Marcinkiewicz-Zygmund inequalities for polynomials in Fock space

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.

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

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

(1)

where $dm(z) = dx dy$ 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, $n\ge n_0$,

$$\begin{aligned} A \Vert p\Vert _{_{\mathcal {F}^2}}^2 \le \sum _{\lambda \in \Lambda _n} \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )} \le B \Vert p\Vert _{_{\mathcal {F}^2}}^2 \qquad \text { for all } p\in \mathcal {P}_n\, . \end{aligned}$$

(2)

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 $A\Vert f\Vert _{\mathcal {H}}^{2} \le \sum _{\lambda \in \Lambda } |f(\lambda )|^2 k(\lambda , \lambda )^{-1} \le B \Vert f\Vert _{\mathcal {H}}^2$ 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

$$\begin{aligned} A\Vert f\Vert _{\mathcal {F}^2}\le \sum _{\lambda \in \Lambda } |f(\lambda )|^2 e^{-\pi |\lambda |^2} \le B \Vert f\Vert _{_{\mathcal {F}^2}}^2 \qquad \text { for all } f\in \mathcal {F}^2 \, . \end{aligned}$$

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 Sect. 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 $\Vert p\Vert _{\mathcal {F}^2}^2 \le A \Vert a\Vert _2^2$.

Theorem 1.2

(i) Assume that $\Lambda \subseteq \mathbb {C}$ is a set of interpolation for $\mathcal {F}^2$. For $\tau >0$ define $\rho _n$ by $\pi \rho _n^2 = n-\sqrt{n} (\sqrt{2\log n} + \tau )$. 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 in 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 are studied in [4], and the study of deterministic point processes [1, 2, 5] uses closely related notions. 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 in 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 in the context of random Marcinkiewicz-Zygmund families and determinantal 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 results in [3, 4] are not directly comparable to ours, but the common ground is the construction of point sets that are sampling for polynomials in Fock spaces. Our main interest is the connection to the infinite-dimensional sampling problem in $\mathcal {F}^2$.

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 2.2 and Corollary 3.1 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 $z= (x,\xi )\in \mathbb {R}^2$ with $g_z(t) = e^{2\pi i \xi t} g(t-x)$ for $t,x, \xi \in \mathbb {R}$. The $L^2$-normalized Hermite functions are denoted by $h_n$, in particular $\phi (t) = 2^{-1/4} h_0(t) = e^{-\pi t^2}$ 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 $\{\pi (\lambda )h_0 : \pi |\lambda | ^2 \le n+\sqrt{n}\tau \}$ is a frame for $V_n = \mathrm {span}\, \{h_k : k=0, \dots , n\}$ with bounds independent of n. This means that

$$\begin{aligned} A\Vert f\Vert _2 \le \sum _{\lambda \in \Lambda : \pi |\lambda |^2 \le n+\sqrt{n}\tau } |\langle f, \phi _\lambda \rangle |^2 \le B\Vert f\Vert _2^2 \qquad \text { for all } f\in V_n \, . \end{aligned}$$

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 of Marcinkiewicz-Zygmund families for multivariate Bergman spaces $A^2({\mathbb {B}} _n)$ in n complex variables on the 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

$$\begin{aligned} e_k(z) = \Big ( \frac{\pi ^k}{k!}\Big )^{1/2} \, z^k \end{aligned}$$

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

$$\begin{aligned} k_n(z,w) = \sum _{k=0}^n e_k(z) \overline{e_k(w)} = \sum _{k=0}^n \frac{(\pi z {\bar{w}})^k}{k!} \, . \end{aligned}$$

(3)

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

$$\begin{aligned} k(z,w) = \lim _{n\rightarrow \infty } k_n(z,w) = e^{\pi z {\bar{w}}} \, . \end{aligned}$$

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

$$\begin{aligned} \Gamma (z,a) = \int _{a} ^\infty t^{z-1} e^{-t} \, dt \, \end{aligned}$$

(4)

and

$$\begin{aligned} \gamma (z,a) = \int _0^{a} t^{z-1} e^{-t} \, dt \, . \end{aligned}$$

(5)

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

$$\begin{aligned} \int _{B_\rho } |z|^{2k} e^{-\pi |z|^2} \, dm(z)&= 2\pi \int _0^\rho r^{2k} e^{-\pi r^2} \, rdr \nonumber \\&= \int _0^{\pi \rho ^2} \Big ( \frac{u}{\pi }\Big )^k \, e^{-u} \, du \nonumber \\&= \frac{1}{\pi ^k} \gamma (k+1,\pi \rho ^2) \, . \end{aligned}$$

(6)

Lemma 2.1

We have

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

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

Proof

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

$$\begin{aligned} \frac{1}{n!} \Gamma (n+1, r) = \frac{1}{n!} \int _r^\infty t^n e^{-t}\, dt = \frac{r^n}{n!} e^{-r} + \frac{1}{(n-1)!} \Gamma (n, r) \end{aligned}$$

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

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

$$\begin{aligned} \int _{B_\rho } |p(z)|^2 \, e^{-\pi |z|^2} \, dm(z)&= \sum _{k=0}^n |a_k|^2 \int _{B_\rho } |z|^{2k} e^{-\pi |z|^2}\, dm(z) \\&= \sum _{k=0}^n |a_k|^2 \frac{k!}{\pi ^k} \, \frac{\gamma (k+1,\pi \rho ^2)}{k!} \\&\ge \min _{0\le k\le n} \frac{\gamma (k+1,\pi \rho ^2)}{k!} \, \sum _{k=0}^n |a_k|^2 \frac{k!}{\pi ^k}&\ge \frac{\gamma (n+1,\pi \rho ^2)}{n!} \Vert p\Vert _{_{\mathcal {F}^2}} ^2 \, . \end{aligned}$$

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

Lemma 2.2

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

$$\begin{aligned} \int _{B_\rho ^c} |p(z)|^2 \, e^{-\pi |z|^2} \, dm(z) \le \frac{\Gamma (n+1,\pi \rho ^2)}{n!} \Vert p\Vert _{_{\mathcal {F}^2}} ^2 \, . \end{aligned}$$

(7)

Proof

This follows immediately from the previous estimates via

$$\begin{aligned} \int _{B_\rho ^c} |p(z)|^2 \, e^{-\pi |z|^2} \, dm(z)&= \Vert p\Vert ^2_{_{\mathcal {F}^2}} - \int _{B_\rho } |p(z)|^2 \, e^{-\pi |z|^2} \, dm(z) \nonumber \\&\le (1- \frac{\gamma (n+1,\pi \rho ^2)}{n!} ) \, \Vert p\Vert _{_{\mathcal {F}^2}} ^2 = \frac{\Gamma (n+1,\pi \rho ^2)}{n!} \Vert p\Vert _{_{\mathcal {F}^2}} ^2 \, . \end{aligned}$$

(8)

$\square$

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 $C>0$ such that $C^{-1}f (x) \le g(x) \le C f(x)$ for all x in the domain of f and g, $f \lesssim g$ means $f(x) \le C g(x)$, and $f \sim g$ near $x_\infty$ means that $\lim _{x\rightarrow x_\infty } \frac{f(x)}{g(x)} = 1$.

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

The normalized incomplete gamma function admits the asymptotic expansion

$$\begin{aligned} \frac{\Gamma (a, a + \tau \sqrt{a})}{\Gamma (a)} \sim \tfrac{1}{2} \mathrm {erfc} \, (\frac{\tau }{\sqrt{2}}) + \frac{1}{\sqrt{2\pi a}} e^{-\tau ^2/2} \sum _{n=0}^\infty \frac{C_n(\tau )}{a^{n/2}} \, , \end{aligned}$$

(9)

where $C_0(\tau ) = \frac{\tau ^2 - 1}{3}$ and $\mathrm {erfc}(y) = \frac{2}{\sqrt{\pi }}\int _y^\infty e^{-t^2} \, dt$. A careful interpretation of the zero order approximation implies that there exists a constant C independent of a and $\tau$, such that

$$\begin{aligned} \Big |\frac{\Gamma (a,a + \tau \sqrt{a})}{\Gamma (a)} - \tfrac{1}{2} \mathrm {erfc} (\frac{\tau }{\sqrt{2}})\Big | \le C (|\tau |^2 + 1) e^{-\tau ^2/2} \frac{1}{\sqrt{ a}} \, \end{aligned}$$

(10)

for $|\tau | \le a^{1/6}$.

See [20], Prop. 1.1 and Eq. (3.1).

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

 

• (i) For every $\epsilon >0$ there is $\tau >0$, such that

  $$\begin{aligned} \frac{ \Gamma (n+1,n+\sqrt{ n }\tau )}{n!} < \epsilon \qquad \forall n\ge n_0 \end{aligned}$$

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

  $$\begin{aligned} \frac{ \Gamma (n+1,n+\sqrt{ n }\tau )}{n!} \ge C(\tau ) \qquad \forall n\ge n_0 \, . \end{aligned}$$

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

• (iii) For $\tau >0$

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

• (iv) For every $x\ge 0$

  $$\begin{aligned} \lim _{n\rightarrow \infty } \frac{\Gamma (n+1,x)}{n!} =1 \, . \end{aligned}$$

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

• (v) For all n

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

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 $n_0 \ge \tau ^6$ and the error $C(|\tau |^2 + 1) e^{-\tau ^2/2} \frac{1}{\sqrt{n+1}} < \epsilon /2$ for $n\ge n_0$. By (10) we then have $\frac{ \Gamma (n+1,n+\sqrt{ n }\tau )}{n!} < \epsilon$ for all $n\ge n_0$.

• (ii) Given $\tau >0$ choose $n_0 \ge \tau ^6$ such that the error $C(|\tau ^2| +1) e^{-\tau ^2/2} \frac{1}{\sqrt{n+1}} < \tfrac{1}{4} \text {erfc} (\tau /\sqrt{2})$ for $n\ge n_0$. Then $\frac{ \Gamma (n+1,n+\sqrt{ n }\tau )}{n!} \ge \tfrac{1}{4} \text {erfc} (\tau /\sqrt{2})>0$ for all $n\ge n_0$.

• (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. $\square$

Corollary 3.1

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

We also need an off-diagonal estimate for the kernel $k_n$.

Lemma 3.1

Assume that $\pi |z|^2 < n(1-\epsilon )$ for fixed $\epsilon >0$

and $|z-w | \le \tau$. Then for n large enough depending on $\epsilon$,

$$\begin{aligned} \Big |\frac{\Gamma (n+1, \pi z{\bar{w}}) }{n!} - \frac{\Gamma (n+1, \pi |z|^2 )}{n!} \Big | \le C e^{-\epsilon ^2 n /4 } \, . \end{aligned}$$

(11)

Proof

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

$$\begin{aligned} \Gamma (n+1, \pi z{\bar{w}})&= \Gamma (n+1, \pi r^2 + \pi r u) = \int _{\pi r^2} ^\infty t^n e^{-t} \, dt + \int _{\pi r^2 + \pi ru} ^{\pi r^2} \dots \\&= \Gamma (n+1, \pi r^2) + \int _{\pi r^2 + \pi ru} ^{\pi r^2} \dots \, . \end{aligned}$$

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

$$\begin{aligned} \big |\int _{\pi r^2} ^{\pi r^2 + \pi ru} t^n e^{-t} \, dt \big |&= \big |\pi r u \int _0^1 (\pi r^2 + s \pi ru)^n e^{-\pi r^2 - s \pi ru} \, ds\big | \\&\le \pi r |u| (\pi r^2 + \pi r|u|)^n e^{-\pi r^2+\pi r |u|} \\&\le \pi r \tau (\pi r^2 + \pi r \tau )^n e^{-\pi r^2 + \pi r \tau } \, . \end{aligned}$$

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

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

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

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 $\Lambda '\subseteq \Lambda$ with lower Beurling density $D^-(\Lambda ' ) >1$.

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

  $$\begin{aligned} |f(\lambda )|^2 e^{-\pi |\lambda |^2} \le c_\rho \int _{B(\lambda , \rho )} |f(z)|^2 \, e^{-\pi |z|^2} \, dm(z) \end{aligned}$$

  (12)

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

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

$$\begin{aligned} \sum _{\lambda \in \Lambda , |\lambda |>R} |f(\lambda )|^2 e^{-\pi |\lambda |^2} \le c_\rho K \int _{|z|> R-\rho } |f(z)|^2 \, e^{-\pi |z|^2} \, dm(z) \, . \end{aligned}$$

(13)

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 $k_n(z,z) \le k(z,z) = e^{\pi |z|^2}$, we may replace $k_n$ by k in the sampling inequalities:

$$\begin{aligned} \sum _{\lambda \in \Lambda _n} \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )}&\ge \sum _{\lambda \in \Lambda _n} \frac{|p(\lambda )|^2}{k(\lambda ,\lambda )}\\&= \sum _{\lambda \in \Lambda _n} |p(\lambda )|^2 e^{-\pi |\lambda |^2} = \sum _{\lambda \in \Lambda } - \sum _{\lambda \in \Lambda : |\lambda | > \rho _n} \dots \, . \end{aligned}$$

Since $\Lambda$ is a sampling set for $\mathcal {F}^2$, the first term satisfies $\sum _{\lambda \in \Lambda } |p(\lambda )|^2 e^{-\pi |\lambda |^2} \ge A \Vert p\Vert _{\mathcal {F}^2}^2$. 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):

$$\begin{aligned} \sum _{\lambda \in \Lambda : |\lambda |>\rho _n} |p(\lambda )|^2 e^{-\pi |\lambda |^2}&\le C \int _{|z|> \rho _n-\rho } |p(z)|^2 e^{-\pi |z|^2}\, dm(z) \\&\le C \frac{\Gamma (n+1, \pi (\rho _n-\rho )^2)}{n!} \, \Vert p\Vert _{\mathcal {F}^2}^2 \qquad \text{ for } p\in \mathcal {P}_n\, . \end{aligned}$$

Our choice of $\rho _n$ implies that

$$\begin{aligned} \pi (\rho _n-\rho )^2&= \pi \rho _n^2 - 2\pi \rho _n \rho + \pi \rho ^2 \\&= n+\sqrt{n} \tau + \pi \rho ^2 - 2\sqrt{\pi } \, \sqrt{n+\sqrt{n}\tau } \rho \\&\ge n+\sqrt{n} \tau - 2\sqrt{\pi n} \rho (1+\frac{\tau }{2\sqrt{n}})\\&\ge n+\sqrt{n} \tau ' \end{aligned}$$

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

$$\begin{aligned} \sum _{\lambda \in \Lambda : |\lambda | >\rho _n} |p(\lambda )|^2 e^{-\pi |\lambda |^2} \le C \frac{\Gamma (n+1, n+\sqrt{n}\tau ')}{n!} \Vert p\Vert _{\mathcal {F}^2}^2 \le \frac{A}{2} \Vert p\Vert _{\mathcal {F}^2}\qquad \text { for all } p \in \mathcal {P}_n\, \end{aligned}$$

for large n, $n\ge n_0$, say. Combining the inequalities, we obtain $\sum _{\lambda \in \Lambda _n} \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )} \ge \frac{A}{2}\Vert p\Vert _{\mathcal {F}^2}^2$ for all $p\in \mathcal {P}_n$.

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

$$\begin{aligned} \frac{\Gamma (n+1, n+\sqrt{n}\tau )}{n!} \ge \tfrac{1}{4} \text {erfc} (\tau /\sqrt{2}) =C(\tau ) = C \, \end{aligned}$$

for $n\ge n_0$. This implies that $k_n(\lambda , \lambda )^{-1}\le C^{-1}k(\lambda , \lambda )^{-1}= C ^{-1}e^{-\pi |\lambda |^2}$ for $\pi |\lambda |^2 \le n+\sqrt{n} \tau$, and thus

$$\begin{aligned} \sum _{\lambda \in \Lambda _n} \frac{|p(\lambda )|^2}{k_n(\lambda , \lambda )} \le C^{-1}\sum _{\lambda \in \Lambda , \pi |\lambda |^2 \le n+\sqrt{n}\tau } |p(\lambda )|^2 e^{-\pi |\lambda |^2} \le C^{-1}B \Vert p\Vert _{\mathcal {F}^2}\, \quad \text { for all } p \in \mathcal {P}_n\, , \end{aligned}$$

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

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 4.1

For every $\epsilon >0$ there exist Marcinkiewicz-Zygmund families $(\Lambda _n)$ for $\mathcal {P}_n$ in $\mathcal {F}^2$ with $\# \Lambda _n \le (1+\epsilon ) (n+1)$ 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

$$\begin{aligned} \# \Lambda _n = \# (\Lambda \cap B_{\rho _n}) \le (1+2\mu ) |B_{\rho _n}| = (1+2\mu ) (n+ \sqrt{n}\tau ) < (1+\epsilon ) (n+1)\, . \end{aligned}$$

$\square$

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 $\# (\Lambda _n \cap B_{\sigma _n}^c) \le B \frac{n+1}{(1-\varepsilon )^n}$. This holds also for $\epsilon =0$.

• (ii) Let $B(z,\rho )$ be a disc in $B_{\sigma _n}$. Then

  $$\begin{aligned} \# (\Lambda _n \cap B(z,\rho )) \le C \, . \end{aligned}$$

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 $\pi |w|^2 \ge n$ and $k\le n$, we have

$$\begin{aligned} \frac{(\pi |w|^2)^k}{k!} \le \frac{(\pi |w|^2)^n}{n!} \, , \end{aligned}$$

so the reproducing kernel satisfies the estimate

$$\begin{aligned} k_n(w,w) = \sum _{k=0}^n \frac{(\pi |w|^2)^k}{k!} \le (n+1) \frac{(\pi |w|^2)^n}{n!} \, . \end{aligned}$$

(14)

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

$$\begin{aligned} k_n(z,z) \le \frac{n+1}{(1-\varepsilon )^n} \frac{(\pi |z|^2)^n}{n!}, \end{aligned}$$

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 $\Vert p_n\Vert _{\mathcal {F}^2}= 1$. (14) implies that

$$\begin{aligned} \frac{|p_n(\lambda )|^2}{k_n(\lambda ,\lambda )} \ge \frac{(1-\varepsilon )^n}{n+1} \qquad \text { for } \pi |\lambda |^2 \ge n(1-\varepsilon ) \, , \end{aligned}$$

and therefore

$$\begin{aligned} \frac{(1-\varepsilon )^n}{n+1} \# (\Lambda _n \cap B_{\sigma _n}^c) \le \sum _{\lambda \in \Lambda _n : \pi |\lambda |^2 \ge n(1-\varepsilon )} \frac{|p_n(\lambda )|^2}{k_n(\lambda ,\lambda )} \le B \Vert p_n\Vert _{\mathcal {F}^2}= B \, , \end{aligned}$$

so we obtain $\# (\Lambda _n \cap B_{\sigma _n}^c) \le \frac{B(n+1)}{(1-\varepsilon )^n}$.

(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 $\pi |z|^2 \le n(1-\epsilon )$ we have $\Gamma (n+1,\pi |z|^2)/n! \ge 1/2$ by Proposition 3.1(v). So after substituting the formulas for the kernel, we obtain

$$\begin{aligned} \sum _{\lambda \in \Lambda _n \cap B(z,\rho )} \frac{|\kappa _{n,z}(\lambda )|^2}{k_n(\lambda ,\lambda )}&= \sum _{\lambda \in \Lambda _n \cap B(z,\rho )} \frac{|k_n(z,\lambda )|^2}{k_n(z,z) k_n(\lambda ,\lambda )}\\&= \sum _{\lambda \in \Lambda _n \cap B(z,\rho )} |e^{\pi {\bar{z}}\lambda }|^2 e^{-\pi |z|^2}e^{-\pi |\lambda |^2} \frac{|\Gamma (n+1,\pi {\bar{z}}\lambda )|^2}{\Gamma (n+1,\pi |z|^2) \Gamma (n+1,\pi |\lambda |^2)}\\&= \sum _{\lambda \in \Lambda _n \cap B(z,\rho )} e^{-\pi |\lambda -z|^2} \frac{|\Gamma (n+1,\pi {\bar{z}}\lambda )|^2}{\Gamma (n+1,\pi |z|^2) \Gamma (n+1,\pi |\lambda |^2)} = (*)\, . \end{aligned}$$

Now note that by Lemma 3.1$|\Gamma (n+1,\pi {\bar{z}}\lambda )|^2/n! \ge 1/4$ for n large, whereas $|\Gamma (n+1,\pi |z|^2)/n! \le 1$, so that the last sum finally is bounded below by

$$\begin{aligned} (*) \ge e^{-\pi \rho ^2} \sum _{\lambda \in \Lambda _n \cap B(z,\rho ) } \frac{1}{4} = \frac{1}{4} \, e^{-\pi \rho ^2} \, \# (\Lambda _n \cap B )\, . \end{aligned}$$

Reading backwards, we obtain

$$\begin{aligned} \frac{1}{4} \, e^{-\pi \rho ^2} \, \# (\Lambda _n \cap B ) \le \sum _{\lambda \in \Lambda _n \cap B} \frac{|\kappa _{n,z}(\lambda )|^2}{k_n(\lambda ,\lambda )} \le B \Vert \kappa _{n,z}\Vert _{\mathcal {F}} ^2 = B \, , \end{aligned}$$

which was claimed. $\square$

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

$$\begin{aligned} \# (\Lambda _n \cap C_{n,\tau } ) \lesssim \sqrt{n} e^{\tau ^2/2} \, . \end{aligned}$$

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 $B\subseteq \mathbb {C}$

$$\begin{aligned} \lim _{n\rightarrow \infty } d\big ((\Lambda _n \cap B) \cup \partial B, (\Lambda \cap B) \cup \partial B \big ) = 0 \, , \end{aligned}$$

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

$$\begin{aligned} \sum _{\lambda \in \Lambda _n \cap B} \frac{|f(\lambda )|^2}{k(\lambda ,\lambda )} \rightarrow \sum _{\lambda \in \Lambda \cap B} \frac{|f(\lambda )|^2}{k(\lambda ,\lambda )} m(\lambda )\, , \end{aligned}$$

(15)

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

Theorem 5.1

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 $A\Vert p\Vert _{\mathcal {F}^2}\le \sum _{\lambda \in \Lambda _n} \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )}\le B\Vert p\Vert ^2_{\mathcal {F}^2}$ for all polynomials $p\in \mathcal {P}_n$.

(i) Let $B= {\bar{B}}(w,\rho )$ be a closed disc. By Lemma 5.1(ii) $\# (\Lambda _n \cap {\bar{B}}(w,\rho )) \le C$ 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 $\# (\Lambda \cap {\bar{B}}(w,\rho )) \le C$. 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

$$\begin{aligned} \int _{|z| \ge \sqrt{r/\pi } } |p(z)|^2 e^{-\pi |z|^2} \, dm(z) < \frac{A}{4 c_\delta K} \Vert p\Vert _{_{\mathcal {F}^2}} ^2 \, , \end{aligned}$$

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 $n\ge N$, therefore

$$\begin{aligned} A\Vert p\Vert _{_{\mathcal {F}^2}} ^2&\le \sum _{\lambda \in \Lambda _n} \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )} \\&= \sum _{\lambda \in \Lambda _n, |\lambda |<\nu +\delta } \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )} \\&\qquad + \sum _{\lambda \in \Lambda _n, \nu +\delta \le |\lambda | < \sigma _n } \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )} + \sum _{\lambda \in \Lambda _n, |\lambda | \ge \sigma _n } \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )} = \\&= A_n + B_n + C_n \, . \end{aligned}$$

If $|\lambda | \le \sigma _n$, then $k_n(\lambda ,\lambda ) \ge \frac{1}{2} k(\lambda ,\lambda ) = \frac{1}{2} e^{\pi |\lambda |^2}$ 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

$$\begin{aligned} A \Vert p\Vert ^2 _{\mathcal {F}^2}\le 2 \sum _{\lambda \in \Lambda _n , |\lambda | \le \nu +\delta } |p(\lambda )|^2 e^{-\pi |\lambda |^2} +B_n+C_n. \end{aligned}$$

(16)

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

$$\begin{aligned} \lim _{n\rightarrow \infty } A_n&\le 2 \lim _{n\rightarrow \infty }\sum _{\lambda \in \Lambda _n , |\lambda | \le \nu +\delta } |p(\lambda )|^2 e^{- \pi |\lambda |^2} = \\&= 2 \sum _{\lambda \in \Lambda \cap \overline{B_{\nu +\delta }}} |p(\lambda )|^2 e^{- \pi |\lambda |^2} m(\lambda ) \end{aligned}$$

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

$$\begin{aligned} B_n&\le 2 \sum _{\lambda \in \Lambda _n, \nu +\delta \le |\lambda | < \sigma _n } \frac{|p(\lambda )|^2}{k(\lambda ,\lambda )} \le 2 c_\delta K \int _{|z|> \nu } |p(z)|^2 e^{-\pi |z|^2}\, dm(z) \\&\le 2 c_\delta K \frac{A}{4 c_\delta K} \Vert p\Vert _{\mathcal {F}^2}^2 = \frac{A}{2} \Vert p\Vert _{\mathcal {F}^2}^2\, . \end{aligned}$$

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

$$\begin{aligned} |p(\lambda )|^2 = |\langle p, k_N (\lambda , \cdot )\rangle |^2 \le \Vert p\Vert _{_{\mathcal {F}^2}} ^2 k_N(\lambda , \lambda ) \end{aligned}$$

and substitute into $C_n$ to obtain

$$\begin{aligned} C_n = \sum _{\lambda \in \Lambda _n, |\lambda | > \sigma _n} \frac{|p(\lambda )|^2}{k_n(\lambda ,\lambda )} \le \Vert p\Vert _{_{\mathcal {F}^2}} ^2 \, \# (\Lambda _n\cap B_{\sigma _n}^c) \sup _{|z| \ge \sigma _n} \frac{k_N(z,z)}{k_n(z,z)} \, . \end{aligned}$$

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

$$\begin{aligned} \frac{k_N(z,z)}{k_n(z,z)} = \frac{e^{\pi |z|^2} \Gamma (N+1, \pi |z|^2) n!}{e^{\pi |z|^2} \Gamma (n+1, \pi |z|^2) N!} \, . \end{aligned}$$

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

$$\begin{aligned} \Gamma (n+1,R)&= \int _R ^\infty t^n e^{-t} \, dt \\&\ge R^{n-N} \int _R ^\infty t^N e^{-t} \, dt \\&= R^{n-N} \Gamma (N+1,R) \, , \end{aligned}$$

so that

$$\begin{aligned} \sup _{\pi |z|^2 >n(1-\varepsilon )} \frac{k_N(z,z)}{k_n(z,z)} \le (n(1-\varepsilon ))^{N-n}\frac{n!}{N!} \, . \end{aligned}$$

Altogether

$$\begin{aligned} C_n \le \Vert p\Vert _{_{\mathcal {F}^2}} ^2 \frac{B(n+1)}{(1-\varepsilon )^n} (n(1-\varepsilon ))^{N-n}\frac{n!}{N!}\rightarrow 0 \, , \end{aligned}$$

as $n\rightarrow \infty$ 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

$$\begin{aligned} \sum _{\lambda \in \Lambda } |p(\lambda )|^2 m(\lambda ) e^{-\pi |\lambda |^2}&\ge \sum _{\lambda \in \Lambda , |\lambda | \le \nu +\delta } |p(\lambda )|^2 m(\lambda ) e^{-\pi |\lambda |^2} - \limsup _{n\rightarrow \infty } B_n - \lim _{n\rightarrow \infty } C_n \ge \\&^\ge \frac{A}{2}\Vert p\Vert _{\mathcal {F}^2}^2 \, . \end{aligned}$$

As the multiplicities satisfy $1\le m(\lambda ) \le K$ 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 $\mathcal {F}^2$. $\square$

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 $\kappa _\lambda = k_\lambda / \Vert k_\lambda \Vert _{_{\mathcal {F}^2}} = k_\lambda / k(\lambda ,\lambda )^{1/2}$ is a Riesz sequence, i.e., there exists $A,B>0$, such that

$$\begin{aligned} A \Vert a\Vert _2^2 \le \Vert \sum _{\lambda \in \Lambda } a_\lambda \kappa _\lambda \Vert _{_{\mathcal {F}^2}}^2 \le B \Vert a\Vert _2^2 \end{aligned}$$

(17)

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 $\kappa _{n,\lambda } = k_{n,\lambda } / \Vert k_{n,\lambda }\Vert _{_{\mathcal {F}^2}}$.

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, $n\ge n_0$,

$$\begin{aligned} A \Vert a\Vert _2^2 \le \Vert \sum _{\lambda \in \Lambda _n } a_\lambda \kappa _{n,\lambda } \Vert _{_{\mathcal {F}^2}}^2 \le B \Vert a\Vert _2^2 \qquad \text { for all } a\in \ell ^2(\Lambda _n) \, . \end{aligned}$$

(18)

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

$$\begin{aligned} \frac{p(\lambda )}{k_n(\lambda ,\lambda )^{1/2}} = a_\lambda \qquad \text { and } \qquad \Vert p\Vert _{_{\mathcal {F}^2}} ^2 \le A \Vert a\Vert _2^2 \, . \end{aligned}$$

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 eigenvalue $\lambda _{min} \ge A$ [18, Sect. 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 $\tau >0$ define $\rho _n$ via $\pi \rho _n^2 = n-\sqrt{n} (\sqrt{2\log n} + \tau )$. 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

$$\begin{aligned} \# (\Lambda \cap B_{\rho _n}) \le 1 \cdot |B_{\rho _n}| \le n \end{aligned}$$

for $n\ge n_0$ 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,

$$\begin{aligned} \Vert \sum _{\lambda \in \Lambda _n} a_\lambda (\kappa _\lambda - \kappa _{n,\lambda })\Vert _{_{\mathcal {F}^2}} ^2 \le \frac{A}{4} \Vert a\Vert _2^2. \end{aligned}$$

(19)

Then via the triangle inequality $\tfrac{A}{4} \Vert a\Vert _2^2 \le \Vert \sum _{\lambda \in \Lambda _n} a_\lambda \kappa _{n,\lambda }\Vert _{_{\mathcal {F}^2}} ^2 \le (B+ \tfrac{A}{4} + \sqrt{AB}) \Vert a\Vert _2^2$.

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 $\Vert E\Vert _{\mathrm {op}} \le A/4$. 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

$$\begin{aligned} E_{\lambda ,\lambda }&= \Vert \kappa _\lambda - \kappa _{n,\lambda }\Vert _{_{\mathcal {F}^2}} ^2 \\&= 2 - 2 \, \mathrm {Re}\, \langle \kappa _\lambda , \kappa _{n,\lambda }\rangle \\&= 2\Big (1-\frac{\langle k_\lambda , k_{n,\lambda }\rangle }{k(\lambda ,\lambda )^{1/2} k_n(\lambda ,\lambda )^{1/2}}\Big ) \\&= 2\Big (1-\frac{ k_n (\lambda ,\lambda )^{1/2}}{k(\lambda ,\lambda )^{1/2} }\Big ) \, . \end{aligned}$$

Since $k_n(\lambda ,\lambda ) < k(\lambda ,\lambda )$, the estimate for the diagonal elements simplifies to

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

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

$$\begin{aligned} 1-\frac{\Gamma (n+1,x)}{n!} \le 1-\frac{\Gamma (n+1,n-\sqrt{n}\tau _n)}{n!} \le e^{-\tau _n^2/2} \end{aligned}$$

Combining these observations, we arrive at

$$\begin{aligned} \Vert E\Vert _{\mathrm {op}}&\le 2 \sum _{\lambda \in \Lambda \cap B_{\rho _n}} \Big ( 1 -\frac{\Gamma (n+1,\pi |\lambda |^2)}{n!}\Big )\\&\le 2 n e^{-\tau _n^2/2} \end{aligned}$$

By choosing $\tau _n = \sqrt{2\log n} + \tau$, with $\tau >0$ large enough, we achieve $\Vert E\Vert _{\mathrm {op}} \le A/4$ for $n\ge n_0$. 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. $\square$

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

Corollary 6.1

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

Proof

The proof is similar to the one of Corollary 4.1. $\square$

Theorem 6.2

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 $\lim _{n\rightarrow \infty } \lambda _j^{(n)} = \lambda _j$.

We show that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Big \Vert \sum _{j=1}^L a_{\lambda _j} (\kappa _{\lambda _j} - \kappa _{n,\lambda _j^{(n)} })\Big \Vert _{_{\mathcal {F}^2}} ^2 = 0\, . \end{aligned}$$

(20)

Consequently,

$$\begin{aligned} \Big \Vert \sum _{\lambda \in \Lambda \cap B_{\rho _N}} a_\lambda \kappa _\lambda \Big \Vert _{_{\mathcal {F}^2}} = \lim _{n\rightarrow \infty } \Big \Vert \sum _{j=1}^L a_{\lambda _j} \kappa _{n,\lambda _j^{(n)}} \Big \Vert _{_{\mathcal {F}^2}} \ge A \Big \Vert a\Big \Vert _2^2 \, , \end{aligned}$$

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

$$\begin{aligned} \Vert E\Vert _{\mathrm {op}}&\le \mathrm {tr}\, E = \sum _{j=1}^L \Vert e_j\Vert ^2_{_{\mathcal {F}^2}} = \sum _{j=1}^L \Vert \kappa _{\lambda _j} - \kappa _{n,\lambda _j^{(n)} } \Vert ^2 _{_{\mathcal {F}^2}} \\&= 2 \sum _{j=1}^L \Big (1-\mathrm {Re}\, \langle \kappa _{\lambda _j} , \kappa _{n,\lambda _j^{(n)} }\rangle \Big ) \, . \end{aligned}$$

Consider a single term of this sum and write $\lambda _j^{(n)} = \lambda$ and $\lambda _j= \mu$ for fixed j. Note that $|\lambda - \mu | \le 1$ for n large enough and that $\pi |\mu |^2 \le N$ by the assumption that $\mathrm {supp} \, a \subseteq B_{\rho _N}$. Now

$$\begin{aligned} \mathrm {Re} \, \langle \kappa _{\lambda _j} , \kappa _{n,\lambda _j^{(n)} } \rangle&= \mathrm {Re} \,\frac{k_n(\lambda ,\mu )}{k_n(\lambda ,\lambda )^{1/2} k(\mu ,\mu )^{1/2}} \\&= e^{-\pi |\lambda -\mu |^2/2} \frac{\Gamma (n+1,\pi |\lambda |^2) + \big [ \Gamma (n+1,\pi \lambda {\bar{\mu }}) -\Gamma (n+1,\pi |\lambda |^2)\big ]}{ \Gamma (n+1,\pi |\lambda |^2)^{1/2} \, n!^{1/2}} \\&= e^{-\pi |\lambda -\mu |^2/2} \frac{\Gamma (n+1,\pi |\lambda |^2)^{1/2}}{ n!^{1/2}} + e(n,\lambda ,\mu )\, . \end{aligned}$$

Since

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

for some $\eta >0$ by Lemma 3.1 and $\frac{\Gamma (n+1, \pi |\lambda |^2)}{n!} \rightarrow 1$, the term $e(n,\lambda ,\mu )$ tends to zero, as $n\rightarrow \infty$. By a similar reasoning, as $n\rightarrow \infty$ and thus $\lambda = \lambda _j^{(n)} \rightarrow \lambda _j= \mu$, we have

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

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

Proposition 6.1

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 5.1 any weak limit $\Lambda$ of a Marcinkiewicz-Zygmund family is a sampling set for $\mathcal {F}^2$, and by Theorem 6.2$\Lambda$ 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]. $\square$

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

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

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 $\phi _z(t) = e^{-2\pi i y t} e^{-\pi (t-x)^2}$ denote the time-frequency shift of the Gaussian by $z=x+iy$. Then

  $$\begin{aligned} B\phi _z(w) = k_z(w) = e^{\pi {\bar{z}} w} \end{aligned}$$

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

• (iii) B maps the normalized Hermite functions $h_k$,

  $$\begin{aligned} h_k(t) = c_k e^{\pi t^2} \tfrac{d^k}{dt^k}(e^{-2\pi t^2}), \quad \Vert h_k\Vert _2=1, \end{aligned}$$

  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 $\{\phi _\lambda : \pi |\lambda | ^2 \le n+\sqrt{n}\tau \}$ is a frame for $V_n = \mathrm {span}\, \{h_k : k=0, \dots , n\}$ with bounds independent of n, i.e.,

$$\begin{aligned} A\Vert f\Vert _2 \le \sum _{\lambda \in \Lambda : \pi |\lambda |^2 \le n+\sqrt{n}\tau } |\langle f, \phi _\lambda \rangle |^2 \le B\Vert f\Vert _2^2 \qquad \text { for all } f\in V_n \, . \end{aligned}$$

Proof

The statement is equivalent to Theorem 4.1 via the Bargman transform.²$\square$

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 $\big (\tfrac{n}{e}\big )^n \sqrt{2\pi n}\le n! \le e \sqrt{n} \big (\tfrac{n}{e}\big )^n$, we write

$$\begin{aligned} \frac{\Gamma (n+1, n+\sqrt{n}\tau )}{n!} = \frac{1}{n!}\int _{n+\sqrt{n}\tau } ^\infty t^n e^{-t} \, dt \asymp \frac{1}{\sqrt{n}} \int _{n+\sqrt{n}\tau } ^\infty \Big (\frac{t}{n}\Big )^n e^{n-t} \, dt \, . \end{aligned}$$

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

$$\begin{aligned} \frac{\Gamma (n+1, n+\sqrt{n}\tau )}{n!}&\asymp \sqrt{n} \int _{\tau / \sqrt{n}} ^\infty (1+u)^n e^{-nu}\, du \nonumber \\&= \sqrt{n} \int _{\tau / \sqrt{n}} ^\infty e^{ -n ( u - \ln (1+u))}\, du = \sqrt{n} \Big ( \int _{\tau / \sqrt{n}} ^1 \dots + \int _1^\infty \dots dt \Big ) \, . \end{aligned}$$

(21)

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

$$\begin{aligned} \sqrt{n} \int _1 ^\infty e^{ -n ( u - \ln (1+u))}\, du \le \sqrt{n} \int _1 ^\infty e^{ -n (1-\ln 2)u}\, du \le \frac{\sqrt{n}}{n(1-\ln 2)} = \mathcal {O}\big (\frac{1}{\sqrt{n}} \big ) \, . \end{aligned}$$

(22)

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

$$\begin{aligned} u-\ln (1+u) = \sum _{k=2}^\infty \frac{(-1)^k}{k} u^k \ge \frac{u^2}{2} - \frac{u^3}{3} \ge \frac{u^2}{6} \, . \end{aligned}$$

Consequently,

$$\begin{aligned} \sqrt{n} \int _{\tau / \sqrt{n}} ^1 (1+u)^n e^{-nu}\, du&\le \sqrt{n} \int _{\tau / \sqrt{n}} ^1 e^{-n u^2/6} \, du \\&= \sqrt{n} \sqrt{\frac{6}{n}} \int _{\tau / \sqrt{6}} ^{\sqrt{n/6}} e^{-v^2} \, dv \lesssim \mathrm {erf} (\tau / \sqrt{6}) \le e^{-\tau ^2/6} \, . \end{aligned}$$

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

$$\begin{aligned} \sqrt{n} \int _{\tau / \sqrt{n}} ^1 (1+u)^n e^{-nu}\, du&\ge \sqrt{n} \int _{\tau / \sqrt{n}} ^1 e^{-n u^2/2} \, du \\&= \sqrt{n} \sqrt{\frac{2}{n}} \int _{\tau / \sqrt{2}} ^{\sqrt{n/2}} e^{-v^2} \gtrsim \mathrm {erf} (\tau / \sqrt{2}) \gtrsim \frac{1}{\tau } e^{-\tau ^2/2} \, , \end{aligned}$$

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,

$$\begin{aligned} 1- \frac{\Gamma (n+1, n-\sqrt{n}\tau )}{n!}&\le \frac{1}{\sqrt{2\pi n}} \Big (\frac{e}{n} \Big )^n \int _0 ^{n-\sqrt{n}\tau } t^n e^{-t} \, dt =\\&=\frac{1}{\sqrt{2\pi n}} \int _0 ^{n-\sqrt{n}\tau } \Big (\frac{t}{n}\Big )^n e^{n-t} \, dt \, . \end{aligned}$$

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

$$\begin{aligned}&\frac{1}{\sqrt{2\pi n} } \int _0 ^{n-\sqrt{n}\tau } \Big (\frac{t}{n}\Big )^n e^{n-t} \, dt = \frac{\sqrt{n}}{\sqrt{2\pi }} \int _{\tau /\sqrt{n}}^1 (1-u)^n e^{nu}\, du = \nonumber \\&= \frac{\sqrt{n}}{\sqrt{2\pi }} \int _{\tau /\sqrt{n}} ^1 e^{ n ( u +\ln (1-u))}\, du = \frac{\sqrt{n}}{\sqrt{2\pi }} \Big ( \int _{\tau /\sqrt{n}} ^{1/2} \dots + \int _{1/2} ^1 \dots dt \Big ) \, . \end{aligned}$$

(23)

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

$$\begin{aligned} \frac{\sqrt{n}}{\sqrt{2\pi }} \int _{\tau /\sqrt{n}} ^{1/2} e^{ n ( u + \ln (1-u))}\, du&\le \frac{\sqrt{n}}{\sqrt{2\pi }} \int _{\tau /\sqrt{n}} ^{1/2} e^{ -n u^2/2}\, du \\&= \frac{1}{\sqrt{2\pi }}\int _{\tau } ^{\sqrt{n}/2} e^{ -v^2/2}\, du \le \tfrac{1}{2} e^{-\tau ^2/2} \, . \end{aligned}$$

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

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

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