# From completeness of discrete translates to phaseless sampling of the short-
time
Fourier transform

Philipp Grohs*,†,‡<EMAIL_ADDRESS>,  Lukas Liehr*
<EMAIL_ADDRESS>and  Irina Shafkulovska*,§
<EMAIL_ADDRESS>*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
§Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse
12-14, 1040 Vienna, Austria

###### Abstract.

This article bridges and contributes to two important research areas, namely
the completeness problem for systems of translates in function spaces and the
short-time Fourier transform (STFT) phase retrieval problem. As a first main
contribution, we show that a complex-valued, compactly supported function can
be uniquely recovered from samples of its spectrogram if certain density
properties of an associated system of translates hold true. Secondly, we
derive new completeness results for systems of discrete translates in spaces
of continuous functions on compact sets. We finally combine these findings to
deduce several novel recovery results from spectrogram samples. Our results
hold for a large class of window functions, including Gaussians, all Hermite
functions, as well as the practically highly relevant Airy disk function. To
the best of our knowledge, our results constitute the first recovery
guarantees for the sampled STFT phase retrieval problem with a non-Gaussian
window.

Keywords. phase retrieval, discrete translates, completeness, sampling, short-
time Fourier transform, lattice

AMS subject classifications. 41A30, 42C30, 94A12, 94A20

##  1\. Introduction

The main aim of the present paper is to bridge and contribute to two active
research areas in pure and applied mathematics: the problem of completeness of
discrete translates and the discretization of the short-time Fourier transform
(STFT) phase retrieval problem. Both problems have received significant
attention in recent years:

  * •

The completeness of translates problem constitutes a classical problem at the
intersection of approximation theory, functional analysis, and harmonic
analysis with a series of recent contributions, see for instance [53, 54].

  * •

The study of the STFT phase retrieval problem has seen a surge in the
mathematical community due to its relevance in a wide-ranging number of
applications [42, 73, 1, 19]. The two recent survey papers [28, 22] contain an
extensive list of results and references on this problem. More recent
contributions on both the theoretical and numerical aspects can be found in
[6, 4, 33, 51, 24].

We start by introducing both the completeness of translates problem as well as
the STFT phase retrieval problem, followed by a detailed derivation of their
interrelations.

###  1.1. Completeness of discrete translates

Let $(X,\|\cdot\|)$ be a Banach space over ${\mathbb{C}}$, let $\Lambda$ be an
index set and consider a subset $\\{f_{\lambda}:\lambda\in\Lambda\\}\subseteq
X$. Then $\\{f_{\lambda}:\lambda\in\Lambda\\}$ is said to be complete in $X$
if its linear span is dense in $X$. Determining whether a given system is
complete lies at the very heart of approximation theory and numerical analysis
alike. The traditionally most notable result is due to Weierstrass [68], later
generalized by Stone [64, 65]. For a more modern exposition we refer to
Pinkus’ overview article on the completeness problem in approximation theory
[55]. The practicality aspect further prompts one to look for structured
complete sets. Indeed, suppose from now on that $X$ is a function space on
${{\mathbb{R}}^{d}}$ and let $f_{\lambda}=f(\cdot-\lambda)$ be the
$\lambda$-shift of a function $f\in X$. We define

(1) $\mathfrak{T}(f,\Lambda)\coloneqq\\{f(\cdot-\lambda):\lambda\in\Lambda\\}$

and call $\mathfrak{T}(f,\Lambda)$ the system of translates of $f$ by
$\Lambda$. The _completeness of translates problem_ asks for assumptions on
$X,f$ and $\Lambda$ so that $\mathfrak{T}(f,\Lambda)$ is complete in $X$. The
investigation of this problem produced a vast literature, a possible starting
point being the classical results of Wiener [71]. In recent years, close
attention was drawn to the situation where $X$ is a space of functions defined
on the entire real line and $\Lambda$ is a _discrete_ set, see for instance
[11, 17, 52, 53, 54, 48]. However, the case where $X=C(K)$,
$K\subseteq{{\mathbb{R}}^{d}}$ compact, is the space of continuous functions
on $K$ seems much less studied, notable exceptions include results of Landau
[47], Zalik [72] and Akhiezer [55, Theorem 5.4.]. In this situation, the
choice $\Lambda={{\mathbb{R}}^{d}}$ leads to the theory of mean-periodic
functions, a concept which will play a rather subtle but indispensable role in
this paper.

######  Definition 1.1.

A function $f\in C({{\mathbb{R}}^{d}})$ is said to be mean-periodic if there
exists a compactly supported complex Radon measure $\mu,\,\mu\neq 0$, such
that

(2) $f*\mu=0.$

An application of the Hahn-Banach theorem implies that if $f$ is not mean-
periodic, then $\mathfrak{T}(f,{{\mathbb{R}}^{d}})$ is complete in $C(K)$ for
every compact set $K\subseteq{{\mathbb{R}}^{d}}$. It is precisely this
compactness assumption that will be of central importance in the remainder of
the present article.

###  1.2. Phaseless sampling of the short-time Fourier transform

Moving on to the second half of the title, we again consider a Banach space
$(X,\|\cdot\|)$, typically a Hilbert space
$(X,\left\langle\cdot,\cdot\right\rangle)$. In this setting, we are given a
linear operator acting as

(3) $\mathcal{A}:X\to{\mathbb{C}}^{\Gamma},\quad
f\mapsto\left(\mathcal{A}f(\gamma)\right)_{\gamma\in\Gamma}.$

Assuming $\mathcal{A}$ is injective, the phase retrieval problem asks whether
the phaseless measurements

(4) $\left\lvert\mathcal{A}\right\rvert:X\to{\mathbb{C}}^{\Gamma},\quad
f\mapsto\left(\left\lvert\mathcal{A}f(\gamma)\right\rvert\right)_{\gamma\in\Gamma}$

still define an injective operator. Since $\mathcal{A}$ is linear, for all
complex numbers $\tau\in{\mathbb{T}}$ of modulus one and all $f\in X$ it holds
that $\left\lvert\mathcal{A}(\tau
f)\right\rvert=\left\lvert\mathcal{A}f\right\rvert.$ Hence, we introduce the
equivalence relation for $f,h\in X$

(5) $f\sim h\ \ratio\Leftrightarrow\ \exists\tau\in{\mathbb{T}}:f=\tau h.$

Then the injectivity of $\left\lvert\mathcal{A}\right\rvert$ is to be
understood on the quotient space $X\mathclose{\hbox{$\left/\vbox
to8.5pt{}\right.\n@space$}}\mathopen{}\sim$. The fundamental operator we are
investigating here is the short-time Fourier transform (STFT) with respect to
a window function $g\in{L^{2}({\mathbb{R}}^{d})}$, $g\neq 0$, given by

(6)
$V_{g}f(x,\omega)\coloneqq\int_{{\mathbb{R}}^{d}}f(t)\overline{\,g(t-x)\,}e^{-2\pi
i\omega\cdot t}\,dt=\left\langle
f,M_{\omega}T_{x}g\right\rangle,\quad(x,\omega)\in{\mathbb{R}}^{2d},$

where $T_{x}g(t)=g(t-x)$ and $M_{\omega}g(t)=e^{2\pi i\omega\cdot t}g(t)$
denote the time resp. frequency shift, and $\langle
v,w\rangle\coloneqq\int_{{\mathbb{R}}^{d}}v(t)\overline{\,w(t)\,}\,dt$ denotes
the $L^{2}$-inner product of two functions $v,w\in{L^{2}({\mathbb{R}}^{d})}$,
while $\omega\cdot t=\sum_{j=1}^{d}\omega_{j}t_{j}$ denotes the standard
scalar product on ${{\mathbb{R}}^{d}}$. The square of the absolute value of
the STFT is called the spectrogram of $f$,

(7) $\mathrm{SPEC}_{g}f(x,\omega)=\left\lvert
V_{g}f(x,\omega)\right\rvert^{2}.$

With respect to (STFT) phase retrieval, for a subspace
$X\subseteq{L^{2}({\mathbb{R}}^{d})}$, subset
$\Gamma\subseteq{\mathbb{R}}^{2d}$ and window function
$g\in{L^{2}({\mathbb{R}}^{d})}$, we ask when

(8) $\mathrm{SPEC}_{g}:X\to{\mathbb{C}}^{\Gamma},\quad
f\mapsto\left(\left\lvert
V_{g}f(\gamma)\right\rvert^{2}\right)_{\gamma\in\Gamma}$

is injective on $X\mathclose{\hbox{$\left/\vbox
to8.5pt{}\right.\n@space$}}\mathopen{}\sim$. Note that $\left\lvert
V_{g}f\right\rvert\geq 0$, so having the samples of the spectrogram is
equivalent to having the phaseless STFT samples. This setup fits within the
abstract framework since the STFT is injective on ${L^{2}({\mathbb{R}}^{d})}$.
In fact, there exists an associated Calderón-type reproducing formula. While
this is a good starting point, the separability of ${L^{2}({\mathbb{R}}^{d})}$
motivates the search for discretizations of this representation. In
particular, it would be advantageous to be able to uniquely determine $f$ from
the evaluation of its STFT (and its spectrogram, respectively) on a separated
set $\Gamma$, i.e., a set with

$\inf_{\begin{subarray}{c}\gamma,\gamma^{\prime}\in\Gamma\\\
\gamma\neq\gamma^{\prime}\end{subarray}}\left\lVert\gamma-\gamma^{\prime}\right\rVert_{2}>0.$

A particular type of separated sets are lattices, that is, sets of type
$\Lambda~{}=~{}A{\mathbb{Z}}^{2d}$ for some invertible generating matrix
$A\in\mathrm{GL}(2d,{\mathbb{R}})$. The special structure comes with
additional nice properties that allow us to understand the STFT sampling
problem better through a range of duality theorems [27, 26]. Unavoidably, one
encounters the reciprocal lattice

(9)
$\Gamma^{*}\coloneqq\left\\{\lambda\in{\mathbb{R}}^{2d}:\gamma\cdot\lambda\in{\mathbb{Z}}\
\text{ for all }\gamma\in\Gamma\right\\}$

and the fundamental domain of $\Gamma$, which is a measurable cross-section of
${\mathbb{R}}^{2d}\mathclose{\hbox{$\left/\vbox
to8.5pt{}\right.\n@space$}}\mathopen{}\Gamma$. The volume of the lattice
$\mathrm{vol}(\Gamma)$ is the Lebesgue measure of a fundamental domain of
$\Gamma$ and its density is its reciprocal
$D(\Lambda)=\mathrm{vol}(\Gamma)^{-1}$. With regards to the discretization of
the STFT, the following meta-theorem holds [36].

######  Theorem 1.2.

Let $g$ be a window function that is sufficiently well-localized in the time-
frequency domain. Then for all lattices $\Gamma$ of sufficient density, the
restriction

(10) ${L^{2}({\mathbb{R}}^{d})}\to\ell^{2}(\Gamma),\quad
f\mapsto\left(V_{g}f(\gamma)\right)_{\gamma\in\Gamma}$

is a well-defined, bounded injective operator with a bounded inverse from its
range to ${L^{2}({\mathbb{R}}^{d})}$.

The theorem not only ensures the injectivity of the restriction operator but
also guarantees a stable reconstruction of the sampled STFT. In stark contrast
to this, phase retrieval in infinite-dimensional Hilbert or Banach spaces is
never uniformly stable [7, 18]. Moreover, the uniqueness properties of the
operator $\mathrm{SPEC}$ when sampled on a lattice differs fundamentally from
the setting above: if $X={L^{2}({\mathbb{R}}^{d})}$, then for every
$g\in{L^{2}({\mathbb{R}}^{d})}$ and all lattices
$\Gamma\subseteq{\mathbb{R}}^{2d}$ which frequently appear in time-frequency
analysis the operator $\mathrm{SPEC}_{g}:X\to{\mathbb{C}}^{\Gamma}$ is _not_
injective on $X\mathclose{\hbox{$\left/\vbox
to8.5pt{}\right.\n@space$}}\mathopen{}\sim$ [8, 30, 31].

###  1.3. Objectives of this paper

In an attempt to circumvent the discretization barrier, in the present paper
we restrict ourselves to proper subspaces of ${L^{2}({\mathbb{R}}^{d})}$,
namely various function spaces consisting of compactly supported functions.
The goal of the present exposition is threefold:

  * •

We derive new completeness results of systems of discrete translates for
spaces of continuous functions on compact sets.

  * •

We develop a systematic link between the problem of completeness of discrete
translates and the phaseless sampling problem of the short-time Fourier
transform. More precisely, given a certain function space
$\mathcal{C}\subseteq{L^{2}({\mathbb{R}}^{d})}$ of compactly supported
functions and a window function $g\in{L^{2}({\mathbb{R}}^{d})}$, we show how
completeness of translates of a (suitably chosen) family of related functions
$\\{g_{\omega}:\omega\in\Omega\\}$, $\Omega$ an index set, implies that a
function $f\in\mathcal{C}$ is uniquely determined by phaseless the samples
$\left\lvert V_{g}f(\Gamma)\right\rvert$, provided that $\Gamma$ is a
sufficiently dense lattice. Generalizations to non-uniform sampling sets
$\Gamma$ are also presented.

  * •

Leveraging these results allows us to deduce several new sampling theorems for
the STFT phase retrieval problem, among them the first ever sampling result
for the Airy disk window function, which is of particular practical importance
[16, Chapter 8.5.2].

The proofs are based on a combination of a duality principle with the theory
of mean-periodic functions and uniqueness sets of multivariate entire
functions of exponential type.

###  1.4. Main results

In this section we shall state the main results of the present article.

####  1.4.1. From completeness to phaseless sampling

We start with an abstract result that highlights an intimate relation between
completeness properties of discrete translates and the property of every
function $f\in L^{2}(K),K\subseteq{{\mathbb{R}}^{d}}$ compact, being uniquely
determined up to a global phase by phaseless STFT samples. As in previous
publications, we use the tensor-product notation
$g_{\omega}=(T_{\omega}g)\overline{\,g\,},\,\omega\in{{\mathbb{R}}^{d}}$.
Additionally, $\mathfrak{T}(f,\Lambda)$ denotes the system of
$\Lambda$-translates of a function $f$ as defined in Equation (1). Finally,
$A\pm B$ denotes the standard set sum resp. set difference of
$A,B\subseteq{{\mathbb{R}}^{d}}$,

$A+B=\\{a+b:a\in A,\,b\in B\\},\quad A-B=\\{a-b:a\in A,\,b\in B\\},$

and $PW_{S}^{p}({{\mathbb{R}}^{d}})$ denotes the Paley-Wiener spaces subjected
to a compact set $S\subseteq{{\mathbb{R}}^{d}}$,

$PW_{S}^{p}({{\mathbb{R}}^{d}})=\left\\{f\in
L^{p}({{\mathbb{R}}^{d}}):{\mathrm{supp}}\,\hat{f}\subseteq S\right\\}$

Finally, $V_{g}f$ denotes the short-time Fourier transform as defined in
Equation (6). To state the first main result, we recall the notion of
uniqueness sets. Throughout this paper, by a uniqueness set for a function
space $X$ of continuous functions on ${{\mathbb{R}}^{d}}$ (or
${{\mathbb{C}}^{d}}$) we understand a subset
$\Lambda\subseteq{\mathbb{R}}^{d}$ such that the samples of a function $f\in
X$ at $\Lambda$ uniquely determine $f$, that is, the following implication
holds:

(11) $f(\lambda)=0\ \forall\lambda\in\Lambda\quad\implies\quad f\equiv 0.$

######  Theorem 1.3.

Let $\Lambda,\,\Gamma\subseteq{{\mathbb{R}}^{d}}$ and let
$K\subseteq{{\mathbb{R}}^{d}}$ be a compact set. Let $g\in
C({{\mathbb{R}}^{d}})$ be a window function such that for every $\omega\in
K-K$ the system of translates $\mathfrak{T}(g_{\omega},\Lambda)$ is complete
in $(C(K),\|\cdot\|_{\infty})$. If $\Gamma\subseteq{{\mathbb{R}}^{d}}$ is a
uniqueness set for $PW_{K-K}^{1}({{\mathbb{R}}^{d}})$, then every $f\in
L^{2}(K)$ is determined up to a global phase by $\left\lvert
V_{g}f(\Lambda\times\Gamma)\right\rvert$. In particular, this holds whenever
$\Gamma$ is a lattice such that the difference set $K-K$ is contained in a
fundamental domain of the reciprocal lattice $\Gamma^{*}$.

The previous theorem is restated and proven in Section 3 as Theorem 3.2. In
terms of discretization of STFT phase retrieval, it states the following: if
one can pick a discrete set $\Lambda$ so that the window function $g$ has the
property that $\mathfrak{T}(g_{\omega},\Lambda)$ is complete in
$(C(K),\left\lVert\,\cdot\,\right\rVert_{\infty})$ for every $\omega\in K-K$,
then every $f\in L^{2}(K)$ is automatically determined by discrete samples of
its spectrogram. If $\Lambda$ can be chosen as a lattice, so can the discrete
set that determines $f$ by spectrogram samples as well. It is worth mentioning
that STFT phase retrieval on $L^{2}(K)$ is equivalent to STFT phase retrieval
on $PW^{2}_{K}({{\mathbb{R}}^{d}})$: by the fundamental identity of time-
frequency analysis, for all $x,\omega\in{{\mathbb{R}}^{d}}$ holds

(12) $V_{g}f(x,\omega)=e^{-2\pi
ix\cdot\omega}V_{{\mathcal{F}^{-1}}g}{\mathcal{F}^{-1}}f(\omega,-x).$

Therefore, any compactly supported function $f\in L^{2}(K)$ can be uniquely
determined by $\left\lvert V_{g}f(\Lambda\times\Gamma)\right\rvert$ if and
only if any band-limited function $h\in PW^{2}_{K}({{\mathbb{R}}^{d}})$ can be
uniquely determined by $\left\lvert
V_{{\mathcal{F}^{-1}}g}h(\Gamma\times(-\Lambda))\right\rvert$. Notice that
while $\mathfrak{T}(g,\Lambda)$ may be complete, it is, in general, unclear
how $\mathfrak{T}(g_{\omega},\Lambda)$ behaves. Therefore, as windows, we
consider a class of functions that satisfies completeness properties and is in
addition closed under tensor products.

Figure 1. Left plot: A lattice $\Gamma$ (empty circles) and its dual lattice
$\Gamma^{*}$ (full circles). Right plot: The boundary of a compact set $K$
(full), the boundary of the set difference $K-K$ (dashed), the dual lattice
$\Gamma^{*}$ as in the left plot, and a fundamental domain of $\Gamma^{*}$
containing $K-K$ (gray).

####  1.4.2. The function class $\mathcal{P}_{\alpha,\beta}({\mathbb{C}}^{d})$

The prime objective of the present section is to propose a unifying approach
to known completeness results of discrete translates with Gaussian generators
[72] and band-limited generators [55]. We provide both an extension and a
synthesis of previous completeness results. This will lead to new uniqueness
statements for the STFT phase retrieval problem.

Recall that an entire function $F:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$ is said
to be of exponential type no greater than $\sigma<\infty$ if for every
$\varepsilon>0$ it holds that

$\left\lvert F(z)\right\rvert\lesssim e^{(\sigma+\varepsilon)\left\lVert
z\right\rVert_{1}},\quad z\in{\mathbb{C}}^{d},\quad\left\lVert
z\right\rVert_{1}=\sum_{j=1}^{d}\left\lvert z_{j}\right\rvert.$

The collection of all entire functions of exponential type no greater than
$\sigma$ is denoted by $\mathcal{E}_{\sigma}({\mathbb{C}}^{d})$. Among various
properties, uniqueness sets have been explored in various setups [37, 56, 58,
49]. Observe that uniqueness sets for functions of exponential type can be
chosen to be discrete, in particular, lattices. We now define a related
function space, denoted with $\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$,
where we will draw window functions from.

######  Definition 1.4.

Let $0\leq\alpha,\beta<\infty$. We define the class of functions
$\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$ via

(13) $\displaystyle\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$
$\displaystyle\coloneqq\Big{\\{}z\mapsto p(z)e^{-z^{T}Az-b^{T}z}:$ (14)
$\displaystyle\qquad
p\in\mathcal{E}_{\alpha}({\mathbb{C}}^{d}),A\in{\mathbb{C}}^{d\times
d},\,b\in{{\mathbb{C}}^{d}},\,\big{\|}A+A^{T}\big{\|}_{1}\leq\beta\Big{\\}}.$

We refer to Section 4 for the proofs of all forthcoming results related to
$\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$. Note that this class
contains, for instance, (generalized) Gaussians, Hermite functions, band-
limited functions and products of these. We will often use the same symbol for
a function on ${{\mathbb{R}}^{d}}$ and its holomorphic extension (provided it
exists) on ${\mathbb{C}}^{d}$. Conversely, we do not make a notational
difference between a function on ${\mathbb{C}}^{d}$ and its restriction on
${{\mathbb{R}}^{d}}$. In addition to the previous concepts, we define the
$\ell^{\infty}$-diameter of a compact set $K$ by

$\Delta(K)\coloneqq\max_{t,t^{\prime}\in K}\left\lVert
t-t^{\prime}\right\rVert_{\infty},$

and arrive at the following completeness statement.

######  Theorem 1.5.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set,
$\varphi\in\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$ not mean-periodic,
and

$\sigma\geq\alpha+\tfrac{\beta}{2}\Delta(K).$

If $\Lambda\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$\mathcal{E}_{\sigma}({\mathbb{C}}^{d})$, then $\mathfrak{T}(\varphi,\Lambda)$
is complete in $C(K)$.

There exists a vast literature on uniqueness sets for functions of exponential
type. Consider for example the following classical result of Carlson [37]
which easily transfers to the multivariate setting: ${\mathbb{N}}$ is a
uniqueness set for the class of entire functions
$F:{\mathbb{C}}\to{\mathbb{C}}$ which satisfy the growth condition
$\left\lvert F(z)\right\rvert\lesssim e^{\beta\left\lvert z\right\rvert}$ for
some $\beta<\pi$. With a standard projection to the univariate case, one can
show that $\alpha{\mathbb{N}}^{d}$ is a uniqueness set for
$\mathcal{E}_{\sigma}({{\mathbb{C}}^{d}})$ whenever $\alpha\sigma<\pi$. We
refer to Section 4.2 for a discussion on uniqueness sets. We now arrive at the
application to STFT phase retrieval in which we assume that the window
function $\varphi$ is square-integrable. Note that this is the standard
assumption for the definition of the STFT.

######  Theorem 1.6.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set, let
$\varphi\in\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})\cap{L^{2}({\mathbb{R}}^{d})}$,
$\varphi\neq 0$, and let

$\sigma\geq 2\alpha+\beta\Delta(K).$

If $\Lambda\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$\mathcal{E}_{\sigma}({\mathbb{C}}^{d})$ and if $\Gamma$ is a uniqueness set
for $PW_{K-K}^{1}({{\mathbb{R}}^{d}})$, then every $f\in L^{2}(K)$ is
determined up to a global phase by $\left\lvert
V_{\varphi}f(\Lambda\times\Gamma)\right\rvert$. In particular, this holds if
$K\subseteq[-\kappa,\kappa]^{d}$ for some $\kappa>0$ and if $\sigma\geq
2\alpha+2\beta\kappa.$

The sets $\Lambda$ and $\Gamma$ in the previous theorem can be chosen to be
lattices where $\Lambda$ depends on the growth variables $\alpha,\beta$ as
well as the size of $K$ which is governed by its diameter. For Hermite
functions, we derive the following consequence.

######  Corollary 1.7.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set, let $\mathfrak{h}$ be a
Hermite function and let

$\sigma\geq 2\pi\Delta(K).$

If $\Lambda\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$\mathcal{E}_{\sigma}({\mathbb{C}}^{d})$ and if
$\Gamma\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$PW^{1}_{K-K}({{\mathbb{R}}^{d}})$, then every $f\in L^{2}(K)$ is determined
up to a global phase by $\left\lvert
V_{\mathfrak{h}}f(\Lambda\times\Gamma)\right\rvert$.

####  1.4.3. Band-limited windows and STFT-induced Paley-Wiener subspaces

Let $K^{\prime}\subseteq{{\mathbb{R}}^{d}}$ be a compact set and let $p\in
PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$. It is evident that for $\alpha>0$
sufficiently large, we have that
$p\in\mathcal{E}_{\alpha}({{\mathbb{C}}^{d}})$. Hence, for such $\alpha$ it
holds that
$PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})\subseteq\mathcal{P}_{\alpha,0}({{\mathbb{C}}^{d}})$.
Since band-limited functions are not mean-periodic, Theorem 1.5 applies and
with a bit more effort we obtain the following statement (see Theorem 5.4).

######  Theorem 1.8.

Let $K,K^{\prime}\subseteq{{\mathbb{R}}^{d}}$ be compact subsets and $g\in
PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$, $g\neq 0$. Then every $f\in
L^{2}(K)$ is determined up to a global phase by $|V_{g}f(\mathcal{L})|$
provided that $\mathcal{L}\subseteq{\mathbb{R}}^{2d}$ is a lattice such that
$(K^{\prime}-K^{\prime})\times(K-K)$ is contained in some fundamental domain
of $\mathcal{L}^{*}$.

Notice that a band-limited window function that is omnipresent in the
application side of the STFT phase retrieval problem is the so-called Airy
disk function (see, for instance, [16, Chapter 8.5.2]), which is given by

$\mathbf{A}_{a}:{\mathbb{R}}^{2}\to{\mathbb{R}}_{\geq 0},\quad
a>0,\quad\mathbf{A}_{a}(t)=\left(\frac{aJ_{1}(2\pi\left\lVert
t\right\rVert_{2}a)}{\left\lVert t\right\rVert_{2}}\right)^{2},$

where $J_{1}$ denotes the Bessel function of the first kind. For every $a>0$,
the map $\mathbf{A}_{a}$ is an element of $PW_{K}^{2}({\mathbb{R}}^{2})$ with
$K=\\{x\in{\mathbb{R}}^{2}:\left\lVert x\right\rVert_{2}\leq 2a\\}$. We
discuss this function in more detail in Section 5.

Furthermore, Theorem 1.8 highlights an intimate relation between STFT phase
retrieval with band-limited window functions and phase retrieval theory in
Paley-Wiener spaces: if $f\in L^{2}(K)$ for some compact set
$K\subseteq{{\mathbb{R}}^{d}}$ and $g\in
PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$ for some compact set
$K^{\prime}\subseteq{{\mathbb{R}}^{d}}$, then $V_{g}f$ defines a band-limited
function of $2d$ variables, namely $V_{g}f\in
PW_{K^{\prime}\times(-K)}^{2}({\mathbb{R}}^{2d})$ (see Lemma 5.3). Before we
progress with the discussion, let us make the following definition.

######  Definition 1.9 (STFT-induced Paley-Wiener subspace).

Let $K,K^{\prime}\subseteq{\mathbb{R}}^{2d}$ be compact sets and let

$J\coloneqq K^{\prime}\times(-K).$

We say that a subspace $\mathcal{S}\subseteq PW_{J}^{2}({\mathbb{R}}^{2d})$ is
_STFT-induced by a band-limited window_ if there exists a band-limited
function $g\in PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$ such that

$\mathcal{S}=V_{g}L^{2}(K)=\\{V_{g}f:f\in L^{2}(K)\\}.$

We can now interpret the assertion of Theorem 1.8 as a result on phase
retrieval in Paley-Wiener spaces: if $\mathcal{S}=V_{g}L^{2}(K)$ is an STFT-
induced Paley-Wiener subspace, then every $F\in\mathcal{S}$ is determined up
to a global phase by $|F(\mathcal{L})|$, provided that
$\mathcal{L}\subseteq{\mathbb{R}}^{2d}$ is a lattice such that the Cartesian
product $(K^{\prime}-K^{\prime})\times(K-K)$ is contained in a fundamental
domain of $\mathcal{L}^{*}$. The question of phase retrieval in Paley-Wiener
spaces was considered by several authors during the last years [46, 9, 15, 41,
6, 66]. Notice that usual phase retrieval in Paley-Wiener spaces is impossible
in the following sense: for every $d\in{\mathbb{N}}$ and every compact
$C\subseteq{{\mathbb{R}}^{d}}$ there exists $f,h\in
PW_{C}^{2}({{\mathbb{R}}^{d}})$ such that $|f(t)|=|h(t)|$ for every
$t\in{{\mathbb{R}}^{d}}$ but $f\nsim h$. The entirety of functions $f,h$ with
the previous property can be constructed via the classical zero-flipping
approach, as the fundamental papers of Akutowicz [2, 3], Hofstetter [39], and
Walther [67] reveal. On the other hand, it was shown in [66, 6] that a
restriction to real-valued functions in $PW_{C}^{2}({{\mathbb{R}}^{d}})$
yields uniqueness. However, it is unclear which infinite-dimensional subspaces
consisting of _complex_ -valued function obey the uniqueness property. To the
best of our knowledge, STFT-induced Paley-Wiener subspaces constitute the
first example of such subspaces.

####  1.4.4. Gaussian windows

A classical choice of a window function are Gaussians. Rather remarkably, no
density assumption is required for Gaussian windows, but only a semigroup
structure. This is hard to be expected from other windows as Proposition 4.2
suggests, adding to the Gaussian’s special status within time-frequency
analysis. We recall that the STFT with respect to the Gaussian window is (up
to a weight factor) the Bargmann transform, providing a unitary transformation
from ${L^{2}({\mathbb{R}}^{d})}$ to the Fock space [36]. Moreover, the
Gaussian is the only window with an analytic model space [10]. This connection
to analytic functions leads to the most extensive results in STFT phase
retrieval with Gaussian windows [4, 8, 5, 35, 34, 32, 29] and to the best of
our knowledge, there are no results available for window functions varying
from a Gaussian.

######  Theorem 1.10.

Let $g\in C({{\mathbb{R}}^{d}})$ be a multivariate complex-valued Gaussian

(15) $g(x)=\exp\left(-(x-\nu)^{T}A(x-\nu)\right),$

where $\nu\in{\mathbb{C}}^{d}$ and $A\in{\mathbb{C}}^{d\times d}$ is a
Hermitian matrix mit invertible real part ${\mathrm{Re}\,}A$. If
$K\subseteq{{\mathbb{R}}^{d}}$ is a compact set and $\Lambda$ is a semigroup
in ${\mathbb{R}}^{d}$ that contains a spanning set, then the system
$\mathfrak{T}(g,\Lambda)$ is complete in
$(C(K),\left\lVert\,\cdot\,\right\rVert_{\infty})$. In particular, this holds
when $\Lambda$ is a lattice.

The proof can be found in Section 6. An immediate consequence is the following
uniqueness result for Gabor phase retrieval.

######  Corollary 1.11.

Let $g\in C({{\mathbb{R}}^{d}})$ be a multivariate complex-valued Gaussian

(16) $g(x)=\exp\left(-(x-\nu)^{T}A(x-\nu)\right),$

where $\nu\in{\mathbb{C}}^{d}$ and where $A\in{\mathbb{C}}^{d\times d}$ is a
positive definite Hermitian matrix. Suppose that
$K\subseteq{{\mathbb{R}}^{d}}$ is a compact set. If
$\Lambda,\Gamma\subseteq{{\mathbb{R}}^{d}}$ are lattices such that $K-K$ is
contained in a fundamental domain of the reciprocal lattice $\Gamma^{*}$, then
every $f\in L^{2}(K)$ is determined up to a global phase by $\left\lvert
V_{g}f(\Lambda\times\Gamma)\right\rvert$.

We now combine the completeness properties of discrete Gaussian translates
with the completeness theory of complex exponentials. To that end, we recall a
result of Kahane which states the existence of a sequence
$\mathcal{K}=\\{\kappa_{n}:n\in{\mathbb{Z}}\\}\subseteq{\mathbb{R}}$ with the
property that the system of complex exponentials $\\{e^{2\pi
i\kappa_{n}\cdot}:n\in{\mathbb{Z}}\\}$ is complete in $C(I)$ for every compact
interval $I\subseteq{\mathbb{R}}$ while $\mathcal{K}$ has density zero. We
refer to $\mathcal{K}$ as _Kahane’s sequence_. In the context of STFT phase
retrieval, the following can be said.

######  Theorem 1.12.

Let $g\in C({\mathbb{R}}^{d})$ be a real, multivariate Gaussian window
function. Let $\mathcal{K}\subseteq{\mathbb{R}}$ be Kahane’s sequence and let
$\Lambda\subseteq{\mathbb{R}}\setminus\\{0\\}$ be a sequence of distinct real
numbers of density zero such that
$\sum_{\lambda\in\Lambda}\frac{1}{\left\lvert\lambda\right\rvert}=\infty.$
Further, let

$\mathcal{V}\coloneqq\bigcup_{\kappa>0}L^{4}[-\kappa,\kappa]^{d}.$

Then the following holds:

  1. i)

$\mathcal{V}$ is dense in $L^{2}({{\mathbb{R}}^{d}})$,

  2. ii)

every $f\in\mathcal{V}$ is determined up to a global phase by $\left\lvert
V_{g}f(\Lambda^{d}\times\mathcal{K}^{d})\right\rvert$,

  3. iii)

$\Lambda^{d}\times\mathcal{K}^{d}$ has density zero.

In particular, the theorem’s conclusion holds true if $\Lambda$ is the set of
prime numbers.

Note that there exists no window function such that every
$f\in{L^{2}({\mathbb{R}}^{d})}$ is determined up to a global phase by
spectrogram samples on equidistant parallel hyperplanes of the form
$\Lambda\times{{\mathbb{R}}^{d}}$, $\Lambda\subseteq{{\mathbb{R}}^{d}}$
lattice [31]. In particular, there exists no window function and no separable
lattice such that every $f\in{L^{2}({\mathbb{R}}^{d})}$ is determined up to a
global phase by spectrogram samples on the lattice, irrespective of the
density of the lattice. By contrast, Theorem 1.12 states that uniqueness can
be achieved on a dense subset $\mathcal{V}\subseteq{L^{2}({\mathbb{R}}^{d})}$
from sampling sets which have density zero. This riveting conundrum once more
underlines the sensitivity of the phase retrieval problem and the high
informational value of the phase factor in time-frequency analysis.

####  1.4.5. Semi-discrete sampling sets

The results presented so far deal with discrete sampling sets and require some
form of growth control on the window function. If one transfers to semi-
discrete sampling sets as addressed by the discretization barriers mentioned
above, then very weak assumptions on a function yield uniqueness. This is due
to the fact that the problem of $\mathfrak{T}(g_{\omega},\Lambda)$ being
complete in $(C(K),\left\lVert\,\cdot\,\right\rVert_{\infty})$ significantly
simplifies if $\Lambda={{\mathbb{R}}^{d}}$ (in this case Theorem 1.3 yields a
uniqueness result for semi-discrete sampling sets which are for $d=1$ parallel
lines in the time-frequency plane). For semi-discrete sampling sets, we have
the following statement.

######  Theorem 1.13.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set. Suppose that
$g:{{\mathbb{R}}^{d}}\to{\mathbb{C}}$ is a window function which satisfies the
following two properties:

  1. i)

$g\in C({{\mathbb{R}}^{d}})\cap L^{2}({{\mathbb{R}}^{d}})$,

  2. ii)

for every $\omega\in K-K$ it holds that $g_{\omega}$ does not vanish
identically.

If $\Gamma$ is a lattice such that $K-K$ is contained in a fundamental domain
of the reciprocal lattice $\Gamma^{*}$, then every $f\in L^{2}(K)$ is
determined up to a global phase by $|V_{g}f({{\mathbb{R}}^{d}}\times\Gamma)|$.

###  1.5. Outline

The paper is divided as follows. In Section 2, we go over several elementary
notions. In Section 3 we derive the connection between completeness of
translates and STFT phase retrieval of compactly supported, square-integrable
functions. Section 4 is dedicated to a detailed study of the function class
$\mathcal{P}_{\alpha,\beta}({\mathbb{C}}^{d})$, its completeness properties,
and consequentially, STFT phase retrieval results tied to
$\mathcal{P}_{\alpha,\beta}({\mathbb{C}}^{d})$. We then consider STFT-induced
Paley Wiener spaces and band-limited window functions in Section 1.4.3. We
conclude with Section 6, which deals with the Gaussian window function and
sampled STFT phase retrieval on dense subspaces of
${L^{2}({\mathbb{R}}^{d})}$.

##  2\. Preliminaries and basic definitions

In this section, we shall provide some basic definitions and settle the
notation which is used throughout the remainder of the article.

###  2.1. (Short-time) Fourier transform

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

(17)
${\mathcal{F}}f(\omega)=\hat{f}(\omega)=\int_{{{\mathbb{R}}^{d}}}f(t)e^{-2\pi
i\omega\cdot t}\,dt.$

This defines a bounded operator from $L^{1}({{\mathbb{R}}^{d}})$ to
$L^{\infty}({{\mathbb{R}}^{d}})$ and extends from
$L^{1}({{\mathbb{R}}^{d}})\cap{L^{2}({\mathbb{R}}^{d})}$ to a unitary operator
on ${L^{2}({\mathbb{R}}^{d})}$ by standard density arguments. The short-time
Fourier transform (STFT) $V_{g}f:{\mathbb{R}}^{2d}\to{\mathbb{C}}$ of a
function $f\in{L^{2}({\mathbb{R}}^{d})}$ with respect to a window function
$g\in{L^{2}({\mathbb{R}}^{d})}$ is defined as in Equation (6). The STFT
defines a uniformly continuous function on ${\mathbb{R}}^{2d}$ and satisfies

$V_{g}f(x,\omega)={\mathcal{F}}(f\overline{\,T_{x}g\,})(\omega)=({\mathcal{F}}f*(M_{-x}\overline{\,\mathcal{R}{\mathcal{F}}g\,}))(\omega),$

where $\mathcal{R}u(t)=u(-t)$ denotes the reflection operator. Further, we
denote by $\langle v,w\rangle$ the integral

(18) $\langle
v,w\rangle=\int_{{{\mathbb{R}}^{d}}}v(t)\overline{\,w(t)\,}\,dt,\quad v\in
L^{p}({{\mathbb{R}}^{d}}),\quad w\in L^{q}({{\mathbb{R}}^{d}}),$

where $p,q\in[1,\infty]$ are Hölder conjugate exponents.

###  2.2. Paley-Wiener spaces

A central role in this paper is played by Paley-Wiener spaces and uniqueness
sets for these spaces. We primarily consider the Paley-Wiener spaces of
integrable and square-integrable functions. Therefore, we define for
$p\in\\{1,2\\}$ and for a compact set $K\subseteq{\mathbb{R}}^{d}$ the Paley-
Wiener space $PW_{K}^{p}({\mathbb{R}}^{d})$ via

$PW_{K}^{p}({{\mathbb{R}}^{d}})=\left\\{f\in
L^{p}({{\mathbb{R}}^{d}}):{\mathrm{supp}}({\mathcal{F}}f)\subseteq
K\right\\}.$

The inclusion $PW_{K}^{1}({\mathbb{R}}^{d})\subseteq
PW_{K}^{2}({\mathbb{R}}^{d})$ is well-known [38]. Recall that the classical
theorem of Shannon, Whittaker and Kotel’nikov asserts that if the subset
$\Gamma\subseteq{{\mathbb{R}}^{d}}$ is a lattice,
$K\subseteq{{\mathbb{R}}^{d}}$ a compact set, and $f\in
PW_{K}^{2}({{\mathbb{R}}^{d}})$, then $f$ can be uniquely retrieved from its
samples at $\Gamma$ if and only if $K$ is contained in a fundamental domain
$D$ of the reciprocal lattice $\Gamma^{*}$ (see, for instance, [38, Theorem
14.4] or [45, Theorem 4.4]). In particular, lattices of the previous type are
examples of uniqueness sets for $PW_{K}^{2}({{\mathbb{R}}^{d}})$. As a matter
of fact, $f$ can be reconstructed via

$f(t)=\sum_{\gamma\in\Gamma}f(\gamma)\mathfrak{r}(t-\gamma),$

where the reconstruction function $\mathfrak{r}$ is given by

$\mathfrak{r}(t)\coloneqq\mathrm{vol}(\Gamma^{*})\int_{D}e^{2\pi ix\cdot
t}\,dx.$

The inclusion $PW_{K}^{2}({{\mathbb{R}}^{d}})\subseteq\mathcal{E}_{2\pi
C}({\mathbb{C}}^{d})$, where $C=\max\limits_{x\in K}\left\lVert
x\right\rVert_{\infty}$, relates Paley-Wiener spaces to the spaces of entire
functions of exponential type and uniqueness sets for $\mathcal{E}_{2\pi
C}({\mathbb{C}}^{d})$ to uniqueness sets for $PW_{K}^{2}({{\mathbb{R}}^{d}})$.

###  2.3. Complex measures

For a locally compact Hausdorff space $X$ we denote by $M(X)$ the set of all
Radon measures on $X$. With the usual definition of total variation $|\mu|$ of
$\mu\in M(X)$ [59, Chapter 6], the space $M(X)$ is a Banach space when endowed
with the total variation norm given by $\|\mu\|_{TV}\coloneqq|\mu|(X)$. Let
$C_{0}(X)=(C_{0}(X),\|\cdot\|_{\infty})$ be the Banach space of all complex-
valued, continuous functions on $X$ which vanish at infinity, that is, for
every $\varepsilon>0$ there exists a compact set $K\subseteq X$ such that
$|f(x)|\leq\varepsilon$ for every $x\in K^{c}$. The classical Riesz
representation theorem states that $C_{0}(X)^{*}\simeq M(X)$ in the following
sense: $\Psi\in C_{0}(X)^{*}$ if and only if it can be represented in the form

$\Psi(f)=\int_{X}f\,d\mu,$

where $\mu\in M(X)$ is the unique Radon measure corresponding to $\Psi$.
Moreover, the operator norm of $\Psi$ is the total variation of $\mu$, i.e.,
$\left\lVert\Psi\right\rVert=\left\lVert\mu\right\rVert_{TV}$. If
$K\subseteq{{\mathbb{R}}^{d}}$ is a compact set, then $C_{0}(K)=C(K)$ and
therefore its dual is $C(K)^{*}\simeq M(K)$.

##  3\. Abstract theory

###  3.1. Convolution operators and systems of translates

We dedicate this section to a conceptual take on STFT phase retrieval. Instead
of showing that explicit functions together with (separated) sets can retrieve
compactly supported functions, we provide general sufficient conditions in
terms of completeness properties of discrete translates that can be tested on
diverse families of window functions.

######  Theorem 3.1.

Let $p\in[1,\infty]$, $\Lambda,\Gamma\subseteq{{\mathbb{R}}^{d}}$ and
$K\subseteq{{\mathbb{R}}^{d}}$ be a compact set. Let $g\in
L^{\infty}_{\mathrm{loc}}({{\mathbb{R}}^{d}})$ be a window function which has
the property that for every $\omega\in K-K$ the linear operator

$\mathcal{C}_{\omega}:L^{p}(K)\to{\mathbb{C}}^{\Lambda},\quad
f\mapsto\left(\int_{K}f(t)T_{\lambda}g_{\omega}(t)\,dt\right)_{\lambda\in\Lambda}$

is injective. Further, let $\Gamma$ be a uniqueness set for
$PW_{K-K}^{1}({{\mathbb{R}}^{d}})$. Then every $f\in L^{2p}(K)$ is determined
up to a global phase by $\left\lvert V_{g}f(\Lambda\times\Gamma)\right\rvert$.
In particular, this holds whenever $\Gamma\subseteq{\mathbb{R}}^{d}$ is a
lattice such that $K-K$ is contained in a fundamental domain of the reciprocal
lattice $\Gamma^{*}$.

###### Proof.

Let $f\in L^{2p}(K)$ be a $2p$-integrable function on $K$. Since $p\geq 1$,
and $K$ has bounded measure, we have $f\in L^{2}(K)$. Additionally, $g\in
L^{\infty}_{\mathrm{loc}}({{\mathbb{R}}^{d}})$, so it follows that
$f\overline{\,T_{x}g\,}\in L^{2}(K)$ for every $x\in{{\mathbb{R}}^{d}}$. Using
the convolution identity of the Fourier transform and the fact that
${\mathcal{F}}h={\mathcal{F}^{-1}}\mathcal{R}h$,
$\overline{\,{\mathcal{F}}h\,}={\mathcal{F}^{-1}}\overline{\,h\,}$ for every
$h\in{L^{2}({\mathbb{R}}^{d})}$, we can represent $\left\lvert
V_{g}f(x,\cdot)\right\rvert^{2}$ as

$\begin{split}\left\lvert
V_{g}f(x,\cdot)\right\rvert^{2}&={\mathcal{F}}(f\,\overline{\,T_{x}g\,})\,\overline{\,{\mathcal{F}}(f\,\overline{\,T_{x}g\,})\,}={\mathcal{F}^{-1}}\mathcal{R}(f\,\overline{\,T_{x}g\,})\,{\mathcal{F}^{-1}}(\overline{\,f\,}\,T_{x}g)\\\
&={\mathcal{F}^{-1}}\left(\mathcal{R}\left(f\,\overline{\,T_{x}g\,}\right)*\left(\overline{\,f\,}\,T_{x}g\right)\right).\end{split}$

Since $f$ has support in $K$, the latter convolution can be written as

$\begin{split}\mathcal{R}\left(f\,\overline{\,T_{x}g\,}\right)*\left(\overline{\,f\,}\,T_{x}g\right)(s)&=\int_{K}f(-(s-t))\,\overline{\,g(-(s-t)-x)\,}\,\overline{\,f(t)\,}\,g(t-x)\,dt\\\
&=\int_{K}f_{s}(t)\overline{\,T_{x}g_{s}(t)\,}\,dt.\end{split}$

Therefore, we obtain the identity

(19) ${\mathcal{F}}\left(\left\lvert
V_{g}f(x,\cdot)\right\rvert^{2}\right)(\omega)=\int_{K}f_{\omega}(t)\overline{\,T_{x}g_{\omega}(t)\,}\,dt.$

Suppose that $\omega\in{{\mathbb{R}}^{d}}$ is a vector such that
$(K+\omega)\cap K\neq\emptyset$. Then there exist $k,k^{\prime}\in K$ such
that $k+\omega=k^{\prime}$, implying that $\omega\in K-K$. By contraposition,
we obtain the implication

$\omega\notin K-K\implies(K+\omega)\cap K=\emptyset.$

Since ${\mathrm{supp}}(T_{\omega}f)\subseteq K+\omega$ and
${\mathrm{supp}}(f)\subseteq K$, the previous implication shows that
$f_{\omega}=\left(T_{\omega}f\right)\overline{\,f\,}=0$ almost everywhere
provided $\omega\notin K-K$. In view of Equation (19), the map

(20) $\omega\mapsto{\mathcal{F}}\left(\left\lvert
V_{g}f(x,\cdot)\right\rvert^{2}\right)(\omega)$

is supported on $K-K$. Further, since $f\,\overline{\,T_{x}g\,}\in L^{2}(K)$
and the fact that the Fourier transform is an $L^{2}$-isomorphism, we have

(21)
$\left\lvert{\mathcal{F}}(f\overline{\,T_{x}g\,})\right\rvert^{2}=\left\lvert
V_{g}f(x,\cdot)\right\rvert^{2}\in L^{1}({{\mathbb{R}}^{d}}),$

and therefore, $\left\lvert V_{g}f(x,\cdot)\right\rvert^{2}\in
PW_{K-K}^{1}({{\mathbb{R}}^{d}})$. Now let $h\in L^{2p}(K)$ be a second
function such that the spectrogram samples of $f$ and $h$ agree on
$\Lambda\times\Gamma$, i.e.,

$\left\lvert V_{g}f(\Lambda\times\Gamma)\right\rvert=\left\lvert
V_{g}h(\Lambda\times\Gamma)\right\rvert.$

Since $\left\lvert V_{g}f(x,\cdot)\right\rvert^{2},\left\lvert
V_{g}h(x,\cdot)\right\rvert^{2}\in PW_{K-K}^{1}({{\mathbb{R}}^{d}})$ and since
$\Gamma$ is a uniqueness set for $PW_{K-K}^{1}({{\mathbb{R}}^{d}})$, we
conclude

(22) $\left\lvert
V_{g}f(\Lambda\times{{\mathbb{R}}^{d}})\right\rvert=\left\lvert
V_{g}h(\Lambda\times{{\mathbb{R}}^{d}})\right\rvert.$

In particular, Identity (19) yields

$\int_{K}f_{\omega}(t)\overline{\,T_{\lambda}g_{\omega}(t)\,}\,dt=\int_{K}h_{\omega}(t)\overline{\,T_{\lambda}g_{\omega}(t)\,}\,dt\
\ \ \forall\lambda\in\Lambda,\ \forall\omega\in{\mathbb{R}}^{d}.$

Given that $f,h\in L^{2p}(K)$, we have $f_{\omega}-h_{\omega}\in L^{p}(K)$ by
Hölder’s inequality. The injectivity assumption on the operator
$\mathcal{C}_{\omega}$ gives $f_{\omega}=h_{\omega}$ a.e. for all $\omega\in
K-K$. Since $f_{\omega}=h_{\omega}=0$ a.e. for all $\omega\notin K-K$, it
follows that $f_{\omega}=h_{\omega}$ a.e. for every
$\omega\in{{\mathbb{R}}^{d}}$. Further we have $f_{\omega},h_{\omega}\in
L^{1}({{\mathbb{R}}^{d}})$ and obtain
${\mathcal{F}}(f_{\omega})(x)={\mathcal{F}}(h_{\omega})(x)$ for every
$x\in{{\mathbb{R}}^{d}}$ and every $\omega\in{{\mathbb{R}}^{d}}$.
Equivalently, it holds that $V_{f}f$ agrees with $V_{h}h$ everywhere in
${\mathbb{R}}^{2d}$, whence $f\sim h$ by standard results on time-frequency
analysis (see for instance [12, 36]).

If we assume that $\Gamma$ is a lattice and $K-K$ is contained in a
fundamental domain of the reciprocal lattice $\Gamma^{*}$, then Shannon’s
sampling theorem implies that $\Gamma$ is a sampling set for
$PW_{K-K}^{2}({{\mathbb{R}}^{d}})\supseteq PW_{K-K}^{1}({{\mathbb{R}}^{d}})$,
in particular, a uniqueness set for $PW_{K-K}^{1}({{\mathbb{R}}^{d}})$. ∎

Notice that confirming the injectivity of the family of convolution operators
$(\mathcal{C}_{\omega})_{\omega\in K-K}$ is not a trivial matter. By
Steinhaus’ Theorem [63, 69], the set $K-K$ contains an open neighborhood of
zero whenever the compactum $K$ has positive Lebesgue measure, implying that
one should guarantee the injectivity of uncountably many operators. If one
aims at retrieving square-integrable functions, then the injectivity of
$\mathcal{C}_{\omega}$ can be achieved via a completeness property of
translates of $(g_{\omega})_{\omega\in K-K}$. Recall that for a continuous
function $g\in C({{\mathbb{R}}^{d}})$ and a set
$\Lambda\subseteq{{\mathbb{R}}^{d}}$ we denote by
$\mathfrak{T}(g,\Lambda)\coloneqq\left\\{T_{\lambda}g:\lambda\in\Lambda\right\\}$
the system of $\Lambda$-translates of $g$. The relation between the
injectivity conditions of Theorem 3.1 to completeness properties of translates
reads as follows.

######  Theorem 3.2.

Let $p\in[1,\infty]$, let $\Lambda,\Gamma\subseteq{{\mathbb{R}}^{d}}$ and let
$K\subseteq{{\mathbb{R}}^{d}}$ be a compact set. Let $g\in
C({{\mathbb{R}}^{d}})$ be a window function such that for every $\omega\in
K-K$ the system of translates $\mathfrak{T}(g_{\omega},\Lambda)$ is complete
in $(C(K),\|\cdot\|_{\infty})$. If $\Gamma$ is a uniqueness set for
$PW_{K-K}^{1}({{\mathbb{R}}^{d}})$, then every $f\in L^{2}(K)$ is determined
up to a global phase by $\left\lvert V_{g}f(\Lambda\times\Gamma)\right\rvert$.
In particular, this holds whenever $\Gamma$ is a lattice such that $K-K$ is
contained in a fundamental domain of the reciprocal lattice $\Gamma^{*}$.

###### Proof.

According to Theorem 3.1, it suffices to prove that the convolution operator

(23) $\mathcal{C}_{\omega}:L^{1}(K)\to{\mathbb{C}}^{\Lambda},\quad
f\mapsto\left(\int_{K}f(t)T_{\lambda}g_{\omega}(t)\,dt\right)_{\lambda\in\Lambda}$

is injective for all $\omega\in K-K$. To that end, we define for $f\in
L^{1}(K)$ the operator

$\Phi_{f}:C(K)\to{\mathbb{C}},\ \ \Phi_{f}(q)=\int_{K}f(t)q(t)\,dt.$

Then $\Phi_{f}$ is a continuous linear functional on $C(K)$ and by the Riesz
representation theorem, there exists a unique $\mu\in M(K)$ such that the
functional is given by $\Phi_{f}(q)=\int_{K}q\,d\mu$ and
$\left\lVert\Phi_{f}\right\rVert=\left\lVert\mu\right\rVert_{TV}$. By
uniqueness, we have $d\mu=f\,dm$, where $m$ denotes the Lebesgue measure on
${{\mathbb{R}}^{d}}$, and therefore
$\left\lVert\mu\right\rVert_{TV}=\left\lVert f\right\rVert_{L^{1}(K)}$. Thus,
the completeness of $\mathfrak{T}(g_{\omega},\Lambda)$ in
$(C(K),\left\lVert\,\cdot\,\right\rVert_{\infty})$ gives rise to the
implication

$\Phi_{f}(T_{\lambda}g_{\omega})=0\ \
\forall\lambda\in\Lambda\implies\Phi_{f}=0\implies f=0\ \text{a.e.}\,.$

This shows that $\mathcal{C}_{\omega}$ is injective on $L^{1}(K)$. The
statement follows from Theorem 3.2. ∎

######  Remark 3.3 (Multiplication by periodic functions).

Consider the operator $\mathcal{C}_{\omega}:L^{p}(K)\to{\mathbb{C}}^{\Lambda}$
as defined in Theorem 3.1,

$\mathcal{C}_{\omega}(f)=\left(\int_{K}f(t)T_{\lambda}g_{\omega}(t)\,dt\right)_{\lambda\in\Lambda},$

with $g\in L^{\infty}_{\mathrm{loc}}({{\mathbb{R}}^{d}})$ and
$\Lambda\subseteq{{\mathbb{R}}^{d}}$. Suppose that
$\Lambda\subseteq{{\mathbb{R}}^{d}}$ is a lattice, i.e.,
$\Lambda=A{\mathbb{Z}}^{d}$ for some $A\in\mathrm{GL}(d,{\mathbb{R}})$, and
let $\chi\in L^{\infty}({{\mathbb{R}}^{d}})$ be an $\Lambda$-periodic function
(i.e. $T_{\lambda}\chi=\chi$ for all $\lambda\in\Lambda$) such that
$\chi(t)\neq 0$ for almost every $t\in{{\mathbb{R}}^{d}}$. Then the
injectivity of $\mathcal{C}_{\omega}$ is unaffected if $g$ is multiplied by
$\chi$. For, if $\mathcal{C}_{\omega}$ is injective, then

$0=\int_{K}f(t)T_{\lambda}(g\chi)_{\omega}(t)\,dt=\int_{K}f(t)\chi_{\omega}(t)T_{\lambda}g_{\omega}(t)\,dt\quad\forall\lambda\in\Lambda$

and therefore, $f\chi_{\omega}=0$, where we used that $\chi_{\omega}$ is
$\Lambda$-periodic and $f\chi_{\omega}\in L^{p}(K)$ provided $f\in L^{p}(K)$
and $\chi\in L^{\infty}({{\mathbb{R}}^{d}})$. Since for every
$\omega\in{{\mathbb{R}}^{d}}$ it holds that $\chi_{\omega}(t)\neq 0$ for
almost every $t\in{{\mathbb{R}}^{d}}$, we conclude that $f=0$.

######  Remark 3.4 (Hermitian functions).

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a measurable set. A function $f\in
L^{2}(K)$ is called Hermitian if $\mathcal{R}f=\overline{\,f\,}$. In the
following, we shall denote the space of Hermitian functions by
$L^{2}_{\mathcal{H}}(K)$. It is evident that a function is Hermitian if and
only if its Fourier transform is real-valued. Now suppose that
$K\subseteq{{\mathbb{R}}^{d}}$ is a compact set and let $g\in
L^{2}_{\mathcal{H}}({{\mathbb{R}}^{d}})$ be a Hermitian window function such
that $\hat{g}\in L^{1}({{\mathbb{R}}^{d}})$. If $f\in L^{2}_{\mathcal{H}}(K)$,
then the fundamental identity of time-frequency analysis (see for instance,
[36, Equation 3.10]) shows that for every $\omega\in{{\mathbb{R}}^{d}}$ one
has

$\left\lvert V_{g}f(0,\omega)\right\rvert=\left\lvert
V_{\hat{g}}\hat{f}(\omega,0)\right\rvert=\left\lvert(\mathcal{R}\hat{g}*\hat{f})(\omega)\right\rvert,$

where we used that $\hat{g}$ is real-valued. The function
$\mathcal{R}\hat{g}*\hat{f}$ is real-valued, square-integrable (Young’s
inequality), and its Fourier transform has support in $K$. In other words,
$\mathcal{R}\hat{g}*\hat{f}$ is a real-valued function belonging to
$PW_{K}^{2}({{\mathbb{R}}^{d}})$. It is well-known that a function of this
type is determined up to a global sign-factor by its modulus sampled on a
sufficiently dense lattice $\Gamma\subseteq{{\mathbb{R}}^{d}}$, see Thakur’s
theorem for the univariate setting and corresponding multivariate extensions
by Daubechies et al. [66, 5]. Consequently, if $h\in
L^{2}_{\mathcal{H}}({{\mathbb{R}}^{d}})$ is a second function such that

$\left\lvert V_{g}f(\\{0\\}\times\Gamma)\right\rvert=\left\lvert
V_{g}h(\\{0\\}\times\Gamma)\right\rvert,$

then $\mathcal{R}\hat{g}*\hat{f}\sim\mathcal{R}\hat{g}*\hat{h}$. The
assumption $\mathcal{R}g(t)\neq 0$ for almost every $t\in K$ implies that
$f\sim h$. The uniqueness of STFT phase retrieval problem for the space
$L^{2}_{\mathcal{H}}(K)$ follows therefore from the uniqueness of a phaseless
deconvolution problem. A similar observation was done in [9] for the
univariate setting. The uniqueness of this problem is a simple application of
uniqueness results in real Paley-Wiener spaces.

###  3.2. Mean-periodicity

The notion of mean-periodicity goes back to Delsarte [20, 21] and was later
further developed by Schwartz [60]. For an extensive overview of mean-periodic
functions, we refer to Kahane’s exposition [43]. In order to motivate the
notation of mean-periodicity, assume that $f\in C({\mathbb{R}})$ and let
$a>0$. Suppose that $f$ is an $a$-periodic function, that is, $f(x)=f(x-a)$
for all $x\in{\mathbb{R}}$. Clearly, an $a$-periodic function cannot be an
$a{\mathbb{Z}}$-generator for $C(K),\,K\subseteq{\mathbb{R}}$ compact. The
$a$-periodicity of $f$ can be equivalently written as the requirement that

(24) $f*\mu(x)\coloneqq\int_{\mathbb{R}}f(x-y)\,d\mu(y)=0$

for all $x\in{\mathbb{R}}$, where $\mu=\delta_{0}-\delta_{a}$ is a sum of two
Dirac measures. This motivates the following definition.

######  Definition 3.5.

A function $f\in C({{\mathbb{R}}^{d}})$ is said to be mean-periodic if there
exists a compactly supported complex Radon measure $\mu,\,\mu\neq 0$, such
that $f*\mu=0$.

Besides the motivating example, the set of mean-periodic functions is rather
sparse. One can easily verify that exponential polynomials, i.e., products of
type $z\mapsto p(z)e^{\omega^{T}z},$ where $p$ is a multivariate polynomial
and $\omega\in{{\mathbb{C}}^{d}}$, are also mean-periodic. In fact, any mean
periodic function can be given in terms of related exponential polynomials
[13]. Several equivalent definitions of mean-periodicity are known. For our
purposes, the most striking one asserts that $f$ is mean-periodic if and only
if $\mathfrak{T}(f,{{\mathbb{R}}^{d}})$ is _not_ dense in
$C({{\mathbb{R}}^{d}})$ with respect to the topology of uniform convergence on
compact sets [43]. The perhaps most direct way to eliminate the property of a
function being mean-periodic is to require an integrability condition.

######  Proposition 3.6.

Let $f\in C({{\mathbb{R}}^{d}})$ such that $f\neq 0$. If
$f\in\mathcal{S}^{\prime}({{\mathbb{R}}^{d}})$ and its Fourier transform
${\mathcal{F}}f$ is induced by a measurable function, then $f$ is not mean-
periodic. In particular, if $f\in C({{\mathbb{R}}^{d}})\cap
L^{p}({{\mathbb{R}}^{d}})$ for some $p\in[1,2]$, then $f$ is not mean-
periodic.

###### Proof.

A function is understood as a tempered distribution via the induced functional

(25) $L_{f}:\mathcal{S}({{\mathbb{R}}^{d}})\to{\mathbb{C}},\qquad
h\mapsto\int_{{{\mathbb{R}}^{d}}}h(t)f(t)\,dt.$

The definition is compatible with the Fourier transform in the sense that,
whenever the Fourier transform of $f$ exists, e.g., when $f\in
L^{1}({{\mathbb{R}}^{d}})\cup{L^{2}({\mathbb{R}}^{d})}$, then the Fourier
transform of the distribution is given by
${\mathcal{F}}L_{f}=L_{{\mathcal{F}}f}$. The analogous (in restricted form)
holds for products and convolutions and can be found in the textbooks [14, 40,
25].

To prove the above Proposition, we assume for the sake of contradiction that
there exists a non-trivial compactly supported measure $\mu\in
M({{\mathbb{R}}^{d}})$ such that $f*\mu=0$. Due to the compact support and the
bounded variation of $\mu$, its Fourier transform ${\mathcal{F}}\mu$ is a
smooth function, has a holomorphic extension on ${\mathbb{C}}^{d}$ and all its
derivatives on ${{\mathbb{R}}^{d}}$ have at most polynomial growth at infinity
(see [40, Theorem 7.3.1]). Since $f$ induces the tempered distribution
$L_{f}$, it also holds $L_{f}*\mu=0$ as a distribution. The Convolution
Theorem for distributions [14, Theorem 8.5.1] implies

(26)
$0={\mathcal{F}}\left(L_{f}*\mu\right)={\mathcal{F}}L_{f}\cdot{\mathcal{F}}\mu=L_{{\mathcal{F}}f}\cdot{\mathcal{F}}\mu.$

While this is initially to be understood in the distributional sense, since we
already know that ${\mathcal{F}}L_{f}=L_{{\mathcal{F}}f}$, we can also
interpret this in the pointwise a.e. sense, leading us to conclude

(27)
$0=L_{{\mathcal{F}}f}\cdot{\mathcal{F}}\mu=L_{{\mathcal{F}}f\cdot{\mathcal{F}}\mu}.$

This holds if and only if

(28) ${\mathcal{F}}f\cdot{\mathcal{F}}\mu=0\ \text{a.e.}$

Since $f$ does not vanish everywhere and the Fourier transform is an
automorphism on $\mathcal{S}^{\prime}({{\mathbb{R}}^{d}})$, ${\mathcal{F}}f$
also does not vanish identically, implying there exists a set
$\Omega\subseteq{{\mathbb{R}}^{d}}$ of positive Lebesgue measure such that
${\mathcal{F}}\mu|_{\Omega}=0$. Considering that ${\mathcal{F}}\mu$ has a
holomorphic extension, ${\mathcal{F}}\mu$ must vanish identically [30, Lemma
3.4], thereby proving that also $L_{\mu}=0$. This is the desired
contradiction.

For the second part of the statement, suppose that $q\in[2,\infty]$ is the
Hölder conjugate of $p\in[1,2]$. Hausdorff-Young’s inequality states that the
Fourier transform has a continuous extension $L^{p}({{\mathbb{R}}^{d}})\to
L^{q}({{\mathbb{R}}^{d}})$. In particular, the tempered distribution induced
by $f$ has a Fourier transform induced by a measurable function. ∎

We note that the special case when $f\in C({{\mathbb{R}}^{d}}),\,d=1$, is an
integrable function that does not vanish identically was already proven in
[55, Proposition 6.3]. Furthermore, this cannot be trivially extended for
$L^{p}({{\mathbb{R}}^{d}}),\,p>2$. The key role in the proof of Proposition
3.6 is the fact that the support of ${\mathcal{F}}f$ has positive Lebesgue
measure. If $p>2$, then there are $f\in L^{p}({{\mathbb{R}}^{d}})$ whose
(distributional) Fourier transform is not induced by a measurable function. As
a matter of fact, there are functions whose Fourier transform is supported on
a set of measure zero. A nice example is the Fourier transform of the surface
measure $\sigma_{d}$ on the sphere $S^{d-1}$. In that case holds (cf. [25,
Appendix B.4])

(29)
$f(\omega)={\mathcal{F}^{-1}}\sigma_{d}(\omega)={\mathcal{F}}\sigma_{d}(\omega)=2\pi\left\lVert\omega\right\rVert^{-\frac{d-2}{2}}_{2}J_{\frac{d-2}{2}}(2\pi\left\lVert\omega\right\rVert_{2}),$

where $J_{\nu}$ denotes the Bessel function of first kind and order $\nu$.
With standard asymptotic estimates on Bessel functions (see [25, Appendix B.6,
Appendix B.8]), one can easily show $f\in L^{p}({{\mathbb{R}}^{d}})$ as long
as it holds $(p-2)(d-1)>2$.

The previous considerations immediately yield a phase retrieval result related
to unions of hyperplanes.

######  Theorem 3.7.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set. Suppose that
$g:{{\mathbb{R}}^{d}}\to{\mathbb{C}}$ is a window function that satisfies the
following two properties:

  1. i)

$g\in C({{\mathbb{R}}^{d}})\cap L^{2}({{\mathbb{R}}^{d}})$,

  2. ii)

for every $\omega\in K-K$ it holds that $g_{\omega}$ does not vanish
identically.

If $\Gamma$ is a lattice such that $K-K$ is contained in a fundamental domain
of the reciprocal lattice $\Gamma^{*}$, then every $f\in L^{2}(K)$ is
determined up to a global phase by $|V_{g}f({{\mathbb{R}}^{d}}\times\Gamma)|$.

###### Proof.

Since $g\in C({{\mathbb{R}}^{d}})\cap L^{2}({{\mathbb{R}}^{d}})$, it follows
that for every $\omega\in{{\mathbb{R}}^{d}}$ we have $g_{\omega}\in
C({{\mathbb{R}}^{d}})\cap L^{1}({{\mathbb{R}}^{d}})$. By assumption,
$g_{\omega}\neq 0$ for every $\omega\in K-K$. In view of Proposition 3.6,
$\mathfrak{T}(g_{\omega},{{\mathbb{R}}^{d}})$ is complete in $C(K)$ for every
$\omega\in K-K$. Theorem 3.2 yields the statement. ∎

######  Remark 3.8 (Non-uniqueness theory in $L^{2}({{\mathbb{R}}^{d}})$).

It was recently shown in [31, Theorem 1.1] that for every lattice
$\Gamma\subseteq{{\mathbb{R}}^{d}}$ and every window function $g\in
L^{2}({{\mathbb{R}}^{d}})$ there exist $f,h\in L^{2}({{\mathbb{R}}^{d}})$ such
that

$\left\lvert V_{g}f({{\mathbb{R}}^{d}}\times\Gamma)\right\rvert=\left\lvert
V_{g}h({{\mathbb{R}}^{d}}\times\Gamma)\right\rvert\quad\mathrm{and}\quad
f\nsim h.$

Hence, without a support condition on the underlying signal space, the
corresponding phase retrieval problem always fails to be unique if one samples
on ${{\mathbb{R}}^{d}}\times\Gamma$, irrespective of the choice of $\Gamma$
and $g$. In contrast, Theorem 3.7 shows that mild conditions on $g$ imply
unique recovery of compactly supported signals from sampling sets of the form
${{\mathbb{R}}^{d}}\times\Gamma$.

######  Remark 3.9 (Landau’s theorem).

A theorem due to Landau [47] asserts that if $\varphi\in L^{1}({\mathbb{R}})$
extends to an entire function and if
$\Lambda\subseteq[-(\delta-\kappa),\delta-\kappa]$, $0<\kappa<\delta<\infty$,
is a sequence which converges to zero, then $\mathfrak{T}(\varphi,\Lambda)$ is
complete in $C[-\kappa,\kappa]$. Now suppose that $g\in{L^{2}({\mathbb{R}})}$
extends to an entire function. Then for every $\omega\in{\mathbb{R}}$ it holds
that $g_{\omega}\in L^{1}({\mathbb{R}})$. Further, for every
$t\in{\mathbb{R}}$ we have
$g_{\omega}(t)=g(t-\omega)\overline{\,g(\overline{\,t\,})\,},$ where the
right-hand side of the previous equation extends to an entire function.
Consequently, if $\Lambda,\kappa,\delta$ are given as in Landau’s theorem and
if $K=[-(\delta-\kappa),\delta-\kappa]$, then according to Theorem 3.2, every
$f\in L^{2}(K)$ is determined up to a global phase by $\left\lvert
V_{g}f(\Lambda\times\Gamma)\right\rvert$, provided that $\Gamma$ is a
uniqueness set for $PW_{K-K}^{1}({\mathbb{R}})$. Note however that in this
situation $\Lambda$, hence $\Lambda\times\Gamma$, is not separated.

###  3.3. Completeness via duality

In view of Theorem 3.2 and the search for separated sampling sets, we concern
ourselves with criteria for completeness of discrete translates. One option is
to use Stone-Weierstrass-type arguments, which will be pursued in Section 6
dedicated to the Gaussian window function. However, the algebra structure
required in the Stone-Weierstrass theorem is quite a strong one. We,
therefore, consider a duality strategy. To that end, recall that for all
compact sets $K\subseteq{{\mathbb{R}}^{d}}$ there is a one-to-one
correspondence between the complex Radon measures $\mu\in M(K)$ and the dual
space $C(K)^{*}$ via

$L_{\mu}(f)=\int_{K}f\,d\mu,\qquad\left\lVert
L_{\mu}\right\rVert_{C(K)^{*}}=\left\lVert\mu\right\rVert_{TV}.$

To relate this to the problem of completeness of discrete translates, we
define an associated function space $\mathcal{O}_{\varphi}(K)$. For this, let
$\varphi:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$ be a continuous function and
$K\subseteq{{\mathbb{R}}^{d}}$ compact. Then $\mathcal{O}_{\varphi}(K)$ is
defined by

$\mathcal{O}_{\varphi}(K)\coloneqq\left\\{F:{\mathbb{C}}^{d}\to{\mathbb{C}}:F(z)=L_{\mu}(T_{z}\varphi)=\int_{K}T_{z}\varphi\,d\mu,\
\mu\in M(K)\right\\}.$

######  Proposition 3.10.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set,
$\Lambda\subseteq{{\mathbb{R}}^{d}}$ and
$\varphi:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$ be a continuous function which is
not mean-periodic. Then the following are equivalent:

  1. i)

$\mathfrak{T}(\varphi,\Lambda)$ is complete in $C(K)$,

  2. ii)

for every $\mu\in M(K)$ holds the implication

(30) $L_{\mu}(T_{\lambda}\varphi)=0\ \forall\lambda\in\Lambda\implies
L_{\mu}=0,$

  3. iii)

$\Lambda$ is a uniqueness set for $\mathcal{O}_{\varphi}(K)$.

###### Proof.

Step 1: Equivalence of i) and ii). Suppose that
$\mathfrak{T}(\varphi,\Lambda)$ is complete in $C(K)$. It follows that if
$g\in C(K)$, then there exists a sequence
$(g_{k})_{k\in{\mathbb{N}}}\subseteq{\mathrm{span}\,}_{\mathbb{C}}\mathfrak{T}(\varphi,\Lambda)$
such that $g_{k}\to g$ in $C(K)$. Hence, if $L_{\mu}(T_{\lambda}\varphi)=0$
for all $\lambda\in\Lambda$, then by linearity, $L_{\mu}(g_{k})=0$ for all
$k\in{\mathbb{N}}$. Since $L_{\mu}$ is a continuous linear functional, we have
$L_{\mu}(g)=0$. As $g$ was arbitrary, it follows that $L_{\mu}=0$. Suppose on
the other hand that ii) holds, but $\mathfrak{T}(\varphi,\Lambda)$ is not
complete. Then there exists a function $g\in C(K)$ such that

$g\notin
S\coloneqq\overline{\,{\mathrm{span}\,}_{\mathbb{C}}\mathfrak{T}(\varphi,\Lambda)\,}.$

By Hahn-Banach theorem, it follows that there exists a continuous linear
functional $\Psi\in C(K)^{*}$ such that $\left\lVert\Psi\right\rVert=1$ and
$\Psi|_{S}=0$. Since $\Psi=\Psi_{\mu}$ for some $\mu\in M(X)$, we arrive at a
contradiction.

Step 2: Equivalence of ii) and iii). Suppose that condition ii) holds. Let
$F\in\mathcal{O}_{\varphi}(K)$ with

(31) $F(z)=\int_{K}T_{z}\varphi(t)\,d\mu(t)$

for some $\mu\in M(K)$ and assume that $F(\lambda)=0$ for every
$\lambda\in\Lambda$. Since $F(\lambda)=L_{\mu}(T_{\lambda}\varphi)$, it
follows that $L_{\mu}=0$. Observing that $T_{z}\varphi\in C(K)$ for every
$z\in{\mathbb{C}}^{d}$ shows that $L_{\mu}(T_{z}\varphi)=0$ for every
$z\in{{\mathbb{C}}^{d}}$. Therefore, the function $F$ vanishes identically.
Suppose on the other hand that $\Lambda$ is a uniqueness set for
$\mathcal{O}_{\varphi}(K)$ and let $\mu\in M(K)$ such that
$L_{\mu}(T_{\lambda}\varphi)=0$ for every $\lambda\in\Lambda$. If
$F\in\mathcal{O}_{\varphi}(K)$ is given as in Equation (31), then the
uniqueness property of $\Lambda$ implies that $F$ vanishes identically. Now
observe that $F=\mathcal{R}\varphi*\mu$. Since the class of not mean-periodic
functions is invariant under reflections, it follows that $\mathcal{R}\varphi$
is not mean-periodic. Therefore, $\mu=0$, which in turn shows that $L_{\mu}=0$
and the statement is proved. ∎

######  Remark 3.11.

We note that the equivalence (i)$\Leftrightarrow$(ii) is independent of the
mean-periodicity of the generator $f$. However, the valuable criterium for the
following section is the third statement where the non-mean-periodicity is
decisive for the equivalence. As a matter of fact, without assuming anything
on the relationship between $f$ and $K$, then $f$ not being mean-periodic is
both a necessary and sufficient condition for the equivalence chain.

##  4\. The function space $\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$

This section builds up on the equivalences derived in Theorem 3.10. The main
aim is to present the proofs of the results stated in Section 1.4.2.

###  4.1. Completeness

Recall that an entire function $F:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$ is said
to be of exponential type no greater than $\sigma<\infty$ if for every
$\varepsilon>0$ it holds that

$\left\lvert F(z)\right\rvert\lesssim e^{(\sigma+\varepsilon)\left\lVert
z\right\rVert_{1}},\qquad z\in{\mathbb{C}}^{d}.$

The collection of all entire functions of exponential type no greater than
$\sigma$ is denoted by $\mathcal{E}_{\sigma}({\mathbb{C}}^{d})$. For
$0\leq\alpha,\beta<\infty$, we define the class of functions
$\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$ via

(32) $\displaystyle\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$
$\displaystyle\coloneqq\Big{\\{}z\mapsto p(z)e^{-z^{T}Az-b^{T}z}:$ (33)
$\displaystyle\qquad
p\in\mathcal{E}_{\alpha}({\mathbb{C}}^{d}),A\in{\mathbb{C}}^{d\times
d},\,b\in{{\mathbb{C}}^{d}},\,\big{\|}A+A^{T}\big{\|}_{1}\leq\beta\Big{\\}}.$

This class is intended to cover a spectrum of functions, ranging from band-
limited functions $f(z)e^{-z^{T}\cdot 0\cdot z},\,f\in
PW_{K}^{2}({{\mathbb{R}}^{d}})$, to exponential polynomials
$p(z)e^{\omega^{T}z-z^{T}\cdot 0\cdot
z},\,p:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$ a polynomial, Hermite functions, and
further. Notice that for all $z\in{{\mathbb{C}}^{d}}$ and
$A\in{\mathbb{C}}^{d\times d}$ holds

(34)
$z^{T}Az=\tfrac{1}{2}z^{T}Az+\tfrac{1}{2}\left(z^{T}Az\right)^{T}=\tfrac{1}{2}z^{T}(A+A^{T})z,$

so that we can without loss of generality assume that $A$ is symmetric (with
bound $2\left\lVert A\right\rVert_{1}\leq\beta$). The last required parameter
is the diameter of the domain of interest. For a compact set
$K\subseteq{{\mathbb{R}}^{d}}$ denote its $\ell^{\infty}$-diameter by

$\Delta(K)\coloneqq\max_{t,t^{\prime}\in K}\left\lVert
t-t^{\prime}\right\rVert_{\infty}.$

With the previous concepts in hand, we are equipped to prove one of the main
theorems on completeness of discrete translates of the present paper.

######  Theorem 4.1.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set,
$\varphi\in\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$ not mean-periodic,
and

$\sigma\geq\alpha+\tfrac{\beta}{2}\Delta(K).$

If $\Lambda\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$\mathcal{E}_{\sigma}({\mathbb{C}}^{d})$, then $\mathfrak{T}(\varphi,\Lambda)$
is complete in $C(K)$.

###### Proof.

Let $\varphi(t)=p(t)e^{-t^{T}At-b^{T}t}$ for some
$p\in\mathcal{E}_{\alpha}({{\mathbb{C}}^{d}})$, $b\in{{\mathbb{C}}^{d}}$ and
$A\in{\mathbb{C}}^{d\times d}$ a matrix which satisfies the upper bound
$\big{\|}A+A^{T}\big{\|}_{1}\leq\beta$. According to Theorem 3.10, it suffices
to show that $\Lambda$ is a uniqueness set for $\mathcal{O}_{\varphi}(K)$. To
that end, let $F\in\mathcal{O}_{\varphi}(K)$ with

$F(z)=\int_{K}T_{z}\varphi(t)\,d\mu(t)$

for some $\mu\in M(K)$. We have to show that $F$ vanishes identically,
provided that $F$ vanishes at $\Lambda$. Let $t_{0}\in{\mathbb{R}}^{d}$ such
that $K-t_{0}$ lies in the centered cube

(35) $\left[-\frac{\Delta(K)}{2},\frac{\Delta(K)}{2}\right]^{d}.$

Since the diameter is invariant under translations, this is the optimal
centered cube that fits $K$. Observe that

(36) $\displaystyle T_{z}\varphi(t)$
$\displaystyle=p(t-z)e^{-(t-z)^{T}A(t-z)-b^{T}(t-z)}$ (37)
$\displaystyle=p(t-z)e^{(t-t_{0})^{T}(A+A^{T})z}e^{-X(z)}e^{-t^{T}At-b^{T}t},$

where $X(z)=z^{T}Az-t_{0}^{T}(A+A^{T})z-b^{T}z$ is an entire function
independent of $t$. Hence, if $G$ is defined by

(38) $G(z)=\int_{K}p(t-z)e^{(t-t_{0})^{T}(A+A^{T})z}\,d\tilde{\mu}(t)$

with $\tilde{\mu}(t)=e^{-t^{T}At-b^{T}t}\mu(t)\in M(K)$, then $e^{X}F=G$. For
every $\varepsilon>0$ the map $G$ satisfies the estimate

(39) $\begin{split}\left\lvert
G(z)\right\rvert&\leq\left\lVert\tilde{\mu}\right\rVert_{TV}\max_{t\in
K}\left\lvert p(t-z)e^{(t-t_{0})^{T}(A+A^{T})z}\right\rvert\\\
&\lesssim\left\lVert\tilde{\mu}\right\rVert_{TV}\max_{t\in
K}\left(e^{(\alpha+\varepsilon)\left\lVert t-z\right\rVert_{1}}\cdot
e^{\left\lVert t-t_{0}\right\rVert_{\infty}\left\lVert
A+A^{T}\right\rVert_{1}\left\lVert z\right\rVert_{1}}\right)\\\
&\lesssim\left\lVert\tilde{\mu}\right\rVert_{TV}e^{(\alpha+\varepsilon)\left\lVert
z\right\rVert_{1}}\cdot e^{\frac{\beta}{2}\Delta(K)\left\lVert
z\right\rVert_{1}}.\end{split}$

Since $\left\lVert\tilde{\mu}\right\rVert_{TV}$ is independent of $z$, it
follows that $G$ is an entire function of exponential type no great than
$\sigma=\alpha+\frac{\beta}{2}\Delta(K)$. We have therefore shown that
$e^{X}F\in\mathcal{E}_{\sigma}({{\mathbb{C}}^{d}}).$ Since $e^{X(z)}\neq 0$
for every $z$, it follows that $F$ vanishes on $\Lambda$ if and only if
$e^{X}F$ vanishes on $\Lambda$. The assumption on $\Lambda$ implies that
$e^{X}F=0$, whence $F=0$, thereby proving the assertion. ∎

At this stage, it is meaningful to reflect upon the proof approach presented
in Theorem 4.1. To that end, we consider a minimal example with a univariate
Gaussian $\varphi(t)=e^{-t^{2}}$, a centered interval $K=[-\kappa,\kappa]$,
and a set $\Lambda\subseteq{\mathbb{R}}$. Since the Gaussian is integrable,
hence not mean-periodic, by Theorem 3.10, the completeness is equivalent to
showing

(40) $\forall\mu\in M([-\kappa,\kappa]):\ L_{\mu}(T_{\lambda}\varphi)=0\
\forall\lambda\in\Lambda\implies L_{\mu}=0.$

A key step in the proof of Theorem 4.1 is the derivation of equation (38). In
the setting of the univariate Gaussian, the functional equation of the
exponential transforms the problem to considering for all $\lambda\in\Lambda$
the integral

(41) $\begin{split}0&=\sigma(\lambda)\int_{-\kappa}^{\kappa}e^{2\lambda
t}\psi(t)\ d\mu(t)=\sigma(\lambda)\int_{-\kappa}^{\kappa}e^{2\lambda t}\
d\widetilde{\mu}(t),\end{split}$

where $\sigma(\lambda)=e^{-\lambda^{2}}$, $\psi(t)=e^{-t^{2}}$, and
$d\widetilde{\mu}(t)=\psi(t)d\mu(t)$. The Gaussian is a bounded function on
the subset $[-\kappa,\kappa]\subseteq{\mathbb{R}}$, so $\widetilde{\mu}\in
M([-\kappa,\kappa])$. The factor $\sigma$ is not zero, so we are left with
investigating the real zeroes of

(42) $F(z)=\int_{-\kappa}^{\kappa}e^{2zt}\ d\widetilde{\mu}(t),\qquad
z\in{\mathbb{C}}.$

This function is well-defined, holomorphic, and of exponential type. The
problem, therefore, reduces to the search for uniqueness sets for functions of
exponential type. We point out that this method can only be used with the
Gaussian due to its functional equation, as the following statement confirms.

######  Proposition 4.2.

Let $\varphi:{\mathbb{C}}^{d}\to{\mathbb{C}}$ be a non-zero entire function.
There exist entire functions $\psi,\,\sigma:{\mathbb{C}}^{d}\to{\mathbb{C}}$
and a matrix $A\in{\mathbb{C}}^{d\times d}$ satisfying

(43) $\varphi(z+\lambda)=\psi(z)\,\sigma(\lambda)e^{-2z^{T}A\lambda},\quad
z,\lambda\in{\mathbb{C}}^{d}$

if and only if $A$ is symmetric and there exist $b\in{\mathbb{C}}^{d}$ and
$c\in{\mathbb{C}}$ such that $\varphi$ is given by

(44) $\varphi(z)=e^{-z^{T}Az+z^{T}b+c}.$

###### Proof.

See Appendix. ∎

###  4.2. Uniqueness sets for functions of exponential type

We present two distinct criteria for uniqueness sets due to Carlson [37] and
Ronkin [57]. Carlson’s theorem does assume a semigroup structure on the
sampling set, while Ronkin’s theorem holds for arbitrary separated sets.
However, the former allows for a considerably better density condition on the
point configuration.

######  Theorem 4.3 (Carlson [37]).

Suppose that $F:{\mathbb{C}}\to{\mathbb{C}}$ is an entire function which
satisfies the growth condition $\left\lvert F(z)\right\rvert\lesssim
e^{\beta\left\lvert z\right\rvert}$ where $\beta<\pi$. Then
$F|_{\mathbb{N}}=0$ implies that $F=0$.

Notice that the density assumption $\beta<\pi$ in Carlson’s theorem is sharp
as evidenced by $\sin(\pi z)$ which vanishes at ${\mathbb{N}}$ but does not
vanish identically. Further observe that Carlson’s theorem readily transfers
to the multivariate setting, leading to the statement that
$\alpha{\mathbb{N}}^{d}$ is a uniqueness set for
$\mathcal{E}_{\sigma}({{\mathbb{C}}^{d}})$ provided that $\alpha\sigma<\pi$.
The same holds for $\alpha{\mathbb{Z}}^{d}$ and this cannot be improved due to
the same counterexample.

Now let $\Lambda\subseteq{{\mathbb{R}}^{d}}$ be an arbitrary countable set.
Then it comes with two distinct quantities related to its separability and its
density. Firstly, the minimum gap of $\Lambda$ is defined via

$\delta\coloneqq\inf_{\begin{subarray}{c}\lambda,\lambda^{\prime}\in\Lambda\\\
\lambda\neq\lambda^{\prime}\end{subarray}}\left\lVert\lambda-\lambda^{\prime}\right\rVert_{2}.$

In view of this definition, $\Lambda$ is called $\delta$-_separated_ if
$\delta>0$. On the other end, we denote by $\mu_{\Lambda}$ the number of
points of $\Lambda$ which lie in the cube $T_{r}=(-r,r)^{d}$. The _lower_
resp. _upper density_ of a set $\Lambda$ is given by

$D^{-}(\Lambda)\coloneqq\liminf\limits_{r\to\infty}\frac{\mu_{\Lambda}(r)}{(2r)^{d}},\qquad
D^{+}(\Lambda)\coloneqq\limsup\limits_{r\to\infty}\frac{\mu_{\Lambda}(r)}{(2r)^{d}}.$

For the prototypical example of a lattice
$\Lambda=A{\mathbb{Z}}^{d},\,A\in\mathrm{GL}_{d}({\mathbb{R}})$, we have

(45) $D^{-}(\Lambda)=D^{+}(\Lambda)=\mathrm{vol}(\Lambda)^{-1}.$

######  Theorem 4.4 (Ronkin [57]).

Let the countable set $\Lambda\subseteq{\mathbb{R}}^{d}$ be
$\delta$-separated. Furthermore, let the entire function
$F:{{\mathbb{C}}^{d}}\to{\mathbb{C}}$, satisfy the condition $F(\lambda)=0$
for all $\lambda\in\Lambda$. If $F(z)$ is of exponential type no greater than
$\sigma<A_{d}\delta^{d-1}D^{+}(\Lambda)$, where

$A_{d}=\frac{2}{d!}\left(\frac{\pi}{2}\right)^{d-1}\left(\frac{d-1}{2}+e+\sum\limits_{k=0}^{d-2}\left(\left(\frac{29}{3}\right)^{d-2-k}\left(\frac{13}{3}k+\frac{25}{3}e\right)\right)\right)^{-1},$

then $F$ vanishes identically.

###  4.3. Implications for sampled STFT phase retrieval

We are now in a position to cross over from complete systems of translates to
phase retrieval results. We begin with a technical observation.

######  Lemma 4.5.

Let $\varphi\in\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$, $\varphi\neq
0$, and let $\omega\in{{\mathbb{R}}^{d}}$. Then $\varphi_{\omega}$ is the
restriction to ${{\mathbb{R}}^{d}}$ of a non-zero entire function which
belongs to $\mathcal{P}_{2\alpha,2\beta}({{\mathbb{C}}^{d}})$.

###### Proof.

Let $\omega\in{{\mathbb{R}}^{d}}$ and let
$\varphi\in\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$, i.e.,
$\varphi(t)=p(t)e^{-t^{T}At-b^{T}t}$ for a suitable non-zero
$p\in\mathcal{E}_{\alpha}({\mathbb{C}}^{d})$, $b\in{{\mathbb{C}}^{d}}$ and
$A\in{\mathbb{C}}^{d\times d}$. For every real $t\in{{\mathbb{R}}^{d}}$, we
have

(46)
$\begin{split}\varphi_{\omega}(t)&=p(t-\omega)\overline{\,p(\overline{\,t\,})\,}e^{-(t-\omega)^{T}A(t-\omega)-b^{T}(t-\omega)}e^{-t^{T}\overline{\,A\,}t-\overline{\,b\,}^{T}t}\\\
&=p(t-\omega)\overline{\,p(\overline{\,t\,})\,}e^{-2t^{T}({\mathrm{Re}\,}A)t}e^{(\omega^{T}(A+A^{T})-({\mathrm{Re}\,}b)^{T})t}e^{-\omega^{T}A\omega+b^{T}\omega}.\end{split}$

Replacing $t\in{{\mathbb{R}}^{d}}$ with $z\in{{\mathbb{C}}^{d}}$ we obtain for
every $\varepsilon>0$ the growth estimate

(47)
$|e^{-\omega^{T}A\omega+b^{T}\omega}p(z-\omega)\overline{\,p(\overline{\,z\,})\,}|\lesssim
e^{(\alpha+\varepsilon)\|z-\omega\|_{1}}e^{(\alpha+\varepsilon)\|z\|_{1}}\lesssim
e^{(2\alpha+2\varepsilon)\|z\|_{1}},$

Note that the implicit constant in the estimate (47) depends only on $\omega$.
Since
$p(z-\omega)\overline{\,p(\overline{\,z\,})\,}\in\mathcal{E}_{2\alpha}({{\mathbb{C}}^{d}})$,
the last estimate shows that
$\varphi_{\omega}\in\mathcal{P}_{2\alpha,2\beta}({{\mathbb{C}}^{d}})$.
Considering the fact that $p$ does not vanish identically, neither does the
map $z\mapsto p(z-\omega)\overline{\,p(\overline{\,z\,})\,}$. ∎

With Theorem 3.2 in hindsight, we arrive at the following statement on STFT
phase retrieval with window functions belonging to the class
$\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})$.

######  Theorem 4.6.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set,
$\varphi\in\mathcal{P}_{\alpha,\beta}({{\mathbb{C}}^{d}})\cap{L^{2}({\mathbb{R}}^{d})}$,
$\varphi\neq 0$, and

$\sigma\geq 2\alpha+\beta\Delta(K).$

If $\Lambda\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$\mathcal{E}_{\sigma}({\mathbb{C}}^{d})$ and if $\Gamma$ is a uniqueness set
for $PW_{K-K}^{1}({{\mathbb{R}}^{d}})$, then every $f\in L^{2}(K)$ is
determined up to a global phase by $\left\lvert
V_{\varphi}f(\Lambda\times\Gamma)\right\rvert$. In particular, this holds if
$K\subseteq[-\kappa,\kappa]^{d}$ for some $\kappa>0$ and if $\sigma\geq
2\alpha+2\beta\kappa.$

###### Proof.

Let $\omega\in{{\mathbb{R}}^{d}}$. According to Lemma 4.5, it holds that
$\varphi_{\omega}\neq 0$. In addition, the assumption that $\varphi\in
L^{2}({{\mathbb{R}}^{d}})$ implies that $\varphi_{\omega}\in
L^{1}({{\mathbb{R}}^{d}})$. In particular, $\varphi_{\omega}$ is not mean-
periodic. Now let $\omega\in K-K$. According to Lemma 4.5, the map
$\varphi_{\omega}$ extends from ${{\mathbb{R}}^{d}}$ to an entire function
which belongs to $\mathcal{P}_{2\alpha,2\beta}({{\mathbb{C}}^{d}})$. Applying
Theorem 4.1 shows that $\mathfrak{T}(\varphi_{\omega},\Lambda)$ is complete in
$C(K)$. Since $\omega\in K-K$ was arbitrary, an application of Theorem 3.2
implies the assertion. ∎

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

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

Every Hermite function
$\mathfrak{h}=\mathfrak{h}_{k},\,k\in{\mathbb{N}}_{0}^{d}$, arises as the
product of a polynomial with the Gaussian $2^{d/4}e^{-\pi\|t\|_{2}^{2}}$.
Since every polynomial is of exponential type zero it holds that
$\mathfrak{h}\in\mathcal{P}_{0,2\pi}({{\mathbb{C}}^{d}})$. This implies the
next corollary.

######  Corollary 4.7.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set and
$\mathfrak{h}=\mathfrak{h}_{k}$ be an arbitrary $d$-dimensional Hermite
function. If $\Lambda\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$\mathcal{E}_{\sigma}({{\mathbb{R}}^{d}})$ with $\sigma\geq 2\pi\Delta(K)$ and
$\Gamma\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$PW^{1}_{K-K}({{\mathbb{R}}^{d}})$, then every $f\in L^{2}(K)$ is determined
up to a global phase by $\left\lvert
V_{\mathfrak{h}}f(\Lambda\times\Gamma)\right\rvert$. In particular, this holds
if $K\subseteq[-\kappa,\kappa]^{d}$ for some $\kappa>0$ and if $\sigma\geq
4\pi\kappa.$

##  5\. Band-limited window functions

This section is dedicated to the proofs of the statements of Section 1.4.3. We
start by showing that the discrete translates of a band-limited function form
a complete set. If $d=1$, then this is essentially known [55, Proposition
5.5]. A similar argument as in [55] yields a respective result for the
multivariate case. For the convenience of the reader, we shall prove this
statement in the following.

######  Proposition 5.1.

Let $K^{\prime}\subseteq{{\mathbb{R}}^{d}}$ be a compact set and $\varphi\in
PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$, $\varphi\neq 0$. If
$\Lambda\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$PW_{-K^{\prime}}^{2}({{\mathbb{R}}^{d}})$, then
$\mathfrak{T}(\varphi,\Lambda)$ is complete in $C({{\mathbb{R}}^{d}})$ with
respect to the topology of uniform convergence on compact sets. In other
words, $\mathfrak{T}(\varphi,\Lambda)$ is complete in $C(K)$ for every compact
set $K\subseteq{{\mathbb{R}}^{d}}$.

###### Proof.

Let $K\subseteq{{\mathbb{R}}^{d}}$ be a compact set and let $\mu\in M(K)$. In
view of Proposition 3.10, we have to show that the function

$F(z)=\int_{K}T_{z}\varphi(t)\,d\mu(t)$

vanishes identically provided that $F$ vanishes at $\Lambda$. Since
$\varphi\in PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$, we have
$\varphi={\mathcal{F}}g$ for some $g\in L^{2}(-K^{\prime})\subseteq
L^{1}(-K^{\prime})$. Using Fubini’s theorem, we have

(48)
$\begin{split}F(z)&=\int_{K}{\mathcal{F}}g(t-z)\,d\mu(t)=\int_{K}\int_{-K^{\prime}}g(s)e^{-2\pi
is\cdot(t-z)}\,ds\,d\mu(t)\\\\[3.01385pt]
&=\int_{-K^{\prime}}g(s){\mathcal{F}}\mu(s)e^{2\pi is\cdot
z}\,ds={\mathcal{F}}(g{\mathcal{F}}\mu)(-z)={\mathcal{F}^{-1}}(g{\mathcal{F}}\mu)(z).\end{split}$

Since ${\mathcal{F}}\mu$ is an entire function, it holds that
$g{\mathcal{F}}\mu\in L^{2}(-K^{\prime})$ and therefore $F\in
PW_{-K^{\prime}}^{2}({{\mathbb{R}}^{d}})$. The statement follows from the
assumption of $\Lambda$ being a uniqueness set for
$PW_{-K^{\prime}}^{2}({{\mathbb{R}}^{d}})$. ∎

An immediate consequence of the previous statement is a phase retrieval result
for band-limited window functions.

######  Corollary 5.2.

Let $K,K^{\prime}\subseteq{{\mathbb{R}}^{d}}$ be compact sets and let $g\in
PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}}),\,g\neq 0$. If
$\Gamma\subseteq{{\mathbb{R}}^{d}}$ is a uniqueness set for
$PW_{K-K}^{1}({{\mathbb{R}}^{d}})$ and $\Lambda\subseteq{{\mathbb{R}}^{d}}$ is
a uniqueness set for $PW_{K^{\prime}-K^{\prime}}^{2}({{\mathbb{R}}^{d}})$,
then every $f\in L^{2}(K)$ is determined up to a global phase by
$|V_{g}f(\Lambda\times\Gamma)|$.

###### Proof.

If $g\in PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$ does not vanish identically,
then for every $\omega\in{{\mathbb{R}}^{d}}$, the map $\varphi_{\omega}$ does
not vanish identically and extends from ${{\mathbb{R}}^{d}}$ to an entire
function on ${{\mathbb{C}}^{d}}$ which satisfies

$g_{\omega}\in PW_{K^{\prime}-K^{\prime}}^{1}({{\mathbb{R}}^{d}})\subseteq
PW_{-(K^{\prime}-K^{\prime})}^{2}({{\mathbb{R}}^{d}}).$

Combining Proposition 5.1 with Theorem 3.2 yields the statement. ∎

We now consider function spaces of the form

$\mathcal{S}=V_{g}L^{2}(K)=\\{V_{g}f:f\in L^{2}(K)\\},$

where $K,K^{\prime}\subseteq{{\mathbb{R}}^{d}}$ are compact sets and $g\in
PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$. According to the notion settled in
Section 1.4.3, we call $\mathcal{S}$ an STFT-induced Paley-Wiener subspace.
This notion is due to the following observation.

######  Lemma 5.3.

Let $K,K^{\prime}\subseteq{{\mathbb{R}}^{d}}$ be compact sets and let $g\in
PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}}),\,g\neq 0$. If
$J=K^{\prime}\times(-K)$, then $V_{g}L^{2}(K)\subseteq
PW_{J}^{2}({{\mathbb{R}}^{d}})$.

###### Proof.

First observe that $V_{g}f\in L^{2}({\mathbb{R}}^{2d})$ since both $f$ and $g$
are square integrable. Since
$V_{g}f(x,\omega)={\mathcal{F}}(f\overline{\,T_{x}g\,})(\omega)$, we have

${\mathcal{F}}(V_{g}f(x,\cdot))(\omega)=\mathcal{R}(f\overline{\,T_{x}g\,})(\omega)=\overline{\,g(-\omega-x)\,}f(-\omega).$

and the right-hand side vanishes provided that $\omega\in-K$. If
$\psi(x)=\overline{\,g(-\omega-x)\,}$, then we further have that

${\mathcal{F}}\psi(y)=e^{2\pi i\omega\cdot
y}\overline{\,{\mathcal{F}}g(y)\,}.$

Therefore, the $2d$-dimensional Fourier transform of $V_{g}f$ is given by

(49) $\displaystyle{\mathcal{F}}(V_{g}f)(y,\omega)$
$\displaystyle={\mathcal{F}}(x\mapsto{\mathcal{F}}(V_{g}f(x,\cdot))(\omega))(y)$
(50)
$\displaystyle={\mathcal{F}}(x\mapsto\overline{\,g(-\omega-x)\,}f(-\omega))(y)$
(51) $\displaystyle=e^{2\pi i\omega\cdot
y}f(-\omega)\overline{\,{\mathcal{F}}g(y)\,}.$

Since $g\in PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}})$, this implies that the
support of the $2d$-dimensional Fourier transform of $V_{g}f$ is contained in
$K^{\prime}\times(-K)$, thereby proving the assertion. ∎

The previous considerations imply uniqueness results for STFT phase retrieval
with band-limited window functions, and equivalently, for STFT-induced Paley-
Wiener subspaces.

######  Theorem 5.4.

Let $K,K^{\prime}\subseteq{{\mathbb{R}}^{d}}$ be compact sets and let $g\in
PW_{K^{\prime}}^{2}({{\mathbb{R}}^{d}}),\,g\neq 0$. Further, let
$U=(K^{\prime}-K^{\prime})\times(K-K)$ and
$\mathcal{L}\subseteq{\mathbb{R}}^{2d}$ be a uniqueness set for
$PW_{U}^{1}({{\mathbb{R}}^{d}})$. Then every $F\in\mathcal{S}=V_{g}L^{2}(K)$
is determined up to a global phase by $|F(\mathcal{L})|$.

###### Proof.

Let $F\in\mathcal{S}$ such that $F=V_{g}f$ for some $f\in L^{2}(K)$. According
to Lemma 5.3 it holds that $F\in PW_{J}^{2}({\mathbb{R}}^{2d})$ where
$J=K^{\prime}\times(-K)$. With a similar argument as in the previous sections,
we deduce that $|F|^{2}\in PW_{J-J}^{1}({\mathbb{R}}^{2d})$. Since $J-J=U$ and
since $\mathcal{L}$ is a uniqueness set for $PW_{U}^{1}({\mathbb{R}}^{2d})$,
it follows that if $H=V_{g}h\in\mathcal{S},\,h\in L^{2}(K)$, is a second
function such that $|F(\mathcal{L})|=|H(\mathcal{L})|$, then $|F|=|H|$
everywhere on ${\mathbb{R}}^{2d}$. Corollary 5.2 shows that $f\sim h$, and
since $V_{g}$ is a linear transformation, we obtain $F\sim H$. ∎

######  Example 5.5 (Airy disk).

Let $D_{a}\coloneqq\\{x\in{\mathbb{R}}^{2}:|x|\leq a\\}$ be a circular disk of
radius $a>0$ and center $0$ in ${\mathbb{R}}^{2}$. The square of the Fourier
transform of the characteristic function of $D_{a}$ is called the Airy disk of
radius $a$. We have

$\begin{split}{\mathcal{F}}({\bm{1}}_{D_{a}})(\omega)&=\int_{0}^{a}r\int_{0}^{2\pi}e^{-2\pi
ir\left\lVert\omega\right\rVert_{2}\cos(\theta)}\,d\theta
dr=2\pi\int_{0}^{a}rJ_{0}(2\pi\left\lVert\omega\right\rVert_{2}r)\,dr,\end{split}$

where we used polar coordinates as well as the definition of the Bessel
function

$J_{n}(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(x\sin\tau-n\tau)}\,d\tau.$

The identity

$\int_{0}^{a}xJ_{0}(x)\,dx=aJ_{1}(a)$

implies that the Airy disk of radius $a$ is then given by the radial function

$\mathbf{A}_{a}(\omega)=({\mathcal{F}}({\bm{1}}_{D_{a}})(\omega))^{2}=\left(\frac{aJ_{1}(2\pi\left\lVert\omega\right\rVert_{2}a)}{\left\lVert\omega\right\rVert_{2}}\right)^{2}.$

As shown in Figure 2, this map behaves approximately like a Gaussian (except
for the decay at infinity). The Airy disk frequently appears in diffraction
imaging, namely in the situation where an incoming wavefront hits on a
circular aperture (a pinhole) before getting diffracted at an object of
interest. See, for instance, [16, Chapter 8.5.2]. The STFT phase retrieval
problem with Airy window functions appears naturally in this context.

Figure 2. Left plot: contour plot of an Airy disk. Right plot: plot of a one-
dimensional Gaussian $g$ and an Airy disk function
$r\mapsto\mathbf{A}_{a}(r)=\mathbf{A}_{a}(re_{1})=\mathbf{A}_{a}(re_{\theta}),\,e_{\theta}\in\partial
B_{1}(0)$.

##  6\. Gaussian windows: Gabor phase retrieval

###  6.1. Completeness of discrete Gaussian translates

Any Gaussian belongs to the class $\mathcal{P}_{0,\beta}({{\mathbb{C}}^{d}})$
for a suitable $\beta>0$, hence the completeness Theorem 4.1, and
consequentially Theorem 3.7 on phase retrieval, apply with appropriate density
assumptions on the generator set $\Lambda$. As already evidenced by the
Bargmann transform [50, 61, 62], the uncertainty principles and the current
progress on phase retrieval [4, 8, 5, 35, 34, 32, 29], the Gaussian is an
exceptional window function. We underline this once more by showing that the
completeness of $\mathfrak{T}(g,\Lambda)$ does not depend on the density of
the lattice $\Lambda$ when $g$ is a Gaussian generator.

######  Theorem 6.1.

Let $g\in C({{\mathbb{R}}^{d}})$ be a multivariate complex-valued Gaussian

(52) $g(x)=\exp\left(-(x-\nu)^{T}A(x-\nu)\right),$

where $\nu\in{\mathbb{C}}^{d}$ and $A\in{\mathbb{C}}^{d\times d}$ is a
Hermitian matrix mit invertible real part ${\mathrm{Re}\,}A$. If
$K\subseteq{{\mathbb{R}}^{d}}$ is a compact set and $\Lambda$ is a semigroup
in ${\mathbb{R}}^{d}$ that contains a spanning set, then the system
$\mathfrak{T}(g,\Lambda)$ is complete in
$(C(K),\left\lVert\,\cdot\,\right\rVert_{\infty})$. In particular, this holds
when $\Lambda$ is a lattice.

###### Proof.

By expanding the term $-(x-\lambda-\nu)^{T}A(x-\lambda-\nu)$, we obtain the
factorization

$T_{\lambda}G(x)=c(\lambda)a(x)B_{\lambda}(x)$

with

$\begin{split}c(\lambda)&=\exp\left(-(\lambda+\nu)^{T}A(\lambda+\nu)\right),\\\
a(x)&=\exp\left(-x^{T}Ax+2\left(\left({\mathrm{Re}\,}A\right)\,x\right)^{T}\nu\right),\\\
B_{\lambda}(x)&=\exp\left(2\left(\left({\mathrm{Re}\,}A\right)\,x\right)^{T}\lambda\right),\end{split}$

where we used that $A$ is Hermitian. Notice that the completeness of
$\mathfrak{T}(g,\Lambda)$ is unaffected if each $T_{\lambda}G$ is multiplied
by a non-zero constant. Since $c(\lambda)\neq 0$, $\mathfrak{T}(g,\Lambda)$ is
complete if and only if the set $\\{aB_{\lambda}:\lambda\in\Lambda\\}$ is
complete. Moreover, $a$ is a smooth, non-vanishing weighting factor
independent of $\lambda$. Hence, $\\{aB_{\lambda}:\lambda\in\Lambda\\}$ is
complete if and only if $\\{B_{\lambda}:\lambda\in\Lambda\\}$ is complete. Now
set

$S\coloneqq{\mathrm{span}\,}_{\mathbb{C}}\\{B_{\lambda}:\lambda\in\Lambda\\}.$

We show that $S$ satisfies the assumptions of the Stone-Weierstrass theorem.
Since $B_{\lambda}$ is real-valued, the set $S$ is invariant under complex
conjugation. Further, for $\lambda,\lambda^{\prime}\in\Lambda$ holds

$B_{\lambda}(x)B_{\lambda^{\prime}}(x)=B_{\lambda+\lambda^{\prime}}(x),$

which is an element of $S$ since $\lambda+\lambda^{\prime}\in\Lambda$. It
remains to show that $S$ separates points. To that end, let $x\neq y\in K$ and
$\lambda\in\Lambda$. Then the equality $B_{\lambda}(x)=B_{\lambda}(y)$ holds
if and only if

$\left(({\mathrm{Re}\,}A)(x-y)\right)^{T}\lambda=0.$

Because of the spanning property of $\Lambda$, if this equality holds for all
$\lambda\in\Lambda$, then it holds for all $\lambda\in{\mathbb{R}}^{d}$,
implying that

$({\mathrm{Re}\,}A)(x-y)=0.$

Since ${\mathrm{Re}\,}A$ is invertible, we have $x=y$. In conclusion, the
Stone-Weierstrass theorem [23, Theorem 4.51] implies that the
$\left\lVert\cdot\right\rVert_{\infty}$-closure of $S$ is either equal to
$C(K)$ or equal to $\\{f\in C(K):f(x_{0})=0\\}$ for a unique $x_{0}\in K$. The
later can be excluded since $B_{\lambda}(x_{0})>0$ for all $x_{0}\in K$. ∎

###  6.2. Implications for sampled Gabor phase retrieval

As an immediate consequence, we can retrieve compactly supported functions
$f\in L^{2}(K)$ from the phaseless samples $\left\lvert
V_{g}f(\mathcal{L})\right\rvert$ with respect to a Gaussian window $g$ on sets
$\mathcal{L}\subseteq{\mathbb{R}}^{2d}$ of arbitrary density.

######  Theorem 6.2.

Let $g\in C({{\mathbb{R}}^{d}})$ be a multivariate complex-valued Gaussian

(53) $g(x)=\exp\left(-(x-\nu)^{T}A(x-\nu)\right),$

where $\nu\in{\mathbb{C}}^{d}$ and where $A\in{\mathbb{C}}^{d\times d}$ is a
positive definite Hermitian matrix. Suppose that
$K\subseteq{{\mathbb{R}}^{d}}$ is a compact set. If $\Lambda$ is a semigroup
in ${\mathbb{R}}^{d}$ that contains a spanning set and $\Gamma$ is a
uniqueness set in $PW_{K-K}^{1}({{\mathbb{R}}^{d}})$, then every $f\in
L^{2}(K)$ is determined up to a global phase by $\left\lvert
V_{g}f(\Lambda\times\Gamma)\right\rvert$.

###### Proof.

Let $\omega\in K-K$. Since ${\mathrm{Re}\,}A$ is positive definite, the window
$g$ is square-integrable. Following Theorem 3.2, we have to show that the
system $\mathfrak{T}(g_{\omega},\Lambda)$ is complete in
$(C(K),\left\lVert\,\cdot\,\right\rVert_{\infty})$. The map $g_{\omega}$ is
given by

(54)
$g_{\omega}(x)=C\cdot\exp\left(-2\,(x-({\mathrm{Re}\,}\nu)-\tfrac{\omega}{2})^{T}({\mathrm{Re}\,}A)\,(x-({\mathrm{Re}\,}\nu)-\tfrac{\omega}{2})\right),$

where $C=C(A,\omega,\nu)$ is a constant depending only on $A,\omega$ and
$\nu$. In view of Theorem 6.1, $\mathfrak{T}(g_{\omega},\Lambda)$ is complete
in $C(K)$. The statement follows from that. ∎

Since every lattice is a semigroup that contains a spanning set, we obtain the
following important consequence of Theorem 6.2.

######  Corollary 6.3.

Let $g\in C({{\mathbb{R}}^{d}})$ be a multivariate complex-valued Gaussian

(55) $g(x)=\exp\left(-(x-\nu)^{T}A(x-\nu)\right),$

where $\nu\in{\mathbb{C}}^{d}$ and where $A\in{\mathbb{C}}^{d\times d}$ is a
positive definite Hermitian matrix. Suppose that
$K\subseteq{{\mathbb{R}}^{d}}$ is a compact set. If
$\Lambda,\Gamma\subseteq{{\mathbb{R}}^{d}}$ are lattices such that $K-K$ is
contained in a fundamental domain of the reciprocal lattice $\Gamma^{*}$, then
every $f\in L^{2}(K)$ is determined up to a global phase by $\left\lvert
V_{g}f(\Lambda\times\Gamma)\right\rvert$.

######  Remark 6.4 (Zalik’s theorem).

Let $g(t)=e^{-at^{2}+bt+c},a>0,\,b,c\in{\mathbb{R}}$, be a univariate
Gaussian, let $K\subseteq{\mathbb{R}}$ be a compact interval and let
$\Lambda\subseteq{\mathbb{R}}\setminus\\{0\\}$ be a countable set. A theorem
due to Zalik shows that $\mathfrak{T}(g,\Lambda)$ is complete in $L^{2}(K)$ if
and only if

$\sum_{\lambda\in\Lambda}\frac{1}{|\lambda|}=\infty,$

see [72, Theorem 4]. For every $\omega\in{\mathbb{R}}$ it holds that
$g_{\omega}$ is again a Gaussian of the form $t\mapsto e^{-\alpha t^{2}+\beta
t+\gamma}$ for some $\alpha>0\,\beta,\gamma\in{\mathbb{R}}$. By standard
tensor arguments, one can show that a multivariate Gaussian of the form
$g(t)=e^{-at\cdot t+b\cdot t+c}$, the system $\mathcal{T}(g,\Lambda^{d})$ is
complete in $L^{2}(K)$ for all Cartesian products of intervals $K$.
Consequently, $\mathfrak{T}(g_{\omega},\Lambda^{d})$ is complete in $L^{2}(K)$
for every $\omega\in K-K\subseteq{{\mathbb{R}}^{d}}$. A combination of Zalik’s
theorem with Theorem 3.1 therefore implies a uniqueness statement for
functions in $L^{4}(K)$. We refer to the articles [32, 70] for further
references on the appearance of Zalik’s theorem in STFT phase retrieval.

###  6.3. Uniqueness in dense subspaces

Note that Corollary 6.3 allows uniqueness when sampling the phaseless Gabor
transform of a compactly supported $f\in
L^{2}(K)\subseteq{L^{2}({\mathbb{R}}^{d})}$ on arbitrarily sparse lattices
$\Lambda\times\Gamma$, since
$D(\Lambda\times\Gamma)=\mathrm{vol}(\Lambda\times\Gamma)^{-1}=\mathrm{vol}(\Lambda)^{-1}\mathrm{vol}(\Gamma)^{-1}$,
and we have complete freedom when choosing $\Lambda$. This is an improvement
from band-limited windows or Hermite windows, where both $\Lambda$ and
$\Gamma$ need to be sufficiently dense. However, the sampling locations still
depend on the compact set $K$ through the choice of $\Gamma$. We now propose a
trade-off for this dependency with a minor additional assumption on the
compactly supported function $f\in L^{2}({{\mathbb{R}}^{d}})$. We recall the
following theorem due to Kahane [44].

######  Theorem 6.5 (Kahane).

There exists a sequence
$\mathcal{K}=\\{\kappa_{n}:n\in{\mathbb{Z}}\\}\subseteq{\mathbb{R}}$ such that
the following three properties hold:

  1. i)

$\mathcal{K}$ is symmetric, that is, $\kappa_{n}=\kappa_{-n}$ for every
$n\in{\mathbb{Z}}$,

  2. ii)

$\lim_{n\to\infty}\frac{n}{\kappa_{n}}=0$,

  3. iii)

the system of complex exponentials $\\{e^{2\pi
i\kappa_{n}\cdot}:n\in{\mathbb{Z}}\\}$ is complete in $C(I)$ for every compact
interval $I\subseteq{\mathbb{R}}$.

The proof is constructive and the set $\mathcal{K}$ defined in [44] will from
now on be referred to as _Kahane’s sequence._ Notice that property ii) in
Theorem 6.5 asserts that Kahane’s sequence has density zero, where the density
of a set $S\subseteq{{\mathbb{R}}^{d}}$ is defined via

$D(S)\coloneqq\lim_{r\to\infty}\frac{\\#(S\cap B_{r}(0))}{m^{d}(B_{r}(0))},$

assuming that the limit exists. Here $m^{d}$ denotes the $d$-dimensional
Lebesgue measure. By a tensor argument as with Zalik’s theorem above, the
system of multivariate complex exponentials $\\{e^{2\pi
i\kappa\cdot}:\kappa\in\mathcal{K}^{d}\\}$ is complete in $C(K)$ for every
compact set $K\subseteq{{\mathbb{R}}^{d}}$. To that, the density of the
Cartesian product $\mathcal{K}^{d}$ is zero;

(56) $D(\mathcal{K}^{d})=\lim\limits_{r\to\infty}\frac{\\#(\mathcal{K}^{d}\cap
B_{r}(0))}{m^{d}(B_{r}(0))}\lesssim\lim\limits_{r\to\infty}\prod\limits_{j=1}^{d}\frac{\\#(\mathcal{K}\cap[-r,r])}{2r}=0.$

We are now in a position to construct a sampling set of density zero which
achieves phase retrieval in a dense subspace of ${L^{2}({\mathbb{R}}^{d})}$.

######  Theorem 6.6.

Let $g\in C({{\mathbb{R}}^{d}})$ be a real multivariate Gaussian window
function. Let $\mathcal{K}\subseteq{\mathbb{R}}$ be Kahane’s sequence and
$\Lambda\subseteq{\mathbb{R}}\setminus\\{0\\}$ a sequence of distinct real
numbers such that
$\sum_{\lambda\in\Lambda}\frac{1}{\left\lvert\lambda\right\rvert}=\infty$ and
such that $D(\Lambda)=0$. Further, let

$\mathcal{V}\coloneqq\bigcup_{\kappa>0}L^{4}[-\kappa,\kappa]^{d}.$

Then the following holds:

  1. i)

$\mathcal{V}$ is dense in $L^{2}({{\mathbb{R}}^{d}})$,

  2. ii)

every $f\in\mathcal{V}$ is determined up to a global phase by $\left\lvert
V_{g}f(\Lambda^{d}\times\mathcal{K}^{d})\right\rvert$,

  3. iii)

$D(\Lambda^{d}\times\mathcal{K}^{d})=0$.

In particular, the theorem’s conclusion holds true if $\Lambda$ is the set of
prime numbers.

###### Proof.

It is evident that $\mathcal{V}$ is dense in $L^{2}({{\mathbb{R}}^{d}})$.
Therefore, we continue with the proof of the second statement. To that end,
let $K\subseteq{{\mathbb{R}}^{d}}$ be an arbitrary compact set. Since $g$ is a
Gaussian, so is $g_{\omega}$ for every $\omega\in{{\mathbb{R}}^{d}}$. In view
of Zalik’s theorem (see Remark 6.4), the system
$\mathfrak{T}(g_{\omega},\Lambda^{d})$ is complete in $L^{2}(K)$. In
particular, the operator

(57) $\mathcal{C}_{\omega}:L^{2}(K)\to{\mathbb{C}}^{\Lambda^{d}},\quad
f\mapsto\left(\langle
f,\overline{\,T_{\lambda}g_{\omega}\,}\rangle_{L^{2}(K)}\right)_{\lambda\in\Lambda^{d}}$

is injective for every $\omega\in K-K$. According to Kahane’s theorem, the
system $\left\\{e^{2\pi i\kappa\cdot}:\kappa\in\mathcal{K}^{d}\right\\}$ is
complete in $C(K-K)$. In particular, it is complete in $L^{2}(K-K)$, which in
turn implies that $\mathcal{K}^{d}$ is a uniqueness set for
$PW_{K-K}^{2}({{\mathbb{R}}^{d}})\supseteq PW_{K-K}^{1}({{\mathbb{R}}^{d}})$.
The statement follows from the observation that $\Lambda^{d},\mathcal{K}^{d}$
are independent of $K$ and if $f,h\in\mathcal{V}$, then there exists a compact
set $K\subseteq{{\mathbb{R}}^{d}}$ such that
${\mathrm{supp}}(f),\,{\mathrm{supp}}(h)\subseteq K$ and $f,h\in L^{4}(K)$.
The property that $D(\Lambda^{d}\times\mathcal{K}^{d})=0$ follows from

$\frac{\\#((\Lambda^{d}\times\mathcal{K}^{d})\cap
B_{r}(0))}{m^{2d}(B_{r}(0))}\lesssim\frac{\\#(\Lambda^{d}\cap
B_{r}(0))\\#(\mathcal{K}^{d}\cap B_{r}(0))}{r^{2d}}\to 0$

as $r\to\infty$ (where on the left-hand side and right-hand side $B_{r}(0)$
denotes a ball in ${\mathbb{R}}^{2d}$ and in ${\mathbb{R}}^{d}$,
respectively). ∎

##  7\. Appendix: proof of Proposition 4.2

###### Proof.

We divide the proof into six steps. The main part considers the case $A=I$.
From there, we first handle non-singular matrices and finish with the singular
case. Observe that if $\varphi(z)=e^{-z^{T}Az+z^{T}b+c}$, then the statement
holds true, so it suffices to show that the existence of a factorization of
the form (43) for $\varphi$ implies that $\varphi$ has the form $\eqref{eq2}$.

Step 1. We start by showing that $\varphi$ is zero-free. To that end, assume
the contrary, that is, the function $\varphi$ has a zero at
$z_{0}\in{\mathbb{C}}^{d}$. Then $z_{0}$ is also a zero of $\psi$ or $\sigma$.
Without loss of generality, $\psi(z_{0})=0$. Then for all
$z\in{\mathbb{C}}^{d}$ it holds that

(58)
$\varphi(z)=\varphi(z_{0}+z-z_{0})=\psi(z_{0})\sigma(z-z_{0})e^{-2z_{0}^{T}A(z-z_{0})}=0,$

contradicting the assumption that $\varphi$ is not the zero function.
Therefore, none of the functions $\varphi,\psi,\sigma$ has a zero.
Furthermore, observe that it holds $\varphi=\sigma(0)\psi=\psi(0)\sigma$.
Hence, $\psi$ and $\sigma$ are entire functions. This also implies

(59)
$\varphi(z)\varphi(\lambda)e^{-2z^{T}A\lambda}=\sigma(0)\psi(0)\varphi(z+\lambda)=\varphi(\lambda)\varphi(z)e^{-2\lambda^{T}Az},$

which allows us to conclude that all $\lambda,z\in{\mathbb{C}}^{d}$ satisfy
$1=e^{2z^{T}(A^{T}-A)\lambda}$. Clearly, this can only be true if and only if
$A=A^{T}$, i.e., $A$ is symmetric.

Step 2: $A=\mathrm{Id}$. Assume in the following that $A$ is the identity
matrix. Since

(60)
$e^{-2z^{T}\lambda}\psi(z)\sigma(\lambda)=\varphi(z+\lambda)=\varphi(\lambda+z)=\psi(\lambda)\sigma(z)e^{-2\lambda^{T}z}=\psi(\lambda)\sigma(z)e^{-2z^{T}\lambda},$

we conclude that there exists a constant $\eta\in{\mathbb{C}}\setminus\\{0\\}$
such that $\psi=\eta\sigma$. Therefore,

(61)
$\begin{split}\varphi(z+\lambda)&=\psi(z)\sigma(\lambda)e^{-2z^{T}\lambda}=\tfrac{1}{\eta}\psi(z)\psi(\lambda)e^{-2z^{T}\lambda}\\\
&=\tfrac{1}{\sqrt{\eta}}\psi(z)\tfrac{1}{\sqrt{\eta}}\psi(\lambda)e^{-2z^{T}\lambda}.\end{split}$

Replacing $\psi,\,\sigma$ by $\tfrac{1}{\sqrt{\eta}}\psi$ we can assume
without loss of generality that $\eta=1$. This implies that for each
$z\in{{\mathbb{C}}^{d}}$ we have

(62) $\varphi(0)=\varphi(z-z)=\psi(z)\psi(-z)e^{2z^{T}z}=\Psi(z)\Psi(-z),$

where $\Psi(z)\coloneqq\psi(z)e^{z^{T}z}$.

Step 3: Projection on the coordinate axes. Let $e_{j}$ be the $j$-th unit
vector in ${{\mathbb{C}}^{d}}$. Further, let
$\Psi_{j}(t)\coloneqq\Psi(te_{j})$ be the restriction of $\Psi$ to the $j$-th
coordinate axis. Since $\Psi_{j}$ is zero-free and entire, there exists an
entire function $g_{j}$ such that $\Psi_{j}=e^{g_{j}}$. Equation (43) together
with the previous considerations implies that for all $t,\xi\in{\mathbb{C}}$
holds

(63)
$\varphi((t+\xi)e_{j})=\Psi_{j}(t)\Psi_{j}(\xi)e^{-(t+\xi)^{2}}=e^{g_{j}(t)}e^{g_{j}(\xi)}e^{-(t+\xi)^{2}}.$

On the other hand, it follows from the identity
$\varphi=\psi(0)\sigma=\psi(0)\psi$ and the definition of $\Psi_{j}$ that

(64)
$\varphi((t+\xi)e_{j})=\psi(0)\Psi_{j}(t+\xi)e^{-(t+\xi)^{2}}=\psi(0)e^{g_{j}(t+\xi)}e^{-(t+\xi)^{2}}.$

Now fix $\xi\in{\mathbb{C}}$. Combining Equation (63) and (64) shows that
there exists a constant $s\in{\mathbb{C}}$ such that

(65) $s=g_{j}(t)+g_{j}(\xi)-g_{j}(t+\xi)$

for all $t\in{\mathbb{C}}$. Differentiating with respect to $t$ yields

(66) $0=g_{j}^{\prime}(t)+0-g_{j}^{\prime}(t+\xi).$

Thus, $g_{j}^{\prime}$ is $\xi$-periodic, and since $\xi$ was arbitrary,
$g_{j}^{\prime}$ must be a constant function. Therefore, there exist constants
$b_{j},c_{j}\in{\mathbb{C}}$ satisfying

(67) $g_{j}(t)=b_{j}t+c_{j},\quad t\in{\mathbb{C}}.$

In particular, the map $\Psi_{j}$ is given by

(68) $\Psi_{j}(t)=e^{b_{j}t+c_{j}}.$

Since $\Psi(0)=\Psi_{j}(0)=e^{c_{j}}$, it holds $c_{j}-c_{k}\in 2\pi
i{\mathbb{Z}}$ for all $j,k\in\\{1,\dots,d\\}$. Therefore, we can assume that
$c\coloneqq c_{j}$ for all $j\in\\{1,\dots,d\\}$.

Step 4: Derivation of $\varphi$ for $A=\mathrm{Id}$. Decomposing the vector
$z=(z_{1},\dots,z_{d})^{T}\in{\mathbb{C}}^{d}$ with respect to the standard
unit vectors $e_{1},\dots,e_{d}\in{\mathbb{C}}^{d}$, and using the
factorization derived in Equation (61) with $\eta=1$ shows that

$\varphi(z)=\varphi\left(z_{1}e_{1}+\sum_{j=2}^{d}z_{j}e_{j}\right)=\psi(z_{1}e_{1})\psi\left(\sum_{j=2}^{d}z_{j}e_{j}\right).$

Since $\psi(z_{j}e_{j})=e^{-z_{j}^{2}+b_{j}z_{j}+c}$ and
$\varphi=\psi(0)\psi=e^{c}\psi$,

$\varphi(z)=e^{-z_{1}^{2}+b_{1}z_{1}}\,\varphi\left(\sum_{j=2}^{d}z_{j}e_{j}\right).$

It follows via induction that

(69)
$\begin{split}\varphi(z)&=\psi_{d}(z_{d}e_{d})\,\prod\limits_{j=1}^{d-1}e^{-z_{j}^{2}+b_{j}z_{j}}=e^{c}\prod\limits_{j=1}^{d}e^{-z_{j}^{2}+b_{j}z_{j}}\\\
&=\exp\left(-z^{T}z+z^{T}b+c\right).\end{split}$

Step 5: Non-singular $A$. Now assume $A$ is an arbitrary invertible symmetric
matrix. By Takagi’s factorization theorem, there exists an invertible matrix
$B\in{\mathbb{C}}^{d\times d}$ satisfying $B^{T}B=A$. This transforms the
problem to

(70)
$\varphi(z+\lambda)=\psi(z)\,\sigma(\lambda)\,e^{-2z^{T}B^{T}\,B\lambda}=\psi(B^{-1}\,Bz)\,\sigma(B^{-1}\,B\lambda)\,e^{-2(Bz)^{T}(B\lambda)}.$

If we define the auxiliary functions $\varphi_{B},\,\psi_{B}$, and
$\sigma_{B}$ via

$\varphi_{B}\coloneqq\varphi\circ B^{-1},\quad\psi_{B}\coloneqq\psi\circ
B^{-1},\quad\sigma_{B}\coloneqq\sigma\circ B^{-1},$

then it holds that

(71)
$\varphi_{B}(z+\lambda)=\psi(B^{-1}z)\,\sigma(B^{-1}\lambda)e^{-2z\cdot\lambda}=\psi_{B}(z)\,\sigma_{B}(\lambda)\,e^{-2z^{T}\lambda}.$

By Step 4 above, there is a vector $b^{\prime}\in{\mathbb{C}}^{d}$ and a
constant $c\in{\mathbb{C}}$ such that

(72) $\varphi_{B}(z)=\exp\left(-z^{T}z+z^{T}b^{\prime}+c\right)$

for all $z\in{\mathbb{C}}^{d}$. Therefore, denoting $b=B^{T}b^{\prime}$, for
all $z\in{\mathbb{C}}^{d}$ we have

(73) $\varphi(z)=\varphi_{B}(Bt)=\exp\left(-z^{T}Az+z^{T}b+c\right).$

Step 6: General symmetric matrices. Finally, suppose that $A$ is an arbitrary
symmetric matrix. Let $A=U\Sigma U^{T}$ be the singular value decomposition of
$A$ with $U\in{\mathbb{C}}^{d\times d}$ unitary and
$\Sigma\in{\mathbb{R}}^{d\times d}$ diagonal,
$\Sigma_{i,i}\geq\Sigma_{i+1,i+1}$. Further, let $\Omega$ be the diagonal
matrix defined by

(74) $\Omega_{i,i}\coloneqq\begin{cases}0,&\qquad\Sigma_{i,i}\neq 0,\\\
1,&\qquad\Sigma_{i,i}=0.\end{cases}$

Then the matrix

(75) $C\coloneqq A+U\Omega U^{T}=U\left(\Sigma+\Omega\right)U^{T}$

is invertible and it holds

(76) $\underbrace{\varphi(z+\lambda)\,e^{-(z+\lambda)^{T}U\Omega
U^{T}(z+\lambda)}}_{\coloneqq\varphi_{0}(z+\lambda)}=\underbrace{\psi(z)e^{-z^{T}U\Omega
U^{T}z}}_{\eqqcolon\psi_{0}(z)}\underbrace{\sigma(\lambda)e^{-\lambda^{T}U\Omega
U^{T}\lambda}}_{\eqqcolon\sigma_{0}(\lambda)}e^{-2z^{T}C\lambda}.$

By the previous step, there exist $b\in{\mathbb{C}}^{d}$ and
$c\in{\mathbb{C}}$ such that

$\varphi_{0}(z)=\exp\left(-z^{T}Cz+z^{T}b+c\right).$

By the definition of the map $\varphi_{0}$ and the matrix $C$, we obtain

$\varphi(z)=e^{-z^{T}Az+z^{T}b+c}.$

This completes the proof of the statement. ∎

## Acknowledgements

Irina Shafkulovska was supported by the Austrian Science Fund (FWF) project
P33217 and thanks the Austrian Academy of Sciences for hosting an internship
at the Acoustics Research Institute.

## References

  * [1] Abdelmalek, R., Mnasri, Z., and Benzarti, F. Audio signal reconstruction using phase retrieval: Implementation and evaluation.  Multimed. Tools Appl. 81, 11 (2022), 15919–15946.  doi.org/10.1007/s11042-022-12421-1.
  * [2] Akutowicz, E. J. On the Determination of the Phase of a Fourier Integral, I.  Trans. Amer. Math. Soc. 83, 1 (1956), 179–192.  doi.org/10.2307/1992910.
  * [3] Akutowicz, E. J. On the Determination of the Phase of a Fourier Integral, II.  Proc. Amer. Math. Soc. 8, 2 (1957), 234–238.  doi.org/10.2307/2033718.
  * [4] Alaifari, R., Bartolucci, F., Steinerberger, S., and Wellershoff, M. On the connection between uniqueness from samples and stability in Gabor phase retrieval, 2022.  arxiv.org/abs/2205.07013.
  * [5] Alaifari, R., Daubechies, I., Grohs, P., and Thakur, G. Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames.  J. Fourier Anal. Appl. 23 (2017), 1480–1494.  doi.org/10.1007/s00041-016-9513-7.
  * [6] Alaifari, R., Daubechies, I., Grohs, P., and Yin, R. Stable Phase Retrieval in Infinite Dimensions.  Found. Comput. Math. 19 (2019), 869–900.  doi.org/10.1007/s10208-018-9399-7 .
  * [7] Alaifari, R., and Grohs, P. Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces.  SIAM J. Math. Anal. 49 (2016).  doi.org/10.1137/16M1071481.
  * [8] Alaifari, R., and Wellershoff, M. Phase Retrieval from Sampled Gabor Transform Magnitudes: Counterexamples.  J. Fourier Anal. Appl. 28, 1 (2021), 9.  doi.org/10.1016/10.1007/s00041-021-09901-7.
  * [9] Alaifari, R., and Wellershoff, M. Uniqueness of STFT phase retrieval for bandlimited functions.  Appl. Comput. Harmon. Anal. 50 (2021), 34–48.  doi.org/10.1016/j.acha.2020.08.003.
  * [10] Ascensi, G., and Bruna, J. Model Space Results for the Gabor and Wavelet Transforms.  IEEE Trans. Inform. Theory 55 (2009), 2250 – 2259.  doi.org/10.1109/TIT.2009.2016028.
  * [11] Atzmon, A., and Olevskii, A. Completeness of Integer Translates in Function Spaces on $\mathbb{R}$.  J. Approx. Theory 87, 3 (1996), 291–327.  doi.org/10.1006/jath.1996.0106.
  * [12] Auslander, L., and Tolimieri, R. Radar Ambiguity Functions and Group Theory.  SIAM J. Math. Anal. 16, 3 (1985), 577–601.  doi.org/10.1137/0516043.
  * [13] Berenstein, C. A., and Taylor, B. A. Mean-periodic functions.  Internat. J. Math. Math. Sci. 3 (1980), 199–235.  doi.org/10.1155/S0161171280000154.
  * [14] Bhattacharyya, P. K. Distributions.  De Gruyter, Berlin, Boston, 2012.  doi.org/10.1515/9783110269291.
  * [15] Boche, H., Pohl, V., and Yang, F. Phaseless Signal Recovery in Infinite Dimensional Spaces Using Structured Modulations.  J. Fourier Anal. Appl. 20, 6 (2014), 1212–1233.  doi.org/10.1007/s00041-014-9352-3.
  * [16] Born, M., and Wolf, E. Principles of Optics: 60th Anniversary Edition, 7 ed.  Cambridge University Press, 2020.
  * [17] Bruna, J., Olevskii, A., and Ulanovskii, A. Completeness in $L^{1}(\mathbb{R})$ of discrete translates.  Rev. Mat. Iberoam. 22, 1 (2006), 1 – 16.
  * [18] Cahill, J., Casazza, P. G., and Daubechies, I. Phase retrieval in infinite-dimensional Hilbert spaces.  Trans. Amer. Math. Soc. Ser. B 3 (2016), 63–76.  doi.org/10.1090/btran/12.
  * [19] da Silva, J. C., and Menzel, A. Elementary signals in ptychography.  Opt. Express 23, 26 (2015), 33812–33821.  doi.org/10.1364/OE.23.033812.
  * [20] Delsarte, J. Les fonctions moyenne-périodiques.  J. Math. Pures et Appl. 14 (1935), 403–453.
  * [21] Delsarte, J. Theorie des Fonctions Moyenne-Periodiques de Deux Variables.  Ann. of Math 72 (1960), 121–178.
  * [22] Fannjiang, A., and Strohmer, T. The Numerics of Phase Retrieval.  Acta Numer. 29 (2020), 125–228.  doi.org/10.1017/S0962492920000069.
  * [23] Folland, G. B. Real Analysis: Modern Techniques and Their Applications.  Wiley, 1984.
  * [24] Forstner, A., Krahmer, F., Melnyk, O., and Sissouno, N. Well-conditioned ptychographic imaging via lost subspace completion.  Inverse Probl. 36, 10 (2022).  doi.org/10.1088/1361-6420/abaf3a.
  * [25] Grafakos, L. Classical Fourier Analysis.  Springer New York, 2014.  doi.org/10.1007/978-1-4939-1194-3.
  * [26] Gröchenig, K. The Mystery of Gabor Frames.  J. Fourier Anal. Appl. 20 (2014), 865–895.  doi.org/10.1007/s00041-014-9336-3 .
  * [27] Gröchenig, K., and Koppensteiner, S. Gabor Frames: Characterizations and Coarse Structure.  Springer International Publishing, Cham, 2019, pp. 93–120.  doi.org/10.1007/978-3-030-32353-0_4 .
  * [28] Grohs, P., Koppensteiner, S., and Rathmair, M. Phase Retrieval: Uniqueness and Stability.  SIAM Rev. 62, 2 (2020), 301–350.  doi.org/10.1137/19M1256865.
  * [29] Grohs, P., and Liehr, L. Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality.  to appear in: Constr. Approx. (2021).  arxiv.org/abs/2109.02494.
  * [30] Grohs, P., and Liehr, L. Non-uniqueness theory in sampled STFT phase retrieval.  arXiv (2022).  doi.org/10.48550/arXiv.2207.05628.
  * [31] Grohs, P., and Liehr, L. On Foundational Discretization Barriers in STFT Phase Retrieval.  J. Fourier Anal. Appl. 28, 39 (2022).  doi.org/10.1007/s00041-022-09935-5.
  * [32] Grohs, P., and Liehr, L. Injectivity of Gabor phase retrieval from lattice measurements.  Appl. Comput. Harmon. Anal. 62 (2023), 173–193.  doi.org/10.1016/j.acha.2022.09.001.
  * [33] Grohs, P., Liehr, L., and Rathmair, M. Multi-window STFT phase retrieval: lattice uniqueness, 2022.  doi.org/10.48550/ARXIV.2207.10620 .
  * [34] Grohs, P., and Rathmair, M. Stable Gabor Phase Retrieval and Spectral Clustering.  Comm. Pure Appl. Math. 72, 5 (2019), 981–1043.  doi.org/10.1002/cpa.21799.
  * [35] Grohs, P., and Rathmair, M. Stable Gabor phase retrieval for multivariate functions.  J. Eur. Math. Soc. (2021).  doi.org/10.4171/JEMS/1114.
  * [36] Gröchenig, K. Foundations of Time-Frequency Analysis.  Birkhäuser Basel, 2001.  doi.org/10.1007/978-1-4612-0003-1.
  * [37] Hardy, G. H. On two theorems of F. Carlson and S. Wigert.  Acta Math. 42 (1920), 327 – 339.  doi.org/10.1007/BF02404414.
  * [38] Higgins, J. R. Sampling Theory in Fourier and Signal Analysis: Foundations.  Clarendon Press, 2001.
  * [39] Hofstetter, E. Construction of time-limited functions with specified autocorrelation functions.  IEEE Trans. Inform. Theory 10, 2 (1964), 119–126.  doi.org/10.1109/TIT.1964.1053648.
  * [40] Hörmander, L. The analysis of linear partial differential operators I, 2 ed.  Classics in mathematics. Springer, Berlin, Germany, 2003.  doi.org/10.1007/978-3-642-61497-2 .
  * [41] Jaming, P., Kellay, K., and Perez, R. On the effect of zero-flipping on the stability of the phase retrieval problem in the Paley-Wiener class.  Monatsh. Math. 198, 4 (2022), 757–776.  doi.org/10.1007/s00605-022-01716-y.
  * [42] Jiang, Y., Chen, Z., Han, Y., Deb, P., Gao, H., Xie, S., Purohit, P., Tate, M. W., Park, J., Gruner, S. M., Elser, V., and Muller, D. A. Electron ptychography of 2d materials to deep sub-ångström resolution.  Nature 559, 7714 (2018), 343–349.  doi.org/10.1038/s41586-018-0298-5.
  * [43] Kahane, J. Lectures on On Mean Periodic Functions.  Tata Institute of Fundamental Research, 1959.
  * [44] Kahane, J.-P. Sur la totalité des suites d’exponentielles imaginaires.  Ann. Inst. Fourier 8 (1958), 273–275.
  * [45] Kalker, T. On Multidimensional Sampling.  In Digital Signal Processing Fundamentals (2009).  doi.org/10.1201/9781420046076.
  * [46] Lai, C.-K., Littmann, F., and Weber, E. S. Conjugate Phase Retrieval in Paley–Wiener Space.  J. Fourier Anal. Appl. 27, 6 (2021), 89.  doi.org/10.1007/s00041-021-09891-6.
  * [47] Landau, H. On the completeness of a set of translates.  J. Approx. Theory 5, 4 (1972), 438–440.  doi.org/10.1016/0021-9045(72)90011-1.
  * [48] Lev, N., and Olevskii, A. Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomenon.  Ann. of Math. 174 (2011), 519–541.  doi.org/10.4007/annals.2011.174.1.15.
  * [49] Logvinenko, V. N., and Favorov, S. Y. Cartwright-type theorems and real uniqueness sets for entire functions of exponential type.  Math. Notes 51, 3 (1993), 294–299.  doi.org/10.1007/bf01207716.
  * [50] Lyubarskii, Y. Frames in the Bargmann space of entire functions.  In Entire and Subharmonic Functions. American Mathematical Society, Providence, RI, 1992, p. 167–180.
  * [51] Melnyk, O. On connections between Amplitude Flow and Error Reduction for phase retrieval and ptychography.  Sampl. Theory Signal Process. Data Anal. 20, 2 (2022), 16.  doi.org/10.1007/s43670-022-00035-5.
  * [52] Olevskii, A., and Ulanovskii, A. Almost Integer Translates. Do Nice Generators Exist?  J. Fourier Anal. Appl. 10, 1 (2004), 93–104.  doi.org/10.1007/s00041-004-8006-2.
  * [53] Olevskii, A., and Ulanovskii, A. Discrete Translates in Function Spaces.  Anal. Math. 44, 2 (2018), 251–261.  doi.org/10.1007/s10476-018-0209-x.
  * [54] Olevskii, A., and Ulanovskii, A. Discrete translates in $L^{p}(\mathbb{R}$).  Bull. Lond. Math. Soc. 50 (2018), 561–568.  doi.org/10.1112/blms.12159.
  * [55] Pinkus, A. Density in Approximation Theory.  Surv. Approx. Theory 1 (2005), 1–45.  eudml.org/doc/51470.
  * [56] Ronkin, L. I. The completeness of the system of functions $e^{i\langle\lambda,x\rangle}$ and the real sets of uniqueness of entire functions of several variables.  Funct. Anal. Its Appl. 5 (1971), 343 – 344.  doi.org/10.1007/BF01086755.
  * [57] Ronkin, L. I. On real sets of uniqueness for entire functions of several variables and on the completeness of the system of functions $e^{i\langle\lambda,x\rangle}$.  Sib. Math. J. 13 (1972), 439 – 443.  doi.org/10.1007/BF00968121.
  * [58] Ronkin, L. I. Some questions of completeness and uniqueness for functions of several variables.  Funct. Anal. Appl. 7 (1973), 370.1007/BF00969587–55.  doi.org/10.1007/bf01075648.
  * [59] Rudin, W. Real and Complex Analysis, 3rd edition ed.  McGraw-Hill, 1986.
  * [60] Schwartz, L. Theorie Generale des Fonctions Moyenne-Periodiques.  Ann. of Math. 48, 4 (1947), 857–929.  doi.org/10.2307/1969386.
  * [61] Seip, K. Density theorems for sampling and interpolation in the Bargmann – Fock space I.  J. für die Reine und Angew. Math. 429 (1992), 91–106.  doi.org/10.1515/crll.1992.429.107.
  * [62] Seip, K., and Wallstén, R. Density theorems for sampling and interpolation in the Bargmann – Fock space II.  J. für die Reine und Angew. Math. 429 (1992), 107–114.  doi.org/10.1090/S0273-0979-1992-00290-2.
  * [63] Steinhaus, H. Sur les distances des points dans les ensembles de mesure positive.  Fund. Math. 1, 1 (1920), 93–104.  doi.org/10.4064/fm-1-1-93-104.
  * [64] Stone, M. H. Applications of the theory of Boolean rings to general topology.  Trans. Amer. Math. Soc. 41, 3 (1937), 375–481.  doi.org/10.1090/s0002-9947-1937-1501905-7.
  * [65] Stone, M. H. The Generalized Weierstrass Approximation Theorem.  Math. Mag. 21, 4 (1948), 167.  doi.org/10.2307/3029750 .
  * [66] Thakur, G. Reconstruction of Bandlimited Functions from Unsigned Samples.  J. Fourier Anal. Appl. 17 (2011), 720–732.  doi.org/10.1007/s00041-010-9144-3.
  * [67] Walther, A. The Question of Phase Retrieval in Optics.  Optica Acta Int. J. Opt. 10, 1 (1963), 41–49.  doi.org/10.1080/713817747.
  * [68] Weierstrass, K. Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen.  Sitzungsberichte der Königl. Preußischen Akademie der Wissenschaften zu Berlin (1885), 633–639.
  * [69] Weil, A. L’intégration dans les groupes topologiques et ses applications.  Actualités scientifiques et industrielles. Hermann, 1941.
  * [70] Wellershoff, M. Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions, 2021.  arxiv.org/abs/2112.10136.
  * [71] Wiener, N. Tauberian Theorems.  Ann. of Math. 33, 1 (1932), 1–100.  doi.org/10.2307/1968102.
  * [72] Zalik, R. A. On approximations by shifts and a theorem of Wiener.  Trans. Amer. Math. Soc. 243 (1978).  doi.org/10.1090/S0002-9947-1978-0493077-1.
  * [73] Zhang, F., Chen, B., Morrison, G. R., Vila-Comamala, J., Guizar-Sicairos, M., and Robinson, I. K. Phase retrieval by coherent modulation imaging.  Nat. Commun. 7, 1 (2016), 13367.  doi.org/10.1038/ncomms13367.