# Sampling in quasi shift-invariant spaces and Gabor frames generated by
ratios of exponential polynomials

Alexander Ulanovskii  Department of Mathematics and Physics, University of
Stavanger, 4036 Stavanger, Norway<EMAIL_ADDRESS>and  Ilya
Zlotnikov  Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-
Platz 1, A-1090 Vienna, Austria<EMAIL_ADDRESS>
###### Abstract.

We introduce two families of generators (functions) $\mathcal{G}$ that consist
of entire and meromorphic functions enjoying a certain periodicity property
and contain the classical Gaussian and hyperbolic secant generators. Sharp
results are proved on the density of separated sets that provide non-uniform
sampling for the shift-invariant and quasi shift-invariant spaces generated by
elements of these families. As an application, we obtain new sharp results on
the density of semi-regular lattices for the Gabor frames generated by
elements from these families.

###### Key words and phrases:

Non-uniform sampling, Shift-invariant space, Gabor frame, Exponential
polynomial, Quasi shift-invariant space.

I. Z. gratefully acknowledge support from the Austrian Science Fund (FWF)
[10.55776/P33217]

##  1\. Introduction and main results

A countable set $\Gamma\subset{\mathbb{R}}$ is called separated if

$\inf_{\gamma,\gamma^{\prime}\in\Gamma,\gamma\neq\gamma^{\prime}}|\gamma-\gamma^{\prime}|>0.$

Given a generator (function) $\mathcal{G}$ with a ”reasonably fast” decay at
$\pm\infty$ and a number $p,1\leq p\leq\infty$, the shift-invariant space
$V_{\mathbb{Z}}^{p}(\mathcal{G})$ consists of all functions $f$ of the form

$f(x)=\sum_{n\in{\mathbb{Z}}}c_{n}\mathcal{G}(x-n),\quad\\{c_{n}\\}\in
l^{p}({\mathbb{Z}}).$

More generally, given a separated set $\Gamma\subset{\mathbb{R}}$, one may
consider the quasi shift-invariant space $V_{\Gamma}^{p}(\mathcal{G})$ which
consists of all functions of the form

$f(x)=\sum_{\gamma\in\Gamma}c_{\gamma}\mathcal{G}(x-\gamma),\quad\\{c_{\gamma}\\}\in
l^{p}(\Gamma).$

An important class of generators is the Wiener amalgam space $W_{0}$, which
consists of measurable functions $\mathcal{G}:{\mathbb{R}}\to{\mathbb{C}},$
satisfying

(1.1)
$\|\mathcal{G}\|_{W}:=\sum\limits_{k\in{\mathbb{Z}}}\|\mathcal{G}\|_{L^{\infty}(k,k+1)}<\infty.$

Shift-invariant and quasi sift-invariant spaces have important applications in
mathematics and engineering, in particular since they are often used as models
for spaces of signals and images. It is also well known that there is a close
connection between the Gabor frames and sampling sets for the shift-invariant
spaces.

A classical example is the Paley–Wiener space $PW^{2}_{\pi}$ which is exactly
the shift-invariant space $V^{2}_{\mathbb{Z}}(\mathcal{G})$ generated by the
sinc function $\mathcal{G}(x)=\sin(\pi x)/(\pi x).$ The remarkable result in
digital signal processing is the Shannon–Whittaker–Kotelnikov sampling theorem
that states that every $f\in PW^{2}_{\pi}$ can be reconstructed from its
values at the integers:

$f(x)=\sum_{n\in{\mathbb{Z}}}f(n)\,\mbox{sinc}(x-n).$

This implies that the set of integers ${\mathbb{Z}}$ is a sampling set for
$PW^{2}_{\pi}$.

The theory of shift-invariant spaces is by now well developed and a number of
sampling theorems are proved for various generators. Due to the mentioned
relation between Gabor frames and sampling in shift-invariant spaces (see
Lemma 2.3), we may say that certain sampling theorems are available for

  * •

B-splines, see [1, 11],

  * •

Hermite functions, see [6, 7, 19],

  * •

Truncated and symmetric exponential functions, see [13, 14],

  * •

Gaussian kernel (see [20, 21, 22, 23]), hyperbolic secant (see [15] and a very
recent paper [2]), and, more generally, totally positive functions, see [9,
10],

  * •

Rational functions, see [3, 4].

The sampling problem for quasi shift-invariant spaces is significantly more
complicated and results are scarcer. For a totally positive generator
$\mathcal{G}$ of finite type a sufficient condition for stable sampling in
terms of covering (or maximum gap111See also a more general condition
$(C_{r}(\varepsilon))$ in Theorem 16 in [10].) for a quasi shift-invariant
space was obtained in [10, Theorems 2 and 16]. The approach in this paper was
based on an application of Schoenberg and Whitney’s characterization of the
invertibility of a pre-Gramian matrix generated by a totally positive function
of finite type.

In this paper, we introduce two families of generators. Using complex-analytic
methods, we prove sharp results on the density of sampling sets for the
corresponding shift-invariant and quasi shift-invariant spaces. As an
application, we obtain new sharp results on the density of semi-regular
lattices for the Gabor frames with generators from these families.

###  1.1. Sampling sets and Beurling densities

A separated set $\Lambda\subset{\mathbb{R}}$ is called a (stable) sampling set
for $V^{p}_{\Gamma}(\mathcal{G})$ if the following sampling inequalities

$A\|f\|_{p}^{p}\leq\sum_{\lambda\in\Lambda}|f(\lambda)|^{p}\leq
B\|f\|_{p}^{p},\quad 1\leq p<\infty,$ $\|f\|_{\infty}\leq
K\sup\limits_{\lambda\in\Lambda}|f(\lambda)|,\quad p=\infty,$

hold true with some positive constants $A,B,K$ and for every $f\in
V_{\Gamma}^{p}(\mathcal{G}).$

The lower and upper uniform densities of a separated set $\Lambda$ (sometimes
called the Beurling densities) are defined by

(1.2)
$D^{-}(\Lambda):=\lim_{R\to\infty}\inf_{x\in\mathbb{R}}\frac{\\#(\Lambda\cap[x-R,x+R])}{2R},$
(1.3)
$D^{+}(\Lambda):=\lim_{R\to\infty}\sup_{x\in\mathbb{R}}\frac{\\#(\Lambda\cap[x-R,x+R])}{2R}.$

These densities play a key role in the study of sampling and interpolation
sets.

Throughout the paper, $\Gamma$ denotes a separated set of translates. To avoid
trivial remarks, in what follows we always assume that $\Gamma$ is relatively
dense, i.e. $D^{-}(\Gamma)>0.$

We will now introduce two classes of generators.

###  1.2. Class $\mathcal{K}(\alpha)$

Given a number $\alpha>0$ and a rational function $\mathcal{R}=P/Q$, we
consider $2\pi i/\alpha$-periodic generator

(1.4) $\mathcal{G}(z)=\mathcal{R}(e^{\alpha z})=\frac{P(e^{\alpha
z})}{Q(e^{\alpha z})}.$

######  Definition 1.1.

We denote by $\mathcal{K}(\alpha)$ the class of all generators $\mathcal{G}$
defined in (1.4) where $P$ and $Q$ are non-trivial complex polynomials without
common zeros and satisfying the following three conditions:

  1. ($A$)

$1\leq{\rm deg}\,P<{\rm deg}\,Q$;

  2. ($B$)

$P(0)=0$;

  3. ($C$)

$Q(x)\neq 0,x\geq 0$.

One may check that conditions $(A)-(C)$ above are necessary and sufficient for
the generator $\mathcal{G}$ defined in (1.4) to be bounded on ${\mathbb{R}}$,
and that these conditions imply

(1.5) $|g(x)|\leq C\exp\\{-\alpha|x|\\},\quad x\in{\mathbb{R}},$

where $C=C(\mathcal{R})$ is a constant. Hence every
$\mathcal{G}\in\mathcal{K}$ has exponential decay at $\pm\infty.$

Note that the classical hyperbolic secant generator belongs to
$\mathcal{K}(1)$, since it is of the form (1.4), where $\alpha=1$ and
$\mathcal{R}(z)=z/(1+z^{2}).$

Observe that for every rational function $\mathcal{R}$ one may find the
largest integer $k$ such that

$\mathcal{R}(e^{2\pi i/k}z)=c\mathcal{R}(z),\quad z\in{\mathbb{C}},$

where $c$ is a constant. For example,

  * •

if $\mathcal{R}(z)=z/(1+z^{2})$, then $k=2$ and $c=-1$;

  * •

if $\mathcal{R}(z)=z/(1+z^{4})$, then $k=4$ and $c=i$;

  * •

if $\mathcal{R}(z)=z/(1+z)^{2}$, then $k=1$ and $c=1$.

Then, clearly, the generator $\mathcal{G}$ defined in (1.4) satisfies
$\mathcal{G}(z+2\pi i/k\alpha)=c\,\mathcal{G}(z),z\in{\mathbb{C}}.$

######  Definition 1.2.

Assume $\mathcal{G}$ is defined in (1.4).

$(i)$ We denote by $k(\mathcal{G})$ is the greatest integer $k$ such that the
equality

(1.6) $\mathcal{G}(z+2\pi i/k\alpha)=c\mathcal{G}(z),\quad z\in{\mathbb{C}},$

holds with some constant $c\in{\mathbb{C}}$.

$(ii)$ We set $q(\mathcal{G})=$ deg$\,Q$, where $Q$ is the denominator in
(1.4).

###  1.3. Class $\mathcal{C}(\alpha)$

######  Definition 1.3.

We denote by $\mathcal{C}(\alpha)$ the class of all generators $\mathcal{G}$
defined by

(1.7) $\mathcal{G}(z)=e^{-\alpha z^{2}/2}\mathcal{R}(e^{\alpha z})=e^{-\alpha
z^{2}/2}\frac{P(e^{\alpha z})}{Q(e^{\alpha z})},$

where $\alpha>0$, $P,Q$ are non-trivial complex polynomials without common
zeros and $Q(x)\neq 0,x\geq 0$. Again, we set $q(\mathcal{G})=$deg$\,Q$, where
$Q$ is the denominator in (1.7).

Clearly, the assumption $Q(x)\neq 0,x\geq 0,$ is necessary and sufficient for
a generator $\mathcal{G}$ defined in (1.7) to be bounded on ${\mathbb{R}}$.
Moreover, every such $\mathcal{G}$ has ”Gaussian decay” at $\pm\infty$.

The case $P=Q\equiv 1$ corresponds to the classical case of Gaussian generator
and has been studied previously in [20] and [23]. The results below hold true
for this particular case.

Note that all generators from $\mathcal{K}(\alpha)$ and $\mathcal{C}(\alpha)$
defined in (1.4) and (1.7) belong to the Wiener amalgam space $W_{0}$ defined
in (1.1).

######  Remark 1.4.

One may extend the definition of classes $\mathcal{K}(\alpha)$ and
$\mathcal{C}(\alpha)$ by considering complex parameter $\alpha$ satisfying
$\rm{Re\,}\alpha>0$ and prove similar results.

###  1.4. Stability of $\Gamma$-shifts

Given a generator $\mathcal{G}$, a basic property of quasi shift-invariant
space $V^{p}_{\Gamma}(\mathcal{G})$ is that the $\Gamma$-shifts of
$\mathcal{G}$ are $l^{p}$-stable, i.e. there exist positive constants $C_{1}$
and $C_{2}$ such that

(1.8) $C_{1}\|{\bf
c}\|_{p}\leq\left\|\sum_{\gamma\in\Gamma}c_{\gamma}\mathcal{G}(\,\cdot-\gamma)\right\|_{p}\leq
C_{2}\|{\bf c}\|_{p},\quad\mbox{for every }{\bf c}=\\{c_{\gamma}\\}\in
l^{p}(\Gamma).$

This property implies that $V^{p}_{\Gamma}(\mathcal{G})$ is a closed subspace
of $L^{p}({\mathbb{R}})$ and that the system
$\\{\mathcal{G}(\,\cdot-\gamma)\\}_{\gamma\in\Gamma}$ forms an unconditional
basis in this space.

The stability property of ${\mathbb{Z}}$-shits is well-studied. The following
is an immediate corollary of Theorem 3.5 in [16]:

######  Lemma 1.5.

Assume $\mathcal{G}\in W_{0}$. Then the integer-shifts of $\mathcal{G}$ are
$l^{p}$-stable for every $p\in[1,\infty]$ if and only the Fourier transform
$\hat{\mathcal{G}}$ of $\mathcal{G}$ satisfies

(1.9) $\hat{\mathcal{G}}\mbox{ does not vanish on any set }{\mathbb{Z}}+b,\
0\leq b<1.$

Throughout the paper, we consider the standard form of Fourier transform,

$\hat{\mathcal{G}}(t):=\int_{\mathbb{R}}e^{-2\pi ixt}\mathcal{G}(x)\,dx.$

We also mention the paper [12], which proves the $l^{2}$-stability of
$\Gamma$-shifts for certain generators $\mathcal{G}$ under the condition that
$\Gamma$ is a complete interpolating sequence for the Paley–Wiener space
$PW_{\pi}^{2}$.

We would like to get conditions on $\Gamma$ and $\mathcal{G}$ that imply
property (1.8) for every value of $p\in[1,\infty].$ In fact, the right hand-
side inequality in (1.8) is true for every separated set $\Gamma$, every
generator $\mathcal{G}\in W_{0}$ and every $p\in[1,\infty]$, see Lemma 8.1
below. On the other hand, under a mild additional condition on the generator,
it suffices to check that the left hand-side inequality is true for
$p=\infty$:

######  Theorem 1.6.

Let $\Gamma$ be a separated set and $\mathcal{G}\in W_{0}\cap
C({\mathbb{R}})$. If the left hand-side inequality in (1.8) is true for
$p=\infty$, then it is true for every $p\in[1,\infty]$.

In what follows, we will say that $\Gamma$-shifts of $\mathcal{G}$ are stable,
if they are $l^{p}$-stable, for every $p\in[1,\infty].$

Let us now recall Beurling’s notion of the weak limit of a sequence of sets. A
sequence $\\{\Gamma(n):n\in{\mathbb{N}}\\}$ of separated subsets of
${\mathbb{R}}$ is said to converge weakly to a separated set
$\Gamma\subset{\mathbb{R}}$, if for every $R>0$ and $\epsilon>0$, there exist
$n_{\epsilon}\in{\mathbb{N}}$ such that for all $n\geq n_{\epsilon}$ we have

$\Gamma(n)\cap(-R,R)\subset\Gamma+(-\epsilon,\epsilon)\ \mbox{ and }\
\Gamma\cap(-R,R)\subset\Gamma(n)+(-\epsilon,\epsilon).$

Given a separated and relatively dense set $\Gamma\subset{\mathbb{R}}$, it is
easy to check that every real sequence $\\{s_{j}\\}$ contains a subsequence
$\\{s_{j_{n}}\\}$ such that the translates $\Gamma+s_{j_{n}}$ converge weakly
to some non-empty separated set $\Gamma^{\prime}$ as $n\to\infty$. Let
$W(\Gamma)$ denote the collection of all such weak limits. It is well-known
(and easy to check) that

(1.10) $D^{-}(\Gamma^{\prime})\geq D^{-}(\Gamma)\ \mbox{and
}D^{+}(\Gamma^{\prime})\leq D^{+}(\Gamma),\quad\mbox{for every }\
\Gamma^{\prime}\in W(\Gamma).$

As a simple example, we observe that

(1.11) $\mbox{The set }W({\mathbb{Z}})\mbox{ consists of all sets
}{\mathbb{Z}}+a,\ a\in[0,1).$

We will now formulate some sufficient conditions for the stability of
$\Gamma$-shifts for the generators from the families $\mathcal{K}(\alpha)$ and
$\mathcal{C}(\alpha)$ defined above. These will be given in terms of the set
$\Gamma$ and the poles $w_{j}$ of the rational function $\mathcal{R}$ in (1.4)
and (1.7).

Denote by $d(w_{j})$ the order of the pole $w_{j}$ and set
$d:=\max\limits_{j}d(b_{j})$. let ${\rm pol}_{d}(\mathcal{R})$ denote the set
of poles of $\mathcal{R}$ of order $d.$

We will consider the following assumptions:

  1. ($\Xi^{\prime}$)

The set ${\rm pol}_{d}(\mathcal{R})$ consists precisely of one element;

  2. ($\Xi^{\prime\prime}$)

For every $\Gamma^{\prime}\in W(\Gamma)$ there exists $w\in{\rm
pol}_{d}(\mathcal{R})$ and $\gamma^{\prime}\in\Gamma^{\prime}$ such that

(1.12)
$\log(w)-\log(w^{\prime})\notin\alpha(\Gamma^{\prime}-\gamma^{\prime})\quad\text{for
any }w^{\prime}\in{\rm pol}_{d}(\mathcal{R}),\,w\neq w^{\prime}.$

When $\Gamma={\mathbb{Z}}$, it follows easily from (1.11) that (1.12) is
equivalent to the simpler condition

  1. ($\Xi^{\prime\prime\prime}$)

$\log(w)-\log(w^{\prime})\notin\alpha{\mathbb{Z}}$, for every
$w^{\prime}\in{\rm pol}_{d}(\mathcal{R}),\,w\neq w^{\prime}.$

######  Proposition 1.7.

(i) If $\mathcal{G}\in\mathcal{C}(\alpha)$ satisfies $(\Xi^{\prime})$ or
$(\Xi^{\prime\prime\prime})$, then ${\mathbb{Z}}$-shifts of $\mathcal{G}$ are
$l^{\infty}$-stable.

(ii) If $\mathcal{G}\in\mathcal{K}(\alpha)$ satisfies $(\Xi^{\prime})$ or
$(\Xi^{\prime\prime})$, then $\Gamma$-shifts of $\mathcal{G}$ are
$l^{\infty}$-stable.

###  1.5. Sampling for generators from $\mathcal{K}(\alpha)$ and
$\mathcal{C}(\alpha)$

Our first sampling theorem concerns the class of generators
$\mathcal{K}(\alpha)$.

######  Theorem 1.8.

Given a generator $\mathcal{G}\in\mathcal{K}(\alpha),\alpha>0$, and two
separated sets $\Lambda,\Gamma\subset{\mathbb{R}}$. Assume that
$\Gamma$-shifts of $\mathcal{G}$ are stable and that

(1.13) $D^{-}(\Lambda)>\frac{q(\mathcal{G})}{k(\mathcal{G})}D^{+}(\Gamma).$

Then $\Lambda$ is a sampling set for $V^{p}_{\Gamma}(\mathcal{G})$, for every
$p\in[1,\infty]$.

Recall that the numbers $q(\mathcal{G})$ and $k(\mathcal{G})$ are defined in
Definition 1.2.

######  Remark 1.9.

Note that condition (1.13) is independent of parameter $\alpha$. However,
above we assume the stability of $\Gamma$-shifts. This property can be
violated for certain values of $\alpha$.

Let

(1.14) $\mathcal{H}(x):=\frac{e^{x}}{e^{2x}+1}\in\mathcal{K}(1)$

denote the hyperbolic secant generator. Consider the family of all finite
linear combinations

(1.15)
$\mathcal{G}(x)=\sum_{j=1}^{N}a_{j}\mathcal{H}(x-b_{j})=\sum_{j=1}^{N}a_{j}e^{b_{j}}\frac{e^{x}}{e^{2x}+e^{2b_{j}}}\in\mathcal{K}(1),\quad
a_{j}\in{\mathbb{C}},e^{2b_{j}}\not\in(-\infty,0).$

Clearly, every $\mathcal{G}$ in (1.15) satisfies $k(\mathcal{G})=2$ and
$q(\mathcal{G})=2N.$ Assuming the stability of ${\mathbb{Z}}$-shifts of
$\mathcal{G}$, Theorem 1.8 implies that every separated set $\Lambda$
satisfying $D^{-}(\Lambda)>N$ is a sampling set for
$V_{\mathbb{Z}}^{p}(\mathcal{G}),1\leq p\leq\infty.$ This result is sharp:

######  Theorem 1.10.

For every $N\geq 2$ there exist $a_{j}\in{\mathbb{R}},b_{j}>0,j=1,\dots,N,$
and a separated set $\Lambda,D^{-}(\Lambda)=N,$ such that the
${\mathbb{Z}}$-shifts of the function $\mathcal{G}(z)$ in (1.15) are stable
and $\Lambda$ is not a uniqueness set for
$V^{\infty}_{{\mathbb{Z}}}(\mathcal{G})$.

Recall that a set $\Lambda$ is not a uniqueness set for
$V^{\infty}_{{\mathbb{Z}}}(\mathcal{G})$ if there is a non-trivial function
$f\in V^{\infty}_{{\mathbb{Z}}}(\mathcal{G})$ which vanishes on $\Lambda$.
Then, clearly, $\Lambda$ is not a sampling set for
$V^{\infty}_{{\mathbb{Z}}}(\mathcal{G}).$

For the generators $\mathcal{G}$ from the class $\mathcal{C}(\alpha)$ we
consider the integer shifts only. Our main sampling theorem for the shift-
invariant space $V^{2}_{{\mathbb{Z}}}(\mathcal{G})$ is as follows.

######  Theorem 1.11.

  1. (i)

Assume a generator $\mathcal{G}\in\mathcal{C}(\alpha)$ has stable
${\mathbb{Z}}$-shifts. If a separated set $\Lambda\subset{\mathbb{R}}$
satisfies

(1.16) $D^{-}(\Lambda)>q(\mathcal{G})+1,$

then $\Lambda$ is a sampling set for $V^{p}_{\mathbb{Z}}(\mathcal{G})$.

  2. (ii)

For every $N\in{\mathbb{N}}$ there exist a generator
$\mathcal{G}\in\mathcal{C}(\alpha)$ with stable ${\mathbb{Z}}$-shits and
$q(\mathcal{G})=N,$ and a separated set $\Lambda$ satisfying
$D^{-}(\Lambda)=q(\mathcal{G})+1$, such that $\Lambda$ is not a uniqueness set
for $V^{\infty}_{{\mathbb{Z}}}(\mathcal{G}).$

###  1.6. Interpolating sets for quasi shift-invariant spaces

If for every ${\bf c}=\\{c_{\gamma}\\}\in l^{p}(\Gamma)$ there is a function
$f\in V_{\Gamma}^{p}(\mathcal{G})$ such that
$f(\lambda)=c_{\lambda},\lambda\in\Lambda$, then $\Lambda$ is called a set of
interpolation for $V_{\Gamma}^{p}(\mathcal{G})$.

The duality between interpolation and sampling is well-known, see e.g. the
discussion in [9]. The following corollary follows from Theorem 1.8:

######  Corollary 1.12.

Assume that $\Lambda,\Gamma$ and $\mathcal{G}$ satisfy assumptions of Theorem
1.8. Then $\Gamma$ is an interpolation set for
$V^{p}_{\Lambda}(\mathcal{G}),1\leq p\leq\infty.$

See a proof in Section 6.

###  1.7. Gabor frames

Our results for the Gabor frames follow from the sampling theorems formulated
above. We use the connection between the Gabor frames and sampling theorems
for shift-invariant spaces that previously turned out to be very fruitful, see
e.g. [9, 10], and [21] for a multi-dimensional setting.

Fix the standard notation for the operators of translation and modulation:

$M_{\xi}f=e^{2\pi i\psi\,\cdot}f\quad\text{and}\quad
T_{\lambda}f=f(\,\cdot-\lambda),\quad\text{where }f\in
L^{2}({\mathbb{R}}),\,(\lambda,\psi)\in{\mathbb{R}}^{2}.$

Let $\Lambda,\Psi$ be separated real sets. For a generator $\mathcal{G}$ the
Gabor system  $G(\mathcal{G},\Lambda\times\Psi)$ is the collection of time-
frequency shifts

(1.17)
$G(\mathcal{G},\Lambda\times\Psi):=\\{M_{\psi}T_{\lambda}\mathcal{G},\,\,(\lambda,\psi)\in\Lambda\times\Psi\\}.$

The system $G(\mathcal{G},\Lambda\times\Psi)$ forms a frame in
$L^{2}({\mathbb{R}})$ if there exist finite positive constants $A,B$ such that

$A\|f\|^{2}_{2}\leq\sum\limits_{(\lambda,\psi)\in\Lambda\times\Psi}|\langle
f,M_{\psi}T_{\lambda}\mathcal{G}\rangle|^{2}\leq B\|f\|^{2}_{2},$

for every $f\in L^{2}({\mathbb{R}}).$

For the Gabor systems with generators from the families $\mathcal{K}(\alpha)$
and $\mathcal{C}(\alpha)$, we study the case of semi-regular lattices, i.e.
$\Psi={\mathbb{Z}}$. Our main result is as follows

######  Theorem 1.13.

  1. (i)

Assume the Fourier transform $\hat{\mathcal{G}}$ of a generator
$\mathcal{G}\in\mathcal{K}(\alpha),\alpha>0,$ satisfies (1.9). If a separated
set $\Lambda\subset{\mathbb{R}}$ satisfies
$D^{-}(\Lambda)>q(\mathcal{G})/k(\mathcal{G})$, then the system
$G(\mathcal{G},\Lambda\times{\mathbb{Z}})$ is a frame in
$L^{2}({\mathbb{R}})$.

  2. (ii)

There exist a generator $\mathcal{G}\in\mathcal{K}(\alpha),\alpha>0,$
satisfying (1.9) and a separated set $\Lambda$ with the critical density
$D^{-}(\Lambda)=q(\mathcal{G})/k(\mathcal{G})$ such that
$G(\mathcal{G},\Lambda\times{\mathbb{Z}})$ is not a frame in
$L^{2}({\mathbb{R}})$.

  3. (iii)

Assume a generator $\mathcal{G}\in\mathcal{C}(\alpha),\alpha>0,$ satisfies
(1.9). If a separated set $\Lambda$ satisfies
$D^{-}(\Lambda)>q(\mathcal{G})+1$, then the system
$G(\mathcal{G},\Lambda\times{\mathbb{Z}})$ is a frame in
$L^{2}({\mathbb{R}})$.

  4. (iv)

There exist a generator $\mathcal{G}\in\mathcal{C}(\alpha),\alpha>0,$
satisfying (1.9) and a separated set $\Lambda$ with the critical density
$D^{-}(\Lambda)=q(\mathcal{G})+1$ such that
$G(\mathcal{G},\Lambda\times{\mathbb{Z}})$ is not a frame in
$L^{2}({\mathbb{R}})$.

###  1.8. Structure of the paper

The paper is organized as follows. In Section 2 we fix notations and recall
some basic definitions. In Section 3 we study stability of $\Gamma$-shits and
${\mathbb{Z}}$-shits for the generators from $\mathcal{K}(\alpha)$ and
$\mathcal{C}(\alpha).$ The uniqueness sets for the corresponding shift-
invariant and quasi shift-invariant spaces are studied in Section 4. In
Section 5 we present examples of functions that vanish on certain sets of
critical density. Combining results of Section 4 and Section 5 and using
classical technique due to Beurling, we prove the sampling and interpolation
theorems (Theorems 1.8, 1.10, 1.11, and Corollary 1.12) in Section 6. The
results for Gabor frames are discussed in Section 7. Finally, in Section 8 we
prove Theorem 1.6.

##  2\. Preliminaries

Throughout the paper, by $C$ we always denote positive constants. The notation
${\bf 1}_{S}$ stands for the indicator function of the set $S$.

Given a meromorphic function $f(z),z\in{\mathbb{C}}$, denote by

${\rm Zer}(f)=\left\\{z\in{\mathbb{C}}:f(z)=0\right\\}$

and Pol$\,(f)$ the multisets of zeros and poles of $f$, i.e. each element is
counted with its multiplicity. The notations ${\rm zer}(f)$ and ${\rm pol}(f)$
will stand for the set of zeros and poles of $f$, i.e. the elements of these
sets are pairwise distinct.

The next lemma is a simplified version of Proposition A.1 in [8], see also
Lemma 7.5 in [9].

######  Lemma 2.1.

Let $\Lambda$ and $\Gamma$ be separated sets in ${\mathbb{R}}$. Let
$A\in{\mathbb{C}}^{\Lambda\times\Gamma}$ be a matrix such that

$|A_{\lambda,\gamma}|\leq\theta(\lambda-\gamma)\quad\lambda\in\Lambda,\gamma\in\Gamma\quad\text{for
some }\theta\in W_{0}.$

Assume that there exist a $p_{0}\in[1,\infty]$ and $C_{0}>0,$ such that

$\|A{\bf c}\|_{p_{0}}\geq C_{0}\|{\bf c}\|_{p_{0}}\quad\mbox{for all }{\bf
c}\in l^{p_{0}}(\Gamma).$

Then there exists a constant $C>0$ independent of $q$ such that, for all
$q\in[1,\infty]$

$\|A{\bf c}\|_{q}\geq C\|{\bf c}\|_{q},\quad\mbox{for all }{\bf c}\in
l^{q}(\Gamma).$

######  Remark 2.2 (See Remark 8.2 in [8]).

The constant $C$ in Lemma 2.1 depends only on the decay properties of the
envelope $\theta$, the lower bound for the given value of $p_{0}$, and on
upper bounds for the relative separation of the index sets.

To connect sampling in shift-invariant spaces with Gabor frames, we use the
following lemma which is a particular case of a well-known result, see e.g.
[9, Theorem 2.3].

######  Lemma 2.3.

Let $\Lambda\subset{\mathbb{R}}$ be a separated set, and
$\mathcal{G}\in\mathcal{C}(\alpha)\cup\mathcal{K}(\alpha)$. The following are
equivalent:

  1. (i)

The family $G(\mathcal{G},-\Lambda\times{\mathbb{Z}})$ is a frame for
$L^{2}({\mathbb{R}})$.

  2. (ii)

For every $x\in[0,1)$ the set $\Lambda+x$ is a sampling set for
$V^{2}_{{\mathbb{Z}}}(\mathcal{G}).$

##  3\. Stability of $\Gamma$-shifts

In this section, we prove Proposition 1.7 and present examples of generators
$\mathcal{G}$ from $\mathcal{K}(\alpha)$ and $\mathcal{C}(\alpha)$ that do not
have stable shifts.

Below we will need

######  Lemma 3.1.

Given a generator $\mathcal{G}\in W_{0}$ and a separate set $\Gamma$. If the
left hand-side inequality in (1.8) is not true, then there is a set
$\Gamma^{\prime}\in W(\Gamma)$ and non-trivial coefficients ${\bf
c}=\\{c_{\gamma^{\prime}}\\}\in l^{\infty}(\Gamma^{\prime})$ such that

$\sum_{\Gamma^{\prime}}c_{\gamma^{\prime}}\mathcal{G}(x-\gamma^{\prime})\equiv
0.$

######  Remark 3.2.

Observe without proof that the converse statement is also true: If the last
equality holds for some $\Gamma^{\prime}\in W(\Gamma)$ and non-trivial
$\\{c_{\gamma^{\prime}}\\}$, then the left hand-side inequality in (1.8) is
not true.

Observe that somewhat similar results are known, see e.g. Theorem 2.1 (c) in
[9].

###### Proof.

The proof below uses the (by now) standard Beurling’s technique based on
passing to a weak limit of translates of the set $\Gamma$.

Write $\Gamma=\\{...<\gamma_{j}<\gamma_{j+1}<...:j\in{\mathbb{Z}}\\}$. Since
$\Gamma$-shifts of $\mathcal{G}$ are not $l^{\infty}$-stable, for every
$N\in{\mathbb{N}}$ there is a bounded sequence ${\bf
c}(N)=\\{c_{j}(N):j\in{\mathbb{Z}}\\}$ such that $\|{\bf c}(N)\|_{\infty}=1$,
and

$\left\|\sum\limits_{j\in{\mathbb{Z}}}c_{j}(N)\mathcal{G}(x-\gamma_{j})\right\|_{\infty}<1/N.$

Choose any $k=k(N)\in{\mathbb{Z}}$ such that $|c_{k}(N)|>1/2$, and set
$\Gamma(N):=\Gamma-\gamma_{N},$ $d_{j}(N):=c_{k+j}(N)$. Then $0\in\Gamma(N)$
and $|d_{0}(N)|>1/2.$ Clearly,
$\|\\{d_{j}(N):j\in{\mathbb{Z}}\\}\|_{\infty}=1$ and

$\left\|\sum\limits_{j\in{\mathbb{Z}}}d_{j}(N)\mathcal{G}(x-\gamma_{k+j})\right\|_{\infty}<1/N.$

Now, passing to a subsequence, we may assume that $\Gamma(N)$ converges weakly
to some separated set
$\Gamma^{\prime}:=\\{\gamma_{j}^{\prime}:j\in{\mathbb{Z}}\\}\in W(\Gamma)$
which contains the origin, and the sequence $\\{d_{j}(N):j\in{\mathbb{Z}}\\}$
converges to a bounded non-trivial sequence ${\bf
c}=\\{c_{j}:j\in{\mathbb{Z}}\\}$ as $N$ tends to $\infty.$ One can check that
we have

$\sum\limits_{j\in{\mathbb{Z}}}c_{j}\mathcal{G}(x-\gamma^{\prime}_{j})\equiv
0,$

which proves the lemma. ∎

###  3.1. Stability of ${\mathbb{Z}}$-shifts for generators in
$\mathcal{C}(\alpha)$

###### Proof of Proposition 1.7, $(i)$.

The proof is by contradiction. We assume that one of the assumptions
$(\Xi^{\prime})$ or $(\Xi^{\prime\prime\prime})$ is satisfied, but the
${\mathbb{Z}}$-shifts of $\mathcal{G}\in\mathcal{C}(\alpha)$ are not
$l^{\infty}$-stable. Then, by Lemma 3.1 and (1.11), we find a sequence ${\bf
c}=\\{c_{n}\\}\in l^{\infty}({\mathbb{Z}})$ such that

(3.1) $f(x):=\sum_{n\in{\mathbb{Z}}}c_{n}\mathcal{G}(x-n)=e^{-\alpha
x^{2}/2}\sum_{n\in{\mathbb{Z}}}c_{n}e^{-\alpha n^{2}/2}e^{\alpha
xn}\mathcal{R}(e^{\alpha(x-n)})\equiv 0.$

Set $w=e^{\alpha x}.$ Then

(3.2) $h(w):=\sum_{n\in{\mathbb{Z}}}c_{n}e^{-\alpha
n^{2}/2}w^{n}\mathcal{R}(we^{-\alpha n})\equiv 0.$

Let $w_{1},...,w_{m}$ be the poles of $\mathcal{R}$, $d(w_{j})$ the order of
$w_{j}$ and let ${\rm pol}_{d}(\mathcal{R})$ denote the set of poles of
$\mathcal{R}$ of maximal order. Clearly, $\mathcal{R}(we^{-\alpha n})$ has
poles at $w_{j}e^{\alpha n},j=1,...,m$. We may assume that $d(w_{1})=d$. There
are two possibilities:

First, assume that $(\Xi^{\prime})$ is true, i.e. $d(w_{j})<d,j=2,...,m$. Then
clearly, the function $h$ has pole of order $d$ at each point $w_{1}e^{\alpha
n},n\in{\mathbb{Z}}$, which contradicts (3.2).

Second, assume that $(\Xi^{\prime\prime\prime})$ is satisfied: there are $k$
poles whose order is equal to $d,$ say $d(w_{j})=d,j=1,...,k.$ If
$\log(w_{1}/w_{j})\not\in\alpha{\mathbb{Z}},$ for all $2\leq j\leq k,$ then
$w_{1}\neq w_{j}e^{\alpha n},n\in{\mathbb{Z}},$ and so the function $h$ has a
pole of order $d$ at $w_{1}$ (and also at every point $w_{1}e^{\alpha
n},n\in{\mathbb{Z}})$, which again contradicts (3.2). This finishes the proof.
∎

We now present an example of generator from $\mathcal{C}(1)$ whose
${\mathbb{Z}}$-shifts are not stable.

######  Example 3.3.

Set

(3.3)
$H(z):=U(z)e^{-z^{2}/2}:=\left(A+\frac{e^{-1/2}}{e^{z-i}-1}-\frac{e^{i}}{e^{z-1-i}-1}\right)e^{-z^{2}/2},$

where $A$ is a constant chosen such that

(3.4) $\int\limits_{-\infty}^{\infty}H(x)\,dx=0.$

Clearly, $H(z)\in\mathcal{C}(1)$ (see definition (1.7)) and both assumption
$(\Xi^{\prime})$ and $(\Xi^{\prime\prime\prime})$ do not hold.

Let us show that the function $f\in V^{\infty}_{\mathbb{Z}}(H)$ defined by

$f(x):=\sum_{n\in{\mathbb{Z}}}H(x-n)$

vanishes identically. Since $f$ is $1$-periodic, it suffices to prove that its
Fourier coefficients vanish:

######  Lemma 3.4.

We have

(3.5) $\hat{H}(n)=0,\quad n\in{\mathbb{Z}}.$

###### Proof.

Our goal is to show that

$0=\hat{H}(n)=\int\limits_{-\infty}^{\infty}U(x)e^{-x^{2}/2}e^{-2\pi
inx}\,dx=0,\quad n\in{\mathbb{Z}}.$

Clearly,

$\hat{H}(n)=e^{-2\pi^{2}n^{2}}I_{n},\quad
I_{n}:=\int_{-\infty}^{\infty}e^{-(x+2\pi in)^{2}/2}U(x)\,dx.$

Therefore, it suffices to prove that $I_{n}=0,n\in{\mathbb{Z}}.$

The proof is by induction. By (3.4), we have $I_{0}=0$. Assume that $I_{n}=0$
for $|n|\leq k,k\geq 0$. We will show that $I_{k+1}=0$ (the proof of
$I_{-k-1}=0$ is similar).

Let us integrate $e^{-(z+2\pi ik)^{2}/2}U(z)$ over the boundary of rectangle
$\\{z=x+iy:|x|\leq R,0\leq y\leq 2\pi\\}$ (integration is in positive
direction with respect to the rectangle), and then let $R\to\infty$. It is
clear that the integrals over the sides parallel to the imaginary axis tend to
$0$ as $R\to\infty.$ The integral over the bottom side of the rectangle tends
to $0$, since $I_{k}=0.$ Since the function $U(z)$ is $2\pi i$-periodic, the
integral over the upper side tends to $-I_{k+1}$ as $R\to\infty$. Applying
Cauchy’s residue theorem, we obtain

$-I_{k+1}=2\pi i\left(\mathop{\mathrm{Res}}_{z=i}e^{-(z+2\pi
ik)^{2}/2}U(z)+\mathop{\mathrm{Res}}_{z=1+i}e^{-(z+2\pi
ik)^{2}/2}U(z)\right).$

Finally, from

${\mathrm{Res}}_{z=i}e^{-(z+2\pi ik)^{2}/2}U(z)=e^{-(i+2\pi
ik)^{2}/2-1/2}=e^{2\pi k+2\pi^{2}k^{2}},$ ${\mathrm{Res}}_{z=1+i}e^{-(z+2\pi
ik)^{2}/2}U(z)=-e^{-(1+i+2\pi ik)^{2}/2+i}=-e^{2\pi k+2\pi^{2}k^{2}},$

we conclude that $I_{k+1}=0$, which finishes the proof. ∎

Observe that it follows from [9, Theorem 2.1], that ${\mathbb{Z}}$-shifts of
the generator $H$ defined in (3.3) are not $l^{p}$-stable for every
$p\in[1,\infty]$.

###  3.2. Stability in $V^{p}_{\Gamma}(g)$

Let us now finish the proof of Proposition 1.7.

###### Proof of Proposition 1.7, $(ii)$.

The proof is by contradiction and it is similar to the one above, and we will
use the same notations.

Assuming that $\Gamma$-shifts are not $l^{\infty}$-stable, by Lemma 3.1 we get
a set $\Gamma^{\prime}\in W(\Gamma)$ and coefficients ${\bf c}\in
l^{\infty}(\Gamma^{\prime})$ such that

(3.6)
$h(w):=\sum_{\gamma^{\prime}\in\Gamma^{\prime}}c_{\gamma^{\prime}}\mathcal{R}(we^{-\alpha\gamma^{\prime}})\equiv
0.$

Let $w_{1}$ be the pole of $\mathcal{R}$ of the maximal order $d$.

If condition $(\Xi^{\prime})$ is satisfied then $h$ has a pole at each point
$w_{1}e^{-\alpha\gamma^{\prime}},\gamma^{\prime}\in\Gamma^{\prime},$ which
contradicts (3.6).

Assume that $(\Xi^{\prime\prime})$ is true: for any $w^{\prime}\in{\rm
pol}_{d}(\mathcal{R})\setminus\\{w_{1}\\}$ and
$\gamma^{\prime}\in\Gamma^{\prime}$ we have
$w_{1}e^{\alpha\gamma_{0}^{\prime}}\neq w^{\prime}e^{\alpha\gamma^{\prime}}$.
Therefore, the function $h$ has a pole of order $d$ at
$w_{1}e^{\alpha\gamma_{0}}$ which contradicts to (3.6). ∎

For the class $\mathcal{K}(\alpha)$ we also provide an example of generator
$H\in\mathcal{K}(1)$ that does not satisfy assumptions $(\Xi^{\prime})$ and
$(\Xi^{\prime\prime})$, and whose ${\mathbb{Z}}$-shifts are not stable.

######  Example 3.5.

Set

$H(x)=\frac{e^{x}}{e^{2x}+1}-\frac{e^{x+1}}{e^{2x}+e^{2}}\in\mathcal{K}(1).$

Since $H(x)=\mathcal{H}(x)-\mathcal{H}(x-1)\in\mathcal{K}(1),$ where
$\mathcal{H}(x)=e^{x}/(e^{2x}+1)$, it is clear that the function
$\sum_{n\in{\mathbb{Z}}}H(x-n)$ belongs to $V_{\mathbb{Z}}^{\infty}(H)$ and
vanishes identically on ${\mathbb{R}}$. Hence, by [9, Theorem 2.1],
${\mathbb{Z}}$-shifts of $H$ are not $l^{p}$-stable for any $p\in[1,\infty]$.

We will formulate without proof the following

######  Example 3.6.

Choose any sequence $\delta_{n}\to 0,n\to\pm\infty,$ satisfying
$0<|\delta_{n}|<1/2,n\in{\mathbb{Z}}$. Set
$\Gamma:=\\{n+\delta_{n}:n\in{\mathbb{Z}}\\}$. Then $\Gamma$-shifts of the
generator $H$ in Example 3.5 are not $l^{\infty}$-stable.

One may prove the above using e.g. Remark 3.2.

##  4\. Uniqueness Sets

Following Beurling’s approach (see [5]), to prove sampling theorems we first
investigate the uniqueness sets for the spaces
$V^{\infty}_{\Gamma}(\mathcal{G})$.

######  Proposition 4.1.

Given two separated sets $\Lambda,\Gamma\subset{\mathbb{R}}$.

  1. (I)

If $\Lambda$ satisfies condition (1.13), then it is a uniqueness set for
$V^{\infty}_{\Gamma}(\mathcal{G})$.

  2. (II)

If $\Lambda$ satisfies condition (1.16), then it is a uniqueness set for
$V^{\infty}_{\mathbb{Z}}(\mathcal{G})$.

###  4.1. Proof of Proposition 4.1, Part (I)

The proof below does not depend on the shape parameter $\alpha$. So, for
simplicity throughout the proof we assume that $\alpha=1$.

We must show that there is no non-trivial function

(4.1) $f(z)=\sum_{\gamma\in\Gamma}c_{\gamma}\mathcal{R}(e^{z-\gamma})\in
V^{\infty}_{\Gamma}(\mathcal{G}),\quad\\{c_{\gamma}\\}\in l^{\infty}(\Gamma),$

that vanishes on a set $\Lambda\subset{\mathbb{R}}$ satisfying (1.13).

The proof is by contradiction. Let us assume that such a function exists.
Clearly, $f$ admits extension to the complex plane ${\mathbb{C}}$ as a
meromorphic function satisfying $f(0)\neq\infty$. We may assume that $f(0)\neq
0$ (otherwise we consider $g(z)=f(z-a)$ and $\Lambda^{\prime}=\Lambda+a$ for a
suitable $a\in{\mathbb{R}}.$)

Recall that Pol$(f)$ and Zer$(f)$ denote the multisets of poles and zeros of
$f$, respectively. Recall also that we denote by $C$ different positive
constants, and that the numbers $k(\mathcal{G})$ and $q(\mathcal{G})$ are
defined in Definition 1.2.

Set

(4.2)
$W:=\log\left(\mbox{Pol}\,\mathcal{R}\right)\cap\\{z=x+iy\in{\mathbb{C}}:0<y<2\pi\\}.$

This means that $\mathcal{R}(z)=\infty$ if and only if $z=e^{u+iv},u+iv\in W,$
and the number of occurrences of $z$ in $e^{W}$ is equal to the multiplicity
of the pole of $\mathcal{R}$ at $z$. Since $f$ is $2\pi i$-periodic, we have

$\mbox{Pol}(f)\subseteq\\{z\in{\mathbb{C}}:e^{z-\gamma}\in
e^{W},\gamma\in\Gamma\\}=\Gamma+W+2\pi i{\mathbb{Z}}.$

Recall that $f$ vanishes on $\Lambda$. It follows from (1.6) that $f$ also
vanishes on the set $\Lambda+\\{2\pi
ij/k(\mathcal{G}):j=0,...,k(\mathcal{G})-1\\}=\Lambda+(2\pi
i)/k(\mathcal{G})\cdot Z$, where $Z:=\\{0,...,k(\mathcal{G})-1\\}.$ Therefore,

$\mbox{Zer}(f)\supseteq\Lambda+\frac{2\pi i}{k(\mathcal{G})}Z+2\pi
i{\mathbb{Z}}.$

The proof below is based on the following two lemmas:

######  Lemma 4.2.

We have

$n_{0}(t)\geq n_{\infty}(t)+Ct^{2},$

for some $C>0$ and for every sufficiently large $t$.

Above $n_{0}(t)$ and $n_{\infty}(t)$ denote the number of zeros and poles
(counting multiplicities) of $f$ in the circle
$B_{t}:=\\{z\in{\mathbb{C}}:|z|\leq t\\}$, respectively.

######  Lemma 4.3.

There is a sequence $R_{j}\to\infty$ such that

$|f(z)|\leq\|{\bf
c}\|_{\infty}\sum_{\gamma\in\Gamma}\left|\mathcal{R}(e^{z-\gamma})\right|\leq
CR_{j}^{q(\mathcal{G})+1},\quad|z|=R_{j},\ j\in{\mathbb{N}}.$

Let us now check that the lemmas above contradict to the classical Jensen
formula for meromorphic functions (see e.g. [18], Ch. 2.4)

(4.3)
$\int_{0}^{R}\frac{n_{0}(t)-n_{\infty}(t)}{t}dt=\frac{1}{2\pi}\int_{0}^{2\pi}\log|f(Re^{i\theta})|d\theta-\log|f(0)|,$

where we assume that $f(z)\neq 0,\infty$ on the circle $|z|=R$.

Indeed, by Lemma 4.2, the left hand-side of the formula is larger than
$CR^{2}$ as $R\to\infty,$ while by Lemma 4.3, the right-hand size has a
logarithmic growth on a sequence $R=R_{j}\to\infty$. Therefore, to finish the
proof of proposition it remains to prove these lemmas.

###### Proof of Lemma 4.2.

We start with three claims:

Let us denote by ${\rm Re}\,W$ the multiset ${\rm Re}\,W:=\\{{\rm Re}\,z:z\in
W\\}$. Observe that

(4.4) $\\#{\rm Re}\,W=\\#W=q(\mathcal{G}).$

Denote by $\tilde{\Lambda}$ the multiset of points of $\Lambda$, each point
has multiplicity $k(\mathcal{G})$. We use the definition (1.2) to define the
lower density $D^{-}(\tilde{\Lambda})$.

######  Claim 4.4.

We have

$D^{+}(\Gamma+{\rm Re}\,W)=q(\mathcal{G})D^{+}(\Gamma),\quad
D^{-}(\tilde{\Lambda})=k(\mathcal{G})D^{-}(\Lambda).$

######  Claim 4.5.

Let numbers $R>0$ and $a,b\in{\mathbb{C}}$ satisfy

$|{\rm Re}\,a|\leq|{\rm Re}\,b|<R.$

Then

$\\#\left((a+2\pi i{\mathbb{Z}})\cap B_{R}\right)\geq\\#\left((b+2\pi
i{\mathbb{Z}})\cap B_{R}\right)-1.$

######  Claim 4.6.

There is a positive number $\rho$ such that for all open intervals
$I,J\subset{\mathbb{R}}$ of length $|I|=|J|=\rho$ we have

$\\#\left(\tilde{\Lambda}\cap I\right)=k(\mathcal{G})\\#\left(\Lambda\cap
I\right)\geq\\#\left((\Gamma+{\rm Re}\,W)\cap J\right)+k(\mathcal{G}).$

We omit the simple proofs of Claims 4.4 and 4.5. The last claim is a simple
consequence of Claim 4.4, (4.4) and (1.13).

Now, choose a number $r>0$ such that $|\gamma+{\rm Re}\,e^{w}|\geq\rho$, for
every $\gamma\in\Gamma,|\gamma|\geq r$ and every $w\in W$, where $\rho$ is the
number in Claim 4.6. Set $\Gamma^{\prime}:=\Gamma\setminus(-r,r).$ By Claim
4.6 we can find a subset $\Lambda^{\prime}\subset\Lambda$ such that there is a
bijection

$T:\Lambda^{\prime}+\frac{2\pi i}{k(\mathcal{G})}Z\to\Gamma^{\prime}+W$

satisfying $|{\rm Re}\,T(u)|\geq|{\rm Re}\,u|$ and such that

(4.5)
$\\#\left((\Lambda\setminus\Lambda^{\prime})\cap(j\rho,(j+1)\rho)\right)\geq
1,\quad j\in{\mathbb{Z}}.$

Since the set $\Gamma\setminus\Gamma^{\prime}$ is finite, it follows that

$\\#\left(((\Gamma\setminus\Gamma^{\prime})+W+2\pi i{\mathbb{Z}})\cap
B_{R}\right)\leq CR.$

Hence, by Claim 4.5 we get the estimate

$\\#\left((\Lambda^{\prime}+\frac{2\pi i}{k(\mathcal{G})}Z+2\pi
i{\mathbb{Z}})\cap B_{R}\right)\geq\\#\left((\Gamma+W+2\pi i{\mathbb{Z}})\cap
B_{R}\right)-CR.$

On the other hand, by (4.5) for all large enough $R$ one gets the estimate

$\\#\left((\Lambda+\frac{2\pi i}{k(\mathcal{G})}Z+2\pi i{\mathbb{Z}})\cap
B_{R}\right)\geq\\#\left((\Lambda^{\prime}+\frac{2\pi i}{k(\mathcal{G})}Z+2\pi
i{\mathbb{Z}})\cap B_{R}\right)+CR^{2},$

where the constant $C$ depends on $\rho.$ This finishes the proof of Lemma
4.2. ∎

###### Proof of Lemma 4.3.

Recall that $\mathcal{R}=P/Q$ satisfies conditions (A) - (C) in Definition
1.1. It follows that there exist constants $C,L>0$ such that

(4.6) $|\mathcal{R}(z)|\leq C|z|,\quad|z|\leq e^{-L}\ \ \mbox{and }\
|\mathcal{R}(z)|\leq\frac{C}{|z|},\quad|z|\geq e^{L}.$

One can also easily check that

(4.7)
$|\mathcal{R}(z)|\leq\frac{C}{(\mbox{dist}(z,e^{W}))^{q(\mathcal{G})}},\quad
e^{-L}<|z|<e^{L}.$

where $e^{W}$ defined in (4.2).

Let $f$ be defined in (4.1). It is easy to see that there is a constant $C$
such that

$\\#\left(\mbox{Pol}(f)\cap B_{R}\right)\leq CR^{2},\quad R\geq 1.$

Therefore, there is a sequence $R=R_{j}\to\infty$ such that

(4.8) $\mbox{dist}(\mbox{Pol}(f),\partial B_{R})\geq\frac{C}{R}.$

We now fix such a number $R$ and split $\Gamma$ into two sets:

$\Gamma_{1}:=\\{\gamma\in\Gamma:|\gamma|\geq
R+L\\},\,\,\Gamma_{2}:=\Gamma\cap(-R-L,R+L).$

By (4.6),

$|\mathcal{R}(e^{z-\gamma})|\leq Ce^{-|x-\gamma|},\quad\gamma\in\Gamma_{1},\
z=x+iy,\ |z|=R.$

Since $\Gamma$ is a separated set, this gives

$\sup_{|z|=R}\sum_{\gamma\in\Gamma_{1}}\left|c_{\gamma}\mathcal{R}(e^{z-\gamma})\right|\leq
C\|{\bf
c}\|_{\infty}\sup_{x\in{\mathbb{R}}}\sum_{\gamma\in\Gamma_{1}}e^{-|x-\gamma|}\leq
C,$

where the last constant does not depend on $R$.

Further, by (4.6) and (4.7), for every $z,|z|=R$ and $\gamma\in\Gamma_{2},$

$|\mathcal{R}(e^{z-\gamma})|\leq\sup_{|z|=R,w\in
W}\frac{C}{|e^{z-\gamma}-e^{w}|^{q(\mathcal{G})}}+C\leq\sup_{|z|=R,w\in
W}\frac{C}{|e^{z-w-\gamma}-1|^{q(\mathcal{G})}}.$

We now use a simple inequality

$|e^{\zeta}-1|\geq C\delta,\quad\mbox{dist}(\zeta,2\pi
i{\mathbb{Z}})\geq\delta.$

It is clear that $\\#\Gamma_{2}\leq CR.$ Hence, the last inequality and (4.8)
imply

(4.9)
$\sum_{\gamma\in\Gamma_{2}}\left|c_{\gamma}\mathcal{R}(e^{z-\gamma})\right|\leq
C\|{\bf c}\|_{\infty}\\#\Gamma_{2}R^{q(\mathcal{G})}=CR^{q(\mathcal{G})+1}.$

This finishes the proof of Lemma 4.3. ∎

###  4.2. Proof of Proposition 4.1, Part (II)

The proof is similar to the proof of Part (I). However, recall that in this
case, we do not exclude the option ${\rm deg}\,P\geq{\rm deg}\,Q.$

For simplicity, we assume that the shape parameter $\alpha=1$. The proof in
the general case is similar.

We argue by contradiction and assume that there is a non-trivial function

$f(z)=\sum_{n\in{\mathbb{Z}}}c_{n}\mathcal{G}(z-n)=\sum_{n\in{\mathbb{Z}}}c_{n}e^{-(z-n)^{2}/2}\mathcal{R}(e^{z-n})\in
V^{\infty}_{\mathbb{Z}}(\mathcal{G}),\ {\bf c}=\\{c_{n}\\}\in
l^{\infty}({\mathbb{Z}}),$

which vanishes on $\Lambda$. We have to show that this implies a
contradiction.

As in Lemma 4.3 above, let $n_{0}(t)$ and $n_{\infty}(t)$ denote the number of
zeros and poles of $f$ in $\\{z\in{\mathbb{C}}:|z|=t\\}$, respectively.

######  Lemma 4.7.

We have

(i) $n_{0}(t)\geq n_{\infty}(t)+(1+C)t^{2}/2$, for some $C>0$ and all large
enough $t.$

(ii) There is sequence $R=R_{j}\to\infty$ such that

$\log|f(z)|\leq y^{2}/2+C\log R,\quad z=x+iy,|z|=R.$

###### Proof.

Instead of conditions (4.6) and (4.7), we now have the conditions

$|\mathcal{R}(z)|\leq C(1+|z|^{k}),\quad|z|<e^{-L},\ |z|>e^{L}$

and

$|\mathcal{R}(z)|\leq\frac{C}{(\mbox{dist}(z,e^{W}))^{q(\mathcal{G})}},\quad
e^{-L}<|z|<e^{L},$

for some $k\in{\mathbb{N}}$ and $C,L>0.$

Let $R_{j}\to\infty$ be a sequence satisfying condition (4.8). Fix an element
$R=R_{j}$ and set $Z_{1}:=\\{n\in{\mathbb{Z}}:|n|\geq R+L\\}$ and
$Z_{2}:={\mathbb{Z}}\cap(-R-L,R+L)$. Using the first inequality above, we get

$\sum_{n\in Z_{1}}\left|c_{n}e^{-(z-n)^{2}/2}\mathcal{R}(e^{z-n})\right|\leq
Ce^{y^{2}/2}\sum_{n\in Z_{1}}e^{-(x-n)^{2}/2}\left(1+e^{k(x-n)}\right)\leq
Ce^{(y^{2}+k^{2})/2}=Ce^{y^{2}/2}.$

Next, by the second inequality above, similarly to (4.9), we get

$\sum_{n\in Z_{2}}\left|c_{n}e^{-(z-n)^{2}/2}\mathcal{R}(e^{z-n})\right|\leq
C\|{\bf c}\|_{\infty}e^{y^{2}/2}R^{q(\mathcal{G})}\sum_{n\in
Z_{2}}e^{-(x-n)^{2}/2}\leq CR^{q(\mathcal{G})+1}e^{y^{2}/2}.$

We conclude that for every $f\in V_{\mathbb{Z}}^{\infty}(\mathcal{G})$ we have

(4.10) $|f(x+iy)|\leq CR^{q(\mathcal{G})+1}e^{y^{2}/2},\quad|z|=R_{j},$

where $C$ depends only on $f.$ This proves part (ii) of Lemma 4.7.

Given $\mathcal{G}\in\mathcal{C}(1)$ and $f\in
V_{{\mathbb{Z}}}^{\infty}(\mathcal{G})$, observe that

$f(z+2\pi i)=\sum_{n\in{\mathbb{Z}}}c_{n}e^{-(z+2\pi
i-n)^{2}/2}\mathcal{R}(e^{z+2\pi i-n})=e^{-2\pi iz+2\pi^{2}}f(z).$

Hence, since $f(\lambda)=0,\lambda\in\Lambda,$ then $f$ vanishes on the set
$\Lambda+2\pi i{\mathbb{Z}}.$ Therefore, similarly to Section 4.1, we have

$\mbox{Pol}(f)\subseteq{\mathbb{Z}}+W+2\pi
i{\mathbb{Z}},\quad\mbox{Zer}(f)\supseteq\Lambda+2\pi i{\mathbb{Z}}.$

By estimate (1.16), we may split $\Lambda$ into two sets,
$\Lambda=\Lambda_{1}\cup\Lambda_{2}$ satisfying
$D^{-}(\Lambda_{1})>q(\mathcal{G})$ and $D^{-}(\Lambda_{2})>1.$ Then

$n_{0}(t)\geq n_{1}(t)+n_{2}(t):=\\#(\Lambda_{1}+2\pi i{\mathbb{Z}})\cap
B_{t}+\\#(\Lambda_{2}+2\pi i{\mathbb{Z}})\cap B_{t}.$

As in the proof of Lemma 4.2, we have

$n_{1}(t)\geq n_{\infty}(t)+Ct^{2},\quad\mbox{for some }C>0.$

To prove part (i) of Lemma 4.7, it suffices to prove the following

######  Claim 4.8.

We have

$n_{2}(t)\geq t^{2}/2,\quad\mbox{for all large enough }t.$

Given a convex set $I\subset{\mathbb{C}},0\in I,$ and a discrete set of points
$\Delta\subset{\mathbb{C}}$, consider the density of $\Delta$ defined as

$d(I,\Delta):=\lim\inf_{r\to\infty}\frac{\\#\Delta\cap(rI)}{r^{2}|I|},$

where $|I|$ denotes the 2D-measure (area) of $I$. If $d(I,\Delta)>0$, then for
every $\epsilon>0$ we have

(4.11) $\\#\Delta\cap rI\geq(1-\epsilon)r^{2}|I|d(I,\Delta).$

for all sufficiently large $r$.

Assume that $I$ is the square

$I=\\{z=x+iy\in{\mathbb{C}}:\max\\{|x|,|y|\leq 1\\}\\}.$

It is easy to check that

$d(I,\Lambda_{2}+2\pi i{\mathbb{Z}})\geq D^{-}(\Lambda_{2})/2\pi.$

However, the density $d(I,\Delta)$ does not depend on the choice of $I$, see
e.g. [17, Lemma 4]. Since $D^{-}(\Lambda_{2})>1,$ Claim 4.8 follows from
(4.11). ∎

Again, to arrive at contradiction, we use Jensen formula (4.3) for meromorphic
functions. From Lemma 4.7, part (ii), we see that the right hand-side of (4.3)
is bounded above by

$\frac{1}{2\pi}\int_{0}^{2\pi}\log|f(Re^{i\theta})|d\theta\leq\frac{R^{2}}{4\pi}\int_{0}^{2\pi}\sin^{2}(\theta)d\theta+C\log
R=\frac{R^{2}}{4}+C\log R,R=R_{j}\to\infty.$

On the other hand, part (i) of Lemma 4.7 shows that the left hand-side of
(4.3) is larger than $(1+C)R^{2}/4,C>0,$ for all large enough $R$, which is a
contradiction. This finishes the proof of Proposition 4.1.

##  5\. Non-uniqueness Sets

In this section, we study zero sets of functions from shift-invariant spaces
$V^{\infty}_{{\mathbb{Z}}}(\mathcal{G})$ generated by
$\mathcal{G}\in\mathcal{C}(\alpha)$ or $\mathcal{G}\in\mathcal{K}(\alpha)$.
More precisely, we build functions from these shift-invariant spaces that
vanish on sets of critical density.

Let us start with

###### Proof of Theorem 1.10.

Given $N\in{\mathbb{N}}$, $\alpha>0$, and $0<b_{1}<\dots<b_{N}$ such that

(5.1) $b_{j}-b_{k}\notin{\mathbb{Z}},\quad\text{for every }\,\,j\neq k.$

It suffices to find a separated set $\Lambda$ satisfying $D^{-}(\Lambda)=N$,
coefficients $\\{a_{j}\\}_{j=1}^{N}\subset{\mathbb{R}}$, and a non-trivial
function $f$ from $V^{\infty}_{{\mathbb{Z}}}(\mathcal{G})$ such that $f$
vanishes on $\Lambda$, where

$\mathcal{G}(x)=\sum\limits_{j=1}^{N}\frac{a_{j}e^{\alpha(x+b_{j})}}{e^{2\alpha
x}+e^{2\alpha b_{j}}}.$

We will distinguish the cases when $N$ is an even or an odd positive integer.

Case 1. Assume that $N\in 2{\mathbb{N}}.$ Consider the functions

$\varphi_{j}(x)=\sum_{n\in{\mathbb{Z}}}\frac{e^{\alpha(x+b_{j}-n)}}{e^{2\alpha(x-n)}+e^{2\alpha
b_{j}}},\quad j=1,...,N.$

Clearly, these functions are $1$-periodic. Observe that they are also linearly
independent, since it follows from (5.1) that they have different poles.
Therefore, we can find $N$ points $0<x_{1}<...<x_{N}<1$ such that the system
of $N$ equations

(5.2) $\sum_{j=1}^{N}a_{j}\varphi_{j}(x_{l})=(-1)^{l},\quad l=1,...,N,$

has a real solution $a_{1},...,a_{N}.$ This means that the function

(5.3) $f(x):=\sum_{j=1}^{N}a_{j}\varphi_{j}(x)\in
V_{{\mathbb{Z}}}^{\infty}(\mathcal{G})$

has at least $N-1$ sign changes on the interval $(0,1).$ However, since $f$ is
also $1$-periodic, it either vanishes at $0$ or has an even number of sign
changes on $(0,1)$, and so $f$ has at least $N$ distinct zeros on $[0,1).$ We
see that ${\rm Zer}\,(f)$ contains a $1$-periodic set of density $N$. This
finishes the proof for the case $N\in 2{\mathbb{N}}$.

Figure 1. Case 2 for $N=5$

Case 2. Assume that $N\in 2{\mathbb{N}}+1.$ Consider the functions

$\varphi_{j}(x)=\sum_{n\in{\mathbb{Z}}}(-1)^{n}\frac{e^{\alpha(x+b_{j}-n)}}{e^{2\alpha(x-n)}+e^{2b_{j}}}.$

Note that $\\{\varphi_{j}\\}_{j=1}^{N}$ are linearly independent,
$2$-periodic, and $\varphi_{j}(x+1)=-\varphi_{j}(x)$.

Similarly to Case 1, we define $f$ by

(5.4) $f(x):=\sum_{j=1}^{N}a_{j}\varphi_{j}(x)$

and find $N$ points $0<x_{1}<...<x_{N}<1$ such that (5.2) has a real solution
$a_{1},...,a_{N}.$ Clearly, $f\in V_{{\mathbb{Z}}}^{\infty}(\mathcal{G}).$

Set $x_{N+j}:=1+x_{j},j=1,...,N.$ Since $N\in 2{\mathbb{N}}+1$ and
$f(x_{N+j})=f(1+x_{j})=-f(x_{j})$, we see that $f$ must have at least one sign
change (and therefore a zero) on each interval $(x_{j},x_{j+1}),j=1,...,2N-1,$
see Figure 1. This means that $f$ has at least $2N-1$ sign changes on
$(x_{1},x_{2N})$. However, since $f$ is $2$-periodic, it either vanishes at
$0$ or has an even number of sign changes on $[0,2).$ Therefore, it has at
least $2N$ different zeros on $[0,2).$ Since $f$ is $2$-periodic, we see that
${\rm Zer}\,(f)$ contains a set of density $N$. This finishes the proof of
Theorem 1.10. ∎

Next, we prove a similar statement for the class of generators
$\mathcal{C}(\alpha)$.

######  Proposition 5.1.

Given $N\in{\mathbb{N}},\alpha>0$, and $0<b_{1}<\dots<b_{N},$ such that

(5.5) $b_{j}-b_{k}\notin{\mathbb{Z}}\quad\text{for every }j,k=1,\dots,N,j\neq
k.$

Let

$\mathcal{G}(x)=e^{-\alpha
x^{2}/2}\left(a_{0}+\sum_{j=1}^{N}\frac{a_{j}}{e^{\alpha x}+e^{\alpha
b_{j}}}\right),\quad\text{ for some }\,a_{j}\in{\mathbb{R}}.$

Then there exist a separated set $\Lambda,D^{-}(\Lambda)=N+1$, coefficients
$\\{a_{j}\\}_{j=0}^{N}\subset{\mathbb{R}}$, and a non-trivial function $f$
from $V^{\infty}_{{\mathbb{Z}}}(\mathcal{G})$ such that $f$ vanishes on
$\Lambda$.

Clearly, the function $\mathcal{G}$ defined above belongs to the class
$\mathcal{C}(\alpha).$

###### Proof.

The argument follows the proof of Theorem 1.10. Again, we consider the cases
$N$ is even and odd integer separately.

Let us assume that $N\in 2{\mathbb{N}}$ and sketch the proof leaving the
details to the reader. The proof of the second case is also left to the
reader. Set

$\psi_{0}(x)=\sum_{n\in{\mathbb{Z}}}(-1)^{n}e^{-\alpha(x-n)^{2}/2},\quad\psi_{j}(x):=\sum\limits_{n\in{\mathbb{Z}}}(-1)^{n}\frac{e^{-\alpha(x-n)^{2}/2}}{e^{\alpha(x-n)}+e^{\alpha
b_{j}}},\,\,j=1,\dots,N.$

Note that for every $j$ the function $\psi_{j}$ is $2$-periodic and
$\psi_{j}(x+1)=-\psi_{j}(x).$ Using the linear independence of the system
$\\{\psi_{i}\\}_{0}^{N}$ that follows from (5.5), we can find $N+1$ points
$0<x_{0}<\dots<x_{N}<1,$ such that the system of $N+1$ equations

$\sum\limits_{j=0}^{N}a_{j}\psi_{j}(x_{l})=(-1)^{l},\quad l=0,\dots,N.$

has a real solution $a_{0},\dots,a_{N}.$ Therefore, the function

$f(x)=\sum\limits_{j=0}^{N}a_{j}\psi_{j}(x),\quad f\in
V^{\infty}_{{\mathbb{Z}}}(\mathcal{G}),$

has the same alternating properties as the function $f$ defined in (5.4). The
rest of the proof is completely similar to the proof of Case 2 above. ∎

##  6\. Proofs of Sampling and Interpolation Theorems

###  6.1. Proof of Theorem 1.8

We split the proof into two steps.

The proof does not depend on $\alpha$, so we set $\alpha=1$.

Step 1. We start with proving that $\Lambda$ is a sampling set for
$V^{\infty}_{\Gamma}(\mathcal{G}).$ The proof is by contradiction.

Let us assume that condition (1.13) is satisfied and that $\Lambda$ is not a
sampling set for $V^{\infty}_{\Gamma}(\mathcal{G}).$ Hence, for every integer
$n$ there exist functions such that

$f_{n}(x)=\sum\limits_{\gamma\in\Gamma}c^{(n)}_{\gamma}\mathcal{G}(x-\gamma),\quad
f_{n}\in V^{\infty}_{\Gamma}(\mathcal{G}),\quad\|{\bf c}^{(n)}\|_{\infty}=1,$

and $\gamma_{n}\in{\mathbb{R}}$ such that $|c^{(n)}_{\gamma_{n}}|>1/2$ and

$\|f\big{|}_{\Lambda}\|_{\infty}=\sup\limits_{\lambda\in\Lambda}\left|\sum\limits_{\gamma\in\Gamma}c^{(n)}_{\gamma}\mathcal{G}(\lambda-\gamma)\right|<\frac{1}{n}.$

Similarly to the proof of Lemma 3.1, one can find non-empty sets
$\Lambda^{\prime}\in W(\Lambda)$, $\Gamma^{\prime}\in W(\Gamma)$ and non-
trivial coefficients ${\bf d}=\\{d_{\gamma^{\prime}}\\}\in
l^{\infty}(\Gamma^{\prime})$ such that the function

$h(x):=\sum_{\gamma^{\prime}\in\Gamma^{\prime}}d_{\gamma^{\prime}}\mathcal{G}(x-\gamma^{\prime})$

vanishes on the set $\Lambda^{\prime}.$ Using (1.10) and Proposition 4.1, we
arrive at a contradiction. Therefore $\Lambda$ is a sampling set for
$V^{\infty}_{\Gamma}(\mathcal{G}).$

Step 2. We show that if $\Lambda$ is a sampling set for
$V^{\infty}_{\Gamma}(\mathcal{G})$ then $\Lambda$ is a sampling set for
$V^{p}_{\Gamma}(\mathcal{G})$ for any $1\leq p<\infty.$ To this end, we use
the approach developed in [8]. Consider an operator $A:l^{p}(\Gamma)\to
l^{p}(\Lambda)$ given by

(6.1) $A{\bf
c}:=\left\\{\sum\limits_{\gamma\in\Gamma}c_{\gamma}\mathcal{G}(\lambda-\gamma):\lambda\in\Lambda\right\\},\quad{\bf
c}=\\{c_{\gamma}\\}.$

This operator is given by the matrix
$A=\\{\mathcal{G}(\lambda-\gamma)\\}\in{\mathbb{C}}^{\Lambda\times\Gamma}.$

Since $\Lambda$ is a sampling set for $V^{\infty}_{\Gamma}(\mathcal{G})$, the
operator $A$ is bounded from below for $p=\infty$:

$\sup\limits_{\lambda\in\Lambda}\left|\sum\limits_{\gamma\in\Gamma}c_{\gamma}\mathcal{G}(\lambda-\gamma)\right|\geq
C\|{\bf c}\|_{\infty}.$

By Lemma 2.1, we deduce that $A$ is bounded from below in $l^{p}$ for any
$1\leq p<\infty.$ Therefore, for every function

$f(x)=\sum\limits_{\gamma\in\Gamma}c_{\gamma}\mathcal{G}(x-\gamma)\in
V^{p}_{\Gamma}(\mathcal{G})$

we get

$\sum\limits_{\lambda\in\Lambda}|f(\lambda)|^{p}=\|A{\bf c}\|_{p}^{p}\geq
C\|{\bf c}\|_{p}^{p}\geq C\|f\|^{p}_{p},$

where the last inequality follows from Lemma 8.1. This completes the proof.

###  6.2. Proof of Theorem 1.11

The proof of part $(i)$ is similar to the proof of Theorem 1.8 above.

Part (ii) follows from Proposition 5.1.

###  6.3. Proof of Corollary 1.12

We will use the following result from Banach theory: Let $X$ and $Y$ be Banach
spaces. Let $U:X\to Y$ be a bounded operator.

$(i)$ $U$ is onto if and only if the dual operator $U^{\ast}:Y^{\ast}\to
X^{\ast}$ is bounded from below.

$(ii)$ $U$ is bounded from below if and only if $U^{\ast}$ is onto.

Let $A:l^{p}(\Gamma)\to l^{p}(\Lambda)$ be the operator defined in (6.1) The
dual operator $A^{\ast}:l^{p^{\prime}}(\Lambda)\to
l^{p^{\prime}}(\Gamma),1/p+1/p^{\prime}=1,$ is given by

$A^{\ast}{\bf
d}:=\left\\{\sum\limits_{\lambda\in\Lambda}d_{\lambda}\mathcal{G}(\lambda-\gamma):\gamma\in\Gamma\right\\},\quad{\bf
d}=\\{d_{\lambda}\\}.$

Since $\Lambda$ is a sampling set for every space
$V^{p}_{\Gamma}(\mathcal{G}),1\leq p\leq\infty$, it is bounded from below.
Using the above theorem, one may conclude that $A^{\ast}$ is onto for every
$1\leq p^{\prime}\leq\infty$. This means that $\Gamma$ is an interpolation set
for $V^{p^{\prime}}_{\Lambda}(\mathcal{G}).$

##  7\. On Gabor Frames

###### Proof of Theorem 1.13.

The proof is a simple consequence of Lemma 2.3 and our sampling Theorems 1.8
and 1.11.

(i) Let $\mathcal{G}\in\mathcal{K}(\alpha)$ satisfy the assumptions of
Corollary 1.13 (i). Then, by Lemma 1.5, ${\mathbb{Z}}$-shifts of $\mathcal{G}$
are stable. Also, since $D^{-}(-\Lambda+x)=D^{-}(\Lambda)$, it follows from
the assumptions, Theorem 1.8 (with $\Gamma={\mathbb{Z}}$) and Lemma 2.3 that
$G(\mathcal{G},\Lambda\times{\mathbb{Z}})$ is a frame in
$L^{2}({\mathbb{R}})$.

The proof of part (iii) is similar to the proof of part (i).

Parts (ii) and (iv) follow from Lemma 1.5 and Theorems 1.10, 1.11.

∎

##  8\. Stability of $\Gamma$-shifts

###  8.1. Estimate from above

######  Lemma 8.1.

Let $1\leq p\leq\infty$, $\Gamma$ be a separated set and $\mathcal{G}\in
W_{0}$. Then

(8.1)
$\left\|\sum\limits_{\gamma\in\Gamma}c_{\gamma}\mathcal{G}(\cdot-\gamma)\right\|_{p}\leq
N^{1/q}(\Gamma)\|\mathcal{G}\|_{W_{0}}\|{\bf c}\|_{p},\quad\mbox{for every
}{\bf c}\in l^{p}(\Gamma),$

where $q=p/(p-1)$ and $N(\Gamma)$ is the covering constant

$N(\Gamma):=\sup_{x\in{\mathbb{R}}}\sum_{\Gamma}{\bf 1}_{[0,1]}(x+\gamma).$

See Theorem 2.1 in [16] for a slightly more general result for the multi-
dimensional integer-shifts.

###### Proof.

Since the set $\Gamma$ is separated, we have $N(\Gamma)<\infty.$

The proof is obvious when $p=\infty.$ Therefore, we assume that $1\leq
p<\infty$.

Clearly, for every $h\in L^{q}({\mathbb{R}})$ and $k\in{\mathbb{Z}}$, we have

$\sum_{\gamma\in\Gamma}\int_{k+\gamma}^{k+1+\gamma}|h(x)|^{q}\,dx\leq
N(\Gamma)\|h\|_{q}^{q}.$

We will use this inequality at the end of the following calculations:

$\int_{\mathbb{R}}\left|\sum_{\Gamma}c_{\gamma}\mathcal{G}(x-\gamma)h(x)\right|\,dx\leq\sum_{k\in{\mathbb{Z}}}\sum_{\Gamma}|c_{\gamma}|\int_{k}^{k+1}|\mathcal{G}(x)h(x+\gamma)|\,dx\leq$
$\sum_{k\in{\mathbb{Z}}}\sum_{\Gamma}\|\mathcal{G}\|_{L^{\infty}(k,k+1)}|c_{\gamma}|\int_{k+\gamma}^{k+1+\gamma}|h(x)|dx\leq$
$\sum_{k\in{\mathbb{Z}}}\|\mathcal{G}\|_{L^{\infty}(k,k+1)}\sum_{\Gamma}|c_{\gamma}|\left(\int_{k+\gamma}^{k+1+\gamma}|h(x)|^{q}dx\right)^{1/q}\leq
N(\Gamma)^{1/q}\|\mathcal{G}\|_{W_{0}}\|{\bf c}\|_{l^{p}}\|h\|_{q}.$

Finally,

$\left\|\sum_{\Gamma}c_{\gamma}\mathcal{G}(x-\gamma)\right\|_{p}=\sup_{0<\|h\|_{q}<\infty}\frac{1}{\|h\|_{q}}\int_{R}\left|\sum_{\Gamma}c_{\gamma}\mathcal{G}(x-\gamma)h(x)\right|dx\leq
N(\Gamma)^{1/q}\|\mathcal{G}\|_{W_{0}}\|{\bf c}\|_{l^{p}}.$

∎

###  8.2. Estimate from below

Here we prove Theorem 1.6.

We assume that $\Gamma$-shifts of $\mathcal{G}$ are $l^{\infty}$-stable, and
prove that they are $l^{p}$-stable, for every $p\in[1,\infty].$ This latter
means that $\|f\|_{p}\geq K_{p}\|{\bf c}\|_{p}$, for every function $f$ of the
form

$f(x)=\sum\limits_{\gamma}c_{\gamma}\mathcal{G}(x-\gamma),\quad{\bf c}\in
l^{p}(\Gamma).$

Choose any positive number $\delta$ and denote by $\mathcal{S_{\delta}}$ the
family of all sequences

$\Lambda=\\{\lambda_{k}\\}_{k\in{\mathbb{Z}}}=\\{...<\lambda_{k}<\lambda_{k+1}<...:\lambda_{k}\in[(k+1/4)\delta,(k+3/4)\delta],k\in{\mathbb{Z}}\\}.$

For every $\Lambda\in\mathcal{S_{\delta}}$ we denote by $A(\Lambda)$ the
discretization operator acting on the spaces of sequences with indexes from
$\Gamma$ to the space of sequences indexed by $\Lambda$ defined by
$A(\Lambda)_{\lambda,\gamma}=\mathcal{G}(\lambda-\gamma)$, so that

(8.2) $\\{A(\Lambda){\bf
c}\\}:=\\{f(\lambda):\lambda\in\Lambda\\}=\left\\{\sum\limits_{\gamma\in\Gamma}c_{\gamma}\mathcal{G}(\lambda-\gamma):\lambda\in\Lambda\right\\}.$

######  Claim 8.2.

There exists $\delta_{0}>0$ such that $\|A_{\Lambda}{\bf c}\|_{\infty}>K\|{\bf
c}\|_{\infty},$ for some $K>0$ and every ${\bf c}\in l^{\infty}(\Gamma)$,
$\Lambda\in\mathcal{S_{\delta}}$ and $\delta\leq\delta_{0}$.

###### Proof of the Claim 8.2.

Since $\Gamma$-shifts of $\mathcal{G}$ are $l^{\infty}$-stable, we have
$\|f\|_{\infty}\geq K_{\infty}\|{\bf c}\|_{\infty}$, for all ${\bf c}\in
l^{\infty}(\Gamma).$ Then for every $f\in V^{\infty}_{\Gamma}(\mathcal{G})$ we
may find a point $x_{0}$ such that $|f(x_{0})|\geq(K_{\infty}/2)\|{\bf
c}\|_{\infty}.$ Clearly, there is an integer $k$ such that
$|\lambda_{k}-x_{0}|<2\delta.$ This gives

$|f(\lambda_{k})-f(x_{0})|\leq\|{\bf
c}\|_{\infty}\sum_{\gamma}|\mathcal{G}(\lambda_{k}-\gamma)-\mathcal{G}(x_{0}-\gamma)|=\sum_{\gamma:|\gamma-
x_{0}|\leq R}+\sum_{\gamma:|\gamma-x_{0}|>R},$

where we choose $R$ so large that

$\sum_{\gamma:|\gamma-
x_{0}|>R}|\mathcal{G}(\lambda_{k}-\gamma)|+|\mathcal{G}(x_{0}-\gamma)|<K_{\infty}/4.$

On the other hand, since $\mathcal{G}\in C({\mathbb{R}})$, it is uniformly
continuous on $[-R,R]$, we have

$\sum_{\gamma:|\gamma-x_{0}|\leq
R}|\mathcal{G}(\lambda_{k}-\gamma)-\mathcal{G}(x_{0}-\gamma)|<K_{\infty}/4,$

provided $\delta$ is sufficiently small, which proves the claim. ∎

In what follows we assume that $\delta\leq\delta_{0}$. Hence, by the
definition of the family $\mathcal{S_{\delta}}$, Lemma 2.1 and Remark 2.2, we
see that there is a constant $C$ such that

(8.3) $\|A(\Lambda){\bf c}\|_{p}>C\|{\bf c}\|_{p},\mbox{ for every }{\bf c}\in
l^{p}(\Gamma)\mbox{ and every }\Lambda\in\mathcal{S_{\delta}}.$

Choose any $p,1\leq p<\infty$. Given any $f\in V^{p}_{\Gamma}(\mathcal{G})$,
choose points $\lambda_{k}\in[(k+1/4)\delta,(k+3/4)\delta]$ so that
$|f(x)|\geq|f(\lambda_{k})|,x\in[(k+1/4)\delta,(k+3/4)\delta]$. Then

$\Lambda:=\\{\lambda_{k}:k\in{\mathbb{Z}}\\}\in\mathcal{S_{\delta}}.$

Using (8.3), we get

$\|f\|_{p}^{p}\geq(\delta/2)\sum_{k\in{\mathbb{Z}}}|f(\lambda_{k})|^{p}=(\delta/2)\|A(\Lambda){\bf
c}\|_{p}^{p}\geq(\delta/2)C^{p}\|{\bf c}\|_{p}^{p}.$

##  9\. Acknowledgements

The second author is grateful to K. Gröchenig, M. Faulhuber, and I.
Shafkulovska for fruitful discussions.

## References

  * [1] A. Aldroubi and K. Gröchenig.  Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces.  J. Fourier Anal. Appl., 6(1):93–103, 2000.
  * [2] A. Baranov and Y. Belov.  Irregular sampling for hyperbolic secant type functions.  arXiv:2312.10174, 2023.
  * [3] Y. Belov, A. Kulikov, and Y. Lyubarskii.  Irregular Gabor frames of Cauchy kernels.  Appl. Comput. Harmon. Anal., 57:101–104, 2022.
  * [4] Y. Belov, A. Kulikov, and Y. Lyubarskii.  Gabor frames for rational functions.  Invent. Math., 231(2):431–466, 2023.
  * [5] A. Beurling.  Balayage of Fourier–Stieltjes transforms.  In L. Carleson, P. Malliavin, J. Neuberger, and J. Wermer, editors, The collected works of Arne Beurling. Vol. 2, Contemporary Mathematicians, pages xx+389. Birkhäuser Boston, Inc., Boston, MA, 1989.  Harmonic analysis.
  * [6] K. Gröchenig and Y. Lyubarskii.  Gabor frames with Hermite functions.  C. R. Math. Acad. Sci. Paris, 344(3):157–162, 2007.
  * [7] K. Gröchenig and Y. Lyubarskii.  Gabor (super)frames with Hermite functions.  Math. Ann., 345(2):267–286, 2009.
  * [8] K. Gröchenig, J. Ortega-Cerdà, and J. L. Romero.  Deformation of Gabor systems.  Adv. Math., 277:388–425, 2015.
  * [9] K. Gröchenig, J. L. Romero, and J. Stöckler.  Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions.  Invent. Math., 211(3):1119–1148, 2018.
  * [10] K. Gröchenig and J. Stöckler.  Gabor frames and totally positive functions.  Duke Math. J., 162(6):1003–1031, 2013.
  * [11] K. Gröchenig and I. Shafkulovska.  Sampling theorems with derivatives in shift-invariant spaces generated by periodic exponential B-splines.  arXiv:2311.08352, 2023.
  * [12] K. Hamm and J. Ledford.  On the structure and interpolation properties of quasi shift-invariant spaces.  J. Funct. Anal., 274(7):1959–1992, 2018.
  * [13] A. J. E. M. Janssen.  Some Weyl-Heisenberg frame bound calculations.  Indag. Math. (N.S.), 7(2):165–183, 1996.
  * [14] A. J. E. M. Janssen.  On generating tight Gabor frames at critical density.  J. Fourier Anal. Appl., 9(2):175–214, 2003.
  * [15] A. J. E. M. Janssen and T. Strohmer.  Hyperbolic secants yield Gabor frames.  Appl. Comput. Harmon. Anal., 12(2):259–267, 2002.
  * [16] R.-Q. Jia and C. A. Micchelli.  Using the refinement equations for the construction of pre-wavelets II: Powers of two.  In Curves and Surfaces, pages 209–246. Elsevier, 1991.
  * [17] H. J. Landau.  Necessary density conditions for sampling and interpolation of certain entire functions.  Acta Math., 117:37–52, 1967.
  * [18] B. Y. Levin.  Lectures on entire functions, volume 150 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1996.  In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko.
  * [19] Y. Lyubarskii and P. G. b. Nes.  Gabor frames with rational density.  Appl. Comput. Harmon. Anal., 34(3):488–494, 2013.
  * [20] Y. I. Lyubarskiĭ.  Frames in the Bargmann space of entire functions.  In Entire and subharmonic functions, volume 11 of Adv. Soviet Math., pages 167–180. Amer. Math. Soc., Providence, RI, 1992.
  * [21] J. L. Romero, A. Ulanovskii, and I. Zlotnikov.  Sampling in the shift-invariant space generated by the bivariate Gaussian function.  arXiv:2306.13619, 2023.
  * [22] K. Seip.  Density theorems for sampling and interpolation in the Bargmann-Fock space. I.  J. Reine Angew. Math., 429:91–106, 1992.
  * [23] K. Seip and R. Wallstén.  Density theorems for sampling and interpolation in the Bargmann-Fock space. II.  J. Reine Angew. Math., 429:107–113, 1992.