}\,dt=F(\omega),$

where $F$ is defined as in Equation (8). According to Step 2, the function $F$
extends from ${{\mathbb{R}}^{d}}$ to an entire function of $d$ complex
variables and satisfies the growth estimate

$|F(\omega+i\delta)|\lesssim\prod_{j=1}^{d}e^{\frac{2\pi^{2}}{a_{j}}\delta_{j}^{2}}.$

Invoking the previous notations shows that
$F\in\mathcal{O}^{\mathfrak{c}}({{\mathbb{C}}^{d}})$ with

$\mathfrak{c}=2\pi^{2}(\tfrac{1}{a_{1}},\dots,\tfrac{1}{a_{d}})^{T}\in{\mathbb{R}}_{>0}^{d}.$

Step 4: Implications for phaseless sampling of the short-time Fourier
transform. Let $f,h\in{L^{2}({\mathbb{R}}^{d})}$ be such that

$|V_{\varphi}f(z)|=|V_{\varphi}h(z)|\ \ \forall z\in\Lambda.$

We want to show that $f\sim h$. To that end, define $\Psi\coloneqq
M(\sqrt{{\mathbb{Z}}})^{d}$ and $\Gamma\coloneqq N(\sqrt{{\mathbb{Z}}})^{d}$
with $M=\mathrm{diag}(\tau_{1},\dots,\tau_{d})$ and
$N=\mathrm{diag}(\nu_{1},\dots,\nu_{d})$ so that $\Lambda=\Psi\times\Gamma$.

Fix $m\in\Psi$. According to Step 3, the two functions
$|V_{\varphi}f(m,\cdot)|^{2}$ and $|V_{\varphi}h(m,\cdot)|^{2}$ extend from
${{\mathbb{R}}^{d}}$ to entire functions which belong to the space
$\mathcal{O}^{\mathfrak{c}}({{\mathbb{C}}^{d}})$ and which agree on $\Gamma$.
The choice of $\nu_{j}$ combined with Proposition 2.5 implies that

(12) $|V_{\varphi}f(m,\omega)|^{2}=|V_{\varphi}h(m,\omega)|^{2}$

for every $\omega\in{{\mathbb{C}}^{d}}$ (in particular, for every
$\omega\in{{\mathbb{R}}^{d}}$). Since $m\in\Psi$ was arbitrary, it follows
that Equation (12) holds true for every $m\in\Psi$ and every
$\omega\in{{\mathbb{R}}^{d}}$. Applying the Fourier transform in the second
argument combined with Equation (11) shows that

$\langle f_{\omega}-h_{\omega},T_{m}\varphi_{\omega}\rangle=0$

for every $m\in\Psi$ and every $\omega\in{{\mathbb{R}}^{d}}$. Now fix
$\omega\in{{\mathbb{R}}^{d}}$ and consider the map

$Q_{\omega}(x)\coloneqq\langle
f_{\omega}-h_{\omega},T_{x}\varphi_{\omega}\rangle=(f_{\omega}-h_{\omega})*(\mathcal{R}(\varphi_{\omega}))(x).$

According to Step 1, the map $x\mapsto T_{x}\varphi_{\omega}$ extends to an
entire function. Further, we have $f_{\omega}-h_{\omega}\in
L^{1}({{\mathbb{R}}^{d}})$. In view of the estimate derived in Equation (6),
it follows from Lemma 3.2 that $Q_{\omega}$ extends to an entire function
which vanishes at $\Psi$. In addition, Equation (6) shows that

$Q_{\omega}\in\mathcal{O}^{\mathfrak{d}}({{\mathbb{C}}^{d}}),\quad\mathfrak{d}=2(b_{1},\dots,b_{d})^{T}.$

The choice of $\tau_{j}$ combined with an application of Proposition 2.5
implies that $Q_{\omega}$ vanishes identically. Since
$f_{\omega}-h_{\omega}\in L^{1}({{\mathbb{R}}^{d}})$ and
$\mathcal{R}(\varphi_{\omega})\in L^{1}({{\mathbb{R}}^{d}})$, we have that
$Q_{\omega}\in L^{1}({{\mathbb{R}}^{d}})$. Applying the Fourier transform once
more yields the relation

${\mathcal{F}}Q_{\omega}(x)={\mathcal{F}}(f_{\omega}-h_{\omega})(x){\mathcal{F}}(\mathcal{R}(\varphi_{\omega}))(x)=0$

which holds for every $x\in{{\mathbb{R}}^{d}}$. Observe that the function
$\mathcal{R}(\varphi_{\omega})$ has Gaussian decay and since $\varphi\neq 0$
it does not vanish identically. Applying Lemma 3.2 once more shows that the
map $x\mapsto{\mathcal{F}}(\mathcal{R}(\varphi_{\omega}))(x)$ extends from
${{\mathbb{R}}^{d}}$ to an entire function which does not vanish identically.
In particular, the set

$Z=\\{x\in{{\mathbb{R}}^{d}}:{\mathcal{F}}(\mathcal{R}(\varphi_{\omega}))(x)=0\\}$

has $d$-dimensional Lebesgue measure zero (cf. Lemma 2.3). Since
${\mathcal{F}}(f_{\omega}-h_{\omega})$ is continuous, it follows that
${\mathcal{F}}(f_{\omega}-h_{\omega})$ must vanish identically. The
arbitrariness of $\omega\in{{\mathbb{R}}^{d}}$ shows that
${\mathcal{F}}(f_{\omega}-h_{\omega})$ vanishes identically for every
$\omega\in{{\mathbb{R}}^{d}}$. In other words,

$V_{f}f(\omega,y)=V_{h}h(\omega,y)$

for every $(\omega,y)\in{\mathbb{R}}^{2d}$. The statement follows from Lemma
3.1. ∎

We continue with the proof of Corollary 1.2.

###### Proof of Corollary 1.2.

The map $x\mapsto e^{-\gamma\|x\|_{2}^{2}}$ is the restriction to
${{\mathbb{R}}^{d}}$ of the entire function $F(z)=\prod_{j=1}^{d}e^{-\gamma
z_{j}^{2}}$. Defining

$\tilde{\gamma}\coloneqq(\underbrace{\gamma,\dots,\gamma}_{d\
\mathrm{times}})^{T}\in{\mathbb{R}}^{d}_{>0}$

it follows that
$F\in\mathcal{O}_{\tilde{\gamma}}^{\tilde{\gamma}}({{\mathbb{C}}^{d}})$ (cf.
Proposition 2.2(4)). Since $p$ is of exponential type, Lemma 2.1 shows that
$pF\in\mathcal{O}_{\tilde{\gamma}-\tilde{\varepsilon}}^{\tilde{\gamma}+\tilde{\varepsilon}}({{\mathbb{C}}^{d}})$
whenever $\tilde{\varepsilon}$ is of the form

$\tilde{\varepsilon}\coloneqq(\underbrace{\varepsilon,\dots,\varepsilon}_{d\
\mathrm{times}})^{T}\in{\mathbb{R}}^{d}_{>0}$

for some $\varepsilon\in(0,\gamma)$. For a sufficiently small choice of
$\varepsilon$, the assumptions on $\alpha$ and $\beta$ imply that

$\alpha<\sqrt{\frac{1}{2(\gamma+\varepsilon)e}},\quad\beta<\sqrt{\frac{\gamma-\varepsilon}{2\pi^{2}e}}.$

The statement therefore follows from Theorem 1.1. ∎

Notice that Corollary 1.3 follows at once from Corollary 1.2 simply by the
choice $\gamma=\pi$ and the fact that every polynomial is an entire function
of exponential type. We end the paper with a series of remarks.

######  Remark 3.3 (Invariance under metaplectic operators).

An invertible matrix $S\in{\mathbb{R}}^{2d\times 2d}$ is called symplectic if
$S^{T}\mathcal{J}S=\mathcal{J}$ where $\mathcal{J}$ denotes the standard
symplectic matrix

$\mathcal{J}\coloneqq\begin{pmatrix}0&-I_{d}\\\ I_{d}&0\end{pmatrix},$

$I_{d}$ is the identity matrix in ${\mathbb{R}}^{d\times d}$ and $0\coloneqq
0_{d}$ denotes the zero matrix in ${\mathbb{R}}^{d\times d}$. The collection
of all symplectic matrices in ${\mathbb{R}}^{2d\times 2d}$ is called the real
symplectic group and is denoted by $\mathrm{Sp}(2d,{\mathbb{R}})$. The
metaplectic group $\mathrm{Mp}(d)$ is the unitary representation of the double
cover of the symplectic group $\mathrm{Sp}(2d,{\mathbb{R}})$ on the Hilbert
space ${L^{2}({\mathbb{R}}^{d})}$. Briefly speaking, to every symplectic
matrix $S\in\mathrm{Sp}(2d,{\mathbb{R}})$ corresponds a pair of unitary
operators $U_{1},U_{2}:{L^{2}({\mathbb{R}}^{d})}\to{L^{2}({\mathbb{R}}^{d})}$
differing by a sign, $U_{1}=-U_{2}$. In the following, we denote this operator
(defined up to a sign) by $\mu(S)$ and call it the metaplectic operator
corresponding to $S$. For an extensive exposition on metaplectic operators we
refer to the book by Folland [10]. The application of metaplectic operators
constitutes a classical technique in time-frequency analysis if one aims to
transfer statements about separable lattices to non-separable ones. For our
purposes the relation

$|V_{g}f(S(x,\omega))|=|V_{\mu(S^{-1})g}(\mu(S^{-1})f)(x,\omega))|,$

which holds for every
$f,g\in{L^{2}({\mathbb{R}}^{d})},\,S\in\mathrm{Sp}(2d,{\mathbb{R}})$ and
$(x,\omega)\in{\mathbb{R}}^{2d}$, will be of significance [20, Lemma 9.4.3].
Recall that the uniqueness statements derived in Theorem 1.1, Corollary 1.2,
and Corollary 1.3 guarantee uniqueness of the STFT phase retrieval problem via
sampling on certain separable square-root lattices $\Lambda$ and window
functions $g\in\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$. Suppose that
$\Lambda$ and $g$ are chosen as in Theorem 1.1 and consider the additional
assumption that $g$ is an eigenfunction of $\mu(S^{-1})$ for some
$S\in\mathrm{Sp}(2d,{\mathbb{R}})$ with eigenvalue $v\neq 0$. Then the
equality

$|V_{g}f(S\lambda)|=|V_{g}h(S\lambda)|\quad\forall\lambda\in\Lambda$

is equivalent to

$|V_{g}(\mu(S^{-1})f)(\lambda)|=|V_{g}(\mu(S^{-1})h)(\lambda)|\quad\forall\lambda\in\Lambda.$

According to Theorem 1.1 we obtain that $\mu(S^{-1})f\sim\mu(S^{-1})h$ and
since $\mu(S^{-1})$ is linear and invertible on ${L^{2}({\mathbb{R}}^{d})}$ it
holds that $f\sim h$. This shows that the property of $g$ being an
eigenfunction of $\mu(S^{-1})$ leads to new uniqueness results where the
sampling set is a square-root lattice which is deformed under the action of a
symplectic matrix.

To give an explicit example of a window function $g$ and square-root lattice
$\Lambda$ where the above method applies, we assume for simplicity that $d=1$
and consider the rotation matrix $R_{\theta}\in{\mathbb{R}}^{2\times
2},\,\theta\in{\mathbb{R}}$, given by

(13) $R_{\theta}\coloneqq\left({\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\\\ \end{array}}\right).$

Then $R_{\theta}\in\mathrm{Sp}(2,{\mathbb{R}})$ and the corresponding
metaplectic operator $\mu(R_{\theta})$ is the so-called fractional Fourier
transform of order $\theta$ [9, 23]. Further, it holds that
$\mu(R_{\theta}^{-1})=\mu(R_{-\theta})$. Every Hermite function
$\mathfrak{h}=h_{n}\in{L^{2}({\mathbb{R}})},\,n\in{\mathbb{N}}_{0}$, is an
eigenfunction of $\mu(R_{\theta})$ corresponding to an eigenvalue
$v\in{\mathbb{T}}$ [21, 23]. The argument above combined with Corollary 1.3
thereby shows that for every Hermite function $\mathfrak{h}$, every
$\theta\in{\mathbb{R}}$, and every $f,h\in{L^{2}({\mathbb{R}})}$ the following
statements are equivalent:

  1. (1)

$|V_{\mathfrak{h}}f(\lambda)|=|V_{\mathfrak{h}}h(\lambda)|$ for every
$\lambda\in\alpha R_{\theta}(\sqrt{{\mathbb{Z}}})^{2}$,

  2. (2)

$f\sim h$.

######  Remark 3.4 (Analyticity of the short-time Fourier transform).

Let $g$ be a window function and let $V_{g}f(x,\omega)$ be the STFT of $f$
with respect to $g$, evaluated at $(x,\omega)\in{\mathbb{R}}^{2d}$. For
simplicity we assume in this remark that $d=1$. If we write
$z=x+i\omega\in{\mathbb{C}}$, then the STFT gives rise to the map

(14) $V_{g}f:{\mathbb{C}}\to{\mathbb{C}},\quad z=x+i\omega\mapsto
V_{g}f(x,\omega).$

If $g$ is a Gaussian window function, then, modulo a multiplication by a non-
zero weighting factor, Equation (14) defines an antiholomorphic function.
Therefore, this case directly allows for the application of techniques from
complex analysis and constitutes the main reason for the choice of Gaussian
windows in most articles on STFT phase retrieval. A classical result of Asenci
and Bruna shows that the Gaussian is essentially the only function which
renders $V_{g}f$ into an analytic function in the above sense [5, Theorem
2.1]. Observe that the results obtained in the present article hold for a
variety of window functions which are different from Gaussians (in fact,
$\mathcal{O}_{a}^{b}({{\mathbb{C}}^{d}})$ is dense in
${L^{2}({\mathbb{R}}^{d})}$ for $a<b$, as stated in Proposition 2.2). Our
techniques therefore not rely on the intimate relation to analytic functions
induced by Gaussian windows.

######  Remark 3.5 (Support constraints).

Corollary 1.2 states that if the window function is the product of a Gaussian
with an entire function of exponential type, then every square-integrable
function $f\in{L^{2}({\mathbb{R}}^{d})}$ is determined up to a global phase by
its phasless STFT samples located on a separable square-root lattice. For
precisely this choice of a window function, it will be shown in [16] that
ordinary separable lattices
($\Lambda=\alpha{\mathbb{Z}}^{d}\times\beta{\mathbb{Z}}^{d}$) yield uniqueness
in classes of _compactly supported_ functions. Formally, if
$K\subseteq{{\mathbb{R}}^{d}}$ is a compact set and $\varphi$ is chosen as in
Corollary 1.2, then there exist $\alpha,\beta>0$ such that the following two
statements are equivalent for every $f,h\in L^{2}(K)$ (we stress the support
constraint on $f$ and $h$):

  1. (1)

$|V_{\varphi}f(\lambda)|=|V_{\varphi}h(\lambda)|$ for every
$\lambda\in\alpha{\mathbb{Z}}^{d}\times\beta{\mathbb{Z}}^{d}$,

  2. (2)

$f\sim h$.

Acknowledgements. The authors thank Irina Shafkulovska for her useful comments
on the manuscript.

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