A Note on Band-limited Minorants of an Euclidean Ball

TL;DR

This paper investigates the problem of constructing band-limited functions that are bounded above by an indicator function of a Euclidean ball, focusing on the critical support radius for positive integral existence.

Contribution

It provides a computation of the critical radius of Fourier support for band-limited minorants of Euclidean balls, advancing understanding of the Beurling-Selberg problem in this context.

Findings

Determined the critical radius for positive integral of band-limited minorants.

Established conditions under which such minorants exist.

Contributed to the theory of band-limited approximation of indicator functions.

Abstract

We study the Beurling-Selberg problem of finding band-limited $L^1$-functions that lie below the indicator function of an Euclidean ball. We compute the critical radius of the support of the Fourier transform for which such construction can have a positive integral.

Equations

$$\beta (d,r)=\underset{F}{\sup }{\int }_{{\mathbb{R}}^d}F(\mathbf{x})dx,$$

$$\hat{F}(\xi )={\int }_{{\mathbb{R}}^d}F(x)e^{2\pi ix\cdot \xi }\text{d}x,$$

$$r_d=\inf \{r>0:\beta (d,r)>0\}$$

$$J_{\nu}(z)=\Big{(}\frac{z}{2}\Big{)}^{\nu}\sum_{n=0}^{\infty}\frac{(-1)^{n}\big{(}\tfrac{z}{2}\big{)}^{2n}}{n!\,\Gamma(\nu+n+1)}.$$

$$z^2{J}_{\nu }^{\prime \prime }(z)+z{J}_{\nu }^{\prime }(z)+(z^2-{\nu }^2)J_{\nu }(z)=0,$$

$$J_{\nu -1}(z)-J_{\nu +1}(z)$$

$$J_{\nu -1}(z)+J_{\nu +1}(z)$$

$$r_d=\frac{j_{d/2-1,1}}{\pi }.$$

$$\beta (d,r)=\frac{(2/r{)}^d}{|{\mathbb{S}}^{d-1}|}\frac{{\gamma }_{\pi r}}{1+{\gamma }_{\pi r}/d},$$

$$\beta(d,r)=\frac{\pi^{2}2^{d}}{r^{d-1}{|\mathbb{S}^{d-1}|}}\bigg{(}r-\frac{j_{d/2-1,1}}{\pi}\bigg{)}+O_{d}\bigg{(}r-\frac{j_{d/2-1,1}}{\pi}\bigg{)}^{2}$$

$$\begin{array}{r}\beta (d,r)=\frac{|{\mathbb{S}}^{d-1}|}{2}\underset{F}{\sup }{\int }_{\mathbb{R}}F(x)|x{|}^{d-1}dx,\end{array}$$

$$|F(z)|\le C{e}^{2\pi r|\mathrm{Im}z|},z\in \mathbb{C}$$

$$A_{\nu}(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\big{(}\tfrac{1}{2}z\big{)}^{2n}}{n!(\nu+1)(\nu+2)\ldots(\nu+n)}=\Gamma(\nu+1)\left(\tfrac{1}{2}z\right)^{-\nu}J_{\nu}(z)$$

$$B_{\nu}(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\big{(}\tfrac{1}{2}z\big{)}^{2n+1}}{n!(\nu+1)(\nu+2)\ldots(\nu+n+1)}=\Gamma(\nu+1)\left(\tfrac{1}{2}z\right)^{-\nu}J_{\nu+1}(z).$$

$$E_{\nu }(z)=A_{\nu }(z)-i{B}_{\nu }(z),$$

$$|E_{\nu }(\overline{z})|<|E_{\nu }(z)|$$

$$c|x{|}^{2\nu +1}\le |E_{\nu }(x){|}^{-2}\le C|x{|}^{2\nu +1}$$

$$\begin{gathered} {\int }_{\mathbb{R}}|F(x){|}^2|E_{\nu }(x){|}^{-2} \\ dx<\infty . \end{gathered}$$

$$a_{\nu }{\int }_{\mathbb{R}}|F(x){|}^2|E_{\nu }(x){|}^{-2}\text{d}x={\int }_{\mathbb{R}}|F(x){|}^2|x{|}^{2\nu +1}\text{d}x$$

$$\begin{array}{rl}{a}_{\nu }{\int }_{\mathbb{R}}F(x)|E_{\nu }(x){|}^{-2}\text{d}x& =a_{\nu }{\int }_{\mathbb{R}}\{4G_n(x{)}^2-|H_n(x){|}^2\}|E_{\nu }(x){|}^{-2}\text{d}x\\ & ={\int }_{\mathbb{R}}\{4G_n(x{)}^2-|H_n(x){|}^2\}|x{|}^{2\nu +1}\text{d}x\\ & ={\int }_{\mathbb{R}}F(x)|x{|}^{2\nu +1}\text{d}x.\end{array}$$

$$\beta (d,r)=\frac{(2/r{)}^d}{\pi |{\mathbb{S}}^{d-1}|}{\Lambda }_{E_{d/2-1}}^{-}(\pi r),$$

$${\Lambda }_{E_{d/2-1}}^{-}(\pi r)=\underset{F}{\sup }{\int }_{\mathbb{R}}F(x)|E_{d/2-1}(x){|}^{-2}\text{d}x$$

$${\Lambda }_{E_{d/2-1}}^{-}(\pi r)=\frac{\pi {\gamma }_{\pi r}}{1+{\gamma }_{\pi r}/d},$$

$${\Lambda }_{E_{d/2-1}}^{-}(\pi r)={\pi }^{2}r(\pi r-j_{d/2-1,1})+O(\pi r-j_{d/2-1,1}{)}^2$$

$$\beta(d,r)=\frac{\pi^{2}2^{d}}{r^{d-1}|{\mathbb{S}^{d-1}}|}\bigg{(}r-\frac{j_{d/2-1,1}}{\pi}\bigg{)}+O_{d}\bigg{(}r-\frac{j_{d/2-1,1}}{\pi}\bigg{)}^{2}.$$

Full text

1. Introduction

For a given $r>0$ we denote by $B^{d}(r)$ the closed Euclidean ball in $\mathbb{R}^{d}$ centered at the origin with radius $r>0$. We simply write $B^{d}$ when $r=1$. Define the following quantity

[TABLE]

where the supremum is taken among functions $F\in L^{1}(\mathbb{R}^{d})$ such that:

1. (1)

The Fourier Transform of $F(x)$,

[TABLE]

is supported in $B^{d}(r)$; 2. (2)

$F(x)\leq{\boldsymbol{1}}_{B^{d}}(x)$ for all $x\in\mathbb{R}^{d}$.

We call such a function $\beta(d,r)$-admissible. A trivial observation is that $F\equiv 0$ is $\beta(d,r)$-admissible, hence $\beta(d,r)\geq 0$. Heuristically, such function $F(x)$ should exist and its mass should be close to ${\rm vol}(B^{d})$ when $r$ is large. On the other hand, if $r$ is small, the mass of $F(x)$ should be close to zero and a critical $r_{d}>0$ should exist such that no function can beat the identically zero function for $r\leq r_{d}$. For this reason we define

[TABLE]

and it is this critical radius that we want to study in this paper.

The problem stated in (1.1) has its origins with Beurling and Selberg which studied one-sided band-limited approximations for many different functions other than indicator functions with the purpose of using them to derive sharp estimates in analytic number theory (see the introduction of [9] for a nice first view). Although Selberg was one of the first to study the higher dimensional problem, it was first systematically analyzed by Holt and Vaaler in the remarkable paper [7]. They were able to construct non-zero $\beta(d,r)$-admissible functions for any $r>0$ and, most importantly, they established a fascinating connection of the $d$-dimensional problem with the theory of Hilbert spaces of entire functions contructed by de Branges (see [1]). They reduced the higher dimensional problem, after a radialization argument, to a weighted one-dimensional problem where the weight was given by a special function of Hermite-Biehler class, which in turn allowed them to use the machinery of homogeneous de Branges spaces to attack the problem. This new connection established by Holt and Vaaler started a new way of thinking about these kind of problems and ultimately inspired Littmann to completely solve the one-dimensional problem in [8] by using a cleaver argument based on a special structure of certain de Branges spaces. Finally, using the ideas introduced by Littmann in [8], the problem of minorizing the indicator function of a symmetric interval was completely solved in [2] in the de Branges space setting.

This paper was mainly motivated by the related problem where balls are substituted by boxes $Q(r)=[-r,r]^{d}$ and where practically nothing is known (see [3]). The box minorant problem is harder since it is a truly higher dimensional problem, whereas for the ball we can make radial reductions that transform it in a one-dimensional problem. Another interesting similar question, connected with upper bounds for sphere packings in $\mathbb{R}^{d}$, is studied in [6] (see also [4]), where the author constructs a minorant $F(x)\leq{\boldsymbol{1}}_{B^{d}}(x)$ with Fourier transform non-negative and supported in $B^{d}(j_{d/2,1})$ and such that it maximizes $\widehat{F}(0)$ among all functions with these properties111It is not the intention of this paper to give a survey of related articles on the subject, which is very rich and full of subtleties, the purpose here is to draw a strait line between what he have so far and what we want to show..

1.1. Main Result

For any given parameter $\nu>-1$ let $J_{\nu}$ denote the classical Bessel function of the first kind. We also denote by $\{j_{\nu,n}\}_{n\geq 1}$ its positive zeros listed in increasing order. The Bessel function of the first kind $J_{\nu}$ can be defined in a number of ways. We follow the treatise [10] and define it for $\nu>-1$ and $\Re(z)>0$ by

[TABLE]

For these values of $\nu$, one can check that on the half space $\{\Re(z)>0\}$ the Bessel functions defined by (1.2) satisfy the differential equation

[TABLE]

and that the following recursion relations hold

[TABLE]

In particular we have $J_{-1/2}(z)=\sqrt{\frac{2}{\pi z}}\,\cos(z)$ and $J_{1/2}(z)=\sqrt{\frac{2}{\pi z}}\,\sin(z)$, which implies that $j_{-1/2,1}=\pi/2$ and $j_{1/2,1}=\pi$. The following is the main result of this paper.

Theorem 1**.**

We have

[TABLE]

Moreover, if ${j_{d/2-1,1}}<\pi r<{j_{d/2,1}}$ then

[TABLE]

where $\gamma_{\pi r}=-\frac{\pi rJ_{d/2-1}(\pi r)}{J_{d/2}(\pi r)}>0$. In particular we have

[TABLE]

for $r$ close to $\frac{j_{d/2-1,1}}{\pi}$.

Remarks

1. (1)

It is known that $j_{\nu,1}=\nu+1.855757\nu^{1/3}+O(\nu^{-1/3})$ as $\nu\to\infty$ (see [5, Section 1.3]). This implies that $r_{d}=\frac{d}{2\pi}+\frac{1.855757}{2^{1/3}\pi}d^{1/3}+O(d^{-1/3})$ as $d\to\infty$. Heuristically, this means that if one wishes to non-trivially minorate (that is, beat the zero function) the indicator function of a ball of radius of order $\sqrt{d}$ then one needs frequencies of order at least $\sqrt{d}$. 2. (2)

The first $5$ values of $r_{d}$ rounded up to 4 significant digits are the following: $r_{1}=1/2$, $r_{2}=0.7655$, $r_{3}=1$, $r_{4}=1.220$ and $r_{5}=1.431$. 3. (3)

Explicit expressions for $\beta(d,r)$ can also be tracked from [2, Theorem 5], but they involve sums of Bessel functions evaluated at Bessel zeros that can be quite complicated to grasp. Moreover, this is the case only when $\pi r$ is a zero of $J_{d/2-1}(z)$ or $J_{d/2}(z)$. If that is not the case, then writing a formula for $\beta(d,r)$ becomes pointless, since it will involve zeros of more complicated functions related to Bessel functions and this is not the purpose here.

2. Proof of Theorem 1

Step 1.

The first step is to reduce the higher dimensional by considering only radial functions. We can apply [7, Lemmas 18 and 19] to reduce the $d$-dimensional problem to the following weighted one-dimensional problem

[TABLE]

where $|\mathbb{S}^{d-1}|$ denotes the surface area of the unit sphere in $\mathbb{R}^{d}$ and the supremum is taken among functions $F\in L^{1}(\mathbb{R},|x|^{d-1}\text{\rm d}x)$ such that:

1. (1)

$F(x)$ is the restriction to the real axis of an even entire function $F(z)$ of exponential type at most $2\pi r$, that is,

[TABLE]

for some constant $C>0$; 2. (2)

$F(x)\leq{\boldsymbol{1}}_{[-1,1]}(x)$ for all $x\in\mathbb{R}$.

In this framework the problem becomes treatable with the theory de Branges spaces of entire functions. The latter generalize the well known Paley-Wiener spaces by using weighted norms given by Hermite-Biehler functions. In what follows we briefly review the construction of a special family of de Branges spaces called homogeneous spaces which were introduced by de Branges (see [1, Section 50] and [7, Section 5]). We refer to [7, Section 3] for a brief description of the general theory and also to [1, Chapter 2] for the full theory.

Step 2.

Let $\nu>-1$ be a parameter and consider the real entire functions $A_{\nu}(z)$ and $B_{\nu}(z)$ given by

[TABLE]

and

[TABLE]

If we write

[TABLE]

then $E_{\nu}(z)$ is a Hermite–Biehler function, that is, it satisfies the following fundamental inequality

[TABLE]

for all $z\in\mathbb{C}$ with ${\rm Im}\,z>0$. It is also known that this function does not have real zeros, that $E(iy)\in\mathbb{R}$ for all real $y$ (that is, $B_{\nu}(z)$ is odd and $A_{\nu}(z)$ is even), that $E_{\nu}(z)$ is of bounded type in the upper-half plane (that is, $\log|E_{\nu}(z)|$ has a positive harmonic majorant in the upper-half plane) and $E_{\nu}(z)$ is of exponential type $1$. We also have that

[TABLE]

for all $|x|\geq 1$ and for some $c,C>0$. The homogeneous space $\mathcal{H}(E_{\nu})$ is then defined as the space of entire functions $F(z)$ of exponential type at most $1$ and such that222 As an historical note, de Branges originally defined this space in another way but, in [7, Lemma 16], the authors showed that this is an equivalent definition.

[TABLE]

Using standard asymptotic expansions for Bessel functions one can show that $A_{\nu},B_{\nu}\notin\mathcal{H}(E_{\nu})$. As a particular case, observing that $E_{-1/2}=e^{-iz}$ we can deduce that $\mathcal{H}(E_{-1/2})$ coincides with the Paley-Wiener space of square integrable entire functions of exponential type at most $1$.

These spaces are relevant to our problem since we have the following magical identity

[TABLE]

for each $F\in\mathcal{H}(E_{\nu})$, where $a_{\nu}=\,2^{2\nu+1}\,\Gamma(\nu+1)^{2}/\pi$. For our purposes we will need an identity analogous of (2.2), but which allow us to compute integrals instead of $L^{2}$-norms. It can be derived as follows. Let $F(z)$ be an entire function of exponential type at most $2$ such that $F(x)\leq{\bf 1}_{[-t,t]}(x)$ for some $t>0$ and $F\in L^{1}(\mathbb{R},|x|^{2\nu+1}\text{\rm d}x)$. Since $G_{n}(x)=(\frac{\sin(x/n)}{x/n})^{n}$ belongs to $\mathcal{H}(E_{\nu})$ for large $n$ and it converges to $1$ uniformly in compact sets as $n\to\infty$, we have that $4G_{n}(x)^{2}-F(x)\geq 0$ for all real $x$ (if $n$ is large and even) and this function has exponential type at most $2$. This implies that $4G_{n}(x)^{2}-F(x)=H_{n}(z)\overline{H_{n}(\overline{z})}$ for all $z\in\mathbb{C}$, for some entire function $H_{n}(z)$ of exponential type at most $1$ (see [1, Theorem 13]). We conclude that $H_{n}\in\mathcal{H}(E_{\nu})$ and we obtain

[TABLE]

Step 3.

Taking $\nu=d/2-1$, we can apply the change of variables $x\mapsto x/(\pi r)$ in (2.1) and use identity (2.3) to reduce the problem of minorizing the indicator function of an Euclidean ball to the following final form

[TABLE]

where

[TABLE]

and the supremum is taken among functions $F\in L^{1}(\mathbb{R},|E_{d/2-1}(x)|^{-2}\text{\rm d}x)$ such that:

1. (1)

$F(x)$ is the restriction to the real axis of an even entire function $F(z)$ of exponential type at most $2$; 2. (2)

$F(x)\leq{\boldsymbol{1}}_{[-\pi r,\pi r]}(x)$ for all $x\in\mathbb{R}$.

The above problem was completely solved in [2]. By all the previous discussion in Step $2$, we can apply [2, Theorem 5 (i) and (iv)] to the function $E_{d/2-1}(z)$ (it actually can be applied to any $E_{\nu}(z)$) to derive that $\pi r_{d}=j_{d/2-1,1}$. Moreover, if $j_{d/2-1,1}<\pi r<j_{d/2,1}$ then [2, Theorem 5 (iv)] also give us that

[TABLE]

where $\gamma_{\pi r}=-\frac{\pi rJ_{d/2-1}(\pi r)}{J_{d/2}(\pi r)}>0$. A simple Taylor expansion leads to

[TABLE]

and we finally obtain that

[TABLE]