#  $L^{2}$-stability analysis for Gabor phase retrieval

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

We consider the problem of reconstructing the missing phase information from
spectrogram data $|\mathcal{G}f|,$ with

$\mathcal{G}f(x,y)=\int_{\mathbb{R}}f(t)e^{-\pi(t-x)^{2}}e^{-2\pi ity}dt,$

the Gabor transform of a signal $f\in L^{2}(\mathbb{R})$. More specifically,
we are interested in domains $\Omega\subseteq\mathbb{R}^{2}$, which allow for
stable local reconstruction, that is

$|\mathcal{G}g|\approx|\mathcal{G}f|\quad\text{in}~{}\Omega\quad\Longrightarrow\quad\exists\tau\in\mathbb{T}:\quad\mathcal{G}g\approx\tau\mathcal{G}f\quad\text{in}~{}\Omega.$

In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and
Spectral Clustering. Comm. Pure Appl. Math. (2019)] and [P. Grohs and M.
Rathmair. Stable Gabor phase retrieval for multivariate functions. J. Eur.
Math. Soc. (2021)] we established a characterization of the stability of this
phase retrieval problem in terms of the connectedness of the observed
measurements. The main downside of the aforementioned results is that the
similarity of two spectrograms is measured w.r.t. a first order weighted
Sobolev norm. In this article we remove this flaw and essentially show that
the Sobolev norm may be replaced by the $L^{2}-$norm. Using this result allows
us to show that it suffices to sample the spectrogram on suitable discrete
sampling sets – a property of crucial importance for practical applications.

##  1\. Introduction

Phase retrieval refers to the general problem of determining a signal from
phase-less measurements. Oftentimes, the operator describing these
measurements is linear and nicely invertible. However, due to the absence of
phase information one has to deal with a nonlinear inverse problem, which is
typically significantly harder to analyze.
Perhaps the most prominent example is the _Fourier phase retrieval_ problem,
where one asks to determine a function $f\in L^{2}(\mathbb{R}^{d})$ (up to a
unimodular factor) from the modulus of its Fourier transform, that is from
$\big{|}\hat{f}\big{|}$. This question is obviously badly ill-posed, as one
can attach any arbitrary phase information to $|\hat{f}|$. Even, if the same
question is posed on the space $L^{2}[-1,1]$ of compactly supported functions
there still remains a vast amount of amibiguity [1, 13, 17]. Possible
strategies to overcome these issues of non-uniqueness include incorporating
a-priori information on the signals under consideration (e.g., $f$ is
compactly supported and non-negative) and/or changing the measurement setup in
order to introduce redundancy (e.g., replace the Fourier transform by the
short-time Fourier transform).
Phase retrieval problems are encountered in a number of highly important
applications in various fields such as astronomy [7], diffraction imaging
[16], audio [9] to mention just a few. For more details we refer to our recent
survey article [10].

###  1.1. Problem formulation and related results

The object which is in the spot light of this article is the short time
Fourier transform with Gaussian window as introduced next.

######  Definition 1.1.

The _Gabor transform_ of $f\in L^{2}(\mathbb{R})$ is defined by

(1.1)
$\mathcal{G}f(x,y):=\left(f(\cdot)e^{-\pi(\cdot-x)^{2}}\right)^{\wedge}(y),\quad
x,y\in\mathbb{R}.$

Moreover, we denote the _spectrogram_ by $\mathcal{S}f:=|\mathcal{G}f|^{2}$.

As it is well known, the Gabor transform essentially produces entire
functions. Its holomorphicity makes $\mathcal{G}f$ a rather alluring object to
study, as one can leverage this strong structural feature and draw upon tools
from complex function theory to analyze it. For example, it follows from the
open mapping theorem that $f\in L^{2}(\mathbb{R})$ is uniquely determined (up
to a unimodular constant) by the phase-less information of its spectrogram
$\mathcal{S}f$ on any open set which can be arbitrarily small. The prior
information that the function to be reconstructed is holomorphic has proved to
be fruitful in a number of phase retrieval problems. To name just a few we
mention [1, 13, 17, 14, 15].

When dealing with real world measurement scenarios stability is absolutely
essential, due to the occurrence of measurement errors, imprecision of machine
arithmetic or imperfect modelling. In the context of phase retrieval,
stability is typically formalized as some sort of Lipschitz estimate. For the
sake of concreteness, in our case

(1.2) $\min_{\tau\in\mathbb{T}}\|g-\tau f\|_{L^{2}(\mathbb{R})}\leq
C\||\mathcal{G}g|-|\mathcal{G}f|\|_{L^{2}(\mathbb{R}^{2})},$

with $C$ an absolute constant independent from $f,g\in L^{2}(\mathbb{R})$
would express uniform stability, as any pair of functions possessing similar
spectrograms must be similar themselves. Unfortunately, the situation is more
complicated than that. By choosing two (or more) components $f_{1},f_{2}$
which have their respective Gabor transforms concentrated on disjoint sets in
the plane, meaning that their product $\mathcal{G}f_{1}\cdot\mathcal{G}f_{2}$
essentially vanishes, one routinely produces instabilities by setting
$f=f_{1}+f_{2}$ and $g=f_{1}-f_{2}$ which have very similar spectrograms while
$f$ and $g$ are not similar at all in the above sense [3]. This phenomenon is
intrinsically manifested in any phase retrieval problem in an infinite
dimensional setting. In particular, there is no uniform stability in infinite
dimensions [4, 2].

To take this into account, one could adjust the formulation and ask for local
stability in the sense that given a domain $\Omega\subseteq\mathbb{R}^{2}$ and
a fixed signal $f\in L^{2}(\mathbb{R})$, what can be said about the optimal
constant satisfying

(1.3)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}\leq
C(f,\Omega)\||\mathcal{G}g|-|\mathcal{G}f|\|_{L^{2}(\Omega)},\quad\forall g\in
L^{2}(\mathbb{R})?$

This notion is local in a twofold meaning. Firstly, local in signal $f$ under
consideration, and secondly local in the domain $\Omega$ where the phase
information is to be recovered.

In previous work [11, 12] we established that in the case of Gabor phase
retrieval there is no other type of instability than the multi-component type
as described above.

######  Theorem 1.2 ([11, 12], informal).

Let $\Omega\subseteq\mathbb{R}^{2}$ be a domain and $f\in L^{2}(\mathbb{R})$.
Then it holds for all $g\in L^{2}(\mathbb{R})$ that

(1.4)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{1}(\Omega)}\lesssim
h(f,\Omega)^{-1}\cdot\||\mathcal{G}g|-|\mathcal{G}f|\|_{\mathcal{W}(\Omega)},$

where $\|\cdot\|_{\mathcal{W}(\Omega)}$ denotes a first order Sobolev norm
with a polynomial weight, and where $h(f,\Omega)$ is the _Cheeger constant_
defined by

(1.5)
$h(f,\Omega):=\inf_{E\subseteq\Omega}\frac{\int\limits_{\Omega\cap\partial
E}|\mathcal{G}f|}{\min\left\\{\int_{E}|\mathcal{G}f|,\int_{\Omega\setminus
E}|\mathcal{G}f|\right\\}}$

The Cheeger constant measures connectedness in the transform domain. Morally,
$h(f,\Omega)$ is small if and only if there exists a partition of
$\Omega=E\cup(\Omega\setminus E)$ such that the overall energy is evenly
distributed between the two components, and furthermore $|\mathcal{G}f|$ is
fairly small on the separating boundary. Whereas, if $f$ is single-component –
which loosely speaking means it is not multi-component – the ratio appearing
in the Cheeger constant cannot become too small, and hence its reciprocal not
too large.
Recently Cheng et al.[6] have considered a rather abstract and general setting
for a phase retrieval problem, under the assumption that the global task can
be broken up into a bunch of uniformly stable sub problems. To such a scheme
is naturally associated a graph whose vertices correspond to the individual
sub problems. The weighting on the edges indicates to what extent the sub
problems are connected to each other. Their main result reveals a connection
between the stability of the original global problem in terms of the
connectivity of the associated graph, i.e. its graph Cheeger constant.

###  1.2. This paper

The work presented in this paper has been driven by the desire to improve upon
Theorem 1.2. While characterizing the stability of the phase reconstruction
process by the connectivity of the measurements is a very satisfactory result
the choice of the norm on the right hand side in (1.4) is unpleasant. In order
to guarantee stable recovery, one would need to be able to record
$|\mathcal{G}f|$ and its derivatives on a continuum $\Omega.$
It is the main objective of this paper to develop new proof techniques in
order to alleviate the norm on the right hand side and therefore obtain
stronger stability results for Gabor phase retrieval. The underlying strategy
is very much inspired by the work of Cheng et al.[6] as our approach as well
hinges on identifying local sub problems whose stability we can understand and
then stitch these together in order to obtain more general stability
statements.

####  1.2.1. Unit square as the base case

Initially, we will lay our focus on studying stability on the arguably
simplest domain one can think of, a square of side length $1$. For this case,
we will establish the following stability result in Section 2.2.

######  Theorem A.

Let $Q\subseteq\mathbb{R}^{2}$ be a square of side length $1$. Then it holds
for all $f,g\in L^{2}(\mathbb{R})$ that

(1.6)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(Q)}\lesssim\left(\frac{\|\mathcal{S}f\|_{L^{\infty}(\mathbb{R}^{2})}+\|\mathcal{S}g\|_{L^{\infty}(\mathbb{R}^{2})}}{\|\mathcal{S}f\|_{L^{1}(Q)}}\right)^{1/2}\cdot\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q)}^{1/2}.$

The aforementioned result implies that, once we demand that the signals we
consider satisfy for example that
$\|\mathcal{S}f\|_{L^{\infty}},\|\mathcal{S}g\|_{L^{\infty}}\leq 1$, we have
stability on any unit square which is located in a position in the time-
frequency plane where $f$ has significant portion of its overall energy. That
the stability potentially behaves badly when moving to squares which feature
very little energy, may be considered acceptable as there is not too much to
reconstruct anyways in that case. In fact, the stability can be arbitrarily
bad if the square is located in an unfavourable position. To convince the
reader that Theorem A is a somewhat sharp result we have a closer look at a
concrete example.

######  Example 1.3.

Let $Q$ denote the square of side length $1$ centered at the origin, with its
sides parallel to the axes and let $a>0$. With $\varphi=e^{-\pi\cdot^{2}}$
denoting the Gaussian, we define

(1.7) $f=f_{a}:=\varphi(\cdot+a)+\varphi(\cdot-a),\quad\text{and}\quad
g=g_{a}:=\varphi(\cdot+a)-\varphi(\cdot-a).$

According to Lemma A.1, the ratio between the distance of the signals and the
distance of the measurements deteriorates rather quickly as
$a\rightarrow\infty$, i.e.,

(1.8)
$\frac{\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(Q)}}{\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q)}^{1/2}}\gtrsim
e^{\pi a}.$

In our stability estimate (1.6) the term $\|\mathcal{S}f\|_{L^{1}(Q)}$
appearing in the denominator compensates for this type of degeneration.
In Lemma A.1 we also compute the respective Gabor transforms of $f$ and $g$,
and find that

(1.9)
$\mathcal{G}f(x,y)=\exp\left\\{-\frac{\pi}{2}a^{2}-\frac{\pi}{2}|z|^{2}-\pi\mathrm{i}xy\right\\}\cos(a\pi\mathrm{i}\bar{z}),\quad
z=x+\mathrm{i}y.$

Based on this factorization we make a couple of observations: As
$a\rightarrow\infty$

  1. i)

the energy $\|\mathcal{S}f\|_{L^{1}(Q)}$ on the square under consideration
decreases (exponentially fast),

  2. ii)

the roots of the anti-holomorphic factor $\cos(a\pi\mathrm{i}\bar{z})$ (and
thus also the roots of $\mathcal{G}f$) accumulate at a density of $\sim
a^{-1}$ on the imaginary axis, while its modulus grows like $e^{a\pi|x|}$; at
the same time, $\mathcal{G}f$ becomes strongly oscillatory in the $y$
direction, and thus sustains a loss of smoothness.

Let us put this into perspective with Theorem A. As discussed before, Theorem
A tells us for $f$ to be an instability the square under consideration must
feature little of the overall energy. In that sense, observation (i) is a
necessary property.
It would be interesting to know if the same can be said about observation
(ii), i.e., for $f$ to be an instability, is it necessary that $\mathcal{G}f$
possesses many roots along a line passing through $Q$?

The proof methods employed to establish Theorem A are fundamentally different
to the techniques used in our prior work [11, 12]. The proof of Theorem A
relies on the idea that the recovery of the missing phase information can be
formulated as an extension problem: If we only know certain values of the
modulus of a holomorphic function $F$, this means that we only have access to
values of the tensor $F\otimes\bar{F}(z,\zeta)=F(z)\overline{F(\zeta)}$ on a
subset of the diagonal $\zeta=z$. Phase information can be obtained by
inferring values of the tensor beyond the diagonal. More precisely, we will
make use of the expansion formula

(1.10)
$F\otimes\bar{F}(z,\zeta)=\sum_{k,l\in\mathbb{N}}\frac{(\bar{\partial}^{l}\partial^{k}|F|^{2})(0)}{k!l!}z^{k}\bar{\zeta}^{l},$

which provides an explicit formula for our reconstruction task
$\mathcal{S}f\rightsquigarrow\mathcal{G}f\sim e^{i\alpha}\mathcal{G}f$.
Clearly, in order to obtain a quantitative result from formula (1.10) we have
to analyze the family of differential operators
$(\bar{\partial}^{l}\partial^{k})_{k,l\in\mathbb{N}}$ acting on the squared
modulus of certain analytic functions. If these functions are sufficiently
smooth locally, we will be able to derive useful bounds for the differential
operators. For entire functions the distribution of zeros – and thus local
smoothness – is strongly connected to the global growth. In fact as we shall
see, if growth can be controlled, which can be done in our case, so can
smoothness.

####  1.2.2. Stability on larger domains

Having established stability on unit squares we will move on to analyzing
stability on more general, larger domains. Before we present our main result
into this direction we introduce the concepts we require.

######  Definition 1.4.

A weighted graph consists of a graph
$\mathfrak{G}=(\mathfrak{V},\mathfrak{E})$ with vertex set $\mathfrak{V}$ and
edge set $\mathfrak{E}$, and a pair of functions
$w:\mathfrak{V}\rightarrow\mathbb{R}_{+}$ and
$\sigma:\mathfrak{E}\rightarrow\mathbb{R}_{+}$. By setting $\sigma(e)=0$ if
$e\in\mathfrak{V}\times\mathfrak{V}\setminus\mathfrak{E}$ we may assume that
$\sigma$ is defined on all of $\mathfrak{V}\times\mathfrak{V}$.

To capture the connectedness of a weighted graph we introduce the graph
version of the Cheeger constant.

######  Definition 1.5.

Let $(\mathfrak{G},w,\sigma)$ be a weighted graph. The graph _Cheeger
constant_ of $\mathfrak{G}$ is defined by

(1.11) $h_{\mathfrak{G}}:=\inf_{S\subseteq\mathfrak{V}}\frac{\sigma(\partial
S)}{\min\left\\{w(S),w(\mathfrak{V}\setminus S)\right\\}},$

where the boundary $\partial S$ denotes the set of all edges which have one
endpoint in $S$ and the other one in the complement $\mathfrak{V}\setminus S$,
and where for a set $V\subseteq\mathfrak{V}$ we write $w(V)$ for $\sum_{v\in
V}w(V)$, and likewise for $E\subseteq\mathfrak{E}$ we write $w(E)$ instead of
$\sum_{e\in\mathfrak{E}}\sigma(e)$.

Our general stability result, which will be established in Section 2.3, is
very much reminiscent of Theorem 1.2.

######  Theorem B (Theorem 2.15, simplified).

Let $f\in L^{2}(\mathbb{R})$ and let $\mathcal{Q}$ be a finite family of unit
squares. We set $\Omega:=\bigcup_{Q\in\mathcal{Q}}Q,$ and define a weighted
graph by identifying the vertex set with $\mathcal{Q}$, and by setting

(1.12)
$w(v):=\|\mathcal{S}f\|_{L^{1}(Q)},v\in\mathfrak{V}\quad\text{and}\quad\sigma(e):=\|\mathcal{S}f\|_{L^{1}(Q\cap
Q^{\prime})}^{2},\,e=(v,v^{\prime})\in\mathfrak{E}.$

Then it holds for all $g\in L^{2}(\mathbb{R})$ that

(1.13)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}\lesssim
h_{\mathfrak{G}}^{-2}\cdot\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}^{1/2},$

with the implicit constant mildly depending on $\mathcal{S}f,\mathcal{S}g$ and
$\mathcal{Q}$.

######  Remark 1.6.

At this stage we would like to stress that Theorem B does not represent an
unrestricted improvement on Theorem 1.2. This may not be instantly apparent
from the simplified version above and require a closer look at Theorem 2.15
and its proof. The main limitation of our new results consists in the
restriction to finite domains. To give a concrete example, our prior results
in [11, 12] produce a finite stability constant for the signal $f$ being the
Gaussian, and the domain being the full time-frequency plane, whereas Theorem
B fails this case.

The proof techniques employed to establish the above result are strongly
inspired by the work of Cheng et al.[6]. Since, the sub problems we consider
are not uniformly stable, we need to make some minor adaptations. However,
once the graph is correctly set up, our results follow from essentially
replicating the proofs in the aforementioned article.

####  1.2.3. Discretization

In Section 3 we aim to take the stability analysis one step further and
consider the case where one can only access the spectrogram data on a discrete
set of sampling points. Into this direction, in Section 3 we will establish a
result of the following flavour.

######  Theorem C (Theorem 3.4, simplified).

Assume the conditions of Theorem B. Moreover, suppose
$\Omega\subseteq\mathbb{R}^{2}$ is a square of side length $s>0$. Then, for
all $\varepsilon>0$, there exists a finite set $\Lambda\subseteq\Omega$ with
$|\Lambda|\sim\max\left\\{\ln(1/\varepsilon),s\right\\}^{2}$, with associated
weights $(w_{\lambda})_{\lambda\in\Lambda}\subseteq\mathbb{R}_{+}$ satisfying
that $\sum_{\lambda}w_{\lambda}=s^{2}$ such that it holds for all $g\in
L^{2}(\mathbb{R})$ that

(1.14)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}\lesssim
h_{\mathfrak{G}}^{-2}\left(\|\mathcal{S}f-\mathcal{S}g\|_{\ell^{2}_{w}(\Lambda_{\varepsilon})}^{1/2}+\varepsilon\right),$

with the implicit constant mildly depending on $\mathcal{S}f,\mathcal{S}g$ and
$\mathcal{Q}$.

The above result will be obtained by combining Theorem B with a tailor-made
error estimate for a carefully picked cubature rule. Note that, if we write
out the norm on the right hand side in (1.13), we get an integral over a real
analytic function. Standard estimates for numerical integration schemes
typically relate the error to the smoothness of the function to be integrated.
We shall pursue a different path here, and show that the error can be
controlled in terms of the size of the holomorphic extension of the integrand,
cf. Theorem 3.2. In our case, this holomorphic extension is quite easy to
handle. This approach saves us from estimating high order derivatives of the
integrand $(\mathcal{S}f-\mathcal{S}g)^{2}$, which would have been a rather
tedious mission.

###  1.3. Notation

Throughout we will identify the complex plane $\mathbb{C}$ with
$\mathbb{R}^{2}$ by virtue of $z\simeq x+\mathrm{i}y$. We use _Wirtinger_
derivatives defined by

(1.15) $\partial=\frac{1}{2}\left(\frac{\partial}{\partial
x}-\mathrm{i}\frac{\partial}{\partial
y}\right)\quad\text{and}\quad\bar{\partial}=\frac{1}{2}\left(\frac{\partial}{\partial
x}+\mathrm{i}\frac{\partial}{\partial y}\right).$

The two dimensional Lebesgue measure will be denoted by $A$. We write
$B_{r}=\\{(x,y):\,\|(x,y)\|_{2}<r\\}$ for the open disk of radius $r>0$
centered at the origin. If the center is different from the origin, say at
$z_{0}$ we write $B_{r}(z_{0})$.

##  2\. Fully continuous measurements

This section is devoted to establishing $L^{2}$-stability estimates for Gabor
phase retrieval. As outlined in the introduction at the beginning we consider
very basic domains. A more general result will be obtained by stitching
together these local results in a suitable manner. First and foremost, we
begin with collecting and recalling a couple of facts which will turn out to
be useful for our further analysis.

###  2.1. Prerequisites

As already mentioned, our setting essentially takes place in a space of
analytic functions. We will make this more concrete now.

######  Definition 2.1.

The _Fock space_ $\mathcal{F}^{p}$, $1\leq p\leq\infty$ consists of all entire
functions $F$, such that

(2.1)
$\|F\|_{\mathcal{F}^{p}}:=\left\|F\exp\left\\{-\frac{\pi}{2}|\cdot|^{2}\right\\}\right\|_{L^{p}(\mathbb{C})}<+\infty.$

The Fock spaces form a nested family, that is, if $p<p^{\prime}$ it holds that
$\mathcal{F}^{p}\subseteq\mathcal{F}^{p^{\prime}}$. The space
$\mathcal{F}^{2}$ will be particularly interesting for our purpose. Equipped
with the canonical inner product

(2.2) $\langle
F,G\rangle_{\mathcal{F}^{2}}=\int_{\mathbb{C}}F(z)\overline{G(z)}e^{-\pi|z|^{2}}\,dA(z).$

$\mathcal{F}^{2}$ is a Hilbert space. In fact, it is a reproducing kernel
Hilbert space and the reproducing kernel is given by $e^{\pi z\bar{w}}$,
meaning that for all $F\in\mathcal{F}^{2}$ we have that

(2.3) $F(z)=\int_{\mathbb{C}}F(w)e^{\pi z\bar{w}}e^{-\pi|w|^{2}}\,dA(w),\quad
z\in\mathbb{C}.$

There is a natural identification between $L^{2}(\mathbb{R})$ and
$\mathcal{F}^{2}$. More precisely, the so-called _Bargmann_ transform defined
by

(2.4) $\mathfrak{B}f(z):=\int_{\mathbb{R}}f(t)\exp\left\\{2\pi tz-\pi
t^{2}-\frac{\pi}{2}z^{2}\right\\}\,dt$

is an isometry from $L^{2}(\mathbb{R})$ to $\mathcal{F}^{2}$, and moreover,
for all $f\in L^{2}(\mathbb{R})$ it holds that

(2.5)
$\mathcal{G}f(x,-y)=\mathfrak{B}f(z)\exp\left\\{\pi\mathrm{i}xy-\frac{\pi}{2}|z|^{2}\right\\},\qquad
z=x+\mathrm{i}y\in\mathbb{C}.$

###  2.2. Basic case

This section is devoted to establishing a first stability result, which
formulated in terms of analytic functions in the Fock space reads as follows.

######  Theorem 2.2.

There exists a universal constant $K>0$ such that for all $r>0$ and
$F,G\in\mathcal{F}^{2}$ it holds that

(2.6) $\min_{\tau\in\mathbb{T}}\left\|G-\tau F\right\|_{L^{2}(B_{r})}\leq
Kr^{2}e^{4\pi^{2}r^{2}}\frac{\|F\|_{\mathcal{F}^{\infty}}+\|G\|_{\mathcal{F}^{\infty}}}{\|F\|_{L^{2}(B_{r})}}\||F|^{2}-|G|^{2}\|_{L^{2}((-1/2,1/2)^{2})}^{1/2}.$

From the above result it follows immediately that for all $f,g\in
L^{2}(\mathbb{R})$ and $r>0$ it holds that

(2.7)
$\min_{\tau\in\mathbb{T}}\left\|\mathcal{G}g-\tau\mathcal{G}f\right\|_{L^{2}(B_{r})}\\\
\lesssim
r^{2}e^{4\pi^{2}r^{2}}\left(\frac{\|\mathcal{S}f\|_{L^{\infty}(\mathbb{C})}+\|\mathcal{S}g\|_{L^{\infty}(\mathbb{C})}}{\|\mathcal{S}f\|_{L^{1}(B_{r})}}\right)^{1/2}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}((-1/2,1/2)^{2})}^{1/2}.$

What if we are given spectrogram data on a unit square which is different from
$(-1/2,1/2)^{2}$ though? That is, what if the sides of the square are not
parallel to the axes, and/or the center is different from the origin? To deal
with this general case we make a simple observation.

######  Remark 2.3.

Note that, the Fock space $\mathcal{F}^{2}$ is invariant w.r.t. rotations,
i.e. with $\theta\in[0,2\pi)$ it holds that
$F(e^{\mathrm{i}\theta}\cdot)\in\mathcal{F}^{2}$ if and only if
$F\in\mathcal{F}^{2}.$ Even though $\mathcal{F}^{2}$ cannot just be simply
replaced by $\mathcal{G}L^{2}(\mathbb{R})$ in this statement, it holds true
that there exists a real-valued function $\mu_{\theta}$ such that for all
$f\in L^{2}(\mathbb{R})$ there exists $f_{\theta}\in L^{2}(\mathbb{R})$
satisfying that

(2.8) $\mathcal{G}f(e^{\mathrm{i}\theta}z)=\mathcal{G}f_{\theta}(z)\cdot
e^{\mathrm{i}\mu_{\theta}(z)}.$

The situation is similar with translations. That is, given $\tau\in\mathbb{C}$
there exists a real-valued function $\nu_{\tau}$ such that for all $f\in
L^{2}(\mathbb{R})$ there exists $f_{\tau}\in L^{2}(\mathbb{R})$ satisfying
that

(2.9) $\mathcal{G}f(z-\tau)=\mathcal{G}f_{\tau}(z)\cdot
e^{\mathrm{i}\nu_{\tau}(z)}.$

Remark 2.3 allows us to replace $(-1/2,1/2)^{2}$ by an arbitrary square of
unit size in (2.7) and we obtain

######  Corollary 2.4.

Let $r>0$, $z_{0}\in\mathbb{C}$, $f,g\in L^{2}(\mathbb{R})$ and let $Q$ be a
square of unit size with center at $z_{0}$. There exists a universal constant
$K>0$ (independent of $r,z_{0},f,g,Q$) such that

(2.10)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(B_{r}(z_{0}))}\\\
\leq
Kr^{2}e^{4\pi^{2}r^{2}}\left(\frac{\|\mathcal{S}f\|_{L^{\infty}(\mathbb{C})}+\|\mathcal{S}g\|_{L^{\infty}(\mathbb{C})}}{\|\mathcal{S}f\|_{L^{1}(B_{r}(z_{0}))}}\right)^{1/2}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q)}^{1/2}.$

The proof of Theorem 2.2 is split up into several sub steps. At the heart of
the proof lies a formula that allows us to recover the tensor
$F\otimes\bar{F}$ from $|F|$. Given two functions $F,G\in L^{2}(B_{r})$ we
introduce the quantity

(2.11)
$\delta_{r}(F,G):=\left\|F\otimes\bar{F}-G\otimes\bar{G}\right\|_{L^{2}(B_{r}\times
B_{r})},$

which will serve as an intermediate stepping stone in our stability analysis.

######  Proposition 2.5.

For all $F\in\mathcal{O}(\mathbb{C})$ it holds that

(2.12)
$F\otimes\bar{F}(z,\zeta)=\sum_{k,l\in\mathbb{N}}\frac{\bar{\partial}^{l}\partial^{k}|F|^{2}(0)}{k!\cdot
l!}z^{k}\bar{\zeta}^{l},\quad z,\zeta\in\mathbb{C}.$

Moreover, for $r>0$ if also $G\in\mathcal{O}(\mathbb{C})$ it holds that

(2.13)
$\delta_{r}(F,G)^{2}=\sum_{k,l\in\mathbb{N}}\omega_{k}\omega_{l}\left|\bar{\partial}^{l}\partial^{k}(|F|^{2}-|G|^{2})(0)\right|^{2},$

where the weights are defined by

(2.14)
$\omega_{k}=\omega_{k}(r):=\frac{\|z^{k}\|_{L^{2}(B_{r})}^{2}}{(k!)^{2}}=\frac{\pi
r^{2k+2}}{k!(k+1)!}.$

###### Proof.

Note that $F$ is holomorphic, and therefore $\bar{F}$ is antiholomorphic.
Thus, we get that

(2.15)
$\bar{\partial}^{l}\partial^{k}|F|^{2}=\bar{\partial}^{l}\partial^{k}(F\cdot\bar{F})=F^{(k)}\cdot\overline{F^{(l)}}.$

Now, Taylor expansion yields the first statement since

(2.16)
$F\otimes\bar{F}(z,\zeta)=F(z)\cdot\overline{F(\zeta)}=\left(\sum_{k\in\mathbb{N}}\frac{F^{(k)}(0)}{k!}z^{k}\right)\cdot\overline{\left(\sum_{l\in\mathbb{N}}\frac{F^{(l)}(0)}{l!}\zeta^{l}\right)}.$

To prove the second statement we set $u:=|F|^{2}-|G|^{2}$ and observe that
$\left(z^{k}\bar{\zeta}^{l}\right)_{k,l}$ forms a system of pairwise
orthogonal vectors in $L^{2}(B_{r}\times B_{r})$. Hence, it follows from the
first part that

(2.17) $\displaystyle\delta_{r}(F,G)^{2}$
$\displaystyle=\sum_{k,l\in\mathbb{N}}\left|\bar{\partial}^{l}\partial^{k}u(0)\right|^{2}\cdot\frac{\left\|z^{k}\bar{\zeta}^{l}\right\|_{L^{2}(B_{r}\times
B_{r})}^{2}}{(k!)^{2}\cdot(l!)^{2}}$
$\displaystyle=\sum_{k,l\in\mathbb{N}}\left|\bar{\partial}^{l}\partial^{k}u(0)\right|^{2}\cdot\frac{\|z^{k}\|_{L^{2}(B_{r})}^{2}}{(k!)^{2}}\cdot\frac{\|\zeta^{l}\|_{L^{2}(B_{r})}^{2}}{(l!)^{2}}$
$\displaystyle=\sum_{k,l\in\mathbb{N}}\left|\bar{\partial}^{l}\partial^{k}u(0)\right|^{2}\omega_{k}\omega_{l}.$

∎

In the next step we relate $\delta_{r}(F,G)$ to a more familiar notion of
similarity.

######  Proposition 2.6.

For all $r>0$ and $F,G\in L^{2}(B_{r})$ it holds that

(2.18) $\min_{\tau\in\mathbb{T}}\left\|G-\tau
F\right\|_{L^{2}(B_{r})}^{2}\lesssim\frac{\delta_{r}(F,G)^{2}}{\|F\|_{L^{2}(B_{r})}^{2}},$

with the minimal implicit constant being not greater than $5$.

###### Proof.

To simplify notation we omit the explicit reference to the $L^{2}(B_{r})$ norm
and innner product, respectively and define three numbers

(2.19) $\alpha:=\|F\|,\quad\beta:=\|G\|,\quad\gamma:=|\langle F,G\rangle|.$

Note that Cauchy-Schwarz inequality tells us that $\gamma\leq\alpha\beta$.
Provided that $\gamma\neq 0$ we can estimate

(2.20) $\min_{\tau\in\mathbb{T}}\left\|G-\tau
F\right\|^{2}\leq\left\|G-\frac{\langle
G,F\rangle}{\gamma}F\right\|^{2}=\beta^{2}+\alpha^{2}-2\gamma=(\alpha-\beta)^{2}+2(\alpha\beta-\gamma).$

Obviously, the inequality remains true if $\gamma=0$. We rewrite

(2.21)
$\delta_{r}(F,G)^{2}=\alpha^{4}+\beta^{4}-2\gamma^{2}=(\alpha+\beta)^{2}(\alpha-\beta)^{2}+2(\alpha\beta+\gamma)(\alpha\beta-\gamma).$

Thus, it suffice to show that

(2.22)
$\alpha^{2}\frac{(\alpha-\beta)^{2}+2(\alpha\beta-\gamma)}{(\alpha+\beta)^{2}(\alpha-\beta)^{2}+2(\alpha\beta+\gamma)(\alpha\beta-\gamma)}\leq
5.$

Since we have that the first part of the left hand side

(2.23)
$\alpha^{2}\frac{(\alpha-\beta)^{2}}{(\alpha+\beta)^{2}(\alpha-\beta)^{2}+2(\alpha\beta+\gamma)(\alpha\beta-\gamma)}\leq\frac{\alpha^{2}}{(\alpha+\beta)^{2}}\leq
1,$

it remains to show that

(2.24)
$\frac{2\alpha^{2}(\alpha\beta-\gamma)}{(\alpha+\beta)^{2}(\alpha-\beta)^{2}+2(\alpha\beta+\gamma)(\alpha\beta-\gamma)}\leq
4.$

To prove (2.24) we distinguish two cases depending on a) $\beta>\alpha/2$ and
b) $\beta\leq\alpha/2$.

In case a) we estimate

(2.25)
$\displaystyle\frac{2\alpha^{2}(\alpha\beta-\gamma)}{(\alpha+\beta)^{2}(\alpha-\beta)^{2}+2(\alpha\beta+\gamma)(\alpha\beta-\gamma)}$
$\displaystyle\leq\frac{2\alpha^{2}(\alpha\beta-\gamma)}{2(\alpha\beta+\gamma)(\alpha\beta-\gamma)}$
$\displaystyle\leq\frac{\alpha^{2}}{\alpha\beta+\gamma}$
$\displaystyle\leq\frac{\alpha^{2}}{\alpha^{2}/2}$ $\displaystyle=2.$

In case b) we have that

(2.26)
$\displaystyle\frac{2\alpha^{2}(\alpha\beta-\gamma)}{(\alpha+\beta)^{2}(\alpha-\beta)^{2}+2(\alpha\beta+\gamma)(\alpha\beta-\gamma)}$
$\displaystyle\leq\frac{\alpha^{4}}{(\alpha+\beta)^{2}(\alpha-\beta)^{2}}$
$\displaystyle\leq\frac{\alpha^{4}}{\alpha^{4}/4}$ $\displaystyle=4,$

which settles the proof. ∎

The upcoming Proposition deals with estimating $\delta_{r}(F,G)$.

######  Proposition 2.7.

For all $F,G\in\mathcal{F}^{\infty}$ and $r>0$ it holds that

(2.27) $\delta_{r}(F,G)^{2}\lesssim
r^{4}e^{8\pi^{2}r^{2}}\cdot\left(\|F\|_{\mathcal{F}^{\infty}}^{2}+\|G\|_{\mathcal{F}^{\infty}}^{2}\right)\cdot\||F|^{2}-|G|^{2}\|_{L^{2}(Q)}.$

In order to prove Proposition 2.7 we require two auxiliary results. The first
one tells us that the derivatives of a small function can only be large if the
function is not too smooth.

######  Lemma 2.8.

Suppose that $u\in C^{\infty}(Q)$, where $Q=[-1/2,1/2]^{2}$. Then it holds for
all $k,l\in\mathbb{N}$ that

(2.28)
$\left|\bar{\partial}^{l}\partial^{k}u(0)\right|^{2}\lesssim\|u\|_{L^{2}(Q)}\\\
\times\left(\|\bar{\partial}^{2l}\partial^{2k}u\|_{L^{2}(Q)}+\|\bar{\partial}^{2l}\partial^{2k+2}u\|_{L^{2}(Q)}+\|\bar{\partial}^{2l+2}\partial^{2k}u\|_{L^{2}(Q)}\right).$

###### Proof.

First note that it follows from the fundamental theorem of calculus that for
every function $v\in C^{\infty}(Q)$ it holds that

(2.29) $|v(0)|^{2}\lesssim\|v\|_{L^{2}(Q)}^{2}+\|\nabla v\|_{L^{2}(Q)}^{2}.$

Plug in $v=\bar{\partial}^{l}\partial^{k}u$ to obtain that

(2.30)
$|\bar{\partial}^{l}\partial^{k}u(0)|^{2}\lesssim\|\bar{\partial}^{l}\partial^{k}u\|_{L^{2}(Q)}^{2}+\|\bar{\partial}^{l}\partial^{k+1}u\|_{L^{2}(Q)}^{2}+\|\bar{\partial}^{l+1}\partial^{k}u\|_{L^{2}(Q)}^{2}.$

We expand $u$ in its Fourier series

(2.31) $u(x,y)=\sum_{m,n\in\mathbb{Z}}\hat{u}_{m,n}e^{2\pi\mathrm{i}(mx+ny)},$

and observe that

(2.32) $\displaystyle\partial e^{2\pi\mathrm{i}(mx+ny)}$
$\displaystyle=\pi\mathrm{i}(m-\mathrm{i}n)e^{2\pi\mathrm{i}(mx+ny)},$
$\displaystyle\bar{\partial}e^{2\pi\mathrm{i}(mx+ny)}$
$\displaystyle=\pi\mathrm{i}(m+\mathrm{i}n)e^{2\pi\mathrm{i}(mx+ny)},$

and furthermore that for $p,q\in\mathbb{N}$

(2.33)
$\bar{\partial}^{q}\partial^{p}u(x,y)=\sum_{m,n\in\mathbb{Z}}\hat{u}_{m,n}(\pi\mathrm{i})^{p+q}(m-\mathrm{i}n)^{p}(m+\mathrm{i}n)^{q}e^{2\pi\mathrm{i}(mx+ny)}.$

Thus, using Cauchy-Schwarz inequality allows us to bound

(2.34)
$\displaystyle\left\|\bar{\partial}^{q}\partial^{p}u\right\|_{L^{2}(Q)}^{2}$
$\displaystyle=\sum_{m,n\in\mathbb{Z}}|\hat{u}_{m,n}|^{2}\pi^{2(p+q)}(m^{2}+n^{2})^{p+q}$
$\displaystyle\leq\left(\sum_{m,n\in\mathbb{Z}}|\hat{u}_{m,n}|^{2}\right)^{1/2}\cdot\left(\sum_{m,n\in\mathbb{Z}}|\hat{u}_{m,n}|^{2}\pi^{4(p+q)}(m^{2}+n^{2})^{2(p+q)}\right)^{1/2}$
$\displaystyle=\|u\|_{L^{2}(Q)}\cdot\|\bar{\partial}^{2q}\partial^{2p}u\|_{L^{2}(Q)}.$

Combining the estimate with (2.30) yields the claim. ∎

Next we relate local smoothness of the functions under consideration to their
global growth.

######  Lemma 2.9.

Let $F\in\mathcal{F}^{2}$ and $p\in\mathbb{N}$. With $Q=[-1/2,1/2]^{2}$ it
holds that

(2.35) $\|F^{(p)}\|_{L^{\infty}(Q)}\leq
2^{p+3}\pi^{p+1}\Gamma\left(\frac{p}{2}+1\right)\cdot\|F\|_{\mathcal{F}^{\infty}}.$

###### Proof.

We make use of the reproducing property of $\mathcal{F}^{2}$ which implies
that

(2.36) $F^{(p)}(z)=\int_{\mathbb{C}}F(w)(\pi\bar{w})^{p}e^{\pi
z\bar{w}}e^{-\pi|w|^{2}}\,dA(w).$

With this we estimate

(2.37)
$\left|F^{(p)}(z)\right|\leq\pi^{p}\cdot\|F\|_{\mathcal{F}^{\infty}}\cdot\int_{\mathbb{C}}|w|^{p}\exp\left\\{-\frac{\pi}{2}|w|^{2}+\pi\Re\\{z\bar{w}\\}\right\\}\,dA(w)$

Since $z\in Q$ implies that $|z|\leq 1/\sqrt{2}$ we can further bound the
integral by

(2.38)
$\int_{\mathbb{C}}|w|^{p}\exp\left\\{-\frac{\pi}{2}|w|^{2}+\frac{\pi}{\sqrt{2}}|w|\right\\}\,dA(w)=2\pi\int\limits_{0}^{\infty}r^{p+1}\exp\left\\{-\frac{\pi}{2}r^{2}+\frac{\pi}{\sqrt{2}}r\right\\}\,dr,$

which – according to Lemma A.2 – is bounded by $2\pi\cdot
2^{p+2}\Gamma\left(\frac{p}{2}+1\right)$. ∎

We are now well prepared for the

###### Proof of Proposition 2.7.

We set $u:=|F|^{2}-|G|^{2}$. We combine Proposition 2.5 and Lemma 2.8 to
obtain that

(2.39) $\delta_{r}(F,G)^{2}\lesssim\|u\|_{L^{2}(Q)}\\\
\times\sum_{k,l\in\mathbb{N}}\omega_{k}\omega_{l}\left(\|\bar{\partial}^{2l}\partial^{2k}u\|_{L^{2}(Q)}+\|\bar{\partial}^{2l}\partial^{2k+2}u\|_{L^{2}(Q)}+\|\bar{\partial}^{2l+2}\partial^{2k}u\|_{L^{2}(Q)}\right).$

It follows from Lemma 2.9 that

(2.40) $\displaystyle\|\bar{\partial}^{2l}\partial^{2k}u\|_{L^{2}(Q)}$
$\displaystyle\lesssim\|\bar{\partial}^{2l}\partial^{2k}|F|^{2}\|_{L^{2}(Q)}+\|\bar{\partial}^{2l}\partial^{2k}|G|^{2}\|_{L^{2}(Q)}$
$\displaystyle\lesssim\|F^{(2k)}\|_{L^{\infty}(Q)}\|F^{(2l)}\|_{L^{\infty}(Q)}+\|G^{(2k)}\|_{L^{\infty}(Q)}\|G^{(2l)}\|_{L^{\infty}(Q)}$
$\displaystyle\lesssim\left(\|F\|_{\mathcal{F}^{\infty}}^{2}+\|G\|_{\mathcal{F}^{\infty}}^{2}\right)\cdot(2\pi)^{2k+2l}\cdot
k!\cdot l!$

Likewise, one shows that

(2.41)
$\|\bar{\partial}^{2l}\partial^{2k+2}u\|_{L^{2}(Q)}\lesssim\left(\|F\|_{\mathcal{F}^{\infty}}^{2}+\|G\|_{\mathcal{F}^{\infty}}^{2}\right)\cdot(2\pi)^{2k+2l}\cdot(k+1)!\cdot
l!,$

as well as

(2.42)
$\|\bar{\partial}^{2l+2}\partial^{2k}u\|_{L^{2}(Q)}\lesssim\left(\|F\|_{\mathcal{F}^{\infty}}^{2}+\|G\|_{\mathcal{F}^{\infty}}^{2}\right)\cdot(2\pi)^{2k+2l}\cdot
k!\cdot(l+1)!.$

Hence, we obtain that

(2.43) $\displaystyle\delta_{r}(F,G)^{2}$
$\displaystyle\lesssim\|u\|_{L^{2}(Q)}\cdot\left(\|F\|_{\mathcal{F}^{\infty}}^{2}+\|G\|_{\mathcal{F}^{\infty}}^{2}\right)\cdot\left(\sum_{k\in\mathbb{N}}\omega_{k}\cdot(2\pi)^{2k}(k+1)!\right)^{2}$
$\displaystyle\lesssim\|u\|_{L^{2}(Q)}\cdot\left(\|F\|_{\mathcal{F}^{\infty}}^{2}+\|G\|_{\mathcal{F}^{\infty}}^{2}\right)\cdot
r^{4}e^{8\pi^{2}r^{2}},$

where, in the last inequality we used that

(2.44) $\displaystyle\sum_{k\in\mathbb{N}}\omega_{k}\cdot(2\pi)^{2k}(k+1)!$
$\displaystyle=\sum_{k\in\mathbb{N}}\frac{\pi
r^{2k+2}}{k!(k+1)!}(2\pi)^{2k}(k+1)!$ $\displaystyle=\pi
r^{2}\sum_{k\in\mathbb{N}}\frac{(2\pi r)^{2k}}{k!}$ $\displaystyle=\pi
r^{2}e^{4\pi^{2}r^{2}}.$

∎

To arrive at the main statement of this section it merely remains to combine
our preparatory results.

###### Proof of Theorem 2.2.

The similarity between $F$ and $G$ on $B_{r}$ can be bounded by

(2.45) $\displaystyle\min_{\tau\in\mathbb{T}}\|G-\tau F\|_{L^{2}(B_{r})}^{2}$
$\displaystyle\overset{\text{Prop.
\ref{prop:reldeltadist}}}{\lesssim}\frac{\delta_{r}(F,G)^{2}}{\|F\|_{L^{2}(B_{r})}^{2}}$
$\displaystyle\overset{\text{Prop.
\ref{prop:bounddelta}}}{\lesssim}\frac{r^{4}e^{8\pi^{2}r^{2}}\left(\|F\|_{\mathcal{F}^{\infty}}^{2}+\|G\|_{\mathcal{F}^{\infty}}^{2}\right)\||F|^{2}-|G|^{2}\|_{L^{2}(Q)}}{\|F\|_{L^{2}(B_{r})}^{2}}.$

Taking roots yields the claim. ∎

###  2.3. General case

The aim of the present section is to modify the techniques developed in [6] so
to make them applicable to our setting.

We begin with with recalling a few graph theoretical concepts. Suppose
$(\mathfrak{G},w,\sigma)$ is a weighted graph. Then, the _adjacency matrix_
$A$ is defined by

(2.46) $A(u,v)=\sigma((u,v)),\quad u,v\in\mathfrak{V}.$

The _degree_ of a vertex $v\in\mathfrak{V}$ is given by
$\operatorname{deg}(v)=\sigma\Big{(}\bigcup_{u\in\mathfrak{V}}\\{(v,u)\\}\Big{)}=\sum_{u\in\mathfrak{V}}A(u,v).$
The _graph Laplacian_ is defined as $L:=D-A$, where $D$ is the diagonal matrix
with $D(v,v)=\operatorname{deg}(v),\,v\in\mathfrak{V}$. With these notions at
hand we can introduce yet another concept measuring connectivity.

######  Definition 2.10.

The _algebraic connectivity_ of a weighted graph $(\mathfrak{G},w,\sigma)$,
where we assume that $w$ is summable, is defined by

(2.47) $\lambda_{\mathfrak{G}}:=\inf_{\mathbf{z}\neq
0,\,\langle\mathbf{z},\mathbf{1}\rangle=0}\frac{\mathbf{z}^{*}L\mathbf{z}}{\|\mathbf{z}\|_{\ell^{2}(\mathfrak{V},w)}^{2}},$

where $L$ is the graph Laplacian, $\mathbf{1}$ is the vector consisting of
ones exclusively, and the inner product $\langle\cdot,\cdot\rangle$ is in
$\ell^{2}(\mathfrak{V},w)$.

Cheeger’s inequality connects algebraic connectivity and Cheeger constant of a
graph. For a proof we refer to [6, Theorem A.1].

######  Theorem 2.11 (Cheeger inequality).

Let $(\mathfrak{G},w,\sigma)$ be a weighted graph. Then it holds that

(2.48)
$2h_{\mathfrak{G}}\geq\lambda_{\mathfrak{G}}\geq\frac{h_{\mathfrak{G}}^{2}}{2\delta_{0}},$

where $\delta_{0}:=\sup_{v\in\mathfrak{V}}\frac{\operatorname{deg}(v)}{w(v)}$.

Next, we set up the graph corresponding to recovering the phase information of
a concrete signal $f$ at hand.

######  Definition 2.12.

Let $f\in L^{2}(\mathbb{R})$ and suppose $\mathcal{Q}$ is a finite collection
of squares of side length $1$. With $(f,\mathcal{Q})$ we associate a finite
graph denoted by
$\mathfrak{G}=\mathfrak{G}_{f,\mathcal{Q}}=(\mathfrak{V},\mathfrak{E})$, where
the vertex set $\mathfrak{V}$ is identified with $\mathcal{Q}$, i.e.
$\mathfrak{V}\simeq\mathcal{Q}$, and which is fully connected, i.e.
$\mathfrak{E}=\mathfrak{V}\times\mathfrak{V}.$ Moreover, to obtain a weighted
graph we assign weights to both the vertices and the edges by defining 333here
we implicitly identify $v\sim Q$ and $v^{\prime}\sim Q^{\prime}$

(2.49)
$w(v):=\|\mathcal{G}f\|_{L^{2}(Q)}^{2}=\|\mathcal{S}f\|_{L^{1}(Q)},\quad
v\in\mathfrak{V},$

as well as

(2.50) $\sigma(e):=\|\mathcal{G}f\|_{L^{2}(Q\cap
Q^{\prime})}^{4}=\|\mathcal{S}f\|_{L^{1}(Q\cap Q^{\prime})}^{2},\quad
e=(v,v^{\prime})\in\mathfrak{E}.$

Now that we have settled the terminology and concepts required, we are able to
formulate the main result of the present section.

######  Theorem 2.13.

Let $f\in L^{2}(\mathbb{R})$ and let $\mathcal{Q}$ be a finite family of
squares of side length $1$, and set $\Omega:=\bigcup_{Q\in\mathcal{Q}}Q$.
There exists a universal constant $C>0$ (independent from $f,\mathcal{Q}$)
such that for all $g\in L^{2}(\mathbb{R})$ it holds that

(2.51)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}\\\
\leq
C\left(KM^{1/2}L^{1/2}+\lambda^{-1}K\nu^{3/2}L^{1/2}+\operatorname{vol}(\Omega)^{1/2}\right)^{1/2}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}^{1/2},$

where $\lambda$ denotes the algebraic connectivity of the graph associated to
$(f,\mathcal{Q})$, $\nu=|\mathcal{Q}|$ and where

$\displaystyle
M=M(\mathcal{S}f,\mathcal{Q}):=\sum_{Q\in\mathcal{Q}}\|\mathcal{S}f\|_{L^{1}(Q)}^{-2},$
$\displaystyle
L=L(\mathcal{Q}):=\|\sum_{Q\in\mathcal{Q}}\mathbbm{1}_{Q}\|_{L^{\infty}(\mathbb{R}^{2})},$
$\displaystyle
K=K(\mathcal{S}f,\mathcal{S}g):=\|\mathcal{S}f\|_{L^{\infty}(\mathbb{R}^{2})}+\|\mathcal{S}g\|_{L^{\infty}(\mathbb{R}^{2})}.$

######  Remark 2.14.

The main reason why we cannot directly apply the results established by Cheng
et al.[6], is the absence of uniform stability of the local phase
reconstruction process in our setting. However, Theorem A allows us to control
the local stability constant by
$\|\mathcal{S}f\|_{L^{1}(Q)}^{-1/2}=\|\mathcal{G}f\|_{L^{2}(Q)}^{-1}$,
essentially. We contain the lack of local uniform stability by defining the
edge weights slightly different as compared to [6]: By using (2.50), instead
of $\sigma(e)=\|\mathcal{G}f\|_{L^{2}(u\cap v)}^{2}$, we compensate for the
dependence of the local stability on the energy of the signal on the
respective square.

By making use of Cheeger inequality we may rewrite Theorem 2.13 in terms of
the Cheeger constant of the associated graph.

######  Theorem 2.15.

Let $f\in L^{2}(\mathbb{R})$ and let $\mathcal{Q}$ be a finite collection of
unit squares. Let $\Omega,\nu,M,L,K$ be defined as in Theorem 2.13. Let
$h=h_{\mathfrak{G}}$ be the Cheeger constant of graph associated with
$(f,\mathcal{Q})$ and
$\delta_{0}:=\max_{v\in\mathfrak{V}}\operatorname{deg}(v)/w(v).$
There exists a universal constant $C>0$ such that for all $g\in
L^{2}(\mathbb{R})$ it holds that

(2.52)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}\\\
\leq
C\left(KM^{1/2}L^{1/2}+h^{-2}\delta_{0}K\nu^{3/2}L^{1/2}+\operatorname{vol}(\Omega)^{1/2}\right)^{1/2}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}^{1/2}.$

To establish Theorem 2.13 we closely follow the proof methods in [6]. We
collect a couple of auxiliary results. The first one allows us to drop the
constraint in the similarity measure.

######  Lemma 2.16.

For all $f,g\in L^{2}(\mathbb{R})$ and measurable $D\subseteq\mathbb{R}^{2}$
we have that

(2.53)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(D)}\leq\sqrt{2}\min_{c\in\mathbb{C}}\|\mathcal{G}g-c\mathcal{G}f\|_{L^{2}(D)}+\||\mathcal{G}g|-|\mathcal{G}f|\|_{L^{2}(D)}.$

###### Proof.

We recycle the proof of [6, Lemma 4.2]. One just needs to replace $\ell^{2}$
norms and inner product with the correct $L^{2}$ norms and inner products,
respectively. ∎

Next, we have a closer look at the overlap of two components. This is where it
becomes apparent, why we have defined the edge weight $\sigma$ as in (2.50).

######  Lemma 2.17.

Let $f,g\in L^{2}(\mathbb{R})$ and let $Q_{1},Q_{2}\subseteq\mathbb{R}^{2}$ be
two squares, each of sidelength $1$. Moreover, for $j\in\\{1,2\\}$ let
$c_{j}\in\mathbb{T}$ be a minimizer of
$\|\mathcal{G}g-c_{j}\mathcal{G}f\|_{L^{2}(Q_{j})}$. Then it holds that

(2.54) $|c_{1}-c_{2}|^{2}\|\mathcal{G}f\|_{L^{2}(Q_{1}\cap
Q_{2})}^{4}\lesssim\left(\|\mathcal{S}f\|_{L^{\infty}(\mathbb{R}^{2})}+\|\mathcal{S}g\|_{L^{\infty}(\mathbb{R}^{2})}\right)\cdot\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q_{1}\cup
Q_{2})}.$

###### Proof.

First rewrite the term to be bounded

(2.55) $|c_{1}-c_{2}|^{2}\cdot\||\mathcal{G}f\|_{L^{2}(Q_{1}\cap
Q_{2})}^{4}=\|(c_{1}-c_{2})\mathcal{G}f\|_{L^{2}(Q_{1}\cap
Q_{2})}^{2}\cdot\|\mathcal{G}f\|_{L^{2}(Q_{1}\cap Q_{2})}^{2}.$

The square root of the first factor can be estimated by making use of the
local stability result, Theorem A.

(2.56) $\displaystyle\|(c_{1}-c_{2})\mathcal{G}f\|_{L^{2}(Q_{1}\cap Q_{2})}$
$\displaystyle\leq\|\mathcal{G}g-c_{1}\mathcal{G}f\|_{L^{2}(Q_{1}\cap
Q_{2})}+\|\mathcal{G}g-c_{2}\mathcal{G}f\|_{L^{2}(Q_{1}\cap Q_{2})}$
$\displaystyle\leq\|\mathcal{G}g-c_{1}\mathcal{G}f\|_{L^{2}(Q_{1})}+\|\mathcal{G}g-c_{2}\mathcal{G}f\|_{L^{2}(Q_{2})}$
$\displaystyle\lesssim\left(\frac{\|\mathcal{S}f\|_{L^{\infty}(\mathbb{R}^{2})}+\|\mathcal{S}g\|_{L^{\infty}(\mathbb{R}^{2})}}{\min\left\\{\|\mathcal{S}f\|_{L^{1}(Q_{1})},\|\mathcal{S}f\|_{L^{1}(Q_{2})}\right\\}}\right)^{1/2}$
$\displaystyle\quad\times\left(\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q_{1})}^{1/2}+\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q_{2})}^{1/2}\right).$

Next, observe that

(2.57) $\|\mathcal{G}f\|_{L^{2}(Q_{1}\cap
Q_{2})}^{2}=\|\mathcal{S}f\|_{L^{1}(Q_{1}\cap
Q_{2})}\leq\min\\{\|\mathcal{S}f\|_{L^{1}(Q_{1})},\|\mathcal{S}f\|_{L^{1}(Q_{2})}\\}.$

Moreover, since for $a,b\geq 0$ it holds that
$(a^{1/2}+b^{1/2})^{2}\asymp(a^{2}+b^{2})^{1/2}$ we have that

(2.58)
$\left(\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q_{1})}^{1/2}+\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q_{2})}^{1/2}\right)^{2}\asymp\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q_{1}\cup
Q_{2})},$

which yields the desired estimate. ∎

We now turn towards the

###### Proof of Theorem 2.13.

Let $\mathbf{z}=(z_{v})_{v\in\mathfrak{V}}$ be such that $z_{v}$ is a
minimizer of
$\zeta\in\mathbb{C}\mapsto\|\mathcal{G}g-\zeta\mathcal{G}f\|_{L^{2}(v)}$.

Let $c_{0}\in C$ be arbitrary but fixed. It follows from Lemma 2.16 that

(2.59)
$\displaystyle\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\lesssim\min_{c\in\mathbb{C}}\|\mathcal{G}g-c\mathcal{G}f\|_{L^{2}(\Omega)}^{2}+\||\mathcal{G}g|-|\mathcal{G}f|\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq\|\mathcal{G}g-c_{0}\mathcal{G}f\|_{L^{2}(\Omega)}^{2}+\||\mathcal{G}g|-|\mathcal{G}f|\|_{L^{2}(\Omega)}^{2}.$

We proceed with estimating the first term on the right hand side and get that

(2.60) $\displaystyle\|\mathcal{G}g-c_{0}\mathcal{G}f\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq\sum_{v\in\mathfrak{V}}\|\mathcal{G}g-c_{0}\mathcal{G}f\|_{L^{2}(v)}^{2}$
$\displaystyle\leq\sum_{v\in\mathfrak{V}}\|\mathcal{G}g-z_{v}\mathcal{G}f+z_{v}\mathcal{G}f-c_{0}\mathcal{G}f\|_{L^{2}(v)}^{2}$
$\displaystyle\lesssim\sum_{v\in\mathfrak{V}}\left(\|\mathcal{G}g-z_{v}\mathcal{G}f\|_{L^{2}(v)}^{2}+|z_{v}-c_{0}|^{2}w(v)\right)$

Thus we have for arbitrary $c_{0}\in\mathbb{C}$ that

(2.61)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}^{2}\\\
\lesssim\sum_{v\in\mathfrak{V}}\|\mathcal{G}g-z_{v}\mathcal{G}f\|_{L^{2}(v)}^{2}+\sum_{v\in\mathfrak{V}}|z_{v}-c_{0}|^{2}w(v)+\||\mathcal{G}g|-|\mathcal{G}f|\|_{L^{2}(\Omega)}^{2}$

We proceed by estimating each of the terms on the right hand side of (2.61),
one after another.

First term: We apply Theorem A on each square and use Cauchy-Schwarz, to get

(2.62)
$\displaystyle\sum_{v\in\mathfrak{V}}\|\mathcal{G}g-z_{v}\mathcal{G}f\|_{L^{2}(v)}^{2}$
$\displaystyle\lesssim
K\sum_{v\in\mathfrak{V}}\|\mathcal{S}f\|_{L^{1}(v)}^{-1}\cdot\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(v)}$
$\displaystyle\leq
K\left(\sum_{v\in\mathfrak{V}}\|\mathcal{S}f\|_{L^{1}(v)}^{-2}\right)^{1/2}\cdot\left(\sum_{v\in\mathfrak{V}}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(v)}^{2}\right)^{1/2}$
$\displaystyle\leq
KM^{1/2}L^{1/2}\cdot\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}.$

Second term: First note that it follows from the Definition of the algebraic
connectivity (2.47) that

(2.63)
$\sum_{v\in\mathfrak{V}}|\mathbf{\zeta}_{v}|^{2}w(v)\leq\lambda^{-1}\cdot\mathbf{\zeta}^{*}L\mathbf{\zeta},\quad\forall\mathbf{\zeta}\in\ell^{2}(\mathfrak{V},w)\setminus\\{0\\}:\langle\mathbf{\zeta},\mathbf{1}\rangle=0.$

In particular by picking
$c_{0}:=\langle\mathbf{z},\mathbf{1}\rangle/\langle\mathbf{1},\mathbf{1}\rangle$,
we can apply the above inequality on
$\mathbf{\zeta}:=\mathbf{z}-c_{0}\mathbf{1}$. We therefore get that

(2.64)
$\sum_{v\in\mathfrak{V}}|z_{v}-c_{0}|^{2}w(v)=\sum_{v\in\mathfrak{V}}|\zeta_{v}|^{2}w(v)\leq\lambda^{-1}\cdot\mathbf{\zeta}^{*}L\mathbf{\zeta}.$

Since the adjacency matrix is symmetric and with the help of Lemma 2.17 we
further get that

(2.65) $\displaystyle\mathbf{\zeta}^{*}L\mathbf{\zeta}$
$\displaystyle=\frac{1}{2}\sum_{u,v\in\mathfrak{V}}|\zeta_{u}-\zeta_{v}|^{2}\sigma((u,v))$
$\displaystyle=\frac{1}{2}\sum_{u,v\in\mathfrak{V}}|z_{u}-z_{v}|^{2}\sigma((u,v))$
$\displaystyle\lesssim
K\sum_{u,v\in\mathfrak{V}}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(u\cup v)}$
$\displaystyle\leq
K\left(\sum_{u,v\in\mathfrak{V}}1\right)^{1/2}\cdot\left(\sum_{u,v\in\mathfrak{V}}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(u\cup
v)}^{2}\right)^{1/2}$
$\displaystyle=K\nu\left(\int\limits_{\Omega}(\mathcal{S}f-\mathcal{S}g)^{2}\left(\sum_{u,v\in\mathfrak{V}}\mathbbm{1}_{u\cup
v}\right)\,dA(z)\right)^{1/2}$ $\displaystyle\lesssim
K\nu^{3/2}L^{1/2}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}$

where we used that

(2.66) $\big{\|}\sum_{u,v\in\mathfrak{V}}\mathbbm{1}_{u\cup
v}\big{\|}_{L^{\infty}(\mathbb{R}^{2})}\leq\big{\|}\sum_{u,v\in\mathfrak{V}}\mathbbm{1}_{u}+\mathbbm{1}_{v}\big{\|}_{L^{\infty}(\mathbb{R}^{2})}\leq
2\nu L.$

Third term: The last term can be bounded as follows.

(2.67) $\displaystyle\||\mathcal{G}f|-|\mathcal{G}g|\|_{L^{2}(\Omega)}^{2}$
$\displaystyle=\int\limits_{\Omega}\left(|\mathcal{G}f|-|\mathcal{G}g|\right)^{2}\cdot\frac{(|\mathcal{G}f|+|\mathcal{G}g|)^{2}}{(|\mathcal{G}f|+|\mathcal{G}g|)^{2}}$
$\displaystyle=\int\limits_{\Omega}\frac{(\mathcal{S}f-\mathcal{S}g)^{2}}{(|\mathcal{G}f|+|\mathcal{G}g|)^{2}}$
$\displaystyle\leq\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}\cdot\left(\int\limits_{\Omega}\frac{(\mathcal{S}f-\mathcal{S}g)^{2}}{(|\mathcal{G}f|+|\mathcal{G}g|)^{4}}\right)^{1/2}$
$\displaystyle\lesssim\operatorname{vol}(\Omega)^{1/2}\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}.$

Collecting the esimates gives the desired result

(2.68)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}^{2}\lesssim\left(KM^{1/2}L^{1/2}+\lambda^{-1}K\nu^{3/2}L^{1/2}+\operatorname{vol}(\Omega)^{1/2}\right)\\\
\times\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}.$

∎

##  3\. Discrete measurements

We consider now the case where we only have access to the spectrogram on a
discrete set of sampling points. More precisely, we aim to replace the term
$\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(\Omega)}$ appearing on the right hand
side of our fully continuous measurement setting from Section 2 by

(3.1)
$\|\mathcal{S}f-\mathcal{S}g\|_{\ell^{2}_{w}(\Lambda)}=\left(\sum_{\lambda\in\Lambda}(\mathcal{S}f(\lambda)-\mathcal{S}g(\lambda))^{2}w_{\lambda}\right)^{1/2}$

for a suitable set of sampling points $\Lambda$ with non-negative weights
$(w_{\lambda})_{\lambda\in\Lambda}.$ Since
$\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}}^{2}$ can be realized as the integral of
a smooth function, utilization of cubature rules appears to be a promising
strategy. Following the pattern from Section 2 we will again work with squares
as basic building blocks. Throughout this section we will denote the square of
side length $2s$ centered at the origin by $Q_{s}$, i.e. $Q_{s}=[-s,s]^{2}$.
Moreover, we will use $R_{a,b}=[-a,a]\times[-b,b]$ for rectangles.

###  3.1. Product Gauss cubature rules for Analytic functions

The integration schemes which we will employ arise from univariate Gauss
rules.

######  Definition 3.1.

For $s>0$ and $N\in\mathbb{N}$ let $(\gamma_{k})_{k=1}^{N}$ denote the nodes
of the $N$ point Gauss quadrature rule on $[-s,s]$, and let
$(\omega_{k})_{k=1}^{N}$ denote the corresponding weights. The _Product Gauss
cubature rule_ of degree $N$ on $Q_{s}$ is then defined as the scheme with
node set

(3.2) $\Lambda=\Lambda^{(N,s)}:=\left\\{(\gamma_{k},\gamma_{l}):\,1\leq
k,l\leq N\right\\}$

with associated weights
$w_{\lambda}=w_{\lambda}^{(N,s)}:=\omega_{k}\omega_{l}$.

Given a continuous function $\phi$ on $Q_{s}$ we will define the integration
error by

(3.3)
$\mathcal{E}(\phi)=\mathcal{E}^{(N,s)}(\phi):=\iint\limits_{Q_{s}}\phi(x,y)\,dxdy-\sum_{\lambda\in\Lambda^{(N,s)}}\phi(\lambda)w_{\lambda}.$

If $Q$ is an arbitrary square of side length $s$ in the plane and $\phi$ a
function defined on $Q$, the product rule can be applied to $\phi$ by pulling
the function back to $Q_{s}$ by virtue of an affine transformation $T$ with
unit determinant. In this case we will denote
$\mathcal{E}_{Q}^{(N)}(\phi):=\mathcal{E}^{(N,s)}(\phi\circ T)$.

Typically the accuracy of a cubature formula depends on the smoothness on the
function to be integrated. We follow the approach by Chawla [5] and consider
functions which have holomorphic extensions.

######  Theorem 3.2.

Let $0<s<a$ and $b>0$. Suppose that $\Phi$ is holomorphic on a neighborhood of
$R_{a,b}\times R_{a,b}\subseteq\mathbb{C}^{2}$ and set

(3.4) $\phi(x,y):=\Phi(x+\mathrm{i}0,y+\mathrm{i}0),\quad(x,y)\in Q_{s}.$

Then it holds for all $N\in\mathbb{N}$ that

(3.5) $\left|\mathcal{E}^{(N,s)}(\phi)\right|\\\
\leq\frac{8s(a+b)}{\pi}\left(\frac{\min\\{a-s,b\\}}{s}\right)^{-N}\left(\frac{2(a+b)}{\min\\{a-s,b\\}}+\frac{1}{2}\log\left(\frac{a+s}{a-s}\right)\right)\|\Phi\|_{L^{\infty}(E_{s,a,b})},$

where $E_{s,a,b}:=(R_{a,b}\times[-s,s])\cup([-s,s]\times
R_{a,b})\subseteq\mathbb{C}^{2}.$

###### Proof.

First we prove the statement for the case $s=1$. The general case will then
follow through a scaling argument.

As a starting point, we pick up formula (26’) from [5] which states that

(3.6)
$\mathcal{E}^{(N,1)}(\phi)=\frac{2}{\pi\mathrm{i}}\int_{\gamma}\frac{Q_{N}(z)}{P_{N}(z)}\Phi(z,\zeta)\,dz+\frac{2}{\pi\mathrm{i}}\int_{\gamma^{\prime}}\frac{Q_{N}(u)}{P_{N}(u)}\Phi(\zeta^{\prime},u)\,du,$

where $\gamma,\gamma^{\prime}\subseteq\mathbb{C}$ are closed curves in the
complex plane enclosing the interval $[-1,1]$, and where
$\zeta,\zeta^{\prime}\in[-1,1]$ are suitably chosen. Moreover, $P_{N}$ denotes
the $N$-th Legendre polynomial, and $Q_{N}$ the $N$-th Legendre function of
the second kind, single-valued and analytic in the complex plane with the
interval $[-1,1]$ deleted.
We pick $\gamma=\gamma^{\prime}=\partial R_{a,b}$, which clearly meets the
above requirement. It follows from (3.6) that

(3.7)
$\left|\mathcal{E}^{(N,1)}(\phi)\right|\leq\frac{8(a+b)}{\pi}\|Q_{N}/P_{N}\|_{L^{\infty}(\gamma)}\|\Phi\|_{L^{\infty}(E_{1,a,b})}.$

In [8, section 3.6.2., formula (22)] we find the identity

(3.8)
$Q_{N}(z)=P_{N}(z)\int\limits_{z}^{\infty}\frac{du}{(u^{2}-1)P_{N}(u)^{2}},$

where the path of integration does not cross the cut $[-1,1]$.
We consider now a fixed point $z_{0}\in\partial R_{a,b}$. Let $\beta$ denote
the component of $\partial
R_{a,b}\setminus(\\{z_{0}\\}\cup\\{a+0\mathrm{i}\\})$ with minimal length.
With that, we get that

(3.9) $\displaystyle\left|Q_{N}(z_{0})/P_{N}(z_{0})\right|$
$\displaystyle=\left|\int_{\beta}\frac{dz}{(z^{2}-1)P_{N}(z)}+\int\limits_{a}^{\infty}\frac{dt}{(t^{2}-1)P_{N}(t)}\right|$
$\displaystyle\leq\left(\inf_{z\in\partial
R_{a,b}\cup(a,+\infty)}|P_{N}(z)|\right)^{-1}$
$\displaystyle\quad\times\left(\operatorname{len}(\beta)\left\|(\cdot^{2}-1)^{-1}\right\|_{L^{\infty}(\partial
R_{a,b})}+\int\limits_{a}^{\infty}(t^{2}-1)^{-1}\,dt\right).$

Since $\operatorname{len}(\beta)\leq 2(a+b)$, and
$\|(\cdot^{2}-1)^{-1}\|_{L^{\infty}(\partial
R_{a,b})}\leq\min\\{a-1,b\\}^{-2}$ and
$\int\limits_{a}^{\infty}(t^{2}-1)^{-1}\,dt=\frac{1}{2}\log\left(\frac{a+1}{a-1}\right)$,
we further have that

(3.10) $\left|Q_{N}(z_{0})/P_{N}(z_{0})\right|\leq\left(\inf_{z\in\partial
R_{a,b}\cup(a,+\infty)}|P_{N}(z)|\right)^{-1}\cdot\left(\frac{2(a+b)}{\min\\{a-1,b\\}^{2}}+\frac{1}{2}\log\left(\frac{a+1}{a-1}\right)\right).$

Since $z_{0}\in\partial R_{a,b}$ was arbitrary, we get with the help of Lemma
A.3 that

(3.11)
$\|Q_{N}/P_{N}\|_{L^{\infty}(\gamma)}\leq\min\\{a-1,b\\}^{-N}\cdot\left(\frac{2(a+b)}{\min\\{a-1,b\\}^{2}}+\frac{1}{2}\log\left(\frac{a+1}{a-1}\right)\right).$

Now, combining this with (3.7) yields the claim for $s=1.$

Let now $s>0$ be arbitrary and suppose that $a>s$ and $b>0$. We set
$a^{\prime}:=a/s$ and $b^{\prime}:=b/s$. Moreover, let
$\Phi^{\prime}:=\Phi(s\cdot)$ and
$\phi^{\prime}:=\Phi^{\prime}(\cdot+0\mathrm{i},\cdot+0\mathrm{i})$. Using the
first part we get that

(3.12) $\displaystyle\left|\mathcal{E}^{(N,s)}(\phi)\right|$
$\displaystyle\overset{\text{c.o.v.}}{=}s^{2}\left|\mathcal{E}^{(N,1)}(\phi^{\prime})\right|$
$\displaystyle\overset{\text{part
1}}{\leq}s^{2}\frac{8(a^{\prime}+b^{\prime})}{\pi}\min\\{a^{\prime}-1,b^{\prime}\\}^{-N}$
$\displaystyle\qquad\times\left(\frac{2(a^{\prime}+b^{\prime})}{\min\\{a^{\prime}-1,b^{\prime}\\}}+\frac{1}{2}\log\left(\frac{a^{\prime}+1}{a^{\prime}-1}\right)\right)\|\Phi^{\prime}\|_{L^{\infty}(E_{1,a^{\prime},b^{\prime}})}$
$\displaystyle=\frac{8s(a+b)}{\pi}\left(\frac{\min\\{a-s,b\\}}{s}\right)^{-N}$
$\displaystyle\qquad\times\left(\frac{2(a+b)}{\min\\{a-s,b\\}}+\frac{1}{2}\log\left(\frac{a+s}{a-s}\right)\right)\|\Phi\|_{L^{\infty}(E_{s,a,b})},$

as desired. ∎

###  3.2. Main results of this section

Next we apply the error estimate, Theorem 3.2 to our integrand
$(\mathcal{S}f-\mathcal{S}g)^{2}$.

######  Proposition 3.3.

Let $Q\subseteq\mathbb{R}^{2}$ be a square of side length $s>0.$ Then it holds
for all $N\in\mathbb{N}$ and $f,g\in L^{2}(\mathbb{R})$ that

(3.13)
$\left|\mathcal{E}^{(N)}_{Q}((\mathcal{S}f-\mathcal{S}g)^{2})\right|\leq
3\left(\sqrt{8\pi}s+2\right)^{N+3}N^{-\frac{N-1}{2}}e^{N/2}(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2}).$

Before we prove Proposition 3.3, we discuss its implications.

######  Theorem 3.4.

Assume the setting and conditions of Theorem 2.15. Moreover, assume that
$\Omega$ is a square of side length $s>0$ and let $\varepsilon\in(0,1/2)$
There exist universal constants $C,C^{\prime}>0$ and a finite set
$\Lambda\subseteq\Omega$ with associated weights
$(w_{\lambda})_{\lambda\in\Lambda}\subseteq\mathbb{R}_{+}$ satisfying
$\sum_{\lambda\in\Lambda}w_{\lambda}=s^{2}$, such that

  1. (i)

for all $g\in L^{2}(\mathbb{R})$ it holds that

(3.14)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}\\\
\leq
C\left(KM^{1/2}L^{1/2}+h^{-2}\delta_{0}K\nu^{3/2}L^{1/2}+\operatorname{vol}(\Omega)^{1/2}\right)^{1/2}\left(\|\mathcal{S}f-\mathcal{S}g\|_{\ell^{2}_{w}(\Lambda)}^{1/2}+\varepsilon\right),$

  2. (ii)

the number of sampling points is bounded according to

(3.15) $|\Lambda|\leq
C^{\prime}\cdot\max\left\\{\ln\frac{1}{\varepsilon},\ln\left(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2}\right),s\right\\}^{2}.$

######  Remark 3.5.

The assumption that the underlying domain $\Omega$ is a square is nothing more
than convenient here. In principle one could also derive a similar result if
$\Omega$ can be written as a union of squares, by applying Proposition 3.3 on
each of these squares.

###### Proof of Theorem 3.4.

Once more, we set $\phi:=(\mathcal{S}f-\mathcal{S}g)^{2}$. Unsurprisingly, the
sampling points and weights employed will be coming from the Gauss product
rule, with its degree $N$ yet to be specified.

Theorem 2.15 tells us that

(3.16)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(\Omega)}\\\
\leq
C\left(KM^{1/2}L^{1/2}+h^{-2}\delta_{0}K\nu^{3/2}L^{1/2}+\operatorname{vol}(\Omega)^{1/2}\right)^{1/2}\|\phi\|_{L^{2}(\Omega)}^{1/2}$

Since we have that

(3.17) $\displaystyle\|\phi\|_{L^{2}(\Omega)}^{1/2}$
$\displaystyle=\left(\|\phi\|_{\ell^{2}_{w}(\Lambda)}+\mathcal{E}^{(N)}_{\Omega}(\phi)\right)^{1/4}$
$\displaystyle\leq\left(\|\phi\|_{\ell^{2}_{w}(\Lambda)}+|\mathcal{E}^{(N)}_{\Omega}(\phi)|\right)^{1/4}$
$\displaystyle\leq\|\phi\|_{\ell^{2}_{w}(\Lambda)}^{1/4}+|\mathcal{E}^{(N)}_{\Omega}(\phi)|^{1/4},$

it suffices to show that
$|\mathcal{E}^{(N)}_{\Omega}(\phi)|\leq\varepsilon^{4}.$ According to
Proposition 3.3 (and after taking logarithms), this can be guaranteed by
demanding that

(3.18) $\ln 3+(N+3)\ln(\sqrt{8\pi}s+2)+\ln\kappa-\frac{(N-1)\ln
N}{2}+\frac{N}{2}\overset{!}{\leq}4\ln\varepsilon,$

where we set
$\kappa:=\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2}.$
Rearranging terms implies that (3.18) is equivalent to

(3.19) $N\ln N-N-\ln
N\overset{!}{\geq}8\ln\frac{1}{\varepsilon}+2\ln\kappa+2(N+3)\ln(\sqrt{8\pi}s+2)+2\ln
3.$

Since, $\ln N<N$, we have that $N\ln N-N-\ln N\geq(N-2)\ln N$. From this, one
realizes that

(3.20) $N\geq
C^{\prime}\max\left\\{\ln\frac{1}{\varepsilon},\ln\kappa,s\right\\},$

with $C^{\prime}$ sufficiently large guarantees the desired accuracy. Since
$|\Lambda|=N^{2}$ we get that indeed (3.15) holds true. ∎

###  3.3. Proof of Proposition 3.3

The idea is to apply the error estimate provided in Theorem 3.2. The proof
consists of two crucial steps:

  1. a)

Identify a function (in fact, such a function is unique if it exists) $\Phi$
which is analytic on $\mathbb{C}^{2}$ and which extends our integrand, i.e.,

(3.21)
$\Phi(x+0\mathrm{i},\xi+0\mathrm{i})=(\mathcal{S}f(x,\xi)-\mathcal{S}g(x,\xi))^{2},\quad(x,\xi)\in
Q.$

and estimate $\|\Phi\|_{L^{\infty}(E_{s,a,b})}$, with $E_{s,a,b}$ defined as
in Theorem 3.2.

  2. b)

Apply (3.5) for carefully chosen $a,b$.

Step a) We will find the following result to be rather useful.

######  Lemma 3.6.

Let $f\in L^{2}(\mathbb{R})$. Then

(3.22)
$\mathcal{T}f(z,\zeta):=\int_{\mathbb{R}}f(t)\exp\left\\{-\pi(t-z)^{2}-2\pi\mathrm{i}\zeta
t\right\\}\,dt,\quad(z,\zeta)\in\mathbb{C}^{2}$

is an entire function and satisfies that

(3.23) $\left|\mathcal{T}f(x+\mathrm{i}y,\xi+\mathrm{i}\eta)\right|\leq
2^{-1/4}e^{\pi(y^{2}+2x\eta+\eta^{2})}.$

###### Proof.

With $z=x+\mathrm{i}y$ and $\zeta=\xi+\mathrm{i}\eta$, Cauchy-Schwarz
inequality gives that

(3.24)
$\displaystyle\int_{\mathbb{R}}\left|f(t)\exp\left\\{-\pi(t-z)^{2}-2\pi\mathrm{i}\zeta
t\right\\}\right|\,dt$
$\displaystyle\leq\|f\|_{L^{2}(\mathbb{R})}\left(\int_{\mathbb{R}}\left|\exp\left\\{-\pi(t-z)^{2}-2\pi\mathrm{i}\zeta
t\right\\}\right|^{2}\,dt\right)^{1/2}$
$\displaystyle=\|f\|_{L^{2}(\mathbb{R})}\left(\int_{\mathbb{R}}\exp\left\\{-2\pi(t-x)^{2}+2\pi
y^{2}+4\pi\eta t\right\\}\,dt\right)^{1/2}$
$\displaystyle=\|f\|_{L^{2}(\mathbb{R})}\cdot
2^{-1/4}\exp\left\\{\pi(\eta^{2}+2x\eta+y^{2})\right\\},$

which implies (3.23). Moreover, with this estimate we can justify that
differentiation and integration can be interchanged when differentiating
$\mathcal{T}f$ w.r.t. $z$ and $\zeta$. Thus, $\mathcal{T}f$ is indeed entire.
∎

From its definition it is evident that $\mathcal{T}f$ extends $\mathcal{G}f$,
meaning that $\mathbb{T}f(x+\mathrm{i}0,\xi+\mathrm{i}0)=\mathcal{G}f(x,\xi)$.
Thus,

(3.25)
$\Phi(z,\zeta):=\left(\mathcal{T}f(z,\zeta)\overline{\mathcal{T}f(\bar{z},\bar{\zeta})}-\mathcal{T}g(z,\zeta)\overline{\mathcal{T}g(\bar{z},\bar{\zeta})}\right)^{2}$

is a function with requested property (3.21). Furthermore, making use of
(3.23) yields

(3.26) $\displaystyle|\Phi(z,\zeta)|$
$\displaystyle\leq\left(\left|\mathcal{T}f(z,\zeta)\mathcal{T}f(\bar{z},\bar{\zeta})\right|+\left|\mathbb{T}g(z,\zeta)\mathbb{T}g(\bar{z},\bar{\zeta})\right|\right)^{2}$
$\displaystyle\leq
2\left(\left|\mathcal{T}f(z,\zeta)\mathcal{T}f(\bar{z},\bar{\zeta})\right|^{2}+\left|\mathcal{T}g(z,\zeta)\mathcal{T}g(\bar{z},\bar{\zeta})\right|^{2}\right)$
$\displaystyle\leq 2\cdot
2^{-1/2}e^{4\pi(y^{2}+\eta^{2})}\left(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2}\right)$

Hence we obtain that

(3.27) $\|\Phi\|_{L^{\infty}(E_{s,a,b})}\leq\sqrt{2}e^{4\pi
b^{2}}(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})^{2}}).$

Step b) We can assume w.l.o.g. that $Q=Q_{s}.$ It follows from Theorem 3.2,
together with (3.27) that

(3.28)
$\left|\mathcal{E}^{(N,s)}((\mathcal{S}f-\mathcal{S}g)^{2})\right|\leq\frac{8s(a+b)}{\pi}\left(\frac{\min\\{a-s,b\\}}{s}\right)^{-N}\\\
\times\left(\frac{2(a+b)}{\min\\{a-s,b\\}}+\frac{1}{2}\log\left(\frac{a+s}{a-s}\right)\right)\cdot\sqrt{2}e^{4\pi
b^{2}}(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2}),$

where $a>s$ and $b>0$ are arbitrary. Depending on $N$, we pick
$b:=\sqrt{\frac{N}{8\pi}}$ and $a:=s+b$. With that, we get that

(3.29)
$\left|\mathcal{E}^{(N,s)}((\mathcal{S}f-\mathcal{S}g)^{2})\right|\leq\frac{\sqrt{2}\cdot
8}{\pi}s(s+2b)\left(\frac{b}{s}\right)^{-N}\left(\frac{2(s+2b)}{b}+\frac{1}{2}\log\left(\frac{2s+b}{b}\right)\right)\\\
\times e^{4\pi
b^{2}}(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2}).$

Using that $\log(1+t)\leq t$ for $t\geq 0$ allows us to further bound the
above error by

(3.30) $\frac{\sqrt{2}\cdot
8}{\pi}\left(\frac{2s(s+2b)^{2}}{b}+\frac{s^{2}(s+2b)}{b}\right)\left(\frac{b}{s}\right)^{-N}e^{4\pi
b^{2}}(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2}).$

Using that $b=\sqrt{\frac{N}{8\pi}}\geq(8\pi)^{-1/2}$ we get that

(3.31)
$\left(\frac{2s(s+2b)^{2}}{b}+\frac{s^{2}(s+2b)}{b}\right)\leq\frac{3s(s+2b)^{2}}{b}=3bs\left(\frac{s}{b}+2\right)^{2}<3b\left(\sqrt{8\pi}s+2\right)^{3}.$

Hence the error can be controlled as follows.

(3.32)
$\displaystyle\left|\mathcal{E}^{(N,s)}((\mathcal{S}f-\mathcal{S}g)^{2})\right|$
$\displaystyle\leq\frac{\sqrt{2}\cdot
24}{\pi\cdot\sqrt{8\pi}}\left(\sqrt{8\pi}s+2\right)^{3}(\sqrt{8\pi}s)^{N}N^{-\frac{N-1}{2}}e^{N/2}(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2})$
$\displaystyle<3\left(\sqrt{8\pi}s+2\right)^{N+3}N^{-\frac{N-1}{2}}e^{N/2}(\|f\|_{L^{2}(\mathbb{R})}^{2}+\|g\|_{L^{2}(\mathbb{R})}^{2}),$

and we arrive at the desired result.

## Acknowledgements

MR gratefully acknowledges the support by the Austrian Science Fund (FWF)
through the START-Project Y963-N35.

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##  Appendix A Auxiliary results

We estimate a couple of quantities involving the Gabor transform of a concrete
pair of functions.

######  Lemma A.1.

Let $\varphi=2^{-1/2}e^{-\pi\cdot^{2}}$. For $a>0$ we define

(A.1) $f=f_{a}:=\varphi(\cdot+a)+\varphi(\cdot-a),\quad\text{and}\quad
g=g_{a}:=\varphi(\cdot+a)-\varphi(\cdot-a).$

Then it holds that

  1. i)

with $z=x+\mathrm{i}y$, the Gabor transforms of the two functions satisfy the
identities

(A.2)
$\displaystyle\mathcal{G}f(x,y)=e^{-\frac{\pi}{2}a^{2}}e^{-\frac{\pi}{2}|z|^{2}-\pi\mathrm{i}xy}\cos(a\pi\mathrm{i}\bar{z}),$
(A.3)
$\displaystyle\mathcal{G}g(x,y)=\mathrm{i}e^{-\frac{\pi}{2}a^{2}}e^{-\frac{\pi}{2}|z|^{2}-\pi\mathrm{i}xy}\sin(a\pi\mathrm{i}\bar{z}).$

  2. ii)

for all $a>0$ we have that

(A.4)
$\frac{\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(Q)}}{\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q)}^{1/2}}\gtrsim
e^{\pi a}.$

###### Proof.

The Gabor transform of a shifted Gaussian can be elementary computed, i.e.,

(A.5)
$(\mathcal{G}\varphi(\cdot-\tau))(x,y)=e^{-\frac{\pi}{2}[(x-\tau)^{2}+y^{2}]}e^{-\pi\mathrm{i}(x+\tau)y}.$

Thus, since $f$ and $g$ only differ by the sign between the Gaussian
components we get, that their respective Gabor transforms are equal to

(A.6) $e^{-\frac{\pi}{2}[(x+a)^{2}+y^{2}]}e^{-\pi\mathrm{i}(x-a)y}\pm
e^{-\frac{\pi}{2}[(x-a)^{2}+y^{2}]}e^{-\pi\mathrm{i}(x+a)y}\\\
=e^{-\frac{\pi}{2}a^{2}}e^{-\frac{\pi}{2}(|z|^{2}-2\pi\mathrm{i}xy)}\left(e^{-\pi
a\bar{z}}\pm e^{\pi a\bar{z}}\right),$

which implies the first claim.

For the second assertion, we first estimate the denominator according to

(A.7) $\displaystyle\|\mathcal{S}f-\mathcal{S}g\|_{L^{2}(Q)}^{2}$
$\displaystyle\asymp e^{-2\pi
a^{2}}\left\||cos(a\pi\mathrm{i}\bar{z})|^{2}-|\sin(a\pi\mathrm{i}\bar{z})|^{2}\right\|_{L^{2}(Q)}^{2}$
$\displaystyle\asymp e^{-2\pi
a^{2}}\left\|\left|e^{a\pi\bar{z}}+e^{-a\pi\bar{z}}\right|^{2}-\left|e^{a\pi\bar{z}}-e^{-a\pi\bar{z}}\right|^{2}\right\|_{L^{2}(Q)}^{2}$
$\displaystyle\asymp e^{-2\pi
a^{2}}\left\|\Re\left\\{e^{a\pi\bar{z}}\overline{e^{-a\pi\bar{z}}}\right\\}\right\|_{L^{2}(Q)}^{2}$
$\displaystyle=e^{-2\pi
a^{2}}\left\|\Re\left\\{e^{-2a\pi\mathrm{i}y}\right\\}\right\|_{L^{2}(Q)}^{2}$
$\displaystyle\leq e^{-2\pi a^{2}}.$

The nominator can be bounded from below by the estimate

(A.8)
$\displaystyle\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(Q)}^{2}$
$\displaystyle\asymp e^{-\pi
a^{2}}\min_{\tau\in\mathbb{T}}\|\sin(a\pi\mathrm{i}\bar{z})-\tau\cos(a\pi\mathrm{i}\bar{z})\|_{L^{2}(Q)}^{2}$
$\displaystyle\gtrsim e^{-\pi
a^{2}}\min_{c\in\mathbb{C}}\|\sin(a\pi\mathrm{i}\bar{z})-c\cos(a\pi\mathrm{i}\bar{z})\|_{L^{2}(B_{1/2})}^{2}.$

With $\phi:=\cos(a\pi\mathrm{i}z)$ and $\psi:=\sin(a\pi\mathrm{i}z)$ we have
that

(A.9)
$\displaystyle\min_{c\in\mathbb{C}}\|\sin(a\pi\mathrm{i}\bar{z})-c\cos(a\pi\mathrm{i}\bar{z})\|_{L^{2}(B_{1/2})}^{2}$
$\displaystyle=\min_{c\in\mathbb{C}}\|\psi-c\phi\|_{L^{2}(B_{1/2})}^{2}$
$\displaystyle=\|\psi\|_{L^{2}(B_{1/2})}^{2}-\frac{|\langle\psi,\phi\rangle_{L^{2}(B_{1/2})}|^{2}}{\|\phi\|_{L^{2}(B_{1/2})}^{2}}$

Now from looking at the Taylor expansions of $\phi$ and $\psi$, and since
complex monomials form a orthonormal system in $L^{2}(B_{1/2})$ we find that
$|\langle\psi,\phi\rangle_{L^{2}(B_{1/2})}=0$. Moreover, again due to the
monomials being pairwise orthogonal, we can estimate the norm of $\psi$ as
follows.

(A.10) $\displaystyle\|\psi\|_{L^{2}(B_{1/2})}^{2}$
$\displaystyle\asymp\|e^{a\pi z}-e^{-a\pi z}\|_{L^{2}(B_{1/2})}^{2}$
$\displaystyle=2\|e^{a\pi
z}\|_{L^{2}(B_{1/2})}^{2}-2\Re\left\\{\int_{B_{1/2}}e^{a\pi
z-a\pi\bar{z}}\,dA(z)\right\\}$ $\displaystyle>2\|e^{a\pi
z}\|_{L^{2}(B_{1/2})}^{2}-\frac{\pi}{2}$

Integration then yields that $\|e^{a\pi z}\|_{L^{2}(B_{1/2})}^{2}=0.405..\cdot
e^{2\pi a}$. Thus we get that

(A.11)
$\min_{\tau\in\mathbb{T}}\|\mathcal{G}g-\tau\mathcal{G}f\|_{L^{2}(Q)}^{2}\gtrsim
e^{-\pi a^{2}+2\pi a}.$

Putting the two estimates together implies the claim. ∎

We require a bound for certain integrals.

######  Lemma A.2.

For all $p\in\mathbb{N}$ it holds that

(A.12)
$\int\limits_{0}^{\infty}r^{p+1}\exp\left\\{-\frac{\pi}{2}r^{2}+\frac{\pi}{\sqrt{2}}r\right\\}\,dr<2^{p+2}\cdot\Gamma\left(\frac{p}{2}+1\right).$

###### Proof.

We rewrite the integral

(A.13)
$\displaystyle\int\limits_{0}^{\infty}r^{p+1}\exp\left\\{-\frac{\pi}{2}r^{2}+\frac{\pi}{\sqrt{2}}r\right\\}\,dr$
$\displaystyle=\int\limits_{0}^{\infty}r^{p+r}\exp\left\\{-\frac{\pi}{2}\left(r-\frac{1}{\sqrt{2}}\right)^{2}+\frac{\pi}{4}\right\\}\,dr$
$\displaystyle=e^{\pi/4}\int\limits_{-1/\sqrt{2}}^{\infty}\left(r+\frac{1}{\sqrt{2}}\right)^{p+1}e^{-\frac{\pi}{2}r^{2}}\,dr$
$\displaystyle=e^{\pi/4}\sum\limits_{l=0}^{p+1}{p+1\choose
l}\left(\frac{1}{\sqrt{2}}\right)^{p+1-l}\int\limits_{-1/\sqrt{2}}^{\infty}r^{l}e^{-\frac{\pi}{2}r^{2}}\,dr$

The integral on the right hand side can be bounded by

(A.14) $\int\limits_{-1/\sqrt{2}}^{\infty}r^{l}e^{-\frac{\pi}{2}r^{2}}\,dr\leq
2\int\limits_{0}^{\infty}r^{l}e^{-\frac{\pi}{2}r^{2}}\,dr=\frac{2\Gamma\left(\frac{l+3}{2}\right)}{(l+1)\cdot\left(\frac{\pi}{2}\right)^{\frac{l+1}{2}}}=\frac{\Gamma\left(\frac{l+1}{2}\right)}{\left(\frac{\pi}{2}\right)^{\frac{l+1}{2}}}$

With this we obtain that

(A.15)
$\displaystyle\int\limits_{0}^{\infty}r^{p+1}\exp\left\\{-\frac{\pi}{2}r^{2}+\frac{\pi}{\sqrt{2}}r\right\\}\,dr$
$\displaystyle\leq e^{\pi/4}\sum\limits_{l=0}^{p+1}{p+1\choose
l}\left(\frac{1}{\sqrt{2}}\right)^{p+1-l}\frac{\Gamma\left(\frac{l+1}{2}\right)}{\left(\frac{\pi}{2}\right)^{\frac{l+1}{2}}}$
$\displaystyle\leq
e^{\pi/4}\Gamma\left(\frac{p}{2}+1\right)\cdot\left(\frac{1}{\sqrt{2}}+\sqrt{\frac{2}{\pi}}\right)^{p+1}\cdot\sqrt{\frac{2}{\pi}}$
$\displaystyle\leq 4\cdot 2^{p}\cdot\Gamma\left(\frac{p}{2}+1\right),$

which yields the desired bound. ∎

We need to control Legendre polynomials from below on the certain sets.

######  Lemma A.3.

Suppose that $a>1$ and $b>0$. Let $P_{N}$ denote the $N$-th degree Legendre
polynomial. Then it holds that

(A.16) $\inf_{z\in\partial
R_{a,b}\cup(a,+\infty)}|P_{N}(z)|\geq\min\\{a-1,b\\}^{N}.$

where $R_{a,b}=[-a,a]\times[-b,b]$.

###### Proof.

First, we observe that $P_{N}$ is a polynomial of degree $N$, which has all
its roots in the interval $(-1,1)$, the roots are symmetrically positioned and
$P_{N}(1)=1.$ Thus, we have that

(A.17)
$P_{N}(z)=z^{\varepsilon}\prod_{j=1}^{\lfloor\frac{N}{2}\rfloor}\frac{z^{2}-t_{j}^{2}}{1-t_{j}^{2}},$

where $t_{j}$ are the positive roots of $P_{N}$, and where
$\varepsilon\in\\{0,1\\}$ depending on whether $N$ is even or odd. Moreover,
since $P_{N}$ has all its roots in $(-1,1)$, we have that $x\mapsto|P_{n}(x)|$
is monotonically increasing on $[a,+\infty)$. Thus it suffices to show that

(A.18) $\inf_{z\in\partial R_{a,b}}|P_{N}(z)|\geq\min\\{a-1,b\\}^{N}.$

For $z\in\partial R_{a,b}$ and arbitrary $t\in(0,1)$ we can estimate that

(A.19)
$\left|\frac{z^{2}-t^{2}}{1-t^{2}}\right|=\frac{|z-t|\cdot|z+t|}{1-t^{2}}\geq\min\\{a-1,b\\}^{2}.$

Since $|z|\geq\min\\{a-1,b\\}$ for $z\in\partial R_{a,b}$ we get (A.18). ∎