# On the frame property of Hermite functions and analytic extensions of their
frame sets

Markus Faulhuber  Faculty of Mathematics, University of Vienna
Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria<EMAIL_ADDRESS>,  Irina Shafkulovska<EMAIL_ADDRESS>and  Ilya Zlotnikov
<EMAIL_ADDRESS>
###### Abstract.

We study Gabor frames with Hermite window functions. Gröchenig and Lyubarskii
provided a sufficient density condition for their frame sets, which leads to
what we call the “safety region”. For rectangular lattices and Hermite windows
of order 4 and higher, we enlarge this safety region by providing new points
on the boundary of this region. For this purpose, we employ the Janssen
representation of the frame operator to compare its distance to the identity
in the operator norm. The calculations lead to estimates on series involving
Laguerre polynomials with Gaussian weight functions.

###### Key words and phrases:

Gabor frame, Hermite function, Laguerre polynomial

###### 1991 Mathematics Subject Classification:

42C15; 33C45

This research was funded in whole or in part by the Austrian Science Fund
(FWF) [10.55776/P33217]. I. Z. is grateful to I. Krasikov for fruitful
discussions.

##  1\. Introduction and notation

###  1.1. Introduction

The study of frame sets has seen breakthroughs and surprises in recent years.
For Hermite functions, there was hope that the structure of their frame sets
may be simple and can be fully understood soon. However, The Mystery of Gabor
frames, after the eponymous article [17] by Gröchenig, is an appropriate
description of our understanding of Gabor systems with Hermite functions. Our
results contribute to this puzzle.

In this work, we are concerned with frame sets of Hermite functions. This
means that we seek to find rectangular lattices
$\Lambda\subset{\mathbb{R}}^{2}$ such that the resulting Gabor system is a
frame. The Hermite function $h_{n}$ of order $n$, scaled to unit norm, is (cf.
[13, Chap. 1.7], [20])

(1.1) $h_{n}(t)=c_{n}(-1)^{n}e^{\pi t^{2}}\frac{d^{n}}{dt^{n}}e^{-2\pi
t^{2}},\,t\in{\mathbb{R}},\qquad\text{so }\qquad\lVert
h_{n}\rVert_{L^{2}({\mathbb{R}})}=1,\,n\in{\mathbb{N}}_{0},$

where $c_{n}=2^{1/4}/\sqrt{n!(2\pi)^{n}2^{n}}$. In particular, the $n$-th
Hermite function is a Gaussian function multiplied with the $n$-th Hermite
polynomial and $n=0$ gives the Gaussian function

(1.2) $h_{0}(t)=2^{1/4}e^{-\pi t^{2}}.$

Gabor systems with Hermite window functions were studied, e.g., by Gröchenig
and Lyubarskii [18], [19]. We recall one of the main results from [18].

######  Theorem (Gröchenig, Lyubarskii).

Consider the Gabor system $\mathcal{G}(h_{n},\Lambda)$. If the density of the
lattice $\Lambda$ exceeds $n+1$, then $\mathcal{G}(h_{n},\Lambda)$ is a Gabor
frame for $L^{2}({\mathbb{R}})$.

Existing results, which we will explain shortly, show in particular the
following:

the Gabor system
$\mathcal{G}_{n}=\mathcal{G}(h_{n},(1/\sqrt{n+1}){\mathbb{Z}}^{2})$ is not a
frame for $n=0,1,2,3$.

It seems natural to expect that this pattern continues, but, again, the frame
set of Hermite functions keeps being mysterious. This is underlined by our
upcoming main result.

######  Theorem 1.1 (Main result).

Denote by $\mathcal{G}_{n}=\mathcal{G}(h_{n},(1/\sqrt{n+1}){\mathbb{Z}}^{2})$
the Gabor system for the $n$-th Hermite function and square lattice of density
$n+1$. Then $\mathcal{G}_{n}$ is a Gabor frame for $L^{2}({\mathbb{R}})$ if
and only if $n\geq 4$.

As Theorem 1.1 shows, the opposite of the expected pattern holds true. So, at
least for square lattices, equality can be achieved in the Theorem of
Gröchenig and Lyubarskii once $n\geq 4$. The case $n=4$ is also the first
Gabor system of the form

(1.3)
$\mathcal{G}_{n}=\mathcal{G}(h_{n},(1/\sqrt{n+1}){\mathbb{Z}}^{2})=\\{\pi(\lambda)h_{n}\mid\lambda\in(1/\sqrt{n+1}){\mathbb{Z}}^{2}\\},$

which has not been treated in the literature so far. We will restrict our
attention to the frame set of $h_{n}$ for rectangular lattices of the form
$\Lambda=a{\mathbb{Z}}\times b{\mathbb{Z}}$.

The case of the Gaussian function, i.e., $n=0$, has been completely settled
(for general point sets) independently by Lyubarskii [28] and in the articles
of Seip [33] and Seip and Wallstén [34]. The case of the first Hermite
function (and more generally odd functions) was treated by Lyubarskii and Nes
[29] for rectangular lattices and extended to general lattices by one of the
authors in [6].

In [17], Gröchenig came up with the following question, which is now called
the frame set conjecture for Hermite functions:
$\bullet$ For $h_{n}$ with $n\in 2{\mathbb{N}}$ is the Gabor system
$\mathcal{G}(h_{n},\Lambda)$ always a frame when ${\rm vol}(\Lambda)<1$?
$\bullet$ For $h_{n}$ with $n\in 2{\mathbb{N}}-1$ is the Gabor system
$\mathcal{G}(h_{n},\Lambda)$ always a frame when ${\rm vol}(\Lambda)<1$ and
${\rm vol}(\Lambda)\neq\frac{N}{N+1}$, $N\in{\mathbb{N}}$?

The questions were originally posed for rectangular lattices and are based on
the restriction found in [29]. Due to the extension given in [6] they may be
formulated for arbitrary lattices. However, the frame set conjecture for
Hermite functions was answered negatively by Lemvig [26], who provided first
counter-examples. More exceptions by Horst, Lemvig, and Videbaek [20]
followed. A sufficiency criterion with numerical investigations was presented
by Ghosh and Selvan [14].

This work is structured as follows. In Sec. 1.2 we introduce the general
notation used in this work. In Sec. 2, we recall essential results from Gabor
analysis in Sec. 2.1, provide necessary details on Laguerre polynomials in
Sec. 2.2 and introduce auxiliary results and outline the rough proof strategy
in Sec. 2.3. Sec. 3 is devoted to new results on the frame set of Hermite
functions. Our main result, Theorem 1.1, is proved in Sec. 3.1. The
limitations of our approach are discussed in Sec. 3.2. Finally, in Sec.3.3, we
present more applications of the approach we used to prove Theorem 1.1. We
enlarge what we call the safety region for $h_{15}$ and provide a new
neighborhood in the frame set of $h_{9}$.

###  1.2. Notation

We mainly follow the notation from the textbook of Gröchenig [16]. In this
work, as usual in time-frequency analysis, we use the following Fourier
transform of a suitable function on ${\mathbb{R}}$:

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

This extends to a unitary operator on $L^{2}({\mathbb{R}})$, hence we have
Plancherel’s theorem and Parseval’s identity, respectively, given by

(1.5) $\lVert
f\rVert_{L^{2}({\mathbb{R}})}=\lVert\mathcal{F}f\rVert_{L^{2}({\mathbb{R}})}\quad\text{
and }\quad\langle f,g\rangle=\langle\mathcal{F}f,\mathcal{F}g\rangle.$

The inner product and norm on $L^{2}({\mathbb{R}})$ are, respectively, given
by

(1.6) $\langle f,g\rangle=\int_{\mathbb{R}}f(t)\overline{g(t)}\,dt\quad\text{
and }\quad\lVert f\rVert_{L^{2}({\mathbb{R}})}^{2}=\langle f,f\rangle.$

Sometimes, we will pass an index to the inner product, denoting the specific
Hilbert space, which will be either $L^{2}({\mathbb{R}})$ or
$L^{2}({\mathbb{R}}^{2})$. Note that the Hermite functions are eigenfunctions
of the Fourier transform, in particular we have

(1.7) $\mathcal{F}h_{n}=(-i)^{n}h_{n}.$

A Gabor system is a set of functions of the form

(1.8) $\mathcal{G}(g,\Lambda)=\\{\pi(\lambda)g\mid\lambda\in\Lambda\\},$

where $g\in L^{2}({\mathbb{R}})$ (of course, not the zero function throughout
this work), $\Lambda\subset{\mathbb{R}}^{2}$ discrete and $\pi(z)$ is a time-
frequency shift by $z$, which is a unitary operator, acting by the rule

(1.9) $\pi(z)g=M_{\omega}T_{x}g,\quad z=(x,\omega)\in{\mathbb{R}}^{2}.$

Here, $T_{x}$ and $M_{\omega}$ denote the usual operators of translation
(time-shift) and modulation (frequency-shift), which act on functions by the
rules

(1.10) $T_{x}g(t)=g(t-x)\qquad\text{ and }\qquad M_{\omega}g(t)=e^{2\pi
i\omega t}g(t),$

with the commutation relation

(1.11) $M_{\omega}T_{x}=e^{2\pi i\omega x}T_{x}M_{\omega}.$

Often $\Lambda$ is assumed to be a lattice in the time-frequency plane, i.e.,
a discrete co-compact subgroup of ${\mathbb{R}}^{2}$. We denote the co-volume
of $\Lambda$ simply by ${\rm vol}(\Lambda)$ in this work and will actually
only deal with the case of rectangular lattices, i.e.,
$\Lambda=\Lambda_{a,b}=a{\mathbb{Z}}\times b{\mathbb{Z}}$, with
$a,b\in{\mathbb{R}}_{+}$. We will also identify the lattice $\Lambda_{a,b}$
with the pair of parameters $(a,b)\in{\mathbb{R}}_{+}\times{\mathbb{R}}_{+}$.
Note that for a rectangular lattice we have

(1.12) ${\rm vol}(\Lambda_{a,b})=ab\quad\text{ and its density is
}\quad\delta={\rm vol}(\Lambda)^{-1}=1/(ab).$

For a general lattice $\Lambda=M{\mathbb{Z}}^{2}$, $M\in GL(2,{\mathbb{R}})$,
we have ${\rm vol}(\Lambda)=|\det(M)|$. For the lattice $\Lambda$ we denote by
$\Lambda^{\circ}$ its adjoint lattice, which is $J\Lambda^{\perp}$, where
$\Lambda^{\perp}$ is the classical dual lattice and $J$ the standard
symplectic matrix;

(1.13) $J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$

Note that while the classical dual lattice is defined by the condition

(1.14)
$\Lambda^{\perp}=\\{\lambda^{\perp}\mid\lambda^{\perp}\cdot\lambda\in{\mathbb{Z}}\
\forall\lambda\in\Lambda\\},$

the adjoint lattice may be defined by the commutation relations of time-
frequency shifts

(1.15)
$\Lambda^{\circ}=\\{\lambda^{\circ}\mid\pi(\lambda)\pi(\lambda^{\circ})=\pi(\lambda^{\circ})\pi(\lambda)\
\forall\lambda\in\Lambda\\}.$

This is, by (1.11), equivalent to saying that
$\lambda^{\circ}\in\Lambda^{\circ}$ if and only if
$\sigma(\lambda^{\circ},\lambda)\in{\mathbb{Z}}$, for all $\lambda\in\Lambda$,
where $\sigma(\lambda^{\circ},\lambda)=\lambda^{\circ}\cdot J\lambda$ denotes
the standard symplectic form. Note that for a rectangular lattice
$\Lambda_{a,b}=a{\mathbb{Z}}\times b{\mathbb{Z}}$ the adjoint lattice is given
by $\Lambda_{a,b}^{\circ}=(1/a){\mathbb{Z}}\times(1/b){\mathbb{Z}}$.

The adjoint (or symplectic dual) lattice makes the usage of a symplectic
Poisson summation formula possible. For a (suitable) function $F$ on
${\mathbb{R}}^{2}$, this is

(1.16) $\sum_{\lambda\in\Lambda}F(\lambda+z)={\rm
vol}(\Lambda)^{-1}\sum_{\lambda^{\circ}\in\Lambda^{\circ}}\mathcal{F}_{\sigma}F(\lambda^{\circ})e^{2\pi
i\sigma(z,\lambda^{\circ})},$

where $\mathcal{F}_{\sigma}$ is the symplectic Fourier transform [4]

(1.17) $\mathcal{F}_{\sigma}F(z)=\iint_{{\mathbb{R}}^{2}}F(w)e^{-2\pi
i\sigma(w,z)}\,dw.$

The symplectic Poisson summation formula (1.16) is implicitly present
throughout the whole article. We mainly apply it to the short-time Fourier
Transform (STFT) of Hermite functions. The STFT of a function $f\in
L^{2}({\mathbb{R}})$ with window $g\in L^{2}({\mathbb{R}})$ is given by

(1.18) $V_{g}f(x,w)=\int\limits_{{\mathbb{R}}}f(t)\overline{g(t-x)}e^{-2\pi
iwt}dt=\langle f,\pi(z)g\rangle,\quad z=(x,\omega)\in{\mathbb{R}}^{2}.$

A Gabor system $\mathcal{G}(g,\Lambda)$ is a frame for $L^{2}({\mathbb{R}})$
if and only if there exist positive constants $0<A\leq B<\infty$, such that
the frame inequality holds:

(1.19) $A\lVert
f\rVert_{L^{2}({\mathbb{R}})}^{2}\leq\sum_{\lambda\in\Lambda}|\langle
f,\pi(\lambda)g\rangle|^{2}\leq B\lVert
f\rVert_{L^{2}({\mathbb{R}})}^{2},\quad\forall f\in L^{2}({\mathbb{R}}).$

The constants $A$ and $B$ are called lower and upper frame bound,
respectively, and the sharpest possible bounds are the spectral bounds of the
associated Gabor frame operator $S_{g,\Lambda}$, which acts on $f\in
L^{2}({\mathbb{R}})$ by

(1.20) $S_{g,\Lambda}f=\sum_{\lambda\in\Lambda}\langle
f,\pi(\lambda)g\rangle\pi(\lambda)g.$

The set of all (rectangular) lattices such that the Gabor system
$\mathcal{G}(g,\Lambda)$ is a Gabor frame for $L^{2}({\mathbb{R}})$, is called
the frame set $\mathfrak{F}$ of $g$:

(1.21)
$\mathfrak{F}(g)=\\{\Lambda\subset{\mathbb{R}}^{2}\mid\mathcal{G}(g,\Lambda)\text{
is a frame}\\}.$

This is also the terminology used by Gröchenig in [17]. An important result
for functions in the modulation space $M^{1}({\mathbb{R}})$, defined in Def.
2.1 just below, is that the frame set is an open subset of $\\{\Lambda\mid{\rm
vol}(\Lambda)<1\\}$ and that it contains a neighborhood of 0 [17, Thm. 2.1]
(see also [10], [11]). The latter statement guarantees that Gabor frames with
a window in $M^{1}({\mathbb{R}})$ exist if the density of $\Lambda$ is large
enough. For rectangular lattices $\Lambda_{a,b}=a{\mathbb{Z}}\times
b{\mathbb{Z}}$, we can picture the frame set by identifying a lattice with the
pair of parameters $(a,b)\in{\mathbb{R}}_{+}\times{\mathbb{R}}_{+}$, as done
in Fig. 1. Due to (1.7), the frame set for Hermite functions is symmetric in
the sense that $(a,b)\in\mathfrak{F}(h_{n})$ if and only if
$(b,a)\in\mathfrak{F}(h_{n})$.

Figure 1. For the $n$-th Hermite function we know that the Gabor system
$\mathcal{G}(h_{n},a{\mathbb{Z}}\times b{\mathbb{Z}})$ is a frame if
$ab<1/(n+1)$ and that it is not a frame if $ab\geq 1$. The frame set structure
of $\mathfrak{F}(h_{n})$, $n\geq 1$, between the “safety region” and the “no
frame” region is currently only partially understood.

For the $n$-th Hermite function we call the part of the frame set ${\rm
vol}(\Lambda)<n+1$ the safety region, as we have a frame by the Theorem of
Gröchenig and Lyubarskii [18]. For rectangular lattices
$\Lambda_{a,b}=a{\mathbb{Z}}\times b{\mathbb{Z}}$, $a,b\in{\mathbb{R}}_{+}$,
the safety region in $\mathfrak{F}(h_{n})$ are all pairs of parameters such
that $ab<n+1$.

##  2\. Preliminaries

###  2.1. Results from Gabor analysis

We recall some well-established results from Gabor analysis. The modulation
space $M^{1}({\mathbb{R}})$, which is also called Feichtinger’s algebra due to
its discoverer Feichtinger [8], will provide a convenient set-up for our
purposes. There are many equivalent definitions of this space and we refer to
the treatise by Jakobsen [21] for details. For the definition of the
modulation spaces one needs to fix a window in the Schwartz space
$\mathcal{S}({\mathbb{R}})$, but all windows lead to the same space with
equivalent norms. Hence, we will pick the Gaussian window $h_{0}$ in our
definition. More details can be found in the textbook of Gröchenig [16, Chap.
11–12].

######  Definition 2.1 (Modulation space).

The modulation space $M^{p}({\mathbb{R}})$ consists of all tempered
distributions $f\in\mathcal{S}^{\prime}({\mathbb{R}})$ such that

(2.1) $\lVert f\rVert_{M^{p}({\mathbb{R}})}=\lVert
V_{h_{0}}f\rVert_{L^{p}({\mathbb{R}}^{2})}<\infty.$

Note that $V_{g}g\in M^{1}({\mathbb{R}}^{2})$ if $g\in M^{1}({\mathbb{R}})$
[21, Cor. 5.5]. Hence, Feichtinger’s algebra is a convenient setting for us,
as it contains $h_{n}$. Therefore, all manipulations in the sequel are
justified and we do not run into any convergence issues (all series converge
unconditionally).

The following result was first shown by Janssen [23] for rectangular lattices
and for general lattices by Feichtinger and Luef [12]. We also refer to the
work of Gröchenig and Koppensteiner [15], where all technical details are
clarified.

######  Proposition 2.2 (Fundamental Identity of Gabor Analysis).

Assume we have functions $f_{1},f_{2},g_{1},g_{2}\in M^{1}({\mathbb{R}})$,
then the following identity holds

(2.2)
$\sum_{\lambda\in\Lambda}V_{g_{1}}f_{1}(\lambda)\,\overline{V_{g_{2}}f_{2}(\lambda)}={\rm
vol}(\Lambda)^{-1}\sum_{\lambda^{\circ}\in\Lambda^{\circ}}V_{g_{1}}g_{2}(\lambda^{\circ})\,\overline{V_{f_{1}}f_{2}(\lambda^{\circ})}.$

The proof uses symplectic Poisson summation (1.16) and the orthogonality
relations

(2.3) $\langle
V_{g_{1}}f_{1},V_{g_{2}}f_{2}\rangle_{L^{2}({\mathbb{R}}^{2})}=\langle
f_{1},f_{2}\rangle_{L^{2}({\mathbb{R}})}\overline{\langle
g_{1},g_{2}\rangle}_{L^{2}({\mathbb{R}})}.$

This is the STFT-version of Moyal’s identity for the Wigner distribution
introduced in [30] (see also [4, Chap. 6], [16, Chap. 4.3]). We continue with
the following observation, which is a slight variation of [16, Lem. 4.2.1] in
the textbook of Gröchenig. The proof employs the Cauchy-Schwarz inequality,
and when equality can be achieved.

######  Lemma 2.3.

For $g\in L^{2}({\mathbb{R}})$ we note that $V_{g}g(0,0)\in{\mathbb{R}}$.
Moreover, we have

(2.4) $V_{g}g(0,0)=\lVert g\rVert_{L^{2}({\mathbb{R}})}^{2}$

and the following estimate for all
$(x,\omega)\in{\mathbb{R}}^{2}\setminus\\{(0,0)\\}$:

(2.5) $|V_{g}g(x,\omega)|<V_{g}g(0,0).$

The following notion was introduced in [37]. A function $g$ satisfies
Condition A if

(Condition A)
$\sum_{\lambda^{\circ}\in\Lambda^{\circ}}|V_{g}g(\lambda^{\circ})|<\infty.$

The next result was shown by Tolimieri and Orr [37] under the assumption that
the window function $g$ indeed satisfies (Condition A). We only add the
convenient choice that we have $g\in M^{1}({\mathbb{R}})$, so that we can
directly use the result.

######  Lemma 2.4.

Let $g\in M^{1}({\mathbb{R}})$, then it satisfies (Condition A) and,
consequently, the Gabor system $\mathcal{G}(g,\Lambda)$ has a finite upper
frame bound $B$, which is at most

(2.6) $B\leq{\rm
vol}(\Lambda)^{-1}\sum_{\lambda^{\circ}\in\Lambda^{\circ}}|V_{g}g(\lambda^{\circ})|<\infty.$

The proof of the previous result follows easily from (2.2) and Lemma 2.3.

######  Lemma 2.5 (Janssen representation).

For a window $g\in M^{1}({\mathbb{R}})$ and a lattice $\Lambda$, the frame
operator $S=S_{g,\Lambda}$ can be expressed as

(2.7) $S={\rm
vol}(\Lambda)^{-1}\sum_{\lambda^{\circ}\in\Lambda^{\circ}}\langle
g,\pi(\lambda^{\circ})g\rangle\pi(\lambda^{\circ}).$

The result is well-known and it follows basically from (2.2). The last result
in this section is the following sufficiency criterion from [39], also known
as the Janssen test. As this will be the basis for all our proofs in Sec. 3,
we provide the proof for this result.

######  Lemma 2.6 (Janssen test).

Let $\Lambda$ be a lattice and $g$ a window with $\lVert
g\rVert_{L^{2}({\mathbb{R}})}=1$. If

(2.8)
$\sum\limits_{\lambda^{\circ}\in\Lambda^{\circ}}|V_{g}g(\lambda^{o})|<2,$

then $\mathcal{G}(g,\Lambda)$ constitutes a Gabor frame for
$L^{2}({\mathbb{R}})$.

###### Proof.

We use the proof given in [39] and compare the scaled frame operator ${\rm
vol}(\Lambda)S$, with $g\in M^{1}({\mathbb{R}})$ and
$\Lambda\subset{\mathbb{R}}^{2}$ a lattice, to the identity operator
$I=I_{L^{2}({\mathbb{R}})\to L^{2}({\mathbb{R}})}$. Noting that $\pi(0)=I$,
and using the Janssen representation of $S$, we get

(2.9) $\displaystyle\lVert{\rm vol}(\Lambda)S-I\rVert_{L^{2}({\mathbb{R}})\to
L^{2}({\mathbb{R}})}$
$\displaystyle=\lVert\sum_{\lambda^{\circ}\in\Lambda^{\circ}\setminus\\{0\\}}\langle
g,\pi(\lambda^{\circ})g\rangle\pi(\lambda^{\circ})\rVert_{L^{2}({\mathbb{R}})\to
L^{2}({\mathbb{R}})}$ (2.10)
$\displaystyle\leq\sum_{\lambda^{\circ}\in\Lambda^{\circ}\setminus\\{0\\}}|\langle
g,\pi(\lambda^{\circ})g\rangle|=\widetilde{B}.$

By our assumption (2.8), we see that ${\rm vol}(\Lambda)S$ is less than 1 away
from the identity operator. Therefore we can build the Neumann series of the
inverse and estimate its norm by

(2.11) $\lVert({\rm vol}(\Lambda)S)^{-1}\rVert_{L^{2}({\mathbb{R}})\to
L^{2}({\mathbb{R}})}\leq\sum_{k=0}^{\infty}\widetilde{B}^{k}=\frac{1}{1-\widetilde{B}},$

which shows that $\lVert({\rm vol}(\Lambda)S)^{-1}\rVert\leq
1/(1-\widetilde{B})$ and so the frame operator is invertible. It follows that
the Gabor system has a lower frame bound. The upper frame bound is finite due
to (2.6), hence $\mathcal{G}(g,\Lambda)$ is a frame. ∎

To the best of our knowledge, this tool has so far not been used in rigorous
analytic investigations of frame sets. It was introduced in [38] for finite
Gabor systems in ${\mathbb{C}}^{L}$, $L\in{\mathbb{N}}$ and then presented in
[39, Chap. 3.1, Cor. 3.1] for Gabor systems in $L^{2}({\mathbb{R}})$. In both
cases it was combined with numerical investigations to have heuristic
arguments for certain optimal sampling strategies. It was again discussed in
[7, Sec. 3] for heuristic reasons and in [35, Sec. 6] an extension of the
result was presented for Gabor $g$-frames. It was called “Janssen condition”
in [38, Chap. 4.1] as the criterion arises from the Janssen representation of
the frame operator and we also refer to [9, Sec. 6] for further reading.

###  2.2. Laguerre polynomials

Recall that the $n$-th Laguerre polynomial, denoted by $\mathcal{L}_{n}$, is
defined on the positive real axis ${\mathbb{R}}_{+}$ (but can be extended to
${\mathbb{R}}$) by

(2.12) $\mathcal{L}_{n}(x)=\sum_{k=0}^{n}\binom{n}{k}\frac{(-x)^{k}}{k!}.$

We remark that there is a generalized version $\mathcal{L}_{n}^{\alpha}$,
which does not concern us here, and that
$\mathcal{L}_{n}=\mathcal{L}_{n}^{0}$. For details we refer to the textbook by
Szegő [36] or to [1, Chap. 22]. The Laguerre polynomials are orthogonal on
${\mathbb{R}}_{+}$ with respect to the exponential weight $w(x)=e^{-x}$ [36,
Sec. 5.1, eq. (5.1.1)] and $\mathcal{L}_{n}(x)$ has $n$ distinct roots in
${\mathbb{R}}_{+}$ [36, Chap. 3.3, Chap. 6.31]. The largest root $x_{n}$ of
$\mathcal{L}_{n}$ satisfies [36, Thm.6.31.2]

(2.13) $x_{n}<2n+1+\sqrt{(2n+1)^{2}+1/4},\quad\text{ and }\quad x_{n}\sim 4n.$

Our interest in Laguerre polynomials comes from the following result, which
may be looked up in [13, Chap. 1.9, Thm. 1.104]. Note that $V_{g}f$ is defined
slightly different in [13, Chap. 1.8, eq. (1.88)] (where it is more or less
the ambiguity function), which makes the appearance of a phase factor explicit
in our work.

######  Lemma 2.7.

For the $n$-th Hermite function $h_{n}$, the STFT with window $h_{n}$ is given
by

(2.14) $V_{h_{n}}h_{n}(x,\omega)=e^{-\pi
ix\omega}\mathcal{L}_{n}\left(\pi(x^{2}+\omega^{2})\right)e^{-\frac{\pi}{2}(x^{2}+\omega^{2})}.$

In the sequel, our goal will always be to find $\Lambda$ such that

(2.15)
$\sum_{\lambda^{\circ}\in\Lambda^{\circ}}|V_{h_{n}}h_{n}(\lambda^{\circ})|=\sum_{\lambda^{\circ}\in\Lambda^{\circ}}\left|\mathcal{L}_{n}\left(\pi|\lambda^{\circ}|^{2}\right)\right|e^{-\frac{\pi}{2}|\lambda^{\circ}|^{2}}<2.$

###  2.3. Auxiliary results and proof strategy

In this section, we present some needed auxiliary results and present the
general proof strategy.

Upper bounds for Laguerre polynomials were investigated in a number of papers,
see, e.g., [24], [25], [27]. There is a classical estimate, known as Szegő
bound, which is

(2.16) $|\mathcal{L}_{n}(x)|\leq e^{x/2}\quad\text{ for }x\geq 0,$

proved by Szegő [36, Sec. 7.21]. However, (2.16) will not be sufficient for
our purposes as, using (2.14), it only gives that $|V_{h_{n}}h_{n}|$ is
bounded by 1, which we also know from Lemma 2.3. It is worth mentioning that,
in the other direction, Lemma 2.3 in combination with Lemma 2.14 implies the
Szegő bound. We need more delicate estimates in the sequel.

First, observe that it is easy to obtain a crude polynomial bound for
$\mathcal{L}_{n}$, as we have

(2.17)
$|\mathcal{L}_{n}(x)|\leq\sum_{k=0}^{n}\binom{n}{k}\frac{x^{k}}{k!}\leq(n+1)\binom{n}{\lfloor
n/2\rfloor}x^{n},\quad\forall x\geq 1.$

Here, the notation ${\lfloor m\rfloor}$ stands for the greatest integer less
than or equal to $m$. Note that the binomial coefficient may be huge, but this
estimate becomes useful once $x$ is, say, larger than $x_{n}$, the largest
root of $\mathcal{L}_{n}$. Our estimate (2.17) will be useful in the
calculations to prove Theorem 1.1 for $n$ sufficiently small, which in our
case is $n\leq 36$.

In the other direction, i.e., for large $n$, we use the following estimates,
obtained from contemporary bounds on the values of Laguerre polynomials by
Krasikov [24], [25].

######  Lemma 2.8.

For $n\geq 11$, the following estimates are simultaneously true.

(2.18) $\mathcal{L}_{n}(-\pi(n+1))\leq e^{(\pi+1)(n+1)}.$ (2.19)
$|\mathcal{L}_{n}(\pi(n+1))|\leq e^{\tfrac{\pi}{2}(n+1)}\sqrt{\frac{2}{n}}.$
(2.20) $|\mathcal{L}_{n}(2\pi(n+1))|\leq e^{\pi(n+1)}\sqrt{\frac{2}{\pi}}\
e^{-0.1n-0.86}.$

In addition, for generalized Laguerre polynomials we noticed that we can
painlessly prove the following estimate:

(2.21)
$|\mathcal{L}_{n}^{\alpha}(t)|\leq\sqrt{\frac{(n+\alpha)!}{n!}}e^{t/2}t^{-\alpha/2},\quad
n,\alpha\in{\mathbb{N}}_{0},\,t>0.$

We don’t actually need relation (2.21), but it may be of independent interest.

###### Proof.

The proof of (2.18) is simple:

$\mathcal{L}_{n}(-\pi(n+1))=\sum_{k=0}^{n}\binom{n}{k}\frac{(\pi(n+1))^{k}}{k!}\leq\binom{n}{\left\lfloor{\frac{n}{2}}\right\rfloor}e^{\pi(n+1)}\leq(2e)^{n/2}e^{\pi(n+1)}\leq
e^{(\pi+1)(n+1)}.$

To prove (2.19), we invoke the following estimate from [24] (see also [25,
Thm. 1]): for $n\geq 2$ and $x\in[q^{2},s^{2}]$, where $q=\sqrt{n+1}-\sqrt{n}$
and $s=\sqrt{n+1}+\sqrt{n}$, the inequality

(2.22) $|\mathcal{L}_{n}(x)|\leq
e^{x/2}\sqrt{\frac{s^{2}-q^{2}}{(x-q^{2})(s^{2}-x)}}$

holds true. Note that $x=\pi(n+1)$ belongs to $[q^{2},s^{2}]$, since

$q^{2}=(\sqrt{n+1}-\sqrt{n})^{2}\leq 2n+1\leq\pi(n+1)\leq
4n\leq(\sqrt{n+1}+\sqrt{n})^{2}=s^{2}.$

Next, for $x=\pi(n+1)$ we estimate the value inside the square root as
follows:

(2.23) $\displaystyle\frac{s^{2}-q^{2}}{(x-q^{2})(s^{2}-x)}$
$\displaystyle=\frac{(\sqrt{n+1}+\sqrt{n})^{2}-(\sqrt{n+1}-\sqrt{n})^{2}}{\left(\pi(n+1)-(\sqrt{n+1}-\sqrt{n})^{2}\right)\left(\sqrt{n+1}+\sqrt{n})^{2}-\pi(n+1)\right)}$
(2.24)
$\displaystyle=\frac{4\sqrt{n(n+1)}}{(4\pi-\pi^{2})n^{2}+(6\pi-2\pi^{2})n-(1-\pi)^{2}}\leq\frac{2}{n},$

provided $n$ is large enough. We estimate $\sqrt{n}$ by $\sqrt{n+1}$ in the
enumerator. Then, by cross-multiplying the ratios, it suffices to establish
that

$\tau(n):=(4\pi-\pi^{2}-2)n^{2}+(6\pi-2\pi^{2}-2)n-(1-\pi)^{2}>0$

holds true for $n\geq 11.$ Note that $4\pi-\pi^{2}-2>0$, and the largest root
$n_{+}$ of $\tau(n)$ satisfies

$n_{+}=\frac{(\pi^{2}-3\pi+1)+\sqrt{(3\pi-\pi^{2}-1)^{2}+(1-\pi)^{2}(4\pi-\pi^{2}-2)}}{(4\pi-\pi^{2}-2)}\leq
6.$

Therefore, using (2.22), we arrive at the desired estimate

$|\mathcal{L}_{n}(\pi(n+1))|\leq e^{\tfrac{\pi}{2}(n+1)}\sqrt{\frac{2}{n}}.$

It remains to prove (2.20). We recall that $q=\sqrt{n+1}-\sqrt{n}\leq
s=\sqrt{n+1}+\sqrt{n}$. Clearly, $2\pi(n+1)\geq s^{2}$. Therefore, we can use
Theorem 5 from [25] that asserts the estimate

(2.25) $|\mathcal{L}_{n}(2\pi(n+1))|\leq\frac{s}{\sqrt{2\pi(n+1)}}\
|\mathcal{L}_{n}(s^{2})|\ e^{\pi(n+1)-\frac{s^{2}}{2}}\
e^{-R(s^{2},2\pi(n+1))/2},$

where

(2.26)
$R(s^{2},2\pi(n+1))=\int\limits_{s^{2}}^{2\pi(n+1)}\frac{\sqrt{(q^{2}-t)(s^{2}-t)}}{t}\,dt.$

We start with estimating the function $R$ from below. Note that

(2.27)
$\begin{split}R(s^{2},2\pi(n+1))&=\int\limits_{s^{2}}^{2\pi(n+1)}\frac{\sqrt{(t-q^{2})(t-s^{2})}}{t}\,dt\\\
&\geq\int\limits_{s^{2}}^{2\pi(n+1)}\frac{(t-s^{2})}{t}dt=2\pi(n+1)-s^{2}\left(1+\ln\frac{2\pi(n+1)}{s^{2}}\right).\end{split}$

Since Clearly, $4n+1\leq s^{2}\leq 4n+3\leq 2\pi(n+1)$. Therefore, the
logarithmic expression in (2.27) is positive and we get

(2.28) $R(s^{2},2\pi(n+1))\geq
2\pi(n+1)-(4n+3)\left(1+\ln\frac{2\pi(n+1)}{4n+1}\right).$

Next, one can check that the logarithmic expression in (2.28) is monotonically
decreasing in $n$ for $n\geq 11$, and by evaluating at $11$, we have
$\ln(2\pi(n+1)/(4n+1))\leq 0.52$. Hence,

(2.29) $R(s^{2},2\pi(n+1))\geq n(2\pi-4\cdot 1.52)+2\pi-3\cdot 1.52\geq
0.2n+1.72.$

Now, we return to relation (2.25). Using the classical Szegő estimate (2.16),
and estimate (2.29), we obtain our desired inequality:

(2.30) $|\mathcal{L}_{n}(2\pi(n+1))|\leq
e^{\pi(n+1)}\frac{\sqrt{4(n+1)}}{\sqrt{2\pi(n+1)}}e^{-0.1n-0.86}\leq
e^{\pi(n+1)}\frac{\sqrt{2}}{\sqrt{\pi}}e^{-0.1n-0.86}.$

∎

######  Remark 2.9.

As can be seen in the proof, we only need the assumption $n\geq 11$ in our
final estimate and the other inequalities already hold for smaller $n$. The
estimates will be employed for $n\geq 36$ and so we did not try to find the
optimal $n$.

We conclude the section with the following estimate.

######  Lemma 2.10.

Let $a\geq 2$, $\gamma>\frac{2n+2}{a^{2}\pi}$, and $n\in{\mathbb{N}}$. Then

(2.31) $\sum\limits_{\begin{subarray}{c}(k,l)\in{\mathbb{Z}}^{2},\\\
k^{2}+l^{2}\geq
a^{2}\end{subarray}}(k^{2}+l^{2})^{n}e^{-\frac{\pi\gamma}{2}(k^{2}+l^{2})}\leq\frac{4}{\gamma}a^{2n}e^{-\frac{a^{2}\pi\gamma}{2}}.$

###### Proof.

The series at the right-hand side of (2.31) is majorized by the integral

(2.32)
$\displaystyle\phantom{=}\int\limits_{0}^{2\pi}\int\limits_{a}^{\infty}r^{2n+1}e^{-\frac{\pi}{2}\gamma
r^{2}}drd\varphi=2\pi\int\limits_{a}^{\infty}r^{2n+1}e^{-\frac{\pi}{2}\gamma
r^{2}}dr$ (2.33)
$\displaystyle=\pi\int\limits_{a^{2}}^{\infty}\rho^{n}e^{-\frac{\pi}{2}\gamma\rho}d\rho=\pi\left(\frac{2}{\pi\gamma}\right)^{n+1}\Gamma\left(n+1,\frac{a^{2}\pi\gamma}{2}\right),$

where the notation $\Gamma(s,x)$ stands for the incomplete Gamma function
defined by

(2.34) $\Gamma(s,x)=\int\limits_{x}^{\infty}t^{s-1}e^{-t}dt.$

To estimate $\Gamma(s,x)$ from above, we need the inequality

(2.35) $|\Gamma(s,x)|\leq Bx^{s-1}e^{-x}\quad\text{for
}x>\frac{B}{(B-1)}(s-1),\,s>0,\ B>1.$

established in [31]. Using this estimate with $B=2$ (recall that
$a^{2}\pi\gamma\geq 2n+2$), we get

$\Gamma\left(n+1,\frac{a^{2}\pi\gamma}{2}\right)\leq
2\left(\frac{a^{2}\pi\gamma}{2}\right)^{n}e^{-\frac{a^{2}\pi\gamma}{2}}$

and conclude that

(2.36)
$\displaystyle\phantom{=}\sum\limits_{\begin{subarray}{c}(k,l)\in{\mathbb{Z}}^{2},\\\
|k|^{2}+|l|^{2}\geq
a^{2}\end{subarray}}(k^{2}+l^{2})^{n}e^{-\frac{\pi\gamma}{2}(k^{2}+l^{2})}\leq
2\pi\left(\frac{2}{\pi\gamma}\right)^{n+1}\left(\frac{a^{2}\pi\gamma}{2}\right)^{n}e^{-\frac{a^{2}\pi\gamma}{2}}=\frac{4}{\gamma}a^{2n}e^{-\frac{a^{2}\pi\gamma}{2}}.$

∎

Proof strategy. In Section 3, we will use the Janssen test (2.8) to show that
certain Gabor systems with Hermite functions over a square lattice form a
frame. By Lemma 2.3 and (2.14) this leads to the verification that

(2.37)
$\sum_{k,l\in{\mathbb{Z}}}|\mathcal{L}_{n}(\pi\delta(k^{2}+l^{2})|e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}<2,$

where $\delta$ is the density of the square lattice. The rough strategy is
always to split the series for $|k|,|l|$ relatively small and the rest. After
all, we have a Gaussian weight which guarantees that the contributions for
$|k|,|l|$ large will be tiny. For small orders $n$, the bound of the terms
with a small index will be achieved by a finite number of computations, i.e.,
explicitly computing values of a finite sum, while for large orders $n$ we
employ the bounds based on Lemma 2.8. The estimates on the remainder (large
index) come from the auxiliary results in this section. The splitting will
either involve squares, containing a finite number of points, or computing
values for layers close to the origin, as shown in Fig. 2.

Figure 2. We split ${\mathbb{Z}}^{2}$ into a finite section, where we perform
a finite number of computations, and a complementary region, where the series
can be bounded by $0<\varepsilon\ll 1$ sufficiently small (left). At some
instances, instead of splitting ${\mathbb{Z}}^{2}$ by means of the max-norm,
we use the Euclidean distance and ${\mathbb{Z}}^{2}$ splits into layers
$\ell_{m}=\\{(k,l)\in{\mathbb{Z}}^{2}\mid k^{2}+l^{2}=m\\}$ (right). This also
shows the connection to decomposing an integer into a sum of two squares.

A layer of a lattice $\Lambda$ is a set of points satisfying

(2.38) $\ell_{r}=\\{\lambda\in\Lambda\mid|\lambda|^{2}=r\\}.$

For most $r$ the set will, of course, be empty. The set with the smallest
index $r>0$ which is not empty is called the first layer (it contains the
closest neighbors of the origin), the non-empty set with the second smallest
index is the second layer, and so on. The decomposition of the square lattice
${\mathbb{Z}}^{2}$ into layers naturally leads to the question of how to write
an integer as the sum of two squares. Denote by $r_{2}(n)$ the number of ways
of writing $n\in{\mathbb{N}}_{0}$ as the sum of two squares, allowing zeros
and distinguishing signs and order. For example,

(2.39) $\displaystyle r_{2}(0)$ $\displaystyle=1\quad\text{ as
}0=0^{2}+0^{2},$ (2.40) $\displaystyle r_{2}(1)$ $\displaystyle=4\quad\text{
as }1=0^{2}+1^{2}=0^{2}+(-1)^{2}=1^{2}+0^{2}=(-1)^{2}+0^{2},$ (2.41)
$\displaystyle r_{2}(2)$ $\displaystyle=4\quad\text{ as
}2=1^{2}+1^{2}=1^{2}+(-1)^{2}=1^{2}+(-1)^{2}=(-1)^{2}+(-1)^{2},$ (2.42)
$\displaystyle r_{2}(3)$ $\displaystyle=0\quad\text{ as 3 factors containing
primes $p\equiv 3$ mod 4 of odd power.}$

Note that $|\ell_{m}|=r_{2}(m)$ and that we have the trivial estimate

(2.43) $r_{2}(m)\leq 4m,\quad m\in{\mathbb{N}},$

as we have at most $\sqrt{m}$ possible choices for each summand and 4 choices
to place signs. This bound is quite rough, but sufficient for our purposes.

The last ingredient we need are truncated geometric series and their
derivatives. This will include, e.g., for $|q|<1$,

(2.44) $\displaystyle f_{N}(q)$ $\displaystyle:=\sum_{m\geq
N}q^{m}=q^{N}\sum_{m\geq 0}q^{m}=\frac{q^{N}}{1-q},$ (2.45) $\displaystyle
qf_{N}^{\prime}(q)$ $\displaystyle=\sum_{m\geq
N}mq^{m}=\frac{q^{N}}{(1-q)^{2}}(N-(N-1)q),$ (2.46) $\displaystyle
q^{2}f_{N}^{\prime\prime}(q)$ $\displaystyle=\sum_{m\geq
N}(m^{2}q^{m}-mq^{m})=\sum_{m\geq N}m^{2}q^{m}-q^{2}f_{N}^{\prime}(q)$ (2.47)
$\displaystyle=\frac{q^{N}}{(1-q)^{3}}\left((N^{2}-N)-(2N^{2}+4N)q+(N^{2}-3N+2)q^{2}\right),$

and so on. As these results are obtained by elementary means, but are rather
laborious for high orders, we will not always provide all steps and only give
the final result instead.

##  3\. Results

###  3.1. Proof of Theorem 1.1

We will now prove Theorem 1.1 by settling the cases for $4\leq n\leq 36$
first, which we solve by a finite number of computations and showing that the
cut-off remainders cannot contribute enough mass such that the finite sums
would violate the Janssen test in Lemma 2.6. The case $n\geq 36$ will be
proved separately by providing uniform bounds (in $n$) on the terms in the
first and second layer of the adjoint lattice and bounding the remainder.

######  Proposition 3.1.

Let $n\in{\mathbb{N}}$ with $4\leq n\leq 36$. Then, the Gabor system

(3.1)
$\mathcal{G}_{n}=\mathcal{G}\left(h_{n},\frac{1}{\sqrt{n+1}}\,{\mathbb{Z}}^{2}\right)$

constitutes a frame for $L^{2}({\mathbb{R}})$.

###### Proof.

We use the Janssen test to show that $\mathcal{G}_{n}$ is a frame for
$n\in{\mathbb{N}}$ with $4\leq n\leq 36$. So, we have to show that

(3.2)
$\sum_{(k,l)\in{\mathbb{Z}}^{2}}\left|\mathcal{L}_{n}(\pi(n+1)(k^{2}+l^{2}))\right|e^{-\frac{\pi}{2}(n+1)(k^{2}+l^{2})}<2.$

As $h_{n}$ is in $M^{1}({\mathbb{R}})$, $n\in{\mathbb{N}}_{0}$, (Condition A)
holds and the series converges unconditionally. So, we may split the series in
two parts: a finite part consisting of lattice points with $|k|,|l|\leq 5$ and
a remainder with $\max\\{|k|,|l|\\}\geq 6$.

By (2.17), (2.43), we bound the remainder in a generous way, as we expect it
to be small:

(3.3) $\displaystyle\ \sum_{\max\\{|k|,|l|\\}\geq
6}\left|\mathcal{L}_{n}(\pi(n+1)(k^{2}+l^{2}))\right|e^{-\frac{\pi}{2}(n+1)(k^{2}+l^{2})}$
(3.4) $\displaystyle\leq$ $\displaystyle\ (n+1)\binom{n}{\lfloor
n/2\rfloor}\sum_{\max\\{|k|,|l|\\}\geq
6}(\pi(n+1)(k^{2}+l^{2}))^{n}e^{-\frac{\pi}{2}(n+1)(k^{2}+l^{2})}$ (3.5)
$\displaystyle=$ $\displaystyle\ \pi^{n}(n+1)^{n+1}\binom{n}{\lfloor
n/2\rfloor}\sum_{m\geq 36}r_{2}(m)\ m^{n}e^{-\frac{\pi}{2}(n+1)m}$ (3.6)
$\displaystyle\leq$ $\displaystyle\ 4\pi^{n}(n+1)^{n+1}\binom{n}{\lfloor
n/2\rfloor}\sum_{m\geq 36}m^{n+1}e^{-\frac{\pi}{2}(n+1)m}.$

We first estimate the constant in front of the series, which is clearly
increasing in $n$:

(3.7) $4\pi^{n}(n+1)^{n+1}\binom{n}{\lfloor n/2\rfloor}\leq 4\pi^{36}\cdot
37^{37}\cdot\binom{36}{18}<4\cdot 10^{86}<10^{87},\quad n\leq 36.$

To estimate the series we need the following auxiliary result.

Claim. For $x,y>0$, the function $f(y):=x^{y}e^{-\frac{\pi}{2}yx}$ is
decreasing in $y$, for any fixed $x$. Indeed, the claim follows easily from

(3.8) $f^{\prime}(y)=x^{y}e^{-\frac{\pi}{2}yx}\
\left(\log(x)-\frac{\pi}{2}x\right)<0,\quad x>0,y>0.$

The series can now be estimated by using the claim and plugging in $n=4$:

(3.9) $\displaystyle\sum_{m\geq
36}m^{n+1}e^{-\frac{\pi}{2}(n+1)m}\leq\sum_{m\geq
36}m^{5}q^{m}=\frac{q^{36}}{(1-q)^{6}}\ \cdot\ $ $\displaystyle\Big{(}\ 60\
466\ 176-293\ 453\ 099\ q$ (3.10) $\displaystyle+570\ 164\ 066\ q^{2}-554\
350\ 974\ q^{3}$ (3.11) $\displaystyle+269\ 695\ 826\ q^{4}-52\ 521\ 875\
q^{5}\Big{)}.$

Here, we set $q=e^{-5\pi/2}$ and used Mathematica [40] to symbolically compute
the infinite series. We remark that this can also be achieved by hand by using
the truncated geometric series and finite linear combinations of its
derivatives as indicated in Sec. 2.3. Plugging in numerical values for
$q=e^{-5\pi/2}$ finally leads to the estimate

(3.12) $\sum_{m\geq 36}m^{n+1}e^{-\frac{\pi}{2}(n+1)m}\leq 10^{-115},\quad
n\geq 4.$

Thus, for $4\leq n\leq 36$, the remainder can be estimated by

(3.13) $\sum_{\max\\{|k|,|l|\\}\geq
6}\left|\mathcal{L}_{n}(\pi(n+1)(k^{2}+l^{2}))\right|e^{-\frac{\pi}{2}(n+1)(k^{2}+l^{2})}<10^{-115+87}=10^{-28}.$

Now, we are left with a finite number of computations. Using Mathematica, we
compute the values of the finite part to 5 significant digits after the
decimal point (always rounded up) to see that indeed we have

(3.14) $\sum_{|k|,|l|\leq 5}|\mathcal{L}_{n}(\pi(n+1)(k^{2}+l^{2}))|\
e^{-\frac{\pi}{2}(n+1)(k^{2}+l^{2})}<1.99999=2-10^{-5},\quad 4\leq n\leq 36.$

Again, these calculations could be carried out to the necessary precision by
hand. The precise values of (3.14) for $n\in\\{0,\ldots,36\\}$ are given in
Fig. 3. For $4\leq n\leq 36$ they are all significantly (at least $10^{-3}$)
smaller than 2. With our estimates on the remainder, this gives a complete
proof of the frame property for the Gabor systems $\mathcal{G}_{n}$ and $4\leq
n\leq 36$. ∎

$n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
value | 2.01497 | 2.00000 | 2.00003 | 2.00000 | 1.99390 | 1.97889 | 1.95381 | 1.91844
$n$ | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
value | 1.87308 | 1.81835 | 1.75515 | 1.68451 | 1.60760 | 1.52567 | 1.44006 | 1.35211
$n$ | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23
value | 1.26320 | 1.17470 | 1.08793 | 1.00419 | 1.07535 | 1.14951 | 1.21732 | 1.27788
$n$ | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31
value | 1.33044 | 1.37440 | 1.40932 | 1.43490 | 1.45101 | 1.45770 | 1.45517 | 1.44376
$n$ | 32 | 33 | 34 | 35 | 36 |  |  |
value | 1.42396 | 1.39642 | 1.36187 | 1.32118 | 1.27528 |  |  |
Figure 3. Values of the finite computations part of the Janssen test for the
Gabor systems $\mathcal{G}(h_{n},(1/\sqrt{n+1}){\mathbb{Z}}^{2})$. The values
for $n=1$ and $n=3$ are exact and the other values are given to 5 digits after
the decimal point (rounded up).

As the values in Fig. 3 did not show a monotonic behavior, we carried out
further numerical investigations for $n$ up to 120. The behavior is curious
and visualized in Fig. 4.

Figure 4. Values of the Janssen test for $\mathcal{G}(h_{n},1/\sqrt{(n+1)}\
{\mathbb{Z}}^{2})$ and $n\leq 120$.

We will now treat the case $n\geq 36$. The proof, again, naturally splits into
bounding the mass of the resulting series on a finite set, consisting of the
origin and the first two layers of the adjoint lattice, and a small remainder.
Our bounds are not the best possible and there is room to improve on $n$,
i.e., make it smaller, but in any case we would be left with finitely many
cases which need to be checked by hand (or computer), unless we take more
layers into account for the finite set.

######  Proposition 3.2.

Let $n\geq 36$, then the Gabor system
$\mathcal{G}(h_{n},\frac{1}{\sqrt{n+1}}{\mathbb{Z}}\times\frac{1}{\sqrt{n+1}}{\mathbb{Z}})$
is a frame.

###### Proof.

We will use the Janssen test again and show that

(3.15)
$\sum_{\lambda^{\circ}\in\Lambda^{\circ}}|V_{h_{n}}h_{n}(\lambda^{\circ})|<2\quad\text{
for }n\geq 36,$

which by (2.14), is equivalent to showing

(3.16)
$\sum\limits_{k,l\in{\mathbb{Z}}}\left|\mathcal{L}_{n}\left(\pi(n+1)(k^{2}+l^{2})\right)\right|e^{-\frac{\pi(n+1)}{2}(k^{2}+l^{2})}<2,\quad
n\geq 36.$

Denote by $K=\left\\{(0,0),(0,\pm 1),(\pm 1,0),(\pm 1,\pm 1)\right\\}$ the set
which contains the origin, its 4 closest neighbors $\ell_{1}$ and its 4 second
closest neighbors $\ell_{2}$. We split the sum at the left-hand side of (3.16)
into the sum over $K$ and ${\mathbb{Z}}^{2}\setminus K$. Since
$\mathcal{L}_{n}(0)=1$, it suffices to prove

(3.17) $\ell_{1}(n)+\ell_{2}(n)+\ell_{3+}(n)<1,$

where $\ell_{1}(n)$ is the value of the sum over points in the first layer,
$\ell_{2}(n)$ the value for the second layer and $\ell_{3+}(n)$ the value of
the sum over all remaining layers, i.e.,

(3.18) $\displaystyle\ell_{1}(n)$ $\displaystyle=4\mathcal{L}_{n}(\pi(n+1))\
e^{-\frac{\pi}{2}(n+1)},\qquad\qquad\ell_{2}(n)=4\mathcal{L}_{n}(2\pi(n+1))\
e^{-\pi(n+1)},$ (3.19) $\displaystyle\ell_{3+}(n)$
$\displaystyle=\sum\limits_{(k,l)\in{\mathbb{Z}}^{2}\setminus{K}}\left|\mathcal{L}_{n}\left(\pi(n+1)(k^{2}+l^{2})\right)\right|\
e^{-\frac{\pi(n+1)}{2}(k^{2}+l^{2})}.$

By (2.19) and (2.20), we have

(3.20)
$|\ell_{1}(n)|<4\sqrt{\frac{2}{n}}\quad\text{and}\quad|\ell_{2}(n)|<4\sqrt{\frac{2}{\pi}}\,e^{-0.1n-0.86}.$

Since the coefficients of Laguerre polynomials have alternating signs, we have

(3.21)
$\ell_{3+}(n)\leq\sum\limits_{(k,l)\in{\mathbb{Z}}^{2}\setminus{K}}(k^{2}+l^{2})^{n}\,|\mathcal{L}_{n}\left(-\pi(n+1)\right)|e^{-\frac{\pi(n+1)}{2}(k^{2}+l^{2})}.$

Using (2.18) and the fact that $n+1\leq\frac{\pi(n+1)}{2}$, we arrive at

(3.22) $\displaystyle\ell_{3+}(n)$
$\displaystyle\leq\sum\limits_{(k,l)\in{\mathbb{Z}}^{2}\setminus{K}}(k^{2}+l^{2})^{n}\,e^{(\pi+1)(n+1)-\frac{\pi(n+1)}{2}(k^{2}+l^{2})}$
(3.23)
$\displaystyle<\sum\limits_{(k,l)\in{\mathbb{Z}}^{2}\setminus{K}}(k^{2}+l^{2})^{n}e^{-\frac{\pi(n+1)}{2}(k^{2}+l^{2}-3)}=e^{\frac{3\pi}{2}(n+1)}\sum\limits_{(k,l)\in{\mathbb{Z}}^{2}\setminus{K}}(k^{2}+l^{2})^{n}e^{-\frac{\pi(n+1)}{2}(k^{2}+l^{2})}.$

To estimate the series at the right-hand side of (3.23), we use Lemma 2.10
with $\gamma=n+1$ and $a=2$:

(3.24)
$\sum\limits_{(k,l)\in{\mathbb{Z}}^{2}\setminus{K}}(k^{2}+l^{2})^{n}e^{-\frac{\pi(n+1)}{2}(k^{2}+l^{2})}\leq\frac{4^{2(n+1)}}{n+1}e^{-2\pi(n+1)}.$

Therefore,

(3.25) $l_{3+}(n)\leq e^{-0.184n}.$

Combining (3.20) and (3.25), we finally arrive at

(3.26)
$\ell_{1}(n)+\ell_{2}(n)+\ell_{3+}(n)<4\sqrt{\frac{2}{n}}+4\sqrt{\frac{2}{\pi}}\,e^{-0.1n-0.86}+\
e^{-0.184n}<1$

as long as $n$ is sufficiently large. Clearly, the expression above is
decreasing in $n$ and the inequality is satisfied for $n=36$. This finishes
the proof. ∎

###  3.2. Failure of the Janssen test

####  3.2.1. The case of small $n$

For the Gabor systems $\mathcal{G}(h_{n},\delta^{-1/2}{\mathbb{Z}}^{2})$ and
$n\in\\{0,1,2,3\\}$ we know that we do not have a frame for $\delta=n+1$. In
particular, the lower frame bound vanishes and the bound in the Janssen test
must yield a value of at least 2. For $\delta>n+1$ we are in the safety region
and have a Gabor frame. For small $n$, in particular up to 3, the Janssen test
does not shed any new light on the frame set of

(3.27)
$\mathcal{G}_{n}(\delta)=\mathcal{G}(h_{n},\delta^{-1/2}{\mathbb{Z}}^{2}).$

In accordance with the previous sections, we set
$\mathcal{G}_{n}=\mathcal{G}_{n}(n+1)$.

For $\mathcal{G}_{0}$, i.e., the Gaussian function $h_{0}$ and the integer
lattice ${\mathbb{Z}}^{2}$, the Janssen test fails:

(3.28)
$\sum_{k,l\in{\mathbb{Z}}}|V_{h_{0}}h_{0}(k,l)|=\sum_{k,l\in{\mathbb{Z}}}e^{-\frac{\pi}{2}(k^{2}+l^{2})}>\sum_{k,l\in\\{-1,0,1\\}}e^{-\frac{\pi}{2}(k^{2}+l^{2})}>2.004.$

So, the Janssen test is inconclusive, but it is a well-known fact that the
lower frame bound vanishes in this case [28], [33], [34] and we do not have a
frame. Similarly, for $\mathcal{G}_{2}$ we have

(3.29)
$\displaystyle\sum_{k,l\in{\mathbb{Z}}}|V_{h_{2}}h_{2}(\sqrt{3}\,k,\sqrt{3}\,l)|$
$\displaystyle>\sum_{k,l\in\\{-1,0,1\\}}\left|\mathcal{L}_{3}\left(3\pi(k^{2}+l^{2})\right)\right|e^{-\frac{3\pi}{2}(k^{2}+l^{2})}>2.00001.$

The lower frame bound vanishes, as shown by J. Lemvig [26, Thm. 7], and we do
not have a frame with the Gabor system $\mathcal{G}_{2}$.

Next, we consider the Gabor systems $\mathcal{G}_{n}$ for $n\in\\{1,3\\}$. The
results of Lyubarskii and Nes show that the lower bound vanishes if $n=1$. For
$n=3$ we have again a result by Lemvig [26, Thm. 8] which shows that the lower
frame bound is zero. We remark that Boon, Zak, and Zucker [3] already studied
the series considered in [26] in the context of coherent states and provided
zeros (see also [2]).

It is quite remarkable that for $n$ being 1 or 3 the Janssen test yields
exactly 2. In this regard, these Gabor systems are closest possible to being a
frame, but they still just fail.

######  Proposition 3.3.

Let $n\in\\{1,3\\}$. Then, the following hold:

(3.30)
$\sum_{k,l\in{\mathbb{Z}}}|V_{h_{n}}h_{n}(\sqrt{n+1}\,k,\sqrt{n+1}\,l)|=\sum_{k,l\in{\mathbb{Z}}}\left|\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)\right|e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}=2$

and

(3.31)
$\sum_{k,l\in{\mathbb{Z}}}V_{h_{n}}h_{n}(\sqrt{n+1}\,k,\sqrt{n+1}\,l)=\sum_{k,l\in{\mathbb{Z}}}\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}=0.$

###### Proof.

Note that for any $n\in{\mathbb{N}}$, the term $(k,l)=(0,0)$ contributes 1.
Hence, our claim is

(3.32)
$\sum_{(k,l)\in{\mathbb{Z}}^{2}\setminus\\{0,0\\}}\left|\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)\right|e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}=1,\quad\text{
for }n=1,3.$

By the results of Lyubarskii and Nes [29] and Lemvig [26] we know that the
Gabor system $\mathcal{G}(h_{n},(n+1)^{-1/2}{\mathbb{Z}}^{2})$ is not a frame
as the lower frame bound vanishes.

Now, on the one hand, we know that this implies (the Janssen test must fail)
that

(3.33)
$\sum_{(k,l)\in{\mathbb{Z}}^{2}\setminus\\{(0,0)\\}}\left|\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)\right|e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}\geq
1.$

On the other hand, we know from the results by Janssen [22] (see also [5])
that the minimum of the Fourier series

(3.34) $F_{g,\Lambda}(z)={\rm
vol}(\Lambda)^{-1}\sum_{\lambda^{\circ}\in\Lambda^{\circ}}V_{g}g(\lambda^{\circ})e^{2\pi
i\sigma(\lambda^{\circ},z)},\quad z=(x,\omega)\in{\mathbb{R}}^{2},$

gives the sharp lower frame bound of the Gabor system $\mathcal{G}(g,\Lambda)$
for ${\rm vol}(\Lambda)^{-1}\in{\mathbb{N}}$. Thus, we have
$F_{g,\Lambda}(x,\omega)\geq 0$. Setting $\Lambda_{n}=1/\sqrt{n+1}\
{\mathbb{Z}}^{2}$ for the moment, this implies

(3.35) $\displaystyle-1\leq{\rm vol}(\Lambda)F_{h_{n},\Lambda_{n}}(0,0)-1$
$\displaystyle=\sum_{(k,l)\in{\mathbb{Z}}^{2}\setminus\\{0,0\\}}V_{h_{n}}h_{n}(\sqrt{n+1}\,k,\sqrt{n+1}\,l)$
(3.36)
$\displaystyle=\sum_{(k,l)\in{\mathbb{Z}}^{2}\setminus\\{0,0\\}}\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}.$

Note that the phase factor, imposed by (2.14), is 1 as the lattice density is
$n+1$, which is even in this case. Next, we look at the Laguerre polynomials
$\mathcal{L}_{1}(x)$ and $\mathcal{L}_{3}(x)$. They are

(3.37) $\mathcal{L}_{1}(x)=1-x\quad\text{ and
}\quad\mathcal{L}_{3}(x)=\frac{1}{6}\left(6-18x+9x^{2}-x^{3}\right).$

The distance of the closest lattice point to the origin is $(n+1)\pi$, so we
see that $\mathcal{L}_{1}(x)$ and $\mathcal{L}_{3}(x)$ are negative when
evaluated at lattice points (except for the origin). Thus, for all
$(k,l)\in{\mathbb{Z}}^{2}\setminus\\{(0,0)\\}$ we have

(3.38)
$\left|\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)\right|e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}=-\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}.$

Hence, relation (3.33) can equivalently be written as

(3.39)
$\sum_{(k,l)\in{\mathbb{Z}}^{2}\setminus\\{0,0\\}}\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}\leq-1$

which ultimately gives the equality

(3.40)
$\sum_{(k,l)\in{\mathbb{Z}}^{2}\setminus\\{0,0\\}}\mathcal{L}_{n}\left((n+1)\pi(k^{2}+l^{2})\right)e^{-\frac{(n+1)\pi}{2}(k^{2}+l^{2})}=-1.$

The claims follow by adding the value at the origin, which contributes 1, to
the sum. ∎

For the proof of this result, we made use of the fact that considered systems
do not form frames. It would be nice to find a proof which does not need this
fact.

As $h_{1}$ remains the only case where the frame set conjecture is still open
(no further restrictions than the ones from [29] are known), we considered it
worthwhile to apply the Janssen test in this case. The result is that we do
not get any new insights.

######  Lemma 3.4.

For the Gabor system $\mathcal{G}(h_{1},\delta^{-1/2}{\mathbb{Z}}^{2})$,
$\delta\geq 1$, the Janssen test decreases, i.e.,

(3.41)
$\delta\mapsto\sum_{k,l}|\mathcal{L}_{1}(\pi\delta(k^{2}+l^{2}))|e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}$

is a decreasing function in $\delta$.

###### Proof.

We introduce the function $j_{1}$, which we define, for $\delta\geq 1$, to be

(3.42)
$j_{1}(\delta)=\sum_{k,l\in{\mathbb{Z}}}|\mathcal{L}_{1}(\pi\delta(k^{2}+l^{2}))|e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}=\sum_{k,l\in{\mathbb{Z}}}|1-\pi\delta(k^{2}+l^{2})|e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}.$

All terms have a negative contribution, except for the constant term, which is
1, provided by the origin. So, we have

(3.43)
$j_{1}(\delta)=1+\sum_{k,l\in{\mathbb{Z}}\setminus\\{0\\}}(\pi\delta(k^{2}+l^{2})-1)e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})},$

which is clearly smooth on $(1,\infty)$ and continuous at 1. Taking the
derivative, we obtain

(3.44) $\displaystyle j_{1}^{\prime}(\delta)$
$\displaystyle=\sum_{k,l\in{\mathbb{Z}}\setminus\\{0\\}}\left(\pi(k^{2}+l^{2})e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}-\frac{\pi(k^{2}+l^{2})}{2}(\pi\delta(k^{2}+l^{2})-1)e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}\right)$
(3.45)
$\displaystyle=\frac{\pi}{2}\sum_{k,l\in{\mathbb{Z}}\setminus\\{0\\}}(k^{2}+l^{2})\left(3-\pi\delta(k^{2}+l^{2})\right)e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}<0.$

∎

Proposition 3.3 together with Lemma 3.4 in particular shows that the Janssen
test cannot be used to enlarge the frame set of $h_{1}$ for Gabor systems of
the form $\mathcal{G}(h_{1},\delta^{-1/2}{\mathbb{Z}}^{2})$ for $1<\delta<2$.
We remark that it also seems to be true that the Janssen test is inconclusive
for rectangular Gabor systems of the form
$\mathcal{G}(h_{1},\delta^{-1/2}(a{\mathbb{Z}}\times(1/a){\mathbb{Z}}))$,
$a\in{\mathbb{R}}_{+}$ and $1<\delta<2$.

###  3.3. Outside the safety region

In this section, we show that the frame set of the Hermite function $h_{n}$ on
the main diagonal $a=b$ may have a rather surprising structure. In particular,
$\mathfrak{F}(h_{n})$ may contain a large segment on the main diagonal outside
of the safety region. Our results do not rule out the option that all pairs
$(a,a)$, $a<1$, belong to $\mathfrak{F}(h_{n})$, $n\geq 4$, except for the
points ruled out by Lemvig [26, Thm. 7, Thm. 8] and Lyubarskii and Nes [29],
but we do not dare to come up with a new conjecture.

####  3.3.1. The 15-th Hermite function

For illustrative reasons, we decided to pick the particular case $n=15$. We
prove the following result.

######  Proposition 3.5.

The Gabor system $\mathcal{G}(h_{15},1/\sqrt{\delta}\ {\mathbb{Z}}^{2})$ is a
frame for $L^{2}({\mathbb{R}})$ for $\delta\geq 11$.

###### Proof.

We follow our approach in the proof of Theorem 1.1 and use the Janssen test
with sharper estimates on the Laguerre polynomials for the particular case
$n=15$. By Lemma 2.6 and (2.14), it suffices to show that

(3.46)
$\sum\limits_{k,l\in{\mathbb{Z}}}\left|\mathcal{L}_{15}\left(\pi\delta(k^{2}+l^{2})\right)\right|e^{-\frac{\pi\delta}{2}(k^{2}+l^{2})}<2\quad\text{for
}11\leq\delta\leq 16.$

We split the summation and, for $\varepsilon\in(0,1)$, reduce the problem to
the verification of

(3.47) $\displaystyle
4|\mathcal{L}_{15}(\pi\delta)|e^{-\frac{\pi}{2}\delta}+4|\mathcal{L}_{15}(2\pi\delta)|e^{-\pi\delta}+4|\mathcal{L}_{15}(4\pi\delta)|e^{-2\pi\delta}+4|\mathcal{L}_{15}(5\pi
t)|e^{-\frac{5\pi}{2}\delta}$ $\displaystyle<1-\varepsilon$ (3.48)
$\displaystyle\sum\limits_{k^{2}+l^{2}\geq
8}\left|\mathcal{L}_{15}\left(\pi\delta(k^{2}+l^{2})\right)\right|e^{-\frac{\pi\delta}{2}(k^{2}+l^{2})}$
$\displaystyle<\varepsilon.$

First, we deal with the tail estimate. By (2.17), we have

(3.49) $\sum\limits_{k^{2}+l^{2}\geq
8}\left|\mathcal{L}_{15}\left(\pi\delta(k^{2}+l^{2})\right)\right|e^{-\frac{\pi\delta}{2}(k^{2}+l^{2})}\leq
16\binom{15}{7}\pi^{15}\delta^{15}\sum\limits_{k^{2}+l^{2}\geq
8}(k^{2}+l^{2})^{15}e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}.$

Next, we apply Lemma 2.10 with $\gamma=\delta$, $a=\sqrt{8}$, and get

(3.50) $\sum\limits_{k^{2}+l^{2}\geq
8}(k^{2}+l^{2})^{15}e^{-\frac{\pi}{2}\delta(k^{2}+l^{2})}\leq
4\,\frac{8^{15}}{\delta}e^{-4\pi\delta}.$

Since

$2^{4+2+45}\binom{15}{7}\pi^{15}\leq
10^{27}\quad\text{and}\quad\delta^{14}e^{-4\pi\delta}<10^{-39}\quad\text{for
}11\leq\delta\leq 16,$

we deduce that

(3.51) $\sum\limits_{k^{2}+l^{2}\geq
8}\left|\mathcal{L}_{15}\left(\pi\delta(k^{2}+l^{2})\right)\right|e^{-\frac{\pi\delta}{2}(k^{2}+l^{2})}\leq
10^{-12}.$

For the remaining layers we use the following estimates for
$\delta\in[11,16]$:

$|\mathcal{L}_{15}(\pi\delta)e^{-\frac{\pi\delta}{2}}|\leq
0.15,\quad|\mathcal{L}_{15}(2\pi\delta)e^{-\pi\delta}|\leq 0.04,$
$|\mathcal{L}_{15}(4\pi\delta)e^{-2\pi\delta}|\leq
10^{-10},\quad|\mathcal{L}_{15}(5\pi\delta)e^{-\frac{5\pi}{2}\delta}|\leq
10^{-16}.$

We support these bounds with Fig. 5. However, they can be verified by hand, as
$\mathcal{L}_{15}$ is a fixed polynomial and evaluated on a compact interval,
and its roots (given, e.g., in [32]) and critical points can be calculated
with any required precision.

Combining these estimates with (3.51), we arrive at

$\sum\limits_{k^{2}+l^{2}\geq 8}\left|\mathcal{L}_{15}\left(\pi
t(k^{2}+l^{2})\right)\right|e^{-\frac{\pi t}{2}(k^{2}+l^{2})}\leq 1+4\cdot
0.15+4\cdot 0.04+4\cdot 10^{-10}+4\cdot 10^{-16}<1.8.$

∎

(a) $m=1$ (b) $m=2$

(c) $m=4$ (d) $m=5$

Figure 5. The functions $|\mathcal{L}_{15}(m\pi\delta)|e^{-m\pi\delta/2}$ for
$m=1,2,4,5$.

######  Remark 3.6.

Numerical investigations (see Fig. 6) suggest that the Gabor system
constitutes a frame in $L^{2}({\mathbb{R}})$ in a neighborhood of
$\delta=3.06$, which is very far from the safety region. This can be checked
with the methods used in Proposition 3.1. We have

(3.52) $\sum_{|k|,|l|\leq 5}|\mathcal{L}_{15}(3.06\
\pi(k^{2}+l^{2}))|e^{-\frac{3.06\ \pi}{2}(k^{2}+l^{2})}\leq 1.96933$

and the remainder term is again small:

(3.53) $\displaystyle\sum_{\max{|k|,|l|}\geq 6}|\mathcal{L}_{15}(3.06\
\pi(k^{2}+l^{2}))|e^{-\frac{3.06\ \pi}{2}(k^{2}+l^{2})}$ $\displaystyle\leq
16\binom{16}{8}(3.06\pi)^{15}\sum_{m\geq
36}m^{16}\left(e^{-\frac{3.06\pi}{2}}\right)^{m}$ (3.54)
$\displaystyle<10^{21-50}=10^{-29}.$

Hence, the Gabor system $\mathcal{G}(h_{15},1/\sqrt{3.06}\ {\mathbb{Z}}^{2})$
is a frame and there is an open neighborhood $U$ around
$(1/\sqrt{3.06},1/\sqrt{3.06})$ such that
$\mathcal{G}(h_{15},a_{U}{\mathbb{Z}}\times b_{U}{\mathbb{Z}})$ is a frame for
$(a_{U},b_{U})\in U$.

Figure 6. The Janssen test for the Gabor system
$\mathcal{G}(h_{15},\delta^{-1/2}{\mathbb{Z}}^{2})$. For the square lattice of
density $\delta>16$, we are in the safety region. However, the Janssen test
provides a sufficient criterion way earlier, e.g., from $\delta\geq 10$. It is
inconclusive around $\delta=7.1$, but suffices around $\delta=3.06$. The gray
area is for visual aid where the value of the test is above or below 2.

######  Remark 3.7.

As can be seen from Fig. 6, the safety region for $h_{15}$ and
$\delta^{-1/2}{\mathbb{Z}}^{2}$ can be extend to, say, $\delta=9.6$. As the
frame set of a function in $M^{1}({\mathbb{R}})$ is open [17], this also means
that there is a neighborhood around the pairs
$(1/\sqrt{\delta},1/\sqrt{\delta})$ for $\delta\geq 9.6$ which belongs to
$\mathfrak{F}(h_{15})$. The above estimates, however, need refinements for
$\delta\in[9.6,11]$ as $\mathcal{L}_{15}(\delta\pi)$ has a zero close to
$\delta=10$ ($\mathcal{L}_{15}(x)$ has a root at $x=31.407519169754$ [32]
which gives $\delta\approx 9.99732$). The mass of the first layer is small,
while the second layer contributes more mass. By dividing the interval
$[9.6,11]$ into a finite number of sub-intervals and performing estimates
similar to the ones above, we get the desired extension. Moreover, with a
finite number of computations, it seems possible to enlarge the safety region
for $h_{15}$ for rectangular lattices $a{\mathbb{Z}}\times b{\mathbb{Z}}$ to,
say, $ab<1/11$, for $a,b\in[1,1.3]$, as shown in Fig. 7.

Figure 7. The safety region for $h_{15}$ and a numerical extension provided by
the Janssen test. For say $a,b\in[0,1.3]$, which are the depicted parameters,
parts of this extensions should be verifiable by hand, e.g., for $ab\leq
1/11$.

####  3.3.2. The 9-th Hermite function

In [20], the existence of pairs $(a,b)$ with $ab=1/3$ which do not belong to
the frame set of $h_{n}$, $n\geq 3$ was shown. We consider the example of
$h_{9}$ and the square lattice of density 3, i.e., $1/\sqrt{3}\
{\mathbb{Z}}^{2}$ and show that
$\mathcal{G}(h_{9},1/\sqrt{3}{\mathbb{Z}}^{2})$ is a frame.

######  Proposition 3.8.

The Gabor system $\mathcal{G}(h_{9},1/\sqrt{3}\ {\mathbb{Z}}^{2})$ is a frame.

###### Proof.

We start with a finite number of computations, which gives

(3.55) $\sum_{|k|,|l|\leq
5}\mathcal{L}_{9}(3\pi(k^{2}+l^{2}))e^{-\frac{3\pi}{2}(k^{2}+l^{2})}\leq
1.76496.$

The bound on the remainder is derived analogously to Proposition 3.1. We
estimate

(3.56) $\displaystyle\sum_{\max{|k|,|l|}\geq
6}\mathcal{L}_{9}(3\pi(k^{2}+l^{2}))e^{-\frac{3\pi}{2}(k^{2}+l^{2})}$
$\displaystyle\leq\sum_{\max{|k|,|l|}\geq
6}10\binom{10}{5}(3\pi(k^{2}+l^{2}))^{9}e^{-\frac{3\pi}{2}(k^{2}+l^{2})}$
(3.57) $\displaystyle<10^{13-58}=10^{-45}.$

∎

For $h_{0}$ and $h_{1}$ we know that density 3 is sufficient as we are in the
safety region [18]. For $h_{2}$ we know that $1/\sqrt{3}\ {\mathbb{Z}}^{2}$
does not provide a frame [26]. Curiously, the only cases $n\geq 3$ where the
Janssen test seems to imply that $\mathcal{G}(h_{n},1/\sqrt{3}\
{\mathbb{Z}}^{2})$ is a frame (see Fig. 8), are $n=4$ (where the value does
not exceed 1.59338) and $n=9$, which we just proved. Due to [26, Thm. 7, Thm.
8], only Hermite functions of order $4m$ and $4m+1$ can constitute a frame
with a square lattice of density 3, in particular
$\mathcal{G}(h_{15},1/\sqrt{3}\ {\mathbb{Z}}^{2})$ is not a frame.

Figure 8. Values of the Janssen test for $\mathcal{G}(h_{n},1/\sqrt{3}\
{\mathbb{Z}}^{2})$, $n=0,\ldots,40$. Only for $n\in\\{0,1,4,9\\}$ the Janssen
test suffices to see that we have a frame.

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# Assessing the Quality of Gridded Population Data for Quantifying the
Population Living in Deprived Communities

Agatha C. Hennigen de Mattos
School of Computer Science
University College Dublin
Ireland
<EMAIL_ADDRESS>
Gavin McArdle
School of Computer Science
University College Dublin
Ireland
<EMAIL_ADDRESS>
Michela Bertolotto
School of Computer Science
University College Dublin
Ireland
<EMAIL_ADDRESS>

###### Abstract

Over a billion people live in slums in settlements that are often located in
ecologically sensitive areas and hence highly vulnerable. This is a problem in
many parts of the world, but it is more prominent in low-income countries,
where in 2014 on average 65% of the urban population lived in slums. As a
result, building resilient communities requires quantifying the population
living in these deprived areas and improving their living conditions. However,
most of the data about slums comes from census data, which is only available
at aggregate levels and often excludes these settlements. Consequently,
researchers have looked at alternative approaches. These approaches, however,
commonly rely on expensive high-resolution satellite imagery and field-
surveys, which hinders their large-scale applicability. In this paper, we
investigate a cost-effective methodology to estimate the slum population by
assessing the quality of gridded population data. We evaluate the accuracy of
the WorldPOP and LandScan population layers against ground-truth data composed
of 1,703 georeferenced polygons that were mapped as deprived areas and which
had their population surveyed during the 2010 Brazilian census. While the
LandScan data did not produce satisfactory results for most polygons, the
WorldPOP estimates were less than 20% off for 67% of the polygons and the
overall error for the totality of the studied area was only -5.9%. This small
error margin demonstrates that population layers with a resolution of at least
a 100m, such as WorldPOP’s, can be useful tools to estimate the population
living in slums111Code available at: https://github.com/ml-labs-crt/popgrids.

##  1 Introduction

More than half of the world population live in cities and it is estimated that
over a billion live in slums [5]. Though many definitions of slums exist [9],
in this paper the term “slum” refers to material conditions of poverty, such
as lack of access to public services, water and sanitation for example, and
lack of house durability. This is a problem in many parts of the world, but it
is more prominent in low-income countries, where in 2014 on average 65% of the
urban population lived in slums, as opposed to 44% in lower-middle-income and
24% in upper-middle-income countries [1]. Slums are often located in areas
that are ecologically sensitive, critical for biodiversity, close to water
bodies or important for flood protection and, as a consequence, a city’s risks
and vulnerability are highly concentrated in these settlements [12]. In the
aftermath of a disaster, uncertainty over population figures and demographic
information constitutes one of the main barriers to accurate needs assessment
[11] and the absence of such data makes it impossible to adequately measure
the impact of the disaster. Hence, to build resilient communities, it is
essential that slums and their population are mapped and that their living
conditions are improved [12]. In this context, the United Nations ratified
goal 11.1 of the Sustainable Development Goals, which states that adequate,
safe and affordable housing should be accessible for all and that slums should
be upgraded by 2030 [5]. Countries, therefore, are expected to be able to
report the progress towards this goal, which includes providing reliable
disaggregated data about the percentage of the population living in slums at
national and sub-national levels [5]. However, most of the information about
the population living in these areas comes from census data, which is only
made available at aggregated levels [7], and updated at lengthy intervals,
making the information obsolete and unsuitable to capture the dynamism of
slums [10]. Moreover, some governments are reluctant to recognize slums, which
are often excluded from official statistics [5]. As a result, other approaches
to provide the percentage of the population living in slums have been
developed, such as using high-resolution satellite imagery and ground-truth
field surveys to train convolutional neural networks to classify slum imagery
patches according to their residential occupancy levels [7]. Yet, the cost for
acquiring and processing high-resolution imagery as well as conducting field
surveys hinders the application of these methods at scale by governments in
low-income countries or not-for-profit organizations. An alternative would be
to map areas using freely available satellite imagery and employ other free
sources of population data to quantify the number of slum dwellers. In this
work, we investigate this latter cost-effective approach by assessing the
suitability of gridded population datasets to estimate the population living
in areas that have been mapped as slums in the metropolitan region of São
Paulo, Brazil. More specifically, we compare the population estimates provided
by WorldPOP and LandScan’s gridded population data with population data
obtained through field surveys. In the next section, the methodology is
described. Section 3 discusses the results and Section 4 presents conclusions
and future work.

##  2 Methodology

Gridded population data are freely available and are usually produced by
private companies or research institutions using machine learning, statistics
and census data. In this study, we evaluate two of these population layers,
WorldPOP and LandScan, as they are the only ones available for 2010, the year
that our ground-truth data was collected. The WorldPOP data were developed
with the support of many organizations [15], while the LandScan data were
created by the Oak Ridge National Laboratory, sponsored by the U.S. Department
of Energy. The gridded population data consist of a raster file with values
that represent the number of people living in a small square area, also called
a grid, with a resolution that can go from a 3 arc-second (100 m), in the case
of WorldPOP, to 30 arc-second (1 km) for LandScan (see Figure 1 for an example
of the data for our area of interest).

Figure 1: LandScan Gridded Population Data (2010) for the metropolitan region
of São Paulo and georeferenced polygons of deprived areas (ground-truth data).

Gridded population data are generated by complex models that use a top-down
approach to disaggregate census data. LandScan uses official census data and
projections calculated by the US Census Bureau International Programs and
WorldPOP utilises the Gridded Population of the World 4.10 [13]. Additionally,
WorldPOP offers two options of gridded population layers, one that is adjusted
using United Nations statistics and one that is not adjusted. In this work, we
chose the latter version as our ground-truth is from the same year that the
census was conducted. As mentioned previously, census data and projection
statistics are only an input to these models that also employ many other
parameters such as information on roads, land cover, built structures, cities
or urban areas, infrastructure, environmental data, protected areas, water
bodies and, in the case of WorldPOP, night-time lights [16] to disaggregate
census population counts into grids. WorldPOP uses random forests to select
the geospatial features that will be used in the model and also makes its code
open source [15]. On the other hand, LandScan’s model is not completely open
[16] and is adapted to each region to account for specific geographical nature
and data conditions [8].

The ground-truth data used to assess the accuracy of the two gridded
population data can be seen in Figure 1 and is composed of 1,703 polygons that
were mapped as deprived areas in the 2010 Brazilian census through field-
surveys [6]. Each polygon has information about the name of the settlement,
geographical coordinates of the area and the number of inhabitants. The
polygons are contained in the metropolitan region of São Paulo, which in 2010
was home to 18.9% of the country’s population living in slum-like conditions.
The criteria used for mapping these communities was that the majority of
households in the surveyed area lacked access to at least one basic public
service, tenure of the land, either at the moment of the survey or during the
last 10 years, and that the area did not meet construction standards. As
mentioned in the introduction, many definitions of slums exist and, as a
consequence, any criteria can introduce biases by including or excluding
households depending on their housing conditions. For instance, the criteria
used in our ground-truth data does not include areas in which only a few
households lack access to public services. Figure 2 summarizes the methodology
used in our experiments and our code is available online1. Firstly, the
ground-truth data was pre-processed to extract population information, filter
our region of interest and convert the coordinates to the same coordinate
system of the gridded population data. Secondly, population layers were
clipped in mini-rasters matching the location of the polygons in the ground-
truth data. Lastly, a comparison of the results against the ground-truth data
was obtained.

Figure 2: Methodology used to extract the population of each slum area in the
metropolitan region of São Paulo.

##  3 Results and Discussion

The comparison of the estimates obtained with WorldPOP [15] and LandScan data
[2] and the ground-truth data shows that satisfactory results could not be
obtained using LandScan’s population layer, but that WorldPOP produced an
estimate of the population living in slums that was only 5.9% less than the
official data (it estimated 2,035,866 people living in the deprived areas
whereas the ground-truth data indicate 2,162,368). When it comes to the
performance at polygon level, the WorldPOP estimated the population of 67% of
the polygons within a relative error between 0% and $\pm$20%, and 24% of the
polygons within an error ranging from $\pm$20 to $\pm$100% (see Table 1). The
relative error was calculated for each polygon by dividing the difference
between the estimated population and the ground-truth population by the
ground-truth population. LandScan’s poor performance can be attributed to its
coarse resolution in comparison to the area of the polygons, which leads to a
situation in which any strategy to count the values of the pixels can produce
errors. That is, if pixels that only partially overlap the polygons are
included, they produce an overestimation of the population whereas if they are
not included, many polygons’ population estimates result in null values.

Even though the literature of the use of gridded population data to estimate
slum dwellers is, to the best of our knowledge, non-existent, WorldPOP results
are in line with the work of Sliuzas et al. [14]. Sliuzas et al. [14] found
that Global Human Settlement Layer products tended to underestimate the built-
up area in deprived areas such as slums in Kampala, Uganda, when compared to
the built-up area obtained using high-resolution satellite images. Since
WorldPOP also employs built-up structures in their modelling, it is possible
that the reason behind the underestimation is associated with missing built
structures in slum areas. Indeed, WorldPOP has already been found to overlook
built structures in rural areas [16]. Other gridded population data, such as
the High Resolution Settlement Layer [4], uses satellite imagery with higher
resolution than WorldPOP in their model, which could potentially lead to
better results. However, such datasets are not available for the year 2010,
when the ground-truth data was collected, and for this reason were not
included in this analysis. Regarding the differences in relative error of
certain polygons, for instance 22 polygons had errors superior to 100%,
further examination using high-resolution satellite imagery can provide
insight into the reasons behind it. On the positive side, the methodology
proposed in this paper can easily be replicated for other regions. Still, both
WorldPOP and LandScan make use of a top-down approach that does not interact
with the local communities, something that has been argued by researchers to
be desirable [7].

| WorldPOP | LandScan
---|---|---
Relative Error | # of polygons | % of polygons | # of polygons | % of polygons
0% to $\pm$20% | 1135 | 67% | 7 | 0%
+20 to +100% | 138 | 8% | 12 | 1%
-20 to -100% | 268 | 16% | 27 | 2%
>100% | 22 | 1% | 50 | 3%
Non-Applicable | 140 | 8% | 1607 | 94%
Table 1: Relative error of the population estimates using WorldPOP and
LandScan gridded population data.

##  4 Conclusions and Future Work

In this paper, we assess the quality of two gridded population data (WorldPOP
and LandScan) for estimating the population living in slums. Whilst LandScan
did not produce satisfactory results, WorldPOP provided an estimate that was
just 5.9% less than the ground-truth data, which suggests that gridded
population data with a resolution of at least 100m can be a useful tool for
estimating the population living in slums.

In the future, we would like to extend this work to countries where the
ground-truth was collected in the year 2015, since more gridded population
dataset have estimates for that year. Also, we would like to contemplate
regions whose census is shared at a lower granularity than Brazil, as these
countries may have less accurate gridded population estimates. Ideally, we
would like to compare the results of top-down approaches, such as the
population layers described in this paper, with bottom-up ones, such as the
Geo-Referenced Infrastructure and Demographic Data for Development (GRID3),
currently only available for four countries, and the World Settlement
Footprint 2015, which is at the moment under restricted access. Finally, to
achieve goal 11.1 of the Sustainable Development Goals, the analysis pipeline
developed in this study needs to be extended to include a cost-effective
method to detect slum areas and the conditions in these settlements improved.

## Broader Impact

This research raises many ethical issues that should be considered. Firstly,
it is important to be aware of the many definitions of the word “slum” [9] and
that, despite it being used in the Sustainable Development Goals and United
Nations Habitat reports [5], it can be considered a derogatory term [3].
Secondly, machine learning applications, like gridded population data and
scripts to extract information from these applications (such as the
methodology proposed in this paper) can be useful to inform public policy and
monitor progress towards goals, but they do not substitute field-surveys and
interaction with the affected communities. In fact, our research will be of
best use if adopted in conjunction with official census data collection to
address quality issues that might arise in the data collection process [10] or
the reluctance of some governments to include slums in official statistics
[5]. Finally, for the research that quantifies the population living in
deprived areas to promote positive social impact, its output needs to be used
with the interest of the poor in mind.

## Acknowledgments

Science Foundation Ireland supported this research through the SFI Centre for
Research Training in Machine Learning (18/CRT/6183).

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