# Marcinkiewicz-Zygmund Inequalities for Polynomials in Fock Space

Karlheinz Gröchenig  Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
A-1090 Vienna, Austria<EMAIL_ADDRESS>and  Joaquim Ortega-
Cerdà  Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes, 585
08007, Barcelona, Spain<EMAIL_ADDRESS>
###### Abstract.

We study the relation between Marcinkiewicz-Zygmund families for polynomials
in a weighted $L^{2}$-space and sampling theorems for entire functions in the
Fock space and the dual relation between uniform interpolating families for
polynomials and interpolating sequences. As a consequence we obtain a
description of signal subspaces spanned by Hermite functions by means of Gabor
frames.

###### Key words and phrases:

Marcinkiewicz-Zygmund inequalities, Fock space, reproducing kernel, Hermite
function, incomplete gamma function

###### 2010 Mathematics Subject Classification:

30E05,30H20,41A10,42B30

K. G. was supported in part by the project P31887-N32 of the Austrian Science
Fund (FWF). J.O.C. has been partially supported by the Generalitat de
Catalunya (grant 2017 SGR 359) and the Spanish Ministerio de Ciencia,
Innovación y Universidades (project MTM2017-83499-P)

##  1\. Introduction

We study sampling and interpolation in Fock space and the relation to sampling
and interpolation of polynomials. The Fock space $\mathcal{F}^{2}$ consists of
all entire functions with finite norm

(1)
$\|f\|_{\mathcal{F}^{2}}=\Big{(}\int_{\mathbb{C}}|f(z)|^{2}\,e^{-\pi|z|^{2}}dm(z)\Big{)}^{1/2}\,,$

where $dm(z)=dxdy$ is the Lebesgue measure on
$\mathbb{C}\simeq\mathbb{R}^{2}$.

We denote by $\mathcal{P}_{n}$ the holomorphic polynomials of degree at most
$n$. A sequence of (finite) subsets $\Lambda_{n}\subseteq\mathbb{C}$ is called
a Marcinkiewicz-Zygmund family for the $\mathcal{P}_{n}$ in Fock space
$\mathcal{F}^{2}$, if there exist constants $A,B>0$, such that for all $n$
large, $n\geq n_{0}$,

(2) $\qquad
A\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\leq\sum_{\lambda\in\Lambda_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}\leq
B\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\qquad\text{ for all }p\in\mathcal{P}_{n}\,.$

Here $k_{n}$ is the reproducing kernel of $\mathcal{P}_{n}$, when endowed with
the inner product inherited from $\mathcal{F}^{2}$.

This notion corresponds to the standard definition of sampling in a
reproducing kernel Hilbert space $\mathcal{H}$. Let
$k_{\lambda}(z)=k(z,\lambda)$ be the reproducing kernel of $\mathcal{H}$,
i.e., $f(\lambda)=\langle f,k_{\lambda}\rangle_{\mathcal{H}}$ at the point
$\lambda$. Then a sequence $\Lambda$ is a sampling set for $\mathcal{H}$, if
the normalized reproducing kernels
$\Big{\\{}\frac{k(z,\lambda)}{\sqrt{k(\lambda,\lambda)}}:\lambda\in\Lambda\Big{\\}}$
constitute a frame for $\mathcal{H}$. Equivalently, the sampling inequality
$A\|f\|_{\mathcal{H}}^{2}\leq\sum_{\lambda\in\Lambda}|f(\lambda)|^{2}k(\lambda,\lambda)^{-1}\leq
B\|f\|_{\mathcal{H}}^{2}$ holds for all $f\in\mathcal{H}$.

In the Fock space $\mathcal{F}^{2}$ the reproducing kernel is $k(z,w)=e^{\pi
z\overline{w}}$, and a sequence $\Lambda\subseteq\mathbb{C}$ is sampling in
$\mathcal{F}^{2}$, if and only if

$A\|f\|_{\mathcal{F}^{2}}\leq\sum_{\lambda\in\Lambda}|f(\lambda)|^{2}e^{-\pi|\lambda|^{2}}\leq
B\|f\|_{{}_{\mathcal{F}^{2}}}^{2}\qquad\text{ for all }f\in\mathcal{F}^{2}\,.$

In this article we compare the notion of Marcinkiewicz-Zygmund families for
$\mathcal{P}_{n}$ with sampling sequences for the Fock space
$\mathcal{F}^{2}$. We will see that both notions are intimately connected.
Roughly speaking, suitable finite sections of a sampling set for
$\mathcal{F}^{2}$ yield a Marcinkiewicz-Zygmund family for the polynomials
$\mathcal{P}_{n}$ in $\mathcal{F}^{2}$, and suitable limits of a
Marcinkiewicz-Zygmund family yield a sampling set for $\mathcal{F}^{2}$.

A precise formulation is contained in our main result. (See Section 5 for an
explanation of weak limits.)

######  Theorem 1.1.

(i) Assume that $\Lambda\subseteq\mathbb{C}$ is a sampling set for
$\mathcal{F}^{2}$. For $\tau>0$ set $\rho_{n}$ such that
$\pi\rho_{n}^{2}=n+\sqrt{n}\tau$ and let $B_{\rho_{n}}$ be the centered disk
of radius $\rho_{n}$. Then for $\tau>0$ large enough, the sets
$\Lambda_{n}=\Lambda\cap B_{\rho_{n}}$ form a Marcinkiewicz-Zygmund family for
$\mathcal{P}_{n}$ in $\mathcal{F}^{2}$.

(ii) Conversely, every weak limit of a Marcinkiewicz-Zygmund family
$(\Lambda_{n})$ for $\mathcal{P}_{n}$ in $\mathcal{F}^{2}$ is a sampling set
for $\mathcal{F}^{2}$.

A dual result establishes a similar relationship between interpolating sets
for $\mathcal{F}^{2}$ and uniformly interpolating sets for $\mathcal{P}_{n}$.
A set $\Lambda$ is interpolating in $\mathcal{F}^{2}$, if for every sequence
$(a_{\lambda})_{\lambda\in\Lambda}\in\ell^{2}(\Lambda)$, there exists
$f\in\mathcal{F}^{2}$ such that
$f(\lambda)e^{-\pi|\lambda|^{2}/2}=a_{\lambda}$ for all $\lambda\in\Lambda$.
Of course, for polynomials of degree $n$ every set of $n+1$ points is
interpolating. In analogy to the definition of Marcinkiewicz-Zygmund families
we call a family of (finite) subsets $\Lambda_{n}\subseteq\mathbb{C}$ a
uniform interpolating family, if there exists a constant $A>0$, such that for
every sequence
$a=(a_{\lambda})_{\lambda\in\Lambda_{n}}\in\ell^{2}(\Lambda_{n})$ there exists
a polynomial $p\in\mathcal{P}_{n}$ such that
$p(\lambda)k_{n}(\lambda,\lambda)^{-1/2}=a_{\lambda}$ for
$\lambda\in\Lambda_{n}$ with norm control $\|p\|_{\mathcal{F}^{2}}^{2}\leq
A\|a\|_{2}^{2}$.

######  Theorem 1.2.

(i) Assume that $\Lambda\subseteq\mathbb{C}$ is a set of interpolation for
$\mathcal{F}^{2}$. For $\tau>0$ and $\epsilon>0$ set $\rho_{n}$, such that
$\pi\rho_{n}^{2}=n-\sqrt{n}(\sqrt{2\log n}+\tau)$. Then for every $\tau>0$
large enough, the sets $\Lambda_{n}=\Lambda\cap B_{\rho_{n}}$ form a uniform
interpolating family for $\mathcal{P}_{n}$ in $\mathcal{F}^{2}$.

(ii) Conversely, every weak limit of a uniform interpolating family
$(\Lambda_{n})$ is a set of interpolation for $\mathcal{F}^{2}$.

Marcinkiewicz-Zygmund families and uniform interpolating families arise
several areas of analysis. They can be understood as finite-dimensional
approximations of sampling theorems in reproducing kernel Hilbert spaces.
Theorem 1.1(ii) shows that a Marcinkiewicz-Zygmund family can be used to prove
a sampling theorem in an infinite dimensional space. In approximation theory,
a Marcinkiewicz-Zygmund family for a sequence of nested subspaces gives rise
to a sequence of quadrature rules and function approximation from point
evaluations, see [11] for this aspect. Random constructions of Marcinkiewicz-
Zygmund families occur in the theory of deterministic point processes, e.g.,
[1, 2, 4, 5]. In the recent advances in complexity theory and data analysis
Marcinkiewicz-Zygmund families are implicit in the discretization of norms.
For a nice survey see [15].

This work has several predecessors in different contexts. In [12] we have
studied the analogous problem in the Bergman space and in the Hardy space in
the unit disk. Indeed, our proof strategy for Theorem 1.1 is taken from [12].
Whereas in Bergman space the results can be formulated similarly to Theorem
1.1, the situation is Hardy space is rather different and the construction of
Marcinkiewicz-Zygmund families needed to be based on different principles. In
[10] a sampling theorem for bandlimited functions was derived via
Marcinkiewicz-Zygmund families for trigonometric polynomials. The set-up of
[16] is a compact manifold with a positive line bundle. Marcinkiewicz-Zygmund
families for the space of holomorphic sections in powers of the line bundle
are connected to sampling sequences in the tangent space.

Though line bundles appear much more complicated objects than Fock space,
which even has a closed-form reproducing kernel, Fock space presents some new
difficulties. It lacks compactness that made off-diagonal estimates for the
reproducing kernel easier in [16]. Another source of difficulty is the
behavior under translation. The Fock space $\mathcal{F}^{2}$ is invariant with
respect to Bargmann-Fock shifts, while $\mathcal{P}_{n}$ endowed with the Fock
norm is not.

Finally we mention the extensive work on the asymptotics of reproducing
kernels for weighted polynomials and related construction of _random_
Marcinkiewicz-Zygmund families in the context of determinental point processes
[1] – [5]. In [3, 4] Y. Ameur and his coauthors have studied a similar notion
of sampling polynomials with respect to the discrete norm
$\sum_{\lambda\in\Lambda_{n}}|p(\lambda)|^{2}e^{-\pi n|\lambda|^{2}}$ instead
of
$\sum_{\lambda\in\Lambda_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}$.
Note that all this work uses measures that depend on the polynomial degree
$n$, quite in contrast to our set-up (2). Their choice was motivated by
problems arising in random Gaussian matrix ensembles and models of the
distribution of points in the one component plasma. The common ground is the
construction of point sets that are sampling for polynomials in Fock spaces.
In [3, 4] this is achieved with random processes (determinental point
processes), whereas our construction is purely deterministic and investigates
the connection to the infinite-dimensional sampling problem in
$\mathcal{F}^{2}$. Therefore the results in [3, 4] are not directly comparable
to ours.

One of our basic tools is the size of the reproducing kernel for the
polynomials in a weighted $L^{2}$-space. Since our weight is the Gaussian
weight, the kernel can be expressed explicitly in terms of the incomplete
Gamma function which is a classical and well-studied object. We have collected
the necessary results in the appendix for the sake of being self-contained.
Estimates for the reproducing kernel have been studied in great generality in
[1, 5] with potential theoretic methods — without any reference to the
incomplete Gamma function. Possibly these estimates could also be used in our
context.

The estimates for the intrinsic reproducing kernel $k_{n}$ show that (i) the
$L^{2}$-energy of a polynomial of degree $n$ is concentrated in a disk of
radius $\sqrt{n/\pi}$, the so-called bulk region, and (ii) that the intrinsic
kernel $k_{n}$ for $\mathcal{P}_{n}$ is comparable to the kernel
$k(z,w)=e^{\pi z\bar{w}}$ precisely in the bulk region. See Lemma Lemma 2.2
and Corollary 3.2 for the precise statements.

As a consequence of Theorem 1.1 we mention an application to time-frequency
analysis. It is well-known that all problems about sampling in Fock space
possess an equivalent formulation about Gabor frames in $L^{2}(\mathbb{R})$.
To state this version, we denote the time-frequency shift of a function $g$ by
$z=(x,\xi)\in\mathbb{R}^{2}$ by $(\pi(z)g)(t)=e^{2\pi i\xi t}g(t-x)$ for
$t,x,\xi\in\mathbb{R}$. The $L^{2}$-normalized Hermite functions are denoted
by $h_{n}$, in particular $h_{0}(t)=2^{1/4}e^{-\pi t^{2}}$ is the Gaussian.
Then Theorem 1.1(i) is equivalent to the following statement, which may be of
interest in the time-frequency analysis of signal subspaces [14].

######  Theorem 1.3.

Assume that $\Lambda$ is a sampling set for $\mathcal{F}^{2}$ and $\tau>0$
large enough. Then $\\{\pi(\lambda)h_{0}:\pi|\lambda|^{2}\leq
n+\sqrt{n}\tau\\}$ is a frame for
$V_{n}=\mathrm{span}\,\\{h_{k}:k=0,\dots,n\\}$ with bounds independent of $n$.
This means that

$A\|f\|_{2}\leq\sum_{\lambda\in\Lambda:\pi|\lambda|^{2}\leq
n+\sqrt{n}\tau}|\langle f,\pi_{\lambda}h_{0}\rangle|^{2}\leq
B\|f\|_{2}^{2}\qquad\text{ for all }f\in V_{n}\,.$

Outlook. It is needless to say that the topic of Marcinkiewicz-Zygmund
families and sampling theorems admits dozens of variations. The ultimate goal
is to understand Marcinkiewicz-Zygmund families for polynomials
$\mathcal{P}_{n}$ in a weighted Bergman space on some general domain
$X\subseteq\mathbb{R}^{d}$ (or $\subseteq\mathbb{C}^{d}$). Intermediate
problems would be Marcinkiewicz-Zygmund families for polynomials in Fock
spaces with more general weight $e^{-Q(z)}$, or the construction
Marcinkiewicz-Zygmund families for multivariate Bergman spaces
$A^{2}(\mathbb{B}_{n})$ in $n$ complex variables on unit ball in
$\mathbb{C}^{n}$. Even simple variations of the set-up yield interesting new
questions.

The paper is organized as follows: In Section 2 we recall the basic facts
about the Fock space and the associated reproducing kernels. Section 3
summarizes the required asymptotics of the incomplete Gamma function. In
Section 4 we relate sampling sets for Fock space to Marcinkiewicz-Zygmund
families and prove the first part of Theorem 1.1. Section 5 covers the
converse statement. In Section 6 we deal with uniform interpolating families
and prove Theorem 1.2. The connection to the time-frequency analysis of signal
subspaces is explained in Section 7. Finally, in the appendix we offer some
elementary estimates for the zero-order asymptotics of the incomplete Gamma
function. These are, of course, well-known and added only to make the paper
self-contained.

##  2\. Fock space

The monomials $z\mapsto z^{k}$ are orthogonal in $\mathcal{F}^{2}$, and the
normalized monomials

$e_{k}(z)=\Big{(}\frac{\pi^{k}}{k!}\Big{)}^{1/2}\,z^{k}$

form an orthonormal basis for $\mathcal{F}^{2}$.

Let $\mathcal{P}_{n}$ be the subspace of polynomials of degree at most $n$ in
$\mathcal{F}^{2}$. The reproducing kernel of $\mathcal{P}_{n}$ is given by

(3)
$k_{n}(z,w)=\sum_{k=0}^{n}e_{k}(z)\overline{e_{k}(w)}=\sum_{k=0}^{n}\frac{(\pi
z\bar{w})^{k}}{k!}\,.$

As $n\to\infty$, this kernel converges to the reproducing kernel of
$\mathcal{F}^{2}$:

$k(z,w)=\lim_{n\to\infty}k_{n}(z,w)=e^{\pi z\bar{w}}\,.$

As we have learned in our study of Marcinkiewicz-Zygmund families in Bergman
spaces [12], we will need to understand the relation of the kernel $k_{n}$ to
$k$. For this purpose we will make use of the properties and the asymptotics
of the _incomplete Gamma function_

(4) $\Gamma(z,a)=\int_{a}^{\infty}t^{z-1}e^{-t}\,dt\,$

and

(5) $\gamma(z,a)=\int_{0}^{a}t^{z-1}e^{-t}\,dt\,.$

Denote the centered disc of radius $\rho$ by
$B_{\rho}=\\{z\in\mathbb{C}:|z|\leq\rho\\}$. Then

$\displaystyle\int_{B_{\rho}}|z|^{2k}e^{-\pi|z|^{2}}\,dm(z)$
$\displaystyle=2\pi\int_{0}^{\rho}r^{2k}e^{-\pi r^{2}}\,rdr$
$\displaystyle=\int_{0}^{\pi\rho^{2}}\Big{(}\frac{u}{\pi}\Big{)}^{k}\,e^{-u}\,du$
(6) $\displaystyle=\frac{1}{\pi^{k}}\gamma(k+1,\pi\rho^{2})\,.$

######  Lemma 2.1.

We have

$k_{n}(z,w)=e^{\pi z\bar{w}}\frac{\Gamma(n+1,\pi z\bar{w})}{n!}$

In particular $k_{n}(z,z)=e^{\pi|z|^{2}}\frac{\Gamma(n+1,\pi|z|^{2})}{n!}$.
111Note that we consider $\mathcal{P}_{n}$ as a subspace of $\mathcal{F}^{2}$
and always use the fixed weight $e^{-\pi|z|^{2}}$. The work on determinental
point processes always uses the weight $e^{-\pi n|z|^{2}}$ for
$\mathcal{P}_{n}$. The formulas and the asymptotics are therefore different.

###### Proof.

See [21, 8.4.8], or use the obvious formula

$\frac{1}{n!}\Gamma(n+1,r)=\frac{1}{n!}\int_{r}^{\infty}t^{n}e^{-t}\,dt=\frac{r^{n}}{n!}e^{-r}+\frac{1}{(n-1)!}\Gamma(n,r)$

repeatedly and then use analytic extension and substitute $r=\pi z\bar{w}$.

The energy of a polynomial $p(z)=\sum_{k=0}^{n}a_{k}z^{k}\in\mathcal{P}_{n}$
on a disc $B_{\rho}$ is

$\displaystyle\int_{B_{\rho}}|p(z)|^{2}\,e^{-\pi|z|^{2}}\,dm(z)$
$\displaystyle=\sum_{k=0}^{n}|a_{k}|^{2}\int_{B_{\rho}}|z|^{2k}e^{-\pi|z|^{2}}\,dm(z)$
$\displaystyle=\sum_{k=0}^{n}|a_{k}|^{2}\frac{k!}{\pi^{k}}\,\frac{\gamma(k+1,\pi\rho^{2})}{k!}$
$\displaystyle\geq\min_{0\leq k\leq
n}\frac{\gamma(k+1,\pi\rho^{2})}{k!}\,\sum_{k=0}^{n}|a_{k}|^{2}\frac{k!}{\pi^{k}}$
$\displaystyle\geq\frac{\gamma(n+1,\pi\rho^{2})}{n!}\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\,.$

In the last inequality we have used the fact that
$k\to\frac{\gamma(k+1,\pi\rho^{2})}{k!}$ is decreasing and that
$\|z^{k}\|_{{}_{\mathcal{F}^{2}}}^{2}=k!/\pi^{k}$ by (6).

Consequently, the energy of a polynomial $p\in\mathcal{P}_{n}$ is concentrated
$B_{\rho}$ with $\rho=\sqrt{n/\pi}$, the so-called bulk region.

######  Lemma 2.2.

For every $p\in\mathcal{P}_{n}$ we have

(7)
$\int_{B_{\rho}^{c}}|p(z)|^{2}\,e^{-\pi|z|^{2}}\,dm(z)\leq\frac{\Gamma(n+1,\pi\rho^{2})}{n!}\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\,.$

###### Proof.

This follows immediately from the previous estimates via

$\displaystyle\int_{B_{\rho}^{c}}|p(z)|^{2}\,e^{-\pi|z|^{2}}\,dm(z)$
$\displaystyle=\|p\|^{2}_{{}_{\mathcal{F}^{2}}}-\int_{B_{\rho}}|p(z)|^{2}\,e^{-\pi|z|^{2}}\,dm(z)$
(8)
$\displaystyle\leq(1-\frac{\gamma(n+1,\pi\rho^{2})}{n!})\,\|p\|_{{}_{\mathcal{F}^{2}}}^{2}=\frac{\Gamma(n+1,\pi\rho^{2})}{n!}\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\,.$

##  3\. Asymptotics of the incomplete Gamma function

The asymptotic behavior of the incomplete Gamma function is well understood.
We collect the properties required for Marcinkiewicz-Zygmund families in Fock
space. As usual $f\asymp g$ means that there exists a constant such that
$C^{-1}f(x)\leq g(x)\leq Cf(x)$ for all $x$ in the domain of $f$ and $g$,
$f\lesssim g$ means $f(x)\leq Cg(x)$, and $f\sim g$ near $x_{\infty}$ means
that $\lim_{x\to x_{\infty}}\frac{f(x)}{g(x)}=1$.

The following result has been proved on several levels of generality [9, 27,
28, 19, 20].

_The normalized incomplete Gamma function admits the asymptotic expansion_

(9)
$\frac{\Gamma(a,a+\tau\sqrt{a})}{\Gamma(a)}\sim\tfrac{1}{2}\mathrm{erfc}\,(\frac{\tau}{\sqrt{2}})+\frac{1}{\sqrt{2\pi
a}}e^{-\tau^{2}/2}\sum_{n=0}^{\infty}\frac{C_{n}(\tau)}{a^{n/2}}\,,$

_where $C_{0}(\tau)=\frac{\tau^{2}-1}{3}$ and
$\mathrm{erfc}(y)=\frac{2}{\sqrt{\pi}}\int_{y}^{\infty}e^{-t^{2}}\,dt$. This
implies that_

(10)
$\Big{|}\frac{\Gamma(a,a+\tau\sqrt{a})}{\Gamma(a)}-\tfrac{1}{2}\mathrm{erfc}(\frac{\tau}{\sqrt{2}})\Big{|}\leq\frac{\tau^{2}-1}{3}e^{-\tau^{2}/2}\frac{1}{\sqrt{2\pi
a}}\,.$

We only need these estimates for $a=n+1$ and $\tau>0$, but their validity has
been established for large domains in $\mathbb{C}$.

######  Proposition 3.1.

  1. (i)

For every $\epsilon>0$ there is $\tau>0$, such that

$\frac{\Gamma(n+1,n+\sqrt{n}\tau)}{n!}<\epsilon\qquad\forall n\geq n_{0}$

  2. (ii)

For every $\tau>0$ there is a constant $C(\tau)>0$, such that

$\frac{\Gamma(n+1,n+\sqrt{n}\tau)}{n!}\geq C(\tau)\qquad\forall n\geq
n_{0}\,.$

In fact, $C(\tau)$ can be taken as
$C(\tau)=\frac{1}{4}\mathrm{erf}(\tau/\sqrt{2})$

  3. (iii)

For $\tau>0$

$1-\frac{1}{n!}\Gamma(n+1,n-\sqrt{n}\tau)\leq e^{-\tau^{2}/2}\,.$

  4. (iv)

For every $x\geq 0$

$\lim_{n\to\infty}\frac{\Gamma(n+1,x)}{n!}=1\,.$

The convergence is uniform on bounded sets $\subseteq\mathbb{R}^{+}$ and
exponentially fast.

  5. (v)

For all $n$

$\frac{\Gamma(n+1,n)}{n!}>1/2\,.$

###### Proof.

Items (i) and (ii) follow readily from (10) as follows:

(i) Choose $\tau>0$, such that
$\tfrac{1}{2}\mathrm{erfc}(\tau/\sqrt{2})<\epsilon/2$. Now choose
$n_{0}\in\mathbb{N}$, such that the error
$\frac{\tau^{2}-1}{3}e^{-\tau^{2}/2}\frac{1}{\sqrt{2\pi(n+1)}}<\epsilon/2$ for
$n\geq n_{0}$. By (10) we then have
$\frac{\Gamma(n+1,n+\sqrt{n}\tau)}{n!}<\epsilon$ for all $n\geq n_{0}$.

(ii) Given $\tau>0$ choose $n_{0}$ such that the error
$\frac{\tau^{2}-1}{3}e^{-\tau^{2}/2}\frac{1}{\sqrt{2\pi(n+1)}}<\tfrac{1}{4}\textrm{erfc}(\tau/\sqrt{2})$
for $n\geq n_{0}$. Then
$\frac{\Gamma(n+1,n+\sqrt{n}\tau)}{n!}\geq\tfrac{1}{4}\textrm{erfc}(\tau/\sqrt{2})>0$
for all $n\geq n_{0}$.

(iii) and (iv) are well-known.

(v) is taken from [21], formula 8.10.13.

For completeness we summarize the arguments for the zero order asymptotics in
the appendix. In contrast to the full asymptotics of the incomplete Gamma
function, they are elementary.

######  Corollary 3.2.

If $\pi|z|^{2}\leq n+\sqrt{n}\tau$, then $k_{n}(z,z)\asymp
k(z,z)=e^{\pi|z|^{2}}$ with constant depending only on $\tau$, but not on $n$.

We also need an off-diagonal estimate for the kernel $k_{n}$.

######  Lemma 3.3.

Assume that $\pi|z|^{2}<n(1-\epsilon)$ for fixed $\epsilon>0$ and
$|z-w|\leq\tau$. Then for $n$ large enough depending on $\epsilon$,

(11) $\Big{|}\frac{\Gamma(n+1,\pi
z\bar{w})}{n!}-\frac{\Gamma(n+1,\pi|z|^{2})}{n!}\Big{|}\leq
Ce^{-\epsilon^{2}n/4}\,.$

###### Proof.

Since $\Gamma(n+1,\pi z\bar{w})$ is invariant with respect to the rotation
$(z,w)\to(e^{i\theta}z,e^{i\theta}w)$, we may assume that
$z=r\in\mathbb{R},r>0$ and $w=r+\bar{u}$ with $|u|\leq\tau$. We then write

$\Gamma(n+1,\pi z\bar{w})=\Gamma(n+1,\pi r^{2}+\pi ru)=\int_{\pi
r^{2}}^{\infty}t^{n}e^{-t}\,dt+\int_{\pi r^{2}+\pi ru}^{\pi
r^{2}}\dots=\Gamma(n+1,\pi r^{2})+\int_{\pi r^{2}+\pi ru}^{\pi r^{2}}\dots\,.$

Let $\gamma(s)=\pi r^{2}+s\pi ru$ the line segment from $\pi
r^{2}\in\mathbb{C}$ to $\pi r^{2}+\pi ru$. Then

$\displaystyle\big{|}\int_{\pi r^{2}}^{\pi r^{2}+\pi
ru}t^{n}e^{-t}\,dt\big{|}$ $\displaystyle=\big{|}\pi ru\int_{0}^{1}(\pi
r^{2}+s\pi ru)^{n}e^{-\pi r^{2}-s\pi ru}\,ds\big{|}$ $\displaystyle\leq\pi
r|u|(\pi r^{2}+\pi r|u|)^{n}e^{-\pi r^{2}+\pi r|u|}$ $\displaystyle\leq\pi
r\tau(\pi r^{2}+\pi r\tau)^{n}e^{-\pi r^{2}+\pi r\tau}\,.$

Observe that $r\to(\pi r^{2}+\pi r\tau)^{n}e^{-\pi r^{2}}$ is increasing, as
long as $\pi r^{2}+\pi r\tau\leq n$. Set $x=\tfrac{\pi r^{2}}{n}\leq
1-\epsilon$, then $\pi r\tau=\sqrt{\pi nx}\tau$, and by assumption $x<1$.
Using Sterling’s formula, we continue with

$\displaystyle\frac{1}{n!}\,\pi r\tau(\pi r^{2}+\pi r\tau)^{n}e^{-\pi
r^{2}+\pi r\tau}$ $\displaystyle\leq\frac{\sqrt{\pi}\tau\sqrt{nx}}{\sqrt{2\pi
n}}\,\Big{(}\frac{e}{n}\Big{)}^{n}\Big{(}nx+\sqrt{\pi
nx}\tau\Big{)}^{n}e^{-nx+\sqrt{\pi xn}\tau}$
$\displaystyle\lesssim\sqrt{x}\Big{(}x+\frac{\sqrt{\pi
x}\tau}{\sqrt{n}}\Big{)}^{n}e^{n-nx+\sqrt{\pi xn}\tau}$ $\displaystyle\leq
x^{n}\Big{(}1+\frac{\sqrt{\pi}\tau}{\sqrt{xn}}\Big{)}^{n}e^{n(1-x+\sqrt{\pi
x}\tau/\sqrt{n})}$ $\displaystyle\leq\exp\Big{(}n\big{(}1-x+\ln
x+\ln(1+\frac{\sqrt{\pi}\tau}{\sqrt{nx}})+\frac{\sqrt{\pi}\tau}{\sqrt{nx}}\big{)}\Big{)}\,.$

Since $x\to 1-x+\ln x$ is increasing on $(0,1]$ and $x\leq 1-\epsilon$, we
have $1-x+\ln x\leq\epsilon+\ln(1-\epsilon)\leq-\epsilon^{2}/2$. Choose $n$ so
large that
$\ln(1+\frac{\sqrt{\pi}\tau}{\sqrt{nx}})+\frac{\sqrt{\pi}\tau}{\sqrt{nx}}\leq\epsilon^{2}/4$,
then the latter expression is dominated by $e^{-n\epsilon^{2}/4}$, and this
expression tends to $0$ exponentially fast, as $n\to\infty$. This proves the
claim.

##  4\. Sampling implies Marcinkiewicz-Zygmund inequalities

We summarize the main facts about sampling sets in $\mathcal{F}^{2}$ from the
literature [17, 23, 24, 25].

(i) A set $\Lambda\subseteq\mathbb{C}$ is sampling for $\mathcal{F}^{2}$, if
and only if it contains a uniformly separated set
$\Lambda^{\prime}\subseteq\Lambda$ with lower Beurling density
$D^{-}(\Lambda^{\prime})>1$.

(ii) _Tail estimates._ Let $f\in\mathcal{F}^{2}$ and $\rho>0$. The
subharmonicity of $|f|^{2}$ implies that

(12) $|f(\lambda)|^{2}e^{-\pi|\lambda|^{2}}\leq
c_{\rho}\int_{B(\lambda,\rho)}|f(z)|^{2}\,e^{-\pi|z|^{2}}\,dm(z)$

for all $\lambda\in\mathbb{C}$. The constant is
$c_{\rho}=e^{\pi\rho^{2}}/(\pi\rho^{2})$, but we will not need it.

(ii) If $\Lambda$ is relatively separated, i.e., a finite union of $K$
uniformly discrete subsets of $\mathbb{C}$ with separation $\rho>0$, then

(13)
$\sum_{\lambda\in\Lambda,|\lambda|>R}|f(\lambda)|^{2}e^{-\pi|\lambda|^{2}}\leq
c_{\rho}K\int_{|z|>R-\rho}|f(z)|^{2}\,e^{-\pi|z|^{2}}\,dm(z)$

######  Theorem 4.1.

Assume that $\Lambda\subseteq\mathbb{C}$ is a sampling set for
$\mathcal{F}^{2}$ with bounds $A,B$. For $\tau>0$ set $\rho_{n}$, such that
$\pi\rho_{n}^{2}=n+\sqrt{n}\tau$. Then for $\tau>0$ large enough, the sets
$\Lambda_{n}=\Lambda\cap B_{\rho_{n}}$ form a Marcinkiewicz-Zygmund family for
$\mathcal{P}_{n}$ in $\mathcal{F}^{2}$.

###### Proof.

_Lower bound_ : Since always $k_{n}(z,z)\leq k(z,z)=e^{\pi|z|^{2}}$, we may
replace $k_{n}$ by $k$ in the sampling inequalities:

$\displaystyle\sum_{\lambda\in\Lambda_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}$
$\displaystyle\geq\sum_{\lambda\in\Lambda_{n}}\frac{|p(\lambda)|^{2}}{k(\lambda,\lambda)}$
$\displaystyle=\sum_{\lambda\in\Lambda_{n}}|p(\lambda)|^{2}e^{-\pi|\lambda|^{2}}=\sum_{\lambda\in\Lambda}-\sum_{\lambda\in\Lambda:|\lambda|>\rho_{n}}\dots\,.$

Since $\Lambda$ is a sampling set for $\mathcal{F}^{2}$, the first term
satisfies $\sum_{\lambda\in\Lambda}|p(\lambda)|^{2}e^{-\pi|\lambda|^{2}}\geq
A\|p\|_{\mathcal{F}^{2}}^{2}$. For the second term we observe that $\Lambda$
is a finite union of uniformly discrete sets with separation $\rho>0$ and
apply (13) and (7):

$\displaystyle\sum_{\lambda\in\Lambda:|\lambda|>\rho_{n}}|p(\lambda)|^{2}e^{-\pi|\lambda|^{2}}$
$\displaystyle\leq C\int_{|z|>\rho_{n}-\rho}|p(z)|^{2}e^{-\pi|z|^{2}}\,dm(z)$
$\displaystyle\leq
C\frac{\Gamma(n+1,\pi(\rho_{n}-\rho)^{2})}{n!}\|p\|_{\mathcal{F}^{2}}^{2}\qquad\text{
for }p\in\mathcal{P}_{n}\,.$

Our choice of $\rho_{n}$ implies that

$\displaystyle\pi(\rho_{n}-\rho)^{2}$
$\displaystyle=\pi\rho_{n}^{2}-2\pi\rho_{n}\rho+\pi\rho^{2}$
$\displaystyle=n+\sqrt{n}\tau+\pi\rho^{2}-2\sqrt{\pi}\,\sqrt{n+\sqrt{n}\tau}\rho$
$\displaystyle\geq n+\sqrt{n}\tau-2\sqrt{\pi n}\rho(1+\frac{\tau}{2\sqrt{n}})$
$\displaystyle\geq n+\sqrt{n}\tau^{\prime}$

with $\tau^{\prime}=\tau-3\sqrt{\pi}\rho$ whenever $\sqrt{n}\geq\tau$. Since
$a\mapsto\Gamma(x,a)$ is decreasing, we have
$\frac{\Gamma(n+1,\pi(\rho_{n}-\rho)^{2})}{n!}\leq\frac{\Gamma(n+1,n+\sqrt{n}\tau^{\prime})}{n!}$.
In view of Corollary 3.1(i) we may choose $\tau^{\prime}$ and hence $\tau$ so
that

$\sum_{\lambda\in\Lambda:|\lambda|>\rho_{n}}|p(\lambda)|^{2}e^{-\pi|\lambda|^{2}}\leq
C\frac{\Gamma(n+1,n+\sqrt{n}\tau^{\prime})}{n!}\|p\|_{\mathcal{F}^{2}}^{2}\leq\frac{A}{2}\|p\|_{\mathcal{F}^{2}}\qquad\text{
for all }p\in\mathcal{P}_{n}\,$

for large $n$, $n\geq n_{0}$, say. Combining the inequalities, we obtain
$\sum_{\lambda\in\Lambda_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}\geq\frac{A}{2}\|p\|_{\mathcal{F}^{2}}^{2}$
for all $p\in\mathcal{P}_{n}$.

_Upper inequality:_ For the above choice of $\tau$ Proposition 3.1(ii) says
that

$\frac{\Gamma(n+1,n+\sqrt{n}\tau)}{n!}\geq\tfrac{1}{4}\textrm{erfc}(\tau/\sqrt{2})=C(\tau)=C\,$

for $n\geq n_{0}$. This implies that $k_{n}(\lambda,\lambda)^{-1}\leq
C^{-1}k(\lambda,\lambda)^{-1}=C^{-1}e^{-\pi|\lambda|^{2}}$ for
$\pi|\lambda|^{2}\leq n+\sqrt{n}\tau$, and thus

$\sum_{\lambda\in\Lambda_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}\leq
C^{-1}\sum_{\lambda\in\Lambda,\pi|\lambda|^{2}\leq
n+\sqrt{n}\tau}|p(\lambda)|^{2}e^{-\pi|\lambda|^{2}}\leq
C^{-1}B\|p\|_{\mathcal{F}^{2}}\,\quad\text{ for all }p\in\mathcal{P}_{n}\,,$

because $\Lambda\supseteq\Lambda_{n}$ is a sampling set for $\mathcal{F}^{2}$.

Note that the lower bound in the Marcinkiewicz-Zygmund inequalities matches
the lower bound $A$ of the sampling inequality in $\mathcal{F}^{2}$, whereas
the upper bound is $4\,\mathrm{erfc}\,(\tau/\sqrt{2})^{-1}B$ depends also on
the additional parameter $\tau$.

######  Corollary 4.2.

For every $\epsilon>0$ there exist Marcinkiewicz-Zygmund families
$(\Lambda_{n})$ for $\mathcal{P}_{n}$ in $\mathcal{F}^{2}$ with
$\\#\Lambda_{n}\leq(1+\epsilon)(n+1)$ points.

###### Proof.

Choose $\mu,\delta$ small enough, so that $(1+2\mu)(1+\delta)<1+\epsilon$. Let
$\Lambda\subseteq\mathbb{C}$ be a (uniformly) discrete subset with
$D^{-}(\Lambda)>1$ and $D^{+}(\Lambda)<1+\mu$. Then $\Lambda$ is a sampling
set for $\mathcal{F}$ by the characterizations of Lyubarskii [17] and Seip
[23, 25], and for $\pi\rho_{n}^{2}=n+\sqrt{n}\tau$ the sets $\Lambda\cap
B_{\rho_{n}}$ form a Marcinkiewicz-Zygmund family for $\mathcal{P}_{n}$. For
$n$ large enough and $\tau/\sqrt{n}<\delta$, we find that

$\\#\Lambda_{n}=\\#(\Lambda\cap
B_{\rho_{n}})\leq(1+2\mu)|B_{\rho_{n}}|=(1+2\mu)(n+\sqrt{n}\tau)<(1+\epsilon)(n+1)\,.$

For a Marcinkiewicz-Zygmund family for $\mathcal{P}_{n}$ we need at least
$\mathrm{dim}\,\mathcal{P}_{n}=n+1$ points in each layer $\Lambda_{n}$. The
construction above yields Marcinkiewicz-Zygmund families for Fock space with
nearly optimal cardinality.

##  5\.  Marcinkiewicz-Zygmund inequalities imply sampling

We first formulate a few properties of the distribution of Marcinkiewicz-
Zygmund families.

######  Lemma 5.1.

Assume that $(\Lambda_{n})$ is a Marcinkiewicz-Zygmund family for
$\mathcal{F}^{2}$ with bounds $A,B$. Let $\varepsilon>0$ and
$\pi\sigma_{n}^{2}=n(1-\varepsilon)$.

(i) Then $\\#(\Lambda_{n}\cap B_{\sigma_{n}}^{c})\leq
B\frac{n+1}{(1-\varepsilon)^{n}}$. This holds also for $\epsilon=0$.

(ii) Let $B(z,\rho)$ be a disc in $B_{\sigma_{n}}$. Then

$\\#(\Lambda_{n}\cap B(z,\rho))\leq C\,.$

Consequently, every $\Lambda_{n}\cap B_{\sigma_{n}}$ is a union of at most $L$
separated sets with uniform separation $\delta>0$ independent of $n$.

###### Proof.

(i) For $\pi|w|^{2}\geq n$ and $k\leq n$, we have

$\frac{(\pi|w|^{2})^{k}}{k!}\leq\frac{(\pi|w|^{2})^{n}}{n!}\,,$

so the reproducing kernel satisfies the estimate

(14)
$k_{n}(w,w)=\sum_{k=0}^{n}\frac{(\pi|w|^{2})^{k}}{k!}\leq(n+1)\frac{(\pi|w|^{2})^{n}}{n!}\,.$

If $\pi|z|^{2}\geq(1-\varepsilon)n$, then, using $\sqrt{1-\varepsilon}\,w=z$,
we have $\pi|w|^{2}\geq n$, and as before we obtain:

$k_{n}(z,z)\leq\frac{n+1}{(1-\varepsilon)^{n}}\frac{(\pi|z|^{2})^{n}}{n!},$

To estimate $\\#(\Lambda_{n}\cap B_{\sigma_{n}}^{c})$, we test the
Marcinkiewicz-Zygmund inequalities for the monomial
$p_{n}(z)=\frac{\pi^{n/2}}{n!^{1/2}}z^{n}$. Then
$\|p_{n}\|_{\mathcal{F}^{2}}=1$. (14) implies that

$\frac{|p_{n}(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}\geq\frac{(1-\varepsilon)^{n}}{n+1}\qquad\pi|\lambda|^{2}\geq
n(1-\varepsilon)\,,$

and therefore

$\displaystyle\frac{(1-\varepsilon)^{n}}{n+1}\\#(\Lambda_{n}\cap
B_{\sigma_{n}}^{c})\leq\sum_{\lambda\in\Lambda_{n}:\pi|\lambda|^{2}\geq
n(1-\varepsilon)}\frac{|p_{n}(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}\leq
B\|p_{n}\|_{\mathcal{F}^{2}}=B$

we obtain $\\#(\Lambda_{n}\cap
B_{\sigma_{n}}^{c})\leq\frac{B(n+1)}{(1-\varepsilon)^{n}}$.

(ii) Let $\kappa_{n,z}(w)=k_{n}(w,z)/k_{n}(z,z)^{1/2}$ be the normalized
reproducing kernel of $\mathcal{P}_{n}$ and $B(z,\rho)\subseteq
B_{\sigma_{n}}$ be an arbitrary disc inside $B_{\sigma_{n}}$. Recall that
$k_{n}(z,w)=e^{\pi\bar{z}w}\Gamma(n+1,\pi\bar{z}w)/n!$ and that for
$\pi|z|^{2}\leq n(1-\epsilon)$ we have $\Gamma(n+1,\pi|z|^{2})/n!\geq 1/2$ by
Proposition 3.1(v). So after substituting the formulas for the kernel, we
obtain

$\displaystyle\sum_{\lambda\in\Lambda_{n}\cap
B(z,\rho)}\frac{|\kappa_{n,z}(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}$
$\displaystyle=\sum_{\lambda\in\Lambda_{n}\cap
B(z,\rho)}\frac{|k_{n}(z,\lambda)|^{2}}{k_{n}(z,z)k_{n}(\lambda,\lambda)}$
$\displaystyle=\sum_{\lambda\in\Lambda_{n}\cap
B(z,\rho)}|e^{\pi\bar{z}\lambda}|^{2}e^{-\pi|z|^{2}}e^{-\pi|\lambda|^{2}}\frac{|\Gamma(n+1,\pi\bar{z}\lambda)|^{2}}{\Gamma(n+1,\pi|z|^{2})\Gamma(n+1,\pi|\lambda|^{2})}$
$\displaystyle=\sum_{\lambda\in\Lambda_{n}\cap
B(z,\rho)}e^{-\pi|\lambda-z|^{2}}\frac{|\Gamma(n+1,\pi\bar{z}\lambda)|^{2}}{\Gamma(n+1,\pi|z|^{2})\Gamma(n+1,\pi|\lambda|^{2})}=(*)\,.$

Now note that by Lemma 3.3 $|\Gamma(n+1,\pi\bar{z}\lambda)|^{2}/n!\geq 1/4$
for $n$ large, whereas $|\Gamma(n+1,\pi|z|^{2})/n!\leq 1$, so that the last
sum finally is bounded below by

$(*)\geq e^{-\pi\rho^{2}}\sum_{\lambda\in\Lambda_{n}\cap
B(z,\rho)}\frac{1}{4}=\frac{1}{4}\,e^{-\pi\rho^{2}}\,\\#(\Lambda_{n}\cap
B)\,.$

Reading backwards, we obtain

$\frac{1}{4}\,e^{-\pi\rho^{2}}\,\\#(\Lambda_{n}\cap
B)\leq\sum_{\lambda\in\Lambda_{n}\cap
B}\frac{|\kappa_{n,z}(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}\leq
B\|\kappa_{n,z}\|_{\mathcal{F}}^{2}=B\,,$

which was claimed.

For completeness we mention that the number of points in the transition region
$C_{n,\tau}=\\{z\in\mathbb{C}:n-\sqrt{n}\tau\leq\pi|z|^{2}\leq
n+\sqrt{n}\tau\\}$ is bounded by

$\\#(\Lambda_{n}\cap C_{n,\tau})\lesssim\sqrt{n}e^{\tau^{2}/2}\,.$

This can be shown as above by testing against the monomial $z^{n}$.

Before stating our main theorem, we recall that a sequence of sets
$\Lambda_{n}\subseteq\mathbb{C}$ converges weakly to
$\Lambda\subseteq\mathbb{C}$, if for all compact disks $B\subseteq\mathbb{C}$

$\lim_{n\to\infty}d\big{(}(\Lambda_{n}\cap B)\cup\partial B,(\Lambda\cap
B)\cup\partial B\big{)}=0\,,$

where $d(\cdot,\cdot)$ denotes the Hausdorff distance between two compact sets
in $\mathbb{C}$. If every $\Lambda_{n}$ is the union of at most $K$ uniformly
separated sets with fixed separation $\delta$, then

(15) $\sum_{\lambda\in\Lambda_{n}\cap
B}\frac{|f(\lambda)|^{2}}{k(\lambda,\lambda)}\to\sum_{\lambda\in\Lambda\cap
B}\frac{|f(\lambda)|^{2}}{k(\lambda,\lambda)}m(\lambda)\,,$

with multiplicities $\mu(\lambda)\in\\{1,\dots,K\\}$.

######  Theorem 5.2.

Assume that $(\Lambda_{n})$ is a Marcinkiewicz-Zygmund family for the
polynomials $\mathcal{P}_{n}$ in $\mathcal{F}^{2}$. Let $\Lambda$ be a weak
limit of $(\Lambda_{n})$ or of some subsequence $(\Lambda_{n_{k}})$. Then
$\Lambda$ is a sampling set for $\mathcal{F}^{2}$.

###### Proof.

The assumption that $\Lambda_{n}$ is a Marcinkiewicz Zygmund family for
$\mathcal{P}_{n}$ in $\mathcal{F}^{2}$ means that there exist $A,B>0$ such
that
$A\|p\|_{\mathcal{F}^{2}}\leq\sum_{\lambda\in\Lambda_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}\leq
B\|p\|^{2}_{\mathcal{F}^{2}}$ for all polynomials $p\in\mathcal{P}_{n}$.

(i) Let $B=\bar{B}(w,\rho)$ be a closed disc. By Lemma 5.1(ii)
$\\#(\Lambda_{n}\cap B(w,\rho))\leq C$ for some constant $C$ independent of
$n$ and $B$, provided that $n$ is big enough. Since $\Lambda$ is a weak limit
of $\Lambda_{n}$, we know that $\\#(\Lambda\cap\bar{B}(w,\rho))\leq C$. This
means that $\Lambda$ is a union of $K$ uniformly separated sets with
separation $\delta>0$.

(ii) It follows immediately from (13) that $\Lambda$ satisfies the upper bound
in the sampling inequality for $\mathcal{F}^{2}$.

(iii) Lower bound. Fix a polynomial $p\in\mathcal{P}_{N}$ (of degree $N$) and
choose $r>0$ such that

$\int_{|z|\geq\sqrt{r/\pi}}|p(z)|^{2}e^{-\pi|z|^{2}}\,dm(z)<\frac{A}{4c_{\delta}K}\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\,,$

where $c_{\delta}$ is the constant in (12) for separation $\delta$. To avoid
the ugly notation in subscript, we write $\nu=\sqrt{r/\pi}$,
$\rho_{n}=\sqrt{n/\pi}$, and $\sigma_{n}=\sqrt{n(1-\varepsilon)/\pi}$.

For $p\in\mathcal{P}_{N}$ the Marcinkiewicz-Zygmund inequalities are satisfied
for every $n\geq N$, therefore

$\displaystyle A\|p\|_{{}_{\mathcal{F}^{2}}}^{2}$
$\displaystyle\leq\sum_{\lambda\in\Lambda_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}$
$\displaystyle=\sum_{\lambda\in\Lambda_{n},|\lambda|<\nu+\delta}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}$
$\displaystyle\qquad+\sum_{\lambda\in\Lambda_{n},\nu+\delta\leq|\lambda|<\sigma_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}+\sum_{\lambda\in\Lambda_{n},|\lambda|\geq\sigma_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}=$
$\displaystyle=A_{n}+B_{n}+C_{n}\,.$

If $|\lambda|\leq\sigma_{n}$, then
$k_{n}(\lambda,\lambda)\geq\frac{1}{2}k(\lambda,\lambda)=\frac{1}{2}e^{\pi|\lambda|^{2}}$
as a consequence of Lemma 2.1 and Proposition 3.1(v) Thus in the expressions
for $A_{n}$ and $B_{n}$ we may replace the kernel $k_{n}$ for polynomials by
the kernel $k(z,z)=e^{\pi|z|^{2}}$ for Fock space. Consequently

(16) $\displaystyle A\|p\|^{2}_{\mathcal{F}^{2}}\leq
2\sum_{\lambda\in\Lambda_{n},|\lambda|\leq\nu+\delta}|p(\lambda)|^{2}e^{-\pi|\lambda|^{2}}+B_{n}+C_{n}.$

Since in this sum all points $\lambda$ lie in the compact set
$\bar{B}(0,\nu+\delta)$, the weak convergence (including multiplicities
$m(\lambda)\in\\{1,\dots,K\\}$) implies the convergence to $\Lambda$ and

$\displaystyle\lim_{n\to\infty}A_{n}$ $\displaystyle\leq
2\lim_{n\to\infty}\sum_{\lambda\in\Lambda_{n},|\lambda|\leq\nu+\delta}|p(\lambda)|^{2}e^{-\pi|\lambda|^{2}}=$
$\displaystyle=2\sum_{\lambda\in\Lambda\cap\overline{B_{\nu+\delta}}}|p(\lambda)|^{2}e^{-\pi|\lambda|^{2}}m(\lambda)$

For the term $B_{n}$, we recall that every $\Lambda_{n}\cap B_{\sigma_{n}}$ is
a finite union of at most $K$ uniformly separated sequences with separation
$\delta$ and apply the tail estimate (13). Our choice of $r$ and
$\nu=\sqrt{r/\pi}$ yields

$\displaystyle B_{n}$ $\displaystyle\leq
2\sum_{\lambda\in\Lambda_{n},\nu+\delta\leq|\lambda|<\sigma_{n}}\frac{|p(\lambda)|^{2}}{k(\lambda,\lambda)}\leq
2c_{\delta}K\int_{|z|>\nu}|p(z)|^{2}e^{-\pi|z|^{2}}\,dm(z)$ $\displaystyle\leq
2c_{\delta}K\frac{A}{4c_{\delta}K}\|p\|_{\mathcal{F}^{2}}^{2}=\frac{A}{2}\|p\|_{\mathcal{F}^{2}}^{2}\,.$

To treat $C_{n}$, recall that $p$ has degree $N<n$. We use the trivial
estimate

$|p(\lambda)|^{2}=|\langle
p,k_{N}(\lambda,\cdot)\rangle|^{2}\leq\|p\|_{{}_{\mathcal{F}^{2}}}^{2}k_{N}(\lambda,\lambda)$

and substitute into $C_{n}$ to obtain

$C_{n}=\sum_{\lambda\in\Lambda_{n},|\lambda|>\sigma_{n}}\frac{|p(\lambda)|^{2}}{k_{n}(\lambda,\lambda)}\leq\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\,\\#(\Lambda_{n}\cap
B_{\sigma_{n}}^{c})\sup_{|z|\geq\sigma_{n}}\frac{k_{N}(z,z)}{k_{n}(z,z)}\,.$

By Lemma 5.1(ii) $\\#\Lambda_{n}\cap
B_{\sigma_{n}}^{c}\leq\frac{B(n+1)}{(1-\varepsilon)^{n}}$, whereas the ratio
of the different reproducing kernels is

$\frac{k_{N}(z,z)}{k_{n}(z,z)}=\frac{e^{\pi|z|^{2}}\Gamma(N+1,\pi|z|^{2})n!}{e^{\pi|z|^{2}}\Gamma(n+1,\pi|z|^{2})N!}\,.$

For simplicity set $\pi|z|^{2}=R>n(1-\varepsilon)$. Then

$\displaystyle\Gamma(n+1,R)$ $\displaystyle=\int_{R}^{\infty}t^{n}e^{-t}\,dt$
$\displaystyle\geq R^{n-N}\int_{R}^{\infty}t^{N}e^{-t}\,dt$
$\displaystyle=R^{n-N}\Gamma(N+1,R)\,,$

so that

$\sup_{\pi|z|^{2}>n(1-\varepsilon)}\frac{k_{N}(z,z)}{k_{n}(z,z)}\leq(n(1-\varepsilon))^{N-n}\frac{n!}{N!}\,.$

Altogether

$C_{n}\leq\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\frac{B(n+1)}{(1-\varepsilon)^{n}}(n(1-\varepsilon))^{N-n}\frac{n!}{N!}\to
0\,,$

as $n\to\infty$ by Stirling’s formula, provided that we choose $\varepsilon$
such that $(1-\varepsilon)^{2}>1/e$.

Combining the estimates for $A_{n},B_{n}$, and $C_{n}$ and letting $n$ go to
$\infty$, we obtain the lower sampling inequality

$\displaystyle\sum_{\lambda\in\Lambda}|p(\lambda)|^{2}m(\lambda)e^{-\pi|\lambda|^{2}}$
$\displaystyle\geq\sum_{\lambda\in\Lambda,|\lambda|\leq\nu+\delta}|p(\lambda)|^{2}m(\lambda)e^{-\pi|\lambda|^{2}}-\limsup_{n\to\infty}B_{n}-\lim_{n\to\infty}C_{n}\geq$
${}^{\geq}\frac{A}{2}\|p\|_{\mathcal{F}^{2}}^{2}\,.$

As the multiplicities satisfy $1\leq m(\lambda)\leq K$ for
$\lambda\in\Lambda$, we may omit them by changing the lower sampling constant
to $A/(2K)$.

Since polynomials are dense in $\mathcal{F}^{2}$, this estimate extends to all
of $\mathcal{F}^{2}$.

##  6\. Uniform Interpolation

In a sense the dual problem to sampling is the interpolation of function
values. A set $\Lambda\subseteq\mathbb{C}$ is interpolating for
$\mathcal{F}^{2}$, if for every
$a=(a_{\lambda})_{\lambda\in\Lambda}\in\ell^{2}(\Lambda)$ there exists
$f\in\mathcal{F}^{2}$, such that
$f(\lambda)e^{-\pi|\lambda|^{2}/2}=a_{\lambda}$. Equivalently, the set of
normalized reproducing kernels
$\kappa_{\lambda}=k_{\lambda}/\|k_{\lambda}\|_{{}_{\mathcal{F}^{2}}}=k_{\lambda}/k(\lambda,\lambda)^{1/2}$
is a Riesz sequence, i.e., there exists $A,B>0$, such that

(17)
$A\|a\|_{2}^{2}\leq\|\sum_{\lambda\in\Lambda}a_{\lambda}\kappa_{\lambda}\|_{{}_{\mathcal{F}^{2}}}^{2}\leq
B\|a\|_{2}^{2}$

for all $a\in\ell^{2}(\Lambda)$. It suffices to require (17) only for all $a$
with finite support.

In analogy to Marcinkiewicz-Zygmund families for sampling, we define uniform
families for interpolation as follows. We denote the normalized reproducing
kernels in $\mathcal{P}_{n}$ by
$\kappa_{n,\lambda}=k_{n,\lambda}/\|k_{n,\lambda}\|_{{}_{\mathcal{F}^{2}}}$.

######  Definition 6.1.

A sequence of finite sets $\Lambda_{n}\subseteq\mathbb{C}$ is a uniform
interpolating family for $\mathcal{P}_{n}$ in $\mathcal{F}^{2}$, if there
exist constants $A,B>0$ independent of $n$, such that for $n$ large enough,
$n\geq n_{0}$,

(18)
$A\|a\|_{2}^{2}\leq\|\sum_{\lambda\in\Lambda_{n}}a_{\lambda}\kappa_{n,\lambda}\|_{{}_{\mathcal{F}^{2}}}^{2}\leq
B\|a\|_{2}^{2}\qquad\text{ for all }a\in\ell^{2}(\Lambda_{n})\,.$

Equivalently, for every $a\in\ell^{2}(\Lambda_{n})$ there exists a polynomial
$p\in\mathcal{P}_{n}$, such that

$\displaystyle\frac{p(\lambda)}{k_{n}(\lambda,\lambda)^{1/2}}=a_{\lambda}\qquad\text{
and }\qquad\|p\|_{{}_{\mathcal{F}^{2}}}^{2}\leq A\|a\|_{2}^{2}\,.$

A further equivalent condition is that the associated Gram matrix with entries
$G_{\mu,\lambda}=\langle\kappa_{n,\lambda},\kappa_{n,\mu}\rangle$ has the
smallest eigenvalues $\lambda_{min}\geq A$ [18, §2.3 Lem. 2].

The relation between sets of interpolation for $\mathcal{F}^{2}$ and uniform
interpolating families is similar to the case of sampling.

######  Theorem 6.1.

Assume that $\Lambda\subseteq\mathbb{C}$ is a set of interpolation for
$\mathcal{F}^{2}$. For $\tau>0$ and $\epsilon>0$ set $\rho_{n}$, such that
$\pi\rho_{n}^{2}=n-\sqrt{n}(\sqrt{2\log n}+\tau)$. Then for every $\tau>0$
large enough, the sets $\Lambda_{n}=\Lambda\cap B_{\rho_{n}}$ form a uniform
interpolating family for $\mathcal{P}_{n}$ in $\mathcal{F}^{2}$.

###### Proof.

Since $D^{+}(\Lambda)<1$ is necessary for an interpolating set in
$\mathcal{F}^{2}$ by [23], the definition of $\rho_{n}$ implies that

$\\#(\Lambda\cap B_{\rho_{n}})\leq 1\cdot|B_{\rho_{n}}|\leq n$

for $n\geq n_{0}$ large enough. Consequently $\Lambda_{n}=\Lambda\cap
B_{\rho_{n}}$ contains at most $n$ points.

We show that we can choose $\tau>0$ in such a manner that, for
$a\in\ell^{2}(\Lambda)$ with finite support and all $n\in\mathbb{N}$
sufficiently large,

(19)
$\|\sum_{\lambda\in\Lambda_{n}}a_{\lambda}(\kappa_{\lambda}-\kappa_{n,\lambda})\|_{{}_{\mathcal{F}^{2}}}^{2}\leq\frac{A}{4}\|a\|_{2}^{2}\,.$

Then via the triangle inequality
$\tfrac{A}{4}\|a\|_{2}^{2}\leq\|\sum_{\lambda\in\Lambda_{n}}a_{\lambda}\kappa_{n,\lambda}\|_{{}_{\mathcal{F}^{2}}}^{2}\leq(B+\tfrac{A}{4})\|a\|_{2}^{2}$.
.

Denote the difference of the kernels by
$e_{\lambda}=\kappa_{\lambda}-\kappa_{n,\lambda}$ and the Gram matrix of
$e_{\lambda}$ by $E$ with entries $E_{\lambda,\mu}=\langle
e_{\mu},e_{\lambda}\rangle,\lambda,\mu\in\Lambda_{n}$. Then (19) amounts to
saying the $\|E\|_{\mathrm{op}}\leq A/4$. Since $E$ is positive
(semi-)definite, it suffices to bound the trace of $E$. To do this, consider
the diagonal elements of $E$ first. We see that

$\displaystyle E_{\lambda,\lambda}$
$\displaystyle=\|\kappa_{\lambda}-\kappa_{n,\lambda}\|_{{}_{\mathcal{F}^{2}}}^{2}$
$\displaystyle=2-2\mathrm{Re}\,\langle\kappa_{\lambda},\kappa_{n,\lambda}\rangle$
$\displaystyle=2\Big{(}1-\frac{\langle
k_{\lambda},k_{n,\lambda}\rangle}{k(\lambda,\lambda)^{1/2}k_{n}(\lambda,\lambda)^{1/2}}\Big{)}$
$\displaystyle=2\Big{(}1-\frac{k_{n}(\lambda,\lambda)^{1/2}}{k(\lambda,\lambda)^{1/2}}\Big{)}\,.$

Since $k_{n}(\lambda,\lambda)<k(\lambda,\lambda)$, the estimate for the
diagonal elements simplifies to

$E_{\lambda,\lambda}\leq
2\Big{(}1-\frac{k_{n}(\lambda,\lambda)}{k(\lambda,\lambda)}\Big{)}=2\Big{(}1-\frac{\Gamma(n+1,\pi|\lambda|^{2})}{n!}\Big{)}\,.$

If $x\leq n-\sqrt{n}\tau_{n}^{2}$ (with $\tau_{n}$ depending on $n$), then by
Proposition 3.1(iii).

$1-\frac{\Gamma(n+1,x)}{n!}\leq
1-\frac{\Gamma(n+1,n-\sqrt{n}\tau_{n})}{n!}\leq e^{-\tau_{n}^{2}/2}$

Combining these observations, we arrive at

$\displaystyle\|E\|_{\mathrm{op}}$ $\displaystyle\leq
2\sum_{\lambda\in\Lambda\cap
B_{\rho_{n}}}\Big{(}1-\frac{\Gamma(n+1,\pi|\lambda|^{2})}{n!}\Big{)}$
$\displaystyle\leq 2ne^{-\tau_{n}^{2}/2}$

By choosing $\tau_{n}=\sqrt{2\log n}+\tau$, with $\tau>0$ large enough, we
achieve $\|E\|_{\mathrm{op}}\leq A/4$ for $n\geq n_{0}$. As we have seen, this
suffices to conclude that $\kappa_{n,\lambda}$ is a Riesz sequence in
$\mathcal{P}_{n}$ with lower constant independent of the degree $n$.

Similar to the case of Marcinkiewicz-Zygmund families for sampling, we obtain
uniform families for interpolation with the correct cardinality.

######  Corollary 6.2.

For every $\epsilon>0$ there exist uniform interpolating families
$(\Lambda_{n})$ for $\mathcal{P}_{n}$ in $\mathcal{F}^{2}$ with
$\\#\Lambda_{n}\geq(1-\epsilon)(n+1)$ points.

###### Proof.

The proof is similar to the one of Corollary 4.2.

######  Theorem 6.3.

Assume that $(\Lambda_{n})$ is a uniform interpolating family for the
polynomials $\mathcal{P}_{n}$ in $\mathcal{F}^{2}$. Let $\Lambda$ be a weak
limit of $(\Lambda_{n})$ or of some subsequence $(\Lambda_{n_{k}})$. Then
$\Lambda$ is a set of interpolation for $\mathcal{F}^{2}$.

###### Proof.

Let $\Lambda$ be a weak limit of $\Lambda_{n}$ (or some subsequence), and let
$a\in\ell^{2}(\Lambda)$ with finite support in some disk $B_{\rho_{N}}$ say.
Enumerate $\Lambda\cap B_{\rho_{N}}=\\{\lambda_{j}:j=1,\dots,L\\}$. By weak
convergence, for every $\lambda_{j}\in\Lambda\cap B_{\rho_{N}}$ there is a
sequence $\lambda^{(n)}_{j}\in\Lambda_{n}$, such that
$\lim_{n\to\infty}\lambda_{j}^{(n)}=\lambda_{j}$.

We show that

(20)
$\lim_{n\to\infty}\|\sum_{j=1}^{L}a_{\lambda_{j}}(\kappa_{\lambda_{j}}-\kappa_{n,\lambda_{j}^{(n)}})\|_{{}_{\mathcal{F}^{2}}}^{2}=0\,.$

Consequently,

$\|\sum_{\lambda\in\Lambda\cap
B_{\rho_{N}}}a_{\lambda}\kappa_{\lambda}\|_{{}_{\mathcal{F}^{2}}}=\lim_{n\to\infty}\|\sum_{j=1}^{L}a_{\lambda_{j}^{(n)}}\kappa_{n,\lambda_{j}^{(n)}}\|_{{}_{\mathcal{F}^{2}}}\geq
A\|a\|_{2}^{2}\,,$

because $(\Lambda_{n})$ is a uniform interpolating family. Thus
$\\{\kappa_{\lambda}:\lambda\in\Lambda\\}$ is a Riesz sequence in
$\mathcal{F}^{2}$.

To show (20), we set $e_{j}=\kappa_{\lambda_{j}}-\kappa_{n,\lambda_{j}^{(n)}}$
and consider the associated Gramian with entries $E_{jk}=\langle
e_{k},e_{j}\rangle$. Again we use

$\displaystyle\|E\|_{\mathrm{op}}$
$\displaystyle\leq\mathrm{tr}\,E=\sum_{j=1}^{L}\|e_{j}\|^{2}_{{}_{\mathcal{F}^{2}}}=\sum_{j=1}^{L}\|\kappa_{\lambda_{j}}-\kappa_{n,\lambda_{j}^{(n)}}\|^{2}_{{}_{\mathcal{F}^{2}}}$
$\displaystyle=2\sum_{j=1}^{L}\Big{(}1-\mathrm{Re}\,\langle\kappa_{\lambda_{j}},\kappa_{n,\lambda_{j}^{(n)}}\rangle\Big{)}\,.$

Consider a single term of this sum and write $\lambda_{j}^{(n)}=\lambda$ and
$\lambda_{j}=\mu$ for fixed $j$. Note that $|\lambda-\mu|\leq 1$ for $n$ large
enough and that $\pi|\mu|^{2}\leq N$ by the assumption that
$\operatorname*{supp}a\subseteq B_{\rho_{N}}$. Now

$\displaystyle\mathrm{Re}\,\langle\kappa_{\lambda_{j}},\kappa_{n,\lambda_{j}^{(n)}}\rangle$
$\displaystyle=\mathrm{Re}\,\frac{k_{n}(\lambda,\mu)}{k_{n}(\lambda,\lambda)^{1/2}k(\mu,\mu)^{1/2}}$
$\displaystyle=e^{-\pi|\lambda-\mu|^{2}/2}\frac{\Gamma(n+1,\pi|\lambda|^{2})+\big{[}\Gamma(n+1,\pi\lambda\bar{\mu})-\Gamma(n+1,\pi|\lambda|^{2})\big{]}}{\Gamma(n+1,\pi|\lambda|^{2})^{1/2}\,n!^{1/2}}$
$\displaystyle=e^{-\pi|\lambda-\mu|^{2}/2}\frac{\Gamma(n+1,\pi|\lambda|^{2})^{1/2}}{n!^{1/2}}+e(n,\lambda,\mu)\,.$

Since

$\frac{\Gamma(n+1,\pi\lambda\bar{\mu})-\Gamma(n+1,\pi|\lambda|^{2}))}{n!}\leq
e^{-n\eta}\,$

for some $\eta>0$ by Proposition 3.3 and
$\frac{\Gamma(n+1,\pi|\lambda|^{2})}{n!}\to 1$, the term $e(n,\lambda,\mu)$
tends to zero, as $n\to\infty$. By a similar reasoning, as $n\to\infty$ and
thus $\lambda=\lambda_{j}^{(n)}\to\lambda_{j}=\mu$, we have

$1-e^{-\pi|\lambda-\mu|^{2}/2}\frac{\Gamma(n+1,\pi|\lambda|^{2})^{1/2}}{n!^{1/2}}\to
0$

for finitely many terms. Unraveling the notation, this means that
$\mathrm{tr}\,E\to 0$ and (20) is proved.

######  Proposition 6.4.

There is no Marcinkiewicz-Zygmund family $(\Lambda_{n})$ for $\mathcal{P}_{n}$
in $\mathcal{F}^{2}$ with $\\#\Lambda_{n}=n+1$.

###### Proof.

A set $\Lambda_{n}$ with $n+1$ points is both sampling and interpolating for
$\mathcal{P}_{n}$ with the same constants for interpolation as for sampling.
By Theorem 5.2 any weak limit $\Lambda$ of a Marcinkiewicz-Zygmund family is a
sampling set for $\mathcal{F}^{2}$, and by Theorem 6.3 $\Lambda$ is a set of
interpolation for $\mathcal{F}^{2}$. This is a contradiction, since
$\mathcal{F}^{2}$ does not admit any sets that are simultaneously sampling and
interpolating. See, e.g., [23, Lemma 6.2].

##  7\. Gabor frames for subspaces spanned by Hermite functions

By using the well-known connection between sampling in Fock space and the
theory of Gaussian Gabor frames we may rephrase the main results in the
language of Gabor frames for subspaces.

Recall that the Bargman transform is defined to be

$Bf(z)=2^{1/4}\int_{\mathbb{R}}f(t)e^{2\pi zt-\pi t^{2}}\,dt\,\,e^{-\pi
z^{2}/2}$

for $z\in\mathbb{C}$. It maps functions and distributions on $\mathbb{R}$ to
entire functions.

We use the following properties of the Bargman transform. See e.g., [8].

(i) The Bargman transform is unitary from $L^{2}(\mathbb{R})$ onto Fock space
$\mathcal{F}^{2}$.

(ii) Let $\phi_{z}(t)=e^{-2\pi iyt}e^{-\pi(t-x)^{2}}$ denote the time-
frequency shift of the Gaussian by $z=x+iy$. Then

$B\phi_{z}(w)=k_{z}(w)=e^{\pi\bar{z}w}$

is the reproducing kernel of $\mathcal{F}^{2}$.

(iii) $B$ maps the normalized Hermite functions $h_{k}$,

$h_{k}(t)=c_{k}e^{\pi t^{2}}\tfrac{d^{k}}{dt^{k}}(e^{-2\pi
t^{2}}),\quad\|h_{k}\|_{2}=1,$

to the monomials $e_{k}(z)=\big{(}\frac{\pi^{k}}{k!}\big{)}^{1/2}\,z^{k}$.
With the Bargman transform all questions about the spanning properties of
time-frequency shifts $\phi_{z}$ of the Gaussian can be translated into
questions about the reproducing kernels $k_{z}$ in Fock space. For instance,
$\\{\phi_{\lambda}:\lambda\in\Lambda\\}$ is a frame for $L^{2}(\mathbb{R})$,
if and only if $\Lambda$ is a sampling set for $\mathcal{F}^{2}$. Almost all
statements about Gaussian Gabor frames have been obtained via complex analysis
methods, notably the complete characterization of Gaussian Gabor frames by
Lyubarski [17] and Seip [23] and many subsequent detailed investigations [6,
7]. To this line of thought we add a statement about Gabor frames for
distinguished subspaces spanned by Hermite polynomials. Constructions of this
type have been used in signal processing [14].

######  Theorem 7.1.

Assume that $\Lambda$ is a sampling set for $\mathcal{F}^{2}$, or equivalently
$\mathcal{G}(h_{0},\Lambda)=\\{\phi_{\lambda}:\lambda\in\Lambda\\}$ is Gabor
frame in $L^{2}(\mathbb{R})$, then $\\{\phi_{\lambda}:\pi|\lambda|^{2}\leq
n+\sqrt{n}\tau\\}$ is a frame for
$V_{n}=\mathrm{span}\,\\{h_{k}:k=0,\dots,n\\}$ with bounds independent of $n$,
i.e.,

$A\|f\|_{2}\leq\sum_{\lambda\in\Lambda:\pi|\lambda|^{2}\leq
n+\sqrt{n}\tau}|\langle f,\phi_{\lambda}\rangle|^{2}\leq
B\|f\|_{2}^{2}\qquad\text{ for all }f\in V_{n}\,.$

###### Proof.

The statement is equivalent to Theorem 4.1 via the Bargman transform.222Since
$\phi_{\lambda}\not\in V_{n}$, some authors use the term “pseudoframe” for
this situation.

##  8\. Appendix

For completeness we present some elementary estimates for the zero order
asymptotics of the incomplete Gamma function that imply Corollary 3.1. Using
Stirling’s formula for $\big{(}\tfrac{n}{e}\big{)}^{n}\sqrt{2\pi n}\leq n!\leq
e\sqrt{n}\big{(}\tfrac{n}{e}\big{)}^{n}$, we write

$\frac{\Gamma(n+1,n+\sqrt{n}\tau)}{n!}\asymp\frac{1}{n!}\int_{n+\sqrt{n}\tau}^{\infty}t^{n}e^{-t}\,dt\asymp\frac{1}{\sqrt{n}}\int_{n+\sqrt{n}\tau}^{\infty}\Big{(}\frac{t}{n}\Big{)}^{n}e^{n-t}\,dt\,.$

(The constants in the equivalence are in $[e^{-1},(2\pi)^{-1/2}]$.) Using the
substitution $u=\frac{t}{n}-1$ we obtain

$\displaystyle\frac{\Gamma(n+1,n+\sqrt{n}\tau)}{n!}$
$\displaystyle\asymp\sqrt{n}\int_{\tau/\sqrt{n}}^{\infty}(1+u)^{n}e^{-nu}\,du$
(21)
$\displaystyle=\sqrt{n}\int_{\tau/\sqrt{n}}^{\infty}e^{-n(u-\ln(1+u))}\,du=\sqrt{n}\Big{(}\int_{\tau/\sqrt{n}}^{1}\dots+\int_{1}^{\infty}\dots
dt\Big{)}\,.$

Using the inequality $u-\ln(1+u)\geq(1-\ln 2)u$ valid for $u\in[1,\infty)$,
the latter integral is bounded by

(22)
$\sqrt{n}\int_{1}^{\infty}e^{-n(u-\ln(1+u))}\,du\leq\sqrt{n}\int_{1}^{\infty}e^{-n(1-\ln
2)u}\,du\leq\frac{\sqrt{n}}{n(1-\ln
2)}=\mathcal{O}\big{(}\frac{1}{\sqrt{n}}\big{)}\,.$

In the first integral in (21) we use the power series of $\ln$ and obtain, for
$u\in[0,1]$,

$u-\ln(1+u)=\sum_{k=2}^{\infty}\frac{(-1)^{k}}{k}u^{k}\geq\frac{u^{2}}{2}-\frac{u^{3}}{3}\geq\frac{u^{2}}{6}\,.$

Consequently,

$\displaystyle\sqrt{n}\int_{\tau/\sqrt{n}}^{1}(1+u)^{n}e^{-nu}\,du$
$\displaystyle\leq\sqrt{n}\int_{\tau/\sqrt{n}}^{1}e^{-nu^{2}/6}\,du$
$\displaystyle=\sqrt{n}\sqrt{\frac{6}{n}}\int_{\tau/\sqrt{6}}^{\sqrt{n/6}}e^{-v^{2}}\lesssim\mathrm{erf}(\tau/\sqrt{6})\leq
e^{-\tau^{2}/6}\,.$

From $u-\ln(1+u)\leq u^{2}/2$ we obtain the lower bound

$\displaystyle\sqrt{n}\int_{\tau/\sqrt{n}}^{1}(1+u)^{n}e^{-nu}\,du$
$\displaystyle\geq\sqrt{n}\int_{\tau/\sqrt{n}}^{1}e^{-nu^{2}/2}\,du$
$\displaystyle=\sqrt{n}\sqrt{\frac{2}{n}}\int_{\tau/\sqrt{2}}^{\sqrt{n/2}}e^{-v^{2}}\gtrsim\mathrm{erf}(\tau/\sqrt{2})\gtrsim\frac{1}{\tau}e^{-\tau^{2}/2}\,,$

for $n$ large enough. Items (i) and (ii) of Proposition 3.1 now follow easily
from these estimates.

A weaker version of item (v) follows by setting $\tau=0$ in the above
estimates.

(iii) Similarly,

$\displaystyle 1-\frac{\Gamma(n+1,n-\sqrt{n}\tau)}{n!}$
$\displaystyle\leq\frac{1}{\sqrt{2\pi
n}}\Big{(}\frac{e}{n}\Big{)}^{n}\int_{0}^{n-\sqrt{n}\tau}t^{n}e^{-t}\,dt=$
$\displaystyle=\frac{1}{\sqrt{2\pi
n}}\int_{0}^{n-\sqrt{n}\tau}\Big{(}\frac{t}{n}\Big{)}^{n}e^{n-t}\,dt\,.$

With the substitution $u=1-\frac{t}{n}$ we obtain

$\displaystyle\frac{1}{\sqrt{2\pi
n}}\int_{0}^{n-\sqrt{n}\tau}\Big{(}\frac{t}{n}\Big{)}^{n}e^{n-t}\,dt=\frac{\sqrt{n}}{\sqrt{2\pi}}\int_{\tau/\sqrt{n}}^{1}(1-u)^{n}e^{nu}\,du=$
(23)
$\displaystyle=\frac{\sqrt{n}}{\sqrt{2\pi}}\int_{\tau/\sqrt{n}}^{1}e^{n(u+\ln(1-u))}\,du=\frac{\sqrt{n}}{\sqrt{2\pi}}\Big{(}\int_{\tau/\sqrt{n}}^{1/2}\dots+\int_{1/2}^{1}\dots
dt\Big{)}\,.$

Since $u+\ln(1-u)\leq 1/2-\ln 2<0$ on the interval $[1/2,1]$, the second term
decays exponentially in $n$. For the first term we use
$u+\log(1-u)=-\sum_{k=2}^{\infty}\frac{u^{k}}{k}\leq-u^{2}/2$ and obtain

$\displaystyle\frac{\sqrt{n}}{\sqrt{2\pi}}\int_{\tau/\sqrt{n}}^{1/2}e^{n(u+\ln(1-u))}\,du$
$\displaystyle\leq\frac{\sqrt{n}}{\sqrt{2\pi}}\int_{\tau/\sqrt{n}}^{1/2}e^{-nu^{2}/2}\,du$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{\tau}^{\sqrt{n}/2}e^{-v^{2}/2}\,du\leq\tfrac{1}{2}e^{-\tau^{2}/2}\,.$

(iv) follows from (iii). If $n-\sqrt{n}\tau\geq x$, then

$\frac{\Gamma(n+1,x)}{n!}\geq\frac{\Gamma(n+1,n-\sqrt{n}\tau)}{n!}\geq
1-e^{-\tau^{2}}\,.$

Since this holds arbitrary $\tau$ and $n$ large, we obtain
$\lim_{n\to\infty}\frac{\Gamma(n+1,x)}{n!}=1$.

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