# Zeros of $L$-Functions in Low-Lying Intervals
and de Branges spaces

Antonio Pedro Ramos  SISSA - Scuola Internazionale Superiore di Studi
Avanzati, Via Bonomea 265, 34136 Trieste, Italy<EMAIL_ADDRESS>
###### Abstract.

We consider a variant of a problem first introduced by Hughes and Rudnick
(2003) and generalized by Bernard (2015) concerning conditional bounds for
small first zeros in a family of $L$-functions. Here we seek to estimate the
size of the smallest intervals centered at a low-lying height for which we can
guarantee the existence of a zero in a family of $L$-functions. This leads us
to consider an extremal problem in analysis which we address by applying the
framework of de Branges spaces, introduced in this context by Carneiro,
Chirre, and Milinovich (2022).

###### Key words and phrases:

Families of $L$-functions; low-lying zeros; reproducing kernels; Hilbert
spaces; de Branges spaces

###### 2020 Mathematics Subject Classification:

42A05, 46E22, 11F66, 11M26, 11M41

##  1\. Introduction

This is a paper at the intersection of Analysis and Number Theory. We consider
a question coming from the theory of $L$-functions, which we tackle by
addressing a related extremal problem in analysis. These are the contents of
Theorems 1 and 2. Theorem 2 is in fact a specialization of the more general
Theorem 4, the main result of this paper, which solves a family of extremal
problems in Hilbert spaces of entire functions.

###  1.1. Zeros of _L_ -functions

As a consequence of a one-level density result, Hughes and Rudnick [23]
conditionally proved the existence of small first zeros within the family of
Dirichlet $L$-functions for large values of the conductor. First, they proved
that for even Schwartz functions $\phi$ with ${\rm
supp}\,\widehat{\phi}\subseteq[-2,2]$, we have the _one-level density_ 111In
[23], they use the term _linear statistic_ to mean the same thing. We adopt
the terminology in [24, 25, 26].

$\lim_{q\rightarrow\infty}\frac{1}{q-2}\sum_{\chi\neq\chi_{0}}\sum_{\gamma_{\chi}}\phi\left(\frac{\gamma_{\chi}\log
q}{2\pi}\right)=\int_{\mathbb{R}}\phi(x)\,\text{\rm d}x,$ (1.1)

where the outside is a sum over non-principal Dirichlet characters modulo a
prime $q$ and the inside sum is over their respective non-trivial zeros
$\frac{1}{2}+i\gamma_{\chi}$. They then observed that, under the generalized
Riemann Hypothesis (GRH), if the one-level density (1.1) holds for even
Schwartz functions $\phi$ with ${\rm
supp}\,\widehat{\phi}\subseteq[-\Delta,\Delta]$, one can always obtain a bound
for the height of the first zero in the family in terms of $\Delta>0$.
Explicitly,

$\limsup_{q\rightarrow\infty}\,\min_{\begin{subarray}{c}{\gamma_{\chi}}\\\
{\chi\neq\chi_{0}}\end{subarray}}\left|\frac{\gamma_{\chi}\log
q}{2\pi}\right|\leq\frac{1}{2\Delta},$ (1.2)

which together with the previous observation means that, for large $q$, there
exist zeros in this family within $\frac{1}{4}$ times the average spacing.

Bernard [5], and later Carneiro, Chirre, and Milinovich [9], extended the
analysis of Hughes and Rudnick to more general families of $L$-functions. The
context is the following: we consider families of automorphic objects
$\mathcal{F}$. Each $f\in\mathcal{F}$ has an associated $L$-function

$L(s,f)=\sum_{n=1}^{\infty}\lambda_{f}(n)n^{-s}$

that can be continued analytically to an entire function, and its
corresponding completed $L$-function $\Lambda(s,f)=L_{\infty}(s,f)L(s,f)$
satisfies a functional equation of the type

$\Lambda(s,f)=\varepsilon_{f}\Lambda(1-s,\bar{f}),$

where $|\varepsilon_{f}|=1$ and $L(s,\bar{f})$ is the dual $L$-function with
Dirichlet series coefficients
$\lambda_{\bar{f}}(n)=\overline{\lambda_{f}(n)}$. We denote the analytic
conductor of $L(s,f)$ by $c_{f}$ and write its non-trivial zeros as
$\rho_{f}=\frac{1}{2}+i\gamma_{f}$. Throughout, we assume GRH for these
$L$-functions, meaning $\gamma_{f}\in\mathbb{R}$. We further assume that our
family satisfies the assumptions

$\begin{cases}f\in\mathcal{F}\implies\bar{f}\in\mathcal{F};\\\
L_{\infty}(s,f)\neq 0\text{ for all }\mathrm{Re}\,(s)=\frac{1}{2},\end{cases}$
(1.3)

which in in particular guarantee that if $\frac{1}{2}+i\gamma$ is a non-
trivial zero within this family of $L$-functions, then so is its conjugate
$\frac{1}{2}-i\gamma$.

As conjectured by Katz and Sarnak [25, 26], for each natural family
$\\{L(s,f):f\in\mathcal{F}\\}$ of $L$-functions there is an associated
symmetry group $G=G(\mathcal{F})$ which governs the distribution of its low-
lying zeros. The group $G$ can be either unitary U, symplectic Sp, orthogonal
O, even orthogonal SO(even), or odd orthogonal SO(odd). The idea is to
consider averages over finite subsets of $\mathcal{F}$ ordered by the
conductor

$\big{\\{}f\in\ F:c_{f}=\mathcal{Q}\big{\\}}$

or, depending on the context,

$\big{\\{}f\in\ F:c_{f}\leq\mathcal{Q}\big{\\}}$

as $\mathcal{Q}\to\infty$. In both cases we denote the set under consideration
by $\mathcal{F}(\mathcal{Q})$ in order to unify the presentation. Now, for
each of these symmetry groups, there is a corresponding density $W_{G}$ for
which Katz and Sarnak conjectured the one level-density

$\lim_{\mathcal{Q}\rightarrow\infty}\frac{1}{|\mathcal{F}(\mathcal{Q})|}\sum_{f\in\mathcal{F}(\mathcal{Q})}\sum_{\gamma_{f}}\phi\left(\frac{\gamma_{f}\log
c_{f}}{2\pi}\right)=\int_{\mathbb{R}}\phi(x)\,W_{G}(x)\,\text{\rm d}x$ (1.4)

for even Schwartz functions $\phi$ with compactly supported Fourier transform.
Naturally, the zeros are counted with their multiplicities in the sums above.
These five densities are

$\displaystyle W_{\text{U}}(x)$ $\displaystyle=1;$ $\displaystyle
W_{\text{Sp}}(x)$ $\displaystyle=1-\frac{\sin 2\pi x}{2\pi x};$ $\displaystyle
W_{\text{O}}(x)$ $\displaystyle=1+\frac{1}{2}{\boldsymbol{\delta}}_{0}(x);$
(1.5) $\displaystyle W_{\text{SO(even)}}(x)$ $\displaystyle=1+\frac{\sin 2\pi
x}{2\pi x};$ $\displaystyle W_{\text{SO(odd)}}$ $\displaystyle=1-\frac{\sin
2\pi x}{2\pi x}(x)+{\boldsymbol{\delta}}_{0}(x),$

where $\boldsymbol{\delta}_{0}$ is the Dirac delta at $x=0$. Some examples of
density results which fall within this framework can be found in [2, 3, 4, 7,
11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33,
34], with possible differences in setup and notation. One-level densities for
families of $L$-functions have been used to estimate the proportion of non-
vanishing at the central point $s=\frac{1}{2}$ [3, 7, 12, 13, 16, 17, 19, 22,
24, 25, 29], the average rank of elliptic curves [4, 21, 34], and the height
of small first zeros [5, 9, 23]. It is worth noting that the authors of [9]
considered a generalization of the problem of the proportion of non-vanishing,
moving away from the central point $s=\frac{1}{2}$ to low-lying heights of the
critical line, where height is measured in terms of the analytic conductor.

Regarding small first zeros, the strategy common to [5, 9, 23] is that one-
level density results lead to certain extremal problems in analysis, which in
turn provide estimates of the height of the first zero in a family of
$L$-functions. Bernard [5] tackled these problems via a careful analysis of an
associated Volterra differential equation, while Carneiro, Chirre, and
Milinovich [9] reframed them as a corollary of a more general result within
the theory of de Branges spaces of entire functions. In this paper we take up
the latter approach to provide estimates for first zeros in intervals at low-
lying heights of the critical line. If we consider the normalized zeros
$\tilde{\rho}_{f}=\frac{1}{2}+i\tilde{\gamma}_{f}$ with
$\tilde{\gamma}_{f}=\gamma_{f}\frac{\log c_{f}}{2\pi}$, we want to know, given
an $\alpha\in\mathbb{R}$, the size of the smallest interval of the critical
line centered at $\frac{1}{2}+i\alpha$ for which we can guarantee the
existence of a zero $\tilde{\rho}_{f}$ in a given family of $L$-functions.

###  1.2. Hilbert spaces of entire functions

The upper-bound (1.2) can be cast as the solution of an extremal problem in
Paley–Wiener space, a classical example of a Hilbert space of entire
functions. This is generalized in the case of the other symmetry groups by the
de Branges spaces.

We recall that an entire function $F:\mathbb{C}\to\mathbb{C}$ is said to be of
exponential type if

$\tau(F):=\limsup_{|z|\to\infty}\,|z|^{-1}\log|F(z)|<\infty,$

and, in that case, we say $\tau(F)$ is the exponential type of $F$. The spaces
$\mathcal{H}_{G,\pi\Delta}$ of entire functions of exponential type at most
$\pi\Delta$ with norm

$\|F\|^{2}_{\mathcal{H}_{G,\pi\Delta}}:=\int_{\mathbb{R}}|F(x)|^{2}\,W_{G}(x)\,\text{\rm
d}x<\infty$

constitute a Hilbert space. We point out that
$\mathcal{H}_{\text{U},\pi\Delta}$ equipped with the norm
$\|\cdot\|_{\mathcal{H}_{\text{U},\pi\Delta}}$ is the classical Paley–Wiener
space. As demonstrated in [9, Section 3], an application of a Fourier
uncertainty principle shows that, as sets, the $\mathcal{H}_{G,\pi\Delta}$ are
all equal. This is the set $\mathcal{H}_{\pi\Delta}$ of functions of
exponential type at most $\pi\Delta$ such that their restriction to
$\mathbb{R}$ is square-integrable. The content of the Paley–Wiener theorem is
that $F\in\mathcal{H}_{\pi\Delta}$ if and only if $F\in C(\mathbb{R})\cap
L^{2}(\mathbb{R})$ and
$\mathrm{supp}\,\widehat{F}\subseteq[-\Delta/2,\Delta/2]$.

If one defines

$\mathbb{A}({G,\pi\Delta,0}):=\inf_{0\neq
f\in\mathcal{H}_{\pi\Delta}}\frac{\int_{\mathbb{R}}|xf(x)|^{2}\,W_{G}(x)\,\text{\rm
d}x}{\int_{\mathbb{R}}|f(x)|^{2}\,W_{G}(x)\,\text{\rm d}x},$ (1.6)

then the authors of [9] showed how to obtain Theorem A below. The presence of
the parameter $0$ will become clear soon.

######  Theorem A (cf. [9, Theorem 9]).

Let $\\{L(s,f),\,f\in\mathcal{F}\\}$ be a family of L-functions with an
associated symmetry group $G\in\\{{\rm U,Sp,O,SO(even),SO(odd)}\\}$, and
assume that GRH holds for $L$-functions in this family. In addition, suppose
(1.4) holds for even Schwartz functions $\phi$ with ${\rm
supp}\,\widehat{\phi}\subseteq(-\Delta,\Delta)$ for a fixed $\Delta>0$. Then
for the families with $G\in\\{\rm U,Sp,SO(even)\\}$

$\limsup_{\mathcal{Q}\rightarrow\infty}\,\min_{\begin{subarray}{c}{\gamma_{f}}\\\
{f\in\mathcal{F}(Q)}\end{subarray}}\left|\frac{\gamma_{f}\log
c_{f}}{2\pi}\right|\leq\sqrt{\mathbb{A}({G,\pi\Delta,0})}.$

Moreover, for the families with $G\in\\{\rm O,SO(odd)\\}$, we may exclude the
zeros at the central point $Z(f)$. Thus, denoting $G^{\sharp}={\rm U}$ if
$G={\rm O}$ and $G^{\sharp}={\rm Sp}$ if $G={\rm SO(odd)}$, we have

$\limsup_{\mathcal{Q}\rightarrow\infty}\,\min_{\begin{subarray}{c}{\gamma_{f}\notin
Z(f)}\\\ {f\in\mathcal{F}(Q)}\end{subarray}}\left|\frac{\gamma_{f}\log
c_{f}}{2\pi}\right|\leq\sqrt{\mathbb{A}({G^{\sharp},\pi\Delta,0})}.$

In all of these spaces the evaluation functionals are continuous. In other
words, for all $w\in\mathbb{C}$ the linear map $F\mapsto F(w)$ is continuous.
By applying the Riesz representation theorem, we know that for each
$w\in\mathbb{C}$ there is an associated entire function
$K_{G,\pi\Delta}(w,\cdot)\in\mathcal{H}_{G,\pi\Delta}$ that corresponds to the
evaluation functional at $w$, meaning

$F(w)=\langle
F,K_{G,\pi\Delta(w,\cdot)}\rangle_{\mathcal{H}_{G,\pi\Delta}}=\int_{\mathbb{R}}F(x)\overline{K_{G,\pi\Delta}(w,x)}\,\text{\rm
d}x$

for all $F\in\mathcal{H}_{G,\pi\Delta}$. Such a $K_{G,\pi\Delta}(w,\cdot)$ is
called a reproducing kernel. We thus say these spaces have the _reproducing
kernel property_. Carneiro, Chirre, and Milinovich studied the spaces
$\mathcal{H}_{G,\pi\Delta}$ and found explicit expressions for their
reproducing kernels when $\Delta>0$ for the groups $G\in\\{\rm{U,O}\\}$ and
when $0<\Delta\leq 2$ for $G\in\\{\rm{Sp,SO(even),SO(odd)}\\}$ (see [9,
Theorems 3–7]). They also established that the solution to the extremal
problem posed by (1.6) was given by

######  Theorem B (cf. [9, Theorem 9 and Theorem 14]).

Let $G\in\\{\rm U,Sp,O,SO(even),SO(odd)\\}$, and let $K=K_{G,\pi\Delta}$ be
the reproducing kernel of the Hilbert space $(\mathcal{H}_{\pi\Delta}$,
$\langle\cdot,\cdot\rangle_{G,\pi\Delta})$. Let $\xi_{1}$ be the first
positive zero of the function $x\mapsto\mathrm{Re}\,((1-ix)K(i,x))$. Then

$\mathbb{A}({G,\pi\Delta,0})=\inf_{0\neq
f\in\mathcal{H}_{\pi\Delta}}\frac{\int_{\mathbb{R}}|xf(x)|^{2}\,W_{G}(x)\,\text{\rm
d}x}{\int_{\mathbb{R}}|f(x)|^{2}\,W_{G}(x)\,\text{\rm d}x}=\xi_{1}^{2}.$

Together, Theorem A and Theorem B provide an upper bound for the minimal
height of the first zero in a family of $L$-functions once knows the relevant
reproducing kernels explicitly. Here we address the question of existence of
zeros in small intervals of the critical line at different heights. In analogy
with Theorem A, we define the following sharp constant for a given real number
$\alpha$:

$\mathbb{A}(G,\pi\Delta,\alpha):=\inf_{0\neq
f\in\mathcal{H}_{\pi\Delta}}\frac{\int_{\mathbb{R}}|(x-\alpha)f(x)|^{2}\,W_{G}(x)\,\text{\rm
d}x}{\int_{\mathbb{R}}|f(x)|^{2}\,W_{G}(x)\,\text{\rm d}x}.$ (1.7)

Given this, we establish the following result.

######  Theorem 1.

Fix $\alpha\neq 0$ real and let $\\{L(s,f),\,f\in\mathcal{F}\\}$ be a family
of L-functions with an associated symmetry group $G\in\\{\rm
U,Sp,O,SO(even),SO(odd)\\}$, and assume that GRH holds for $L$-functions in
this family. In addition, suppose (1.4) holds for even Schwartz functions
$\phi$ with ${\rm supp}\,\widehat{\phi}\subseteq(-\Delta,\Delta)$ for a fixed
$\Delta>0$. Then

$\limsup_{\mathcal{Q}\rightarrow\infty}\,\min_{\begin{subarray}{c}{\gamma_{f}}\\\
{f\in\mathcal{F}(Q)}\end{subarray}}\left|\frac{\gamma_{f}\log
c_{f}}{2\pi}-\alpha\right|\leq\sqrt{\mathbb{A}(G,\pi\Delta,\alpha)}.$

Remark: In orthogonal and odd orthogonal families, where, due to the sign of
the functional equation, we may expect at least part of the family to
trivially vanish at the central point, we can further bound

$\limsup_{\mathcal{Q}\rightarrow\infty}\,\min_{\begin{subarray}{c}{\gamma_{f}}\\\
{f\in\mathcal{F}(Q)}\end{subarray}}\left|\frac{\gamma_{f}\log
c_{f}}{2\pi}-\alpha\right|\leq\min\left\\{\sqrt{\mathbb{A}(G,\pi\Delta,\alpha)},|\alpha|\right\\}.$

The main difference with the problem considered previously is that when
$\alpha\neq 0$ the symmetry is in a sense broken. All the measures considered
originally by Katz and Sarnak are symmetric around the origin, as is the
“power weight” $|x|^{2}$. Since this no longer holds for $|x-\alpha|^{2}$, we
must go through a different path to provide an estimate for the sharp
constants $\mathbb{A}(G,\pi\Delta,\alpha)$. We are inspired by the methods of
[10] to prove the analogous of Theorem B below.

######  Theorem 2.

Fix $\alpha$ real. Let $G\in\\{\rm U,Sp,O,SO(even),SO(odd)\\}$, and let
$K=K_{G,\pi\Delta}$ be the reproducing kernel of the Hilbert space
$(\mathcal{H}_{\pi\Delta}$, $\langle\cdot,\cdot\rangle_{G,\pi\Delta})$ Let
$\xi_{1}$ be the first positive zero of the function $x\mapsto
K(\alpha+x,\alpha-x)$. Then

$\mathbb{A}(G,\pi\Delta,\alpha)=\inf_{0\neq
f\in\mathcal{H}_{\pi\Delta}}\frac{\int_{\mathbb{R}}|(x-\alpha)f(x)|^{2}\,W_{G}(x)\,\text{\rm
d}x}{\int_{\mathbb{R}}|f(x)|^{2}\,W_{G}(x)\,\text{\rm d}x}=\xi_{1}^{2}.$

Remark. The above solution for the case $\alpha=0$ is, of course, the same as
that given by Theorem B. The first positive zero of $x\mapsto K(x,-x)$ and
that of $x\mapsto\mathrm{Re}\,((1-ix)K(i,x))$ are both equal to the first
positive zero of the function $x\mapsto A(x)$, an auxiliary function of the
associated de Branges space, which will be defined in the next section.

Remark. Notice from the above formula that the case of unitary symmetry $G=\rm
U$ has the same solution for all $\alpha$ since
$K(\alpha+x,\alpha-x)=\frac{\sin 2\pi\Delta x}{2\pi x}$ (from this we can also
see that
$\mathbb{A}({\mathrm{U},\pi\Delta,\alpha})\equiv\frac{1}{4\Delta^{2}})$. One
can reach the same conclusion by observing that the density $W_{\rm
U}(x)\equiv 1$ is translation invariant.

We illustrate the bounds provided by the above theorems in Figure 1, where,
for the five symmetry groups and $\Delta\in\\{1,\frac{4}{3},\frac{3}{2},2\\}$,
we graph the functions $\alpha\mapsto\sqrt{\mathbb{A}(G,\pi\Delta,\alpha)}$
using the explicit reproducing kernels computed in [9]. Our choice of values
of $\Delta$ represented in the figure is inspired by one-level density results
present in the literature. For instance, the original result of Hughes and
Rudnick for Dirichlet $L$-functions was proven for $\Delta=2$ [23, Theorem
3.1]. In [24], Iwaniec, Luo, and Sarnak establish such results in [24,
Theorems 1.1 and 1.3] for certain orthogonal families when $\Delta=2$, and in
[24, Theorem 1.5] for a particular symplectic family when
$\Delta=\frac{3}{2}$, among other things. As a last example, Fouvry and
Iwaniec [18] considered the family of $L$-functions $L(s,\Psi)$ where $\Psi$
runs over the characters of the ideal class group $\mathcal{C}\ell(K)$ of the
imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D})$, with $D>3$ a square-free
number congruent to $3$ modulo $4$. They showed (1.4) holds for this family
when $\Delta=1$ with $G=\rm Sp$ in their [18, Theorem 1.1] and, by taking a
certain average over $D$, extended it to $\Delta=\frac{4}{3}$ in [18, Theorem
1.2].

Finally, a qualitative behavior that can be inferred from the graphs of Figure
1 is that as $\alpha$ grows large the values $\mathbb{A}(G,\pi\Delta,\alpha)$
get closer and closer to the constant value
$\mathbb{A}({\mathrm{U},\pi\Delta,0})=\frac{1}{4\Delta^{2}}$. Our next result
shows that this is indeed the case.

######  Theorem 3.

For all $G\in\\{\rm U,Sp,O,SO(even),SO(odd)\\}$ we have

$\lim_{|\alpha|\to\infty}\mathbb{A}(G,\pi\Delta,\alpha)=\mathbb{A}({\mathrm{U},\pi\Delta,0})=\frac{1}{4\Delta^{2}}.$

We prove this in Section 3. To show this limit holds we work from the
expression of the densities (1.5) and, through this proof, illustrate the
relevance of Fourier uncertainty principles and of the reproducing kernel
structure of the spaces under investigation.

(a) $\Delta=1.$

(b) $\Delta=\frac{4}{3}$.

(c) $\Delta=\frac{3}{2}.$

(d) $\Delta=2$.

Figure 1. Values of $\sqrt{\mathbb{A}(G,\pi\Delta,\alpha)}$ as $\alpha$
varies, for different symmetry groups and different values of $\Delta$. Note
$K_{\rm O,\pi\Delta}=K_{\rm SO(even),\pi\Delta}=K_{\rm SO(odd),\pi\Delta}$
when $\Delta=1$.

###  1.3. An extremal problem in de Branges spaces

We connect the problem of determining the sharp constants
$\mathbb{A}(G,\pi\Delta,\alpha)$ to the theory of de Branges spaces. Theorem 2
is a consequence of the more general Theorem 4, which is about sharp
inequalities related to the operators of multiplication by $(z-\alpha)^{k}$ in
a de Branges space, with $k\in\mathbb{N}$ and $\alpha\in\mathbb{R}$. This
theorem addresses an extremal problem which is a generalization of the one
studied by Carneiro et al in [10, Section 8], who take up the case $\alpha=0$.
In fact, this discussion parallels that of [10] in many respects. However, as
mentioned previously, the broken symmetry introduces new challenges to
consider.

We first recall some basic notions from this theory. See de Branges’s
monograph [6] for a more detailed exposition. We use the notation
$f^{*}(z)=\overline{f(\bar{z})}$ throughout. A function that is analytic in
the upper half-plane $\mathbb{C}^{+}$ is said to be of _bounded type_ if it
can be expressed as the quotient of two bounded analytic functions in
$\mathbb{C}^{+}$. For such a function $f$, one has [6, Theorems 9 and 10]

$v(f):=\limsup_{y\rightarrow+\infty}\,{y}^{-1}\,{\log|f(iy)|}<\infty,$

and we call $v(f)$ the mean type of $f$.

Let $E:\mathbb{C}\rightarrow\mathbb{C}$ be a _Hermite-Biehler_ function, that
is, an entire function satisfying the inequality

$|E^{*}(z)|<|E(z)|$

for all $z\in\mathbb{C}^{+}$. The de Branges space $\mathcal{H}(E)$ associated
to $E$ is the space of entire functions $f:\mathbb{C}\rightarrow\mathbb{C}$
such that

$\|f\|_{\mathcal{H}(E)}^{2}:=\int_{-\infty}^{\infty}\left|\frac{f(x)}{E(x)}\right|^{2}\text{\rm
d}x<\infty,$

and such that $f/E$ and $f^{*}/E$ are of bounded type with non-positive mean
type. This space forms a reproducing kernel Hilbert space with the inner
product

$\langle
f,g\rangle_{\mathcal{H}(E)}:=\int_{-\infty}^{\infty}\frac{f(x)\overline{g(x)}}{|E(x)|^{2}}\,\text{\rm
d}x.$

To every such $E$ there exist unique real entire functions222We say an entire
function $F:\mathbb{C}\to\mathbb{C}$ is real entire if its restriction to
$\mathbb{R}$ is real-valued. Equivalently, $F(z)=F^{*}(z)$.
$A:\mathbb{C}\rightarrow\mathbb{C}$ and $B:\mathbb{C}\rightarrow\mathbb{C}$
such that $E(z)=A(z)-iB(z)$. These companion functions must be

$A(z)=\frac{1}{2}\left(E(z)+E^{*}(z)\right)\text{ and
}B(z)=\frac{i}{2}\left(E(z)-E^{*}(z)\right),$

and we can write the reproducing kernels for these spaces in terms of these
functions. For every $w\in\mathbb{C}$, the corresponding reproducing kernel is
given by (see [6, Theorem 19])

$K(w,z)=\frac{B(z)A(\overline{w})-A(z)B(\overline{w})}{\pi(z-\overline{w})},$
(1.8)

and, when $w=\bar{z}$,

$K(\bar{z},z)=\frac{1}{\pi}\left(B^{\prime}(z)A(z)-A^{\prime}(z)B(z)\right).$

A classical example is the Paley-Wiener space $\mathcal{H}_{\rm U,\pi\Delta}$,
which is the space $\mathcal{H}(E)$ when $E(z)=e^{-i\pi\Delta z}$. In this
case the auxiliary functions are $A(z)=\cos\pi\Delta z$ and
$B(z)=\sin\pi\Delta z$.

We will be considering the following extremal problem, a generalization of the
one defined in (1.7).

_Extremal Problem_ : Fix $k\in\mathbb{N}$ and let
$\mathcal{X}_{k}:=\\{f\in\mathcal{H}(E):z^{k}f\in\mathcal{H}(E)\\}$. Find for
$\alpha\in\mathbb{R}$

$(\mathbb{EP})(E,k,\alpha):=\inf_{0\neq
f\in\mathcal{X}_{k}}\frac{\|(z-\alpha)^{k}f\|_{\mathcal{H}(E)}^{2}}{\|f\|_{\mathcal{H}(E)}^{2}}.$
(1.9)

The fact that we can determine the solutions of equations of the form
$(z-\alpha)^{k}=\eta$ for $\eta>0$ allows us to adapt the argument of [10] to
pin down a precise a value of $(\mathbb{EP})(E,k,\alpha)$ when the function
$E$ satisfies the following hypotheses:

  1. (C1)

$E$ has no real zeros;

  2. (C2)

the function $z\mapsto E(iz)$ is real entire;

  3. (C3)

$A\notin\mathcal{H}(E)$.

Condition (C2) is equivalent to the fact that the auxiliary functions $A$ and
$B$ are even and odd, respectively, while conditions (C1) and (C3) allow for
the use of the following interpolation formula in de Branges spaces:

$f(z)=\sum_{A(\xi)=0}\frac{f(\xi)}{K(\xi,\xi)}K(\xi,z)$ (1.10)

for all $f\in\mathcal{H}(E)$, with the sum above converging both uniformly in
compact sets and in the norm of $\mathcal{H}(E)$. From (C1) and (C3), we also
have Parseval’s identity

$\|f\|_{\mathcal{H}(E)}^{2}=\sum_{A(\xi)=0}\frac{|f(\xi)|^{2}}{K(\xi,\xi)}.$
(1.11)

For these results consult [6, Theorem 22].

Let

$0<\xi_{1}<\xi_{2}<\ldots$

denote the sequence of positive zeros of $A$ and define the meromorphic
function

$C(z):=\frac{B(z)}{A(z)}.$

We can obtain the following result concerning the extremal problem defined in
(1.9).

######  Theorem 4.

Suppose $E$ is a Hermite-Biehler function satisfying (C1), (C2), (C3) and that
$\mathcal{X}_{k}\neq\\{0\\}$. Setting
$\lambda_{0}:=\left((\mathbb{EP})(E,k,\alpha)\right)^{1/2k}$, we have that
$\lambda_{0}$ is the first positive zero of the equation

$A(\alpha+\lambda)A(\alpha-\lambda)\det\mathcal{V}(\lambda)=0,$ (1.12)

where $\mathcal{V}(\lambda)$ is the $k\times k$ matrix with entries

$\big{(}\mathcal{V}(\lambda)\big{)}_{lj}=\sum_{r=0}^{2k-1}\omega^{-r(l+j-1)}C(\alpha+\omega^{r}\lambda)\qquad(1\leq
l,j\leq k)$

and $\omega:=e^{i\pi/k}$.

We prove the above theorem in Section 4. As in [10], we convert the problem in
$\mathcal{H}(E)$ to a problem in an appropriate space of sequences, which is
Lemma 6. Then we discuss how Theorem 2 may be seen as its consequence in
Corollary 7.

##  2\. Proof of Theorem 1: Existence of Zeros in Low-Lying Intervals

Let us recall briefly the idea of Hughes and Rudnick to find the smallest
interval of the central point for which there exists a zero in
$\\{L(s,f),\,f\in\mathcal{F}\\}$. They construct an even Schwartz function
$\phi$ with ${\rm supp}\,\widehat{\phi}\subseteq(-\Delta,\Delta)$, which
satisfies the properties

  1. (i)

$\phi(x)\leq 0$ for $|x|\leq\beta$,

  2. (ii)

$\phi(x)\geq 0$ for $|x|>\beta$

for a given $\beta>0$ and apply the one-level density (1.4) to it. Such a
$\phi$ is constructed by taking a non-negative $g$ Schwartz with ${\rm
supp}\,\widehat{g}\subseteq(-\Delta,\Delta)$ and setting
$\phi(x)=(x^{2}-\beta^{2})\,g(x)$.

The key insight here is that if we choose an appropriate $\beta>0$, we can
find a $g$ for which

$\int_{\mathbb{R}}\phi(x)\,W_{G}(x)\,\text{\rm
d}x=\int_{\mathbb{R}}(x^{2}-\beta^{2})\,g(x)\,W_{G}(x)\,\text{\rm d}x<0$ (2.1)

and this implies by (1.4) and properties (i) and (ii) that for large values of
$\mathcal{Q}$ the family $\\{L(s,f),\,f\in\mathcal{F}(\mathcal{Q})\\}$ has a
zero in the interval $[-\beta,\beta]$. Of course, it is in our interest to
make the value of $\beta$ as small as possible, and this is where the extremal
problem in an appropriate function space arises, for (2.1) is equivalent to

$\frac{\int_{\mathbb{R}}x^{2}\,g(x)\,W_{G}(x)\,\text{\rm
d}x}{\int_{\mathbb{R}}g(x)\,W_{G}(x)\,\text{\rm d}x}<\beta^{2}.$ (2.2)

From here, we can get to the formulation contemplated in Theorem B. Indeed,
Krein’s decomposition theorem states that non-negative entire functions of
exponential type at most $2\pi\Delta$ can be decomposed into
$g(x)=|f(x)|^{2}$, with $f$ entire of exponential type at most $\pi\Delta$
(see [1, p. 154]), so that inequality (2.2) becomes

$\frac{\int_{\mathbb{R}}x^{2}\,|f(x)|^{2}\,W_{G}(x)\,\text{\rm
d}x}{\int_{\mathbb{R}}|f(x)|^{2}\,W_{G}(x)\,\text{\rm d}x}<\beta^{2}$

and, in trying to minimize $\beta$, we are led to consider the extremal
problem

$\inf_{\begin{subarray}{c}{0\neq f\in\mathcal{H}_{\pi\Delta}}\\\
{f\in\mathcal{S}(\mathbb{R})}\end{subarray}}\frac{\int_{\mathbb{R}}|xf(x)|^{2}\,W_{G}(x)\,\text{\rm
d}x}{\int_{\mathbb{R}}|f(x)|^{2}\,W_{G}(x)\,\text{\rm d}x}.$

By the density of functions of exponential type at most $\pi\Delta$ which are
Schwartz on the real line, we can take the infimum above over the whole space
$\mathcal{H}_{\pi\Delta}$ (see [9, proof of Theorem 2]).

Now one would try to construct, for an arbitrary $\alpha\in\mathbb{R}$, an
even Schwartz function $\phi_{\alpha}$ with ${\rm
supp}\,\widehat{\phi}_{\alpha}\subseteq(-\Delta,\Delta)$ satisfying

  1. (i’)

$\phi_{\alpha}(x)\leq 0$ for $|x-\alpha|\leq\beta$;

  2. (ii’)

$\phi_{\alpha}(x)\geq 0$ for $|x-\alpha|>\beta$,

in the same way. However, given the fact $\phi_{\alpha}$ must be even, the
above properties cannot hold simultaneously when $\alpha\neq 0$. A more
symmetric set of conditions would be

  1. (I)

$\phi_{\alpha}(x)\leq 0$ for
$x\in[\alpha-\beta,\alpha+\beta]\cup[-\alpha-\beta,-\alpha+\beta]$,

  2. (II)

$\phi_{\alpha}(x)\geq 0$ for
$x\notin[\alpha-\beta,\alpha+\beta]\cup[-\alpha-\beta,-\alpha+\beta]$.

Trying to mimic the previous construction for $\alpha\neq 0$ would give us the
candidate $((x-\alpha)^{2}-\beta^{2})((x+\alpha)^{2}-\beta^{2})\,g(x)$, for a
non-negative even Schwartz function $g$ with ${\rm
supp}\,\widehat{g}\subseteq(-\Delta,\Delta)$. This leads to an overly
complicated extremal problem. A simple observation allows us to fix this. On
second inspection, one may notice that property (i), and its analogous (I),
are in fact superfluous to conclude the existence of a zero. Because if (2.1)
holds, only the fact that $\phi(x)\geq 0$ when
$x\notin[\alpha-\beta,\alpha+\beta]\cup[-\alpha-\beta,-\alpha+\beta]$, imply
there must be a zero in the family of $L$-functions in
$[\alpha-\beta,\alpha+\beta]\cup[-\alpha-\beta,-\alpha+\beta]$ given that the
integral is negative.

We may thus propose the following: define
$\varphi_{\alpha}(x)=((x-\alpha)^{2}-\beta^{2}))\,g(x)$, with $g$ non-
negative, Schwartz and with ${\rm
supp}\,\widehat{g}\subseteq(-\Delta,\Delta)$. We do not suppose $g$ is even
this time, but instead take our candidate to be the symmetrized version of
$\varphi_{\alpha}$

$\phi_{\alpha}(x)=\frac{\varphi_{\alpha}(x)+\varphi_{\alpha}(-x)}{2}.$

The function $\phi_{\alpha}$ is, of course, Schwartz with ${\rm
supp}\,\widehat{\phi_{\alpha}}\subseteq(-\Delta,\Delta)$ and satisfies
property (II). We can apply the one-level density, and if we obtained again
that

$\lim_{\mathcal{Q}\rightarrow\infty}\frac{1}{|\mathcal{F}(\mathcal{Q})|}\sum_{f\in\mathcal{F}(\mathcal{Q})}\sum_{\gamma_{f}}\phi_{\alpha}\left(\frac{\gamma_{f}\log
c_{f}}{2\pi}\right)=\int_{\mathbb{R}}\phi_{\alpha}(x)\,W_{G}(x)\,\text{\rm
d}x<0,$ (2.3)

it would imply together with the properties (1.3) that

$\limsup_{\mathcal{Q}\rightarrow\infty}\,\min_{\begin{subarray}{c}{\gamma_{f}}\\\
{f\in\mathcal{F}(Q)}\end{subarray}}\left|\frac{\gamma_{f}\log
c_{f}}{2\pi}-\alpha\right|\leq\beta.$ (2.4)

Let us rewrite the right-hand side of (2.3):

$\displaystyle 0$
$\displaystyle>\int_{\mathbb{R}}\phi_{\alpha}(x)\,W_{G}(x)\,\text{\rm
d}x=\int_{\mathbb{R}}\frac{\varphi_{\alpha}(x)+\varphi_{\alpha}(-x)}{2}\,W_{G}(x)\,\text{\rm
d}x$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}\varphi_{\alpha}(x)\,W_{G}(x)\,\text{\rm
d}x+\frac{1}{2}\int_{\mathbb{R}}\varphi_{\alpha}(-x)\,W_{G}(x)\,\text{\rm d}x$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}\varphi_{\alpha}(x)\,W_{G}(x)\,\text{\rm
d}x+\frac{1}{2}\int_{\mathbb{R}}\varphi_{\alpha}(x)\,W_{G}(-x)\,\text{\rm d}x$
$\displaystyle=\int_{\mathbb{R}}\varphi_{\alpha}(x)\,W_{G}(x)\,\text{\rm d}x$
$\displaystyle=\int_{\mathbb{R}}((x-\alpha)^{2}-\beta^{2}))\,g(x)\,W_{G}(x)\,\text{\rm
d}x,$

by the evenness of the density $W_{G}(x)$. This, in turn, is equivalent to

$\frac{\int_{\mathbb{R}}(x-\alpha)^{2}\,g(x)\,W_{G}(x)\,\text{\rm
d}x}{\int_{\mathbb{R}}g(x)\,W_{G}(x)\,\text{\rm d}x}<\beta^{2},$

so that, again, by Krein’s decomposition and the density of Schwartz
functions, we obtain the relation to our desired extremal problem given by
(1.7). Combining this observation with (2.4), we arrive at the content of
Theorem 1.

##  3\. Proof of Theorem 3: the Limit of $\mathbb{A}(G,\pi\Delta,\alpha)$ as
$|\alpha|\to\infty$

First notice that $f(z)\in\mathcal{H}_{\pi\Delta}$ if and only if
$f(z-\alpha)\in\mathcal{H}_{\pi\Delta}$, so by a simple change of variables
(1.7) can be rewritten to

$\mathbb{A}(G,\pi\Delta,\alpha)=\inf_{0\neq
f\in\mathcal{H}_{\pi\Delta}}\frac{\int_{\mathbb{R}}|xf(x)|^{2}\,W_{G}(x+\alpha)\,\text{\rm
d}x}{\int_{\mathbb{R}}|f(x)|^{2}\,W_{G}(x+\alpha)\,\text{\rm d}x}.$

Moreover, by the evenness of $W_{G}$, we may consider $\alpha>0$ without loss
of generality. We may also write a general formula for the densities (1.5) in
the compact way

$W_{G}(x)=1+\gamma\,\frac{\sin 2\pi x}{2\pi
x}+\eta\,{\boldsymbol{\delta}}_{0}(x),$

with $\gamma\in\\{-1,0,1\\}$ and $\eta\in\\{0,1/2,1\\}$.

Fixing any non-zero $f\in\mathcal{H}_{\pi\Delta}$ such that
$zf(z)\in\mathcal{H}_{\pi\Delta}$, we thus have for all $\alpha$

$\displaystyle\mathbb{A}(G,\pi\Delta,\alpha)\leq\frac{\int_{\mathbb{R}}|xf(x)|^{2}\,\text{\rm
d}x+\gamma\int_{\mathbb{R}}|xf(x)|^{2}\,\frac{\sin
2\pi(x+\alpha)}{2\pi(x+\alpha)}\,\text{\rm
d}x+\eta\,|\alpha\,f(-\alpha)|^{2}}{\int_{\mathbb{R}}|f(x)|^{2}\,\text{\rm
d}x+\gamma\int_{\mathbb{R}}|f(x)|^{2}\,\frac{\sin
2\pi(x+\alpha)}{2\pi(x+\alpha)}\,\text{\rm d}x+\eta\,|f(-\alpha)|^{2}}.$

By the Riemann-Lebesgue lemma, $xf(x)\to 0$ as $|x|\to\infty$ and, by the
dominated convergence theorem,

$\lim_{\alpha\to\infty}\int_{\mathbb{R}}|xf(x)|^{2}\,\frac{\sin
2\pi(x+\alpha)}{2\pi(x+\alpha)}\,\text{\rm
d}x=\lim_{\alpha\to\infty}\int_{\mathbb{R}}|f(x)|^{2}\,\frac{\sin
2\pi(x+\alpha)}{2\pi(x+\alpha)}\,\text{\rm d}x=0.$

Hence,

$\limsup_{\alpha\to\infty}\mathbb{A}(G,\pi\Delta,\alpha)\leq\frac{\int_{\mathbb{R}}|xf(x)|^{2}\,\text{\rm
d}x}{\int_{\mathbb{R}}|f(x)|^{2}\,\text{\rm d}x}$ (3.1)

for all non-zero $f\in\mathcal{H}_{\pi\Delta}$.

We note that

$\liminf_{\alpha\to\infty}\mathbb{A}(G,\pi\Delta,\alpha)>0,$ (3.2)

as a consequence of the equivalence of norms

$c_{1}\|F\|^{2}_{\mathcal{H}_{G,\pi\Delta}}\leq\|F\|^{2}_{\mathcal{H}_{\rm{U},\pi\Delta}}\leq
c_{2}\|F\|^{2}_{\mathcal{H}_{G,\pi\Delta}}$

and the fact that
$\mathbb{A}(\mathrm{U},\pi\Delta,\alpha)\equiv\frac{1}{4\Delta^{2}}$. For a
sequence $\alpha_{n}\to\infty$ such that

$\liminf_{\alpha\to\infty}\mathbb{A}(G,\pi\Delta,\alpha)=\lim_{n\to\infty}\mathbb{A}(G,\pi\Delta,\alpha_{n}),$

we can associate a sequence of extremizers for
$\mathbb{A}(G,\pi\Delta,\alpha_{n})$ denoted by
$f_{n}\in\mathcal{H}_{\pi\Delta}$. We postpone the proof of the existence of
such extremizers to the next section (see Proposition 5), but in fact this
proof works just as well taking near-extremizers instead. We further normalize
these functions to have

$\int_{\mathbb{R}}|xf_{n}(x)|^{2}\,\text{\rm
d}x+\gamma\int_{\mathbb{R}}|xf_{n}(x)|^{2}\,\frac{\sin
2\pi(x+\alpha_{n})}{2\pi(x+\alpha_{n})}\,\text{\rm
d}x+\eta\,|\alpha_{n}\,f_{n}(-\alpha_{n})|^{2}=1.$ (3.3)

By the uncertainty principle of Nazarov333The uncertainty principle of Nazarov
states that there exists a universal constant $C$ such that if
$A,B\subseteq\mathbb{R}$ are sets of finite Lebesgue measure and $F\in
L^{2}(\mathbb{R})$, then $\int_{\mathbb{R}}|F(x)|^{2}\,\text{\rm d}x\leq
Ce^{C|A||B|}\left(\int_{\mathbb{R}\setminus A}|F(x)|^{2}\,\text{\rm
d}x+\int_{\mathbb{R}\setminus B}|\widehat{F}(y)|^{2}\,\text{\rm d}y\right)$.
[28], $\\{zf_{n}\\}$ is a uniformly bounded sequence in the Hilbert space
$\mathcal{H}_{\rm U,\pi\Delta}$, since it implies

$\displaystyle\int_{\mathbb{R}}|xf_{n}(x)|^{2}\text{\rm d}x$
$\displaystyle\ll\int_{|x+\alpha_{n}|\geq 1}|xf_{n}(x)|^{2}\text{\rm d}x$
$\displaystyle\ll\int_{|x+\alpha_{n}|\geq
1}|xf_{n}(x)|^{2}\bigg{\\{}1+\gamma\,\frac{\sin
2\pi(x+\alpha_{n})}{2\pi(x+\alpha_{n})}\bigg{\\}}\text{\rm d}x\leq 1,$ (3.4)

where the implicit constants are uniform on $n$. Moreover, having in mind
(3.2) and

$\int_{\mathbb{R}}|f_{n}(x)|^{2}\,\text{\rm
d}x+\gamma\int_{\mathbb{R}}|f_{n}(x)|^{2}\,\frac{\sin
2\pi(x+\alpha_{n})}{2\pi(x+\alpha_{n})}\,\text{\rm
d}x+\eta\,|\,f_{n}(-\alpha_{n})|^{2}=\frac{1}{\mathbb{A}(G,\pi\Delta,\alpha_{n})},$

we obtain that $\\{f_{n}\\}$ is uniformly bounded by the same reasoning. Thus,
we have the weak convergences in $\mathcal{H}_{\rm U,\pi\Delta}$

$\begin{cases}f_{n}\rightharpoonup f\\\ zf_{n}\rightharpoonup
g\end{cases}\text{ as }n\to\infty,$

which are also pointwise convergences because evaluation functionals are
continuous in this space. In particular, it must be the case that
$g(z)=zf(z)$.

Fix $R>0$ and
$0<r<\limsup_{\alpha\to\infty}\mathbb{A}(G,\pi\Delta,\alpha)^{-1}$. For $n$
big enough,

$\displaystyle
r\leq\int_{\mathbb{R}}|f_{n}(x)|^{2}\bigg{\\{}1+\gamma\,\frac{\sin
2\pi(x+\alpha_{n})}{2\pi(x+\alpha_{n})}+\eta\,{\boldsymbol{\delta}}_{0}(x+\alpha_{n})\bigg{\\}}\text{\rm
d}x$
$\displaystyle\leq\int_{-R}^{R}|f_{n}(x)|^{2}\bigg{\\{}1+\gamma\,\frac{\sin
2\pi(x+\alpha_{n})}{2\pi(x+\alpha_{n})}\bigg{\\}}\text{\rm
d}x+\int_{|x|>R}|f_{n}(x)|^{2}\bigg{\\{}1+\gamma\,\frac{\sin
2\pi(x+\alpha_{n})}{2\pi(x+\alpha_{n})}\bigg{\\}}\text{\rm
d}x+|f_{n}(-\alpha_{n})|^{2}$
$\displaystyle\leq\int_{-R}^{R}|f_{n}(x)|^{2}\bigg{\\{}1+\gamma\,\frac{\sin
2\pi(x+\alpha_{n})}{2\pi(x+\alpha_{n})}\bigg{\\}}\text{\rm
d}x+\frac{C}{R}+|f_{n}(-\alpha_{n})|^{2},$ (3.5)

where from the second to the last line we used (3.4). But by the reproducing
kernel property of $\mathcal{H}_{\rm U,\pi\Delta}$ and Cauchy-Schwarz,

$|xf_{n}(x)|=\left|\int_{\mathbb{R}}tf_{n}(t)\frac{\sin\pi\Delta(t-x)}{\pi(t-x)}\text{\rm
d}t\right|\leq\left(\int_{\mathbb{R}}|tf_{n}(t)|^{2}\,\text{\rm
d}t\right)^{1/2}\left(\int_{\mathbb{R}}\left|\frac{\sin\pi\Delta(t-x)}{\pi(t-x)}\right|^{2}\,\text{\rm
d}t\right)^{1/2}\ll_{\Delta}1$

for all $x\in\mathbb{R}$, and so, for $|x|\geq 1$,

$|f_{n}(x)|\leq\frac{1}{|x|}|xf_{n}(x)|\ll\frac{1}{|x|},$

meaning $|f_{n}(-\alpha_{n})|^{2}=O\left(\frac{1}{\alpha_{n}^{2}}\right)$ as
$n\to\infty$. Sending $n\to\infty$ in (3.5),

$r\leq\int_{|x|\leq R}|f(x)|^{2}\,\text{\rm d}x+\frac{C}{R}.$

Now send $r\to\limsup_{\alpha\to\infty}\mathbb{A}(G,\pi\Delta,\alpha)^{-1}$
and $R\to\infty$ and rearrange to obtain

$\liminf_{\alpha\to\infty}\mathbb{A}(G,\pi\Delta,\alpha)\geq\frac{1}{\int_{\mathbb{R}}|f(x)|^{2}\,\text{\rm
d}x}.$

Thus, if we prove $\int_{\mathbb{R}}|xf(x)|^{2}\,\text{\rm d}x\leq 1$, we are
done, since we also have (3.1). But indeed, by Fatou’s lemma and (3.3),

$\int_{\mathbb{R}}|xf(x)|^{2}\,\text{\rm
d}x\leq\liminf_{n\to\infty}\int_{\mathbb{R}}|xf_{n}(x)|^{2}\bigg{\\{}1+\gamma\,\frac{\sin
2\pi(x+\alpha_{n})}{2\pi(x+\alpha_{n})}\bigg{\\}}\text{\rm d}x\leq 1.$

##  4\. The Extremal Problem in de Branges Spaces

We need a few preliminary results concerning the Extremal Problem defined in
(1.9) before embarking on the proof of Theorem 4.

######  Proposition 5.

Let $E$ be a Hermite-Biehler function and suppose
$\mathcal{X}_{k}=\\{f\in\mathcal{H}(E):z^{k}f\in\mathcal{H}(E)\\}\neq\\{0\\}$.
The following statements hold:

  1. (i)

$(\mathbb{EP})(E,k,\alpha)>0$.

  2. (ii)

There exist extremizers for $(\mathbb{EP})(E,k,\alpha)$.

###### Proof.

Essentially as found in [10], we include it here for the sake of completion.

First observe that $f\in\mathcal{X}_{k}$ if and only if
$(z-\alpha)^{k}f\in\mathcal{H}(E)$. This follows because multiplication by
$(z-\alpha)^{k}$ or $z^{k}$ does not change the mean type of a function and
the integrability condition is in essence the same. Indeed, outside a compact
set the function $|x-\alpha|^{2k}$ is of the order of $|x|^{2k}$, and vice
versa.

Note now that the function $x\mapsto K(x,x)|E(x)|^{-2}$ is continuous, and, in
particular, locally integrable. We can thus find $0<\eta<1$ for which

$\int_{-\eta+\alpha}^{\eta+\alpha}\frac{K(x,x)}{|E(x)|^{2}}\,\text{\rm
d}x\leq\frac{1}{2}.$

Thus, from the reproducing kernel property and Cauchy-Schwarz

$\displaystyle\int_{-\eta+\alpha}^{\eta+\alpha}\left|\frac{f(x)}{E(x)}\right|^{2}\,\text{\rm
d}x=\int_{-\eta+\alpha}^{\eta+\alpha}\left|\frac{\langle
f,K(x,\cdot)\rangle_{\mathcal{H}(E)}}{E(x)}\right|^{2}\,\text{\rm
d}x\leq\|f\|_{\mathcal{H}(E)}^{2}\int_{-\eta+\alpha}^{\eta+\alpha}\frac{K(x,x)}{|E(x)|^{2}}\,\text{\rm
d}x\leq\frac{\|f\|_{\mathcal{H}(E)}^{2}}{2}.$

So that

$\displaystyle\int_{\mathbb{R}}\left|\frac{(x-\alpha)^{k}f(x)}{E(x)}\right|^{2}\,\text{\rm
d}x\geq\int_{|x-\alpha|\geq\eta}\left|\frac{(x-\alpha)^{k}f(x)}{E(x)}\right|^{2}\,\text{\rm
d}x\geq\eta^{2k}\int_{|x-\alpha|\geq\eta}\left|\frac{f(x)}{E(x)}\right|^{2}\,\text{\rm
d}x\geq\eta^{2k}\frac{\|f\|_{\mathcal{H}(E)}^{2}}{2},$

which implies $(\mathbb{EP})(E,k,\alpha)\geq\frac{\eta^{2k}}{2}>0$.

To prove (ii), take a minimizing sequence $\\{f_{n}\\}\subseteq\mathcal{H}(E)$
normalized by $\|(z-\alpha)^{k}f_{n}\|_{\mathcal{H}(E)}^{2}=1$, implying

$\lim_{n\to\infty}\|f_{n}\|^{2}_{\mathcal{H}(E)}=(\mathbb{EP})(E,k,\alpha)^{-1}.$

This is a bounded sequence in $\mathcal{H}(E)$, so by the Banach–Alaoglu
theorem it converges weakly to a $g\in\mathcal{H}(E)$. Since evaluation
functionals are continuous, $f_{n}(z)\to g(z)$ for all $z\in\mathbb{C}$. So
that applying Fatou’s lemma

$\int_{\mathbb{R}}\bigg{|}\frac{(x-\alpha)^{k}g(x)}{E(x)}\bigg{|}^{2}\,\text{\rm
d}x\leq\lim_{n\to\infty}\int_{\mathbb{R}}\bigg{|}\frac{(x-\alpha)^{k}f_{n}(x)}{E(x)}\bigg{|}^{2}\,\text{\rm
d}x=1,$

verifying $g\in\mathcal{X}_{k}.$

If $0<r<(E,k,\alpha)^{-1}$, then for $n$ big enough and for $R>0$,

$\displaystyle r$
$\displaystyle\leq\int_{\mathbb{R}}\bigg{|}\frac{f_{n}(x)}{E(x)}\bigg{|}^{2}\,\text{\rm
d}x$
$\displaystyle=\int_{-R+\alpha}^{R+\alpha}\bigg{|}\frac{f_{n}(x)}{E(x)}\bigg{|}^{2}\,\text{\rm
d}x+\int_{|x-\alpha|\geq
R}\bigg{|}\frac{f_{n}(x)}{E(x)}\bigg{|}^{2}\,\text{\rm d}x$
$\displaystyle\leq\int_{-R+\alpha}^{R+\alpha}\bigg{|}\frac{f_{n}(x)}{E(x)}\bigg{|}^{2}\,\text{\rm
d}x+\frac{1}{R^{2k}}\int_{|x-\alpha|\geq
R}\bigg{|}\frac{(x-\alpha)^{k}f_{n}(x)}{E(x)}\bigg{|}^{2}\,\text{\rm d}x$
$\displaystyle\leq\int_{-R+\alpha}^{R+\alpha}\bigg{|}\frac{f_{n}(x)}{E(x)}\bigg{|}^{2}\,\text{\rm
d}x+\frac{1}{R^{2k}},$

by the normalization $\|(z-\alpha)^{k}f_{n}\|^{2}_{\mathcal{H}(E)}=1$. Sending
$n\to\infty$ and applying the dominated convergence theorem,

$r\leq\int_{-R+\alpha}^{R+\alpha}\bigg{|}\frac{g(x)}{E(x)}\bigg{|}^{2}\,\text{\rm
d}x+\frac{1}{R^{2k}}.$

Now we can send $R\to\infty$ and $r\to(E,k,\alpha)^{-1}$ to conclude $g$ is
the desired extremizer. ∎

Remark. The proof above is valid without assuming any of three conditions
(C1)–(C3) on the Hermite-Biehler function $E$. Hypotheses (C2) and (C3) are
clearly not being used, but even (C1) may be discarded, because possible real
zeros of $|E(x)|^{2}$ are cancelled out by those of $K(x,x)$, so that
$x\mapsto K(x,x)|E(x)|^{-2}$ is in any case continuous.

We now establish an equivalence between the extremal problem in
$\mathcal{H}(E)$ to a corresponding one in an appropriate space of sequences.

######  Lemma 6.

Let $E$ be a Hermite-Biehler function satisfying (C1) and (C3) such that
$\mathcal{X}_{k}\neq\\{0\\}$. Set $c_{n}:=-A^{\prime}(\xi_{n})/B(\xi_{n})$ and
let $X$ be the space of real-valued sequences $\\{a_{n}\\}_{\mathbb{Z}^{*}}$
such that

  1. (i)

$\displaystyle\sum_{\begin{subarray}{c}{n=-\infty}\\\ {n\neq
0}\end{subarray}}^{\infty}c_{n}a_{n}^{2}\xi_{n}^{2k}<\infty$;

  2. (ii)

$\displaystyle\sum_{\begin{subarray}{c}{n=-\infty}\\\ {n\neq
0}\end{subarray}}^{\infty}a_{n}(\xi_{n}-\alpha)^{k-\ell}=0\,\,\text{ for
}\,\,1\leq\ell\leq k$.

Then

$(\mathbb{EP})(E,k,\alpha)=\inf_{\\{a_{n}\\}\in
X\setminus\\{0\\}}\frac{\textstyle\sum_{n\in\mathbb{Z}^{*}}c_{n}a_{n}^{2}(\xi_{n}-\alpha)^{2k}}{\textstyle\sum_{n\in\mathbb{Z}^{*}}c_{n}a_{n}^{2}}.$
(4.1)

###### Proof.

We work with real-valued sequences because an extremizer for
$(\mathbb{EP})(E,k,\alpha)$ can be taken real entire. Indeed, we know by
Proposition 5 that an $f\in\mathcal{X}_{k}$ attaining
$(\mathbb{EP})(E,k,\alpha)$ exists, but we can further decompose
$f(z)=g(z)-ih(z)$, where $g,h\in\mathcal{X}_{k}$ are real entire functions, by
setting $g=(f+f^{*})/2$ and $h=i(f-f^{*})/2$ . One then has

$\frac{\|(z-\alpha)^{k}f\|_{\mathcal{H}(E)}^{2}}{\|f\|_{\mathcal{H}(E)}^{2}}=\frac{\|(z-\alpha)^{k}g\|_{\mathcal{H}(E)}^{2}+\|(z-\alpha)^{k}h\|_{\mathcal{H}(E)}^{2}}{\|g\|_{\mathcal{H}(E)}^{2}+\|h\|_{\mathcal{H}(E)}^{2}},$

which, from elementary considerations, implies $g$ and $h$ must also be
extremizers.

Given a non-identically zero $f\in\mathcal{X}_{k}$ that is real entire, let us
set $a_{n}:=f(\xi_{n})/A^{\prime}(\xi_{n})$. By (C1) and (C3), we may use the
interpolation formula and Parseval’s identity in $\mathcal{H}(E)$ given
respectively by (1.10) and (1.11). In particular, by (1.11) we have

$\|f\|_{\mathcal{H}(E)}^{2}=\pi\sum_{\begin{subarray}{c}{n=-\infty}\\\ {n\neq
0}\end{subarray}}^{\infty}c_{n}a_{n}^{2}\quad\text{and
}\quad\|(z-\alpha)^{k}f\|_{\mathcal{H}(E)}^{2}=\pi\sum_{\begin{subarray}{c}{n=-\infty}\\\
{n\neq 0}\end{subarray}}^{\infty}c_{n}a_{n}^{2}(\xi_{n}-\alpha)^{2k}.$

The fact that the norm $\|(z-\alpha)^{k}f\|_{\mathcal{H}(E)}^{2}$ is finite
implies the sequence $\\{a_{n}\\}$ satisfies condition (i). And indeed we have

$\frac{\|(z-\alpha)^{k}f\|_{\mathcal{H}(E)}^{2}}{\|f\|_{\mathcal{H}(E)}^{2}}=\frac{\textstyle\sum_{n\in\mathbb{Z}^{*}}c_{n}a_{n}^{2}(\xi_{n}-\alpha)^{2k}}{\textstyle\sum_{n\in\mathbb{Z}^{*}}c_{n}a_{n}^{2}}.$
(4.2)

Further, we have that $f\in\mathcal{X}_{k}$ if and only if
$g_{\ell}(z):=(z-\alpha)^{k-\ell+1}f(z)\in\mathcal{H}(E)$ for all
$1\leq\ell\leq k$, which gives the constraints

$g_{1}(\alpha)=\ldots=g_{k}(\alpha)=0,$

and this implies item (ii). In fact, if $\alpha$ is not a zero of $A$, then by
(1.10)

$0=g_{\ell}(\alpha)=\sum_{\begin{subarray}{c}{n=-\infty}\\\ {n\neq
0}\end{subarray}}^{\infty}\frac{g_{\ell}(\xi_{n})}{K(\xi_{n},\xi_{n})}K(\xi_{n},\alpha)=\sum_{\begin{subarray}{c}{n=-\infty}\\\
{n\neq
0}\end{subarray}}^{\infty}\frac{g_{\ell}(\xi_{n})}{A^{\prime}(\xi_{n})}\frac{A(\alpha)}{\alpha-\xi_{n}}=-\sum_{\begin{subarray}{c}{n=-\infty}\\\
{n\neq
0}\end{subarray}}^{\infty}\frac{f(\xi_{n})}{A^{\prime}(\xi_{n})}(\xi_{n}-\alpha)^{k-\ell}A(\alpha),$

and since $A(\alpha)\neq 0$, this is equivalent to

$\sum_{\begin{subarray}{c}{n=-\infty}\\\ {n\neq
0}\end{subarray}}^{\infty}a_{n}(\xi_{n}-\alpha)^{k-\ell}=0$ (4.3)

for all $1\leq\ell\leq k$. If $\alpha$ is a zero of $A$, then
$A^{\prime}(\alpha)\neq 0$ since by condition (C1) the zeros of $A$ are
simple. Moreover, for any $g\in\mathcal{H}(E)$ its representation (1.10)
converges uniformly in compact sets, which allows us to differentiate inside
the integral to obtain

$\displaystyle g^{\prime}(z)=\sum_{\begin{subarray}{c}{n=-\infty}\\\ {n\neq
0}\end{subarray}}^{\infty}\frac{g(\xi_{n})}{K(\xi_{n},\xi_{n})}\frac{\text{\rm
d}}{\text{\rm d}z}K(\xi_{n},z)=\sum_{\begin{subarray}{c}{n=-\infty}\\\ {n\neq
0}\end{subarray}}^{\infty}\frac{g(\xi_{n})}{A^{\prime}(\xi_{n})}\bigg{\\{}\frac{A^{\prime}(z)(z-\xi_{n})-A(z)}{(z-\xi_{n})^{2}}\bigg{\\}}.$

Applying the above formula to

$g^{\prime}_{\ell}(z)=g_{\ell+1}(z)+(z-\alpha)g^{\prime}_{\ell+1}(z)$

evaluated at $\alpha$, we obtain

$\displaystyle
g_{\ell+1}(\alpha)=g^{\prime}_{\ell}(\alpha)=-\frac{g_{\ell}(\alpha)}{A^{\prime}(\alpha)}\frac{A^{\prime\prime}(\alpha)}{2}+\sum_{\begin{subarray}{c}{n=-\infty}\\\
{n\neq 0}\\\
{\xi_{n}\neq\alpha}\end{subarray}}^{\infty}\frac{g_{\ell}(\xi_{n})}{A^{\prime}(\xi_{n})}\frac{A^{\prime}(\alpha)}{(\alpha-\xi_{n})}=-\sum_{\begin{subarray}{c}{n=-\infty}\\\
{n\neq 0}\\\
{\xi_{n}\neq\alpha}\end{subarray}}^{\infty}\frac{f(\xi_{n})}{A^{\prime}(\xi_{n})}(\xi_{n}-\alpha)^{k-\ell}A^{\prime}(\alpha)$

for $1\leq\ell\leq k$, which is again equivalent to (4.3), establishing that
$\\{a_{n}\\}\in X$.

Conversely, if one has a non-zero sequence $\\{a_{n}\\}\in X$, the summability
condition (i) allows us to construct $g\in\mathcal{H}(E)$ using (1.10) by
setting $g(\xi_{n})=(\xi_{n}-\alpha)^{k}a_{n}A^{\prime}(\xi_{n})$. Condition
(ii) in turn guarantees that $g$ has a zero of order at least $k$ at $\alpha$.
So taking $f(z)=(z-\alpha)^{-k}g(z)$, this entire function will occur in
$\mathcal{X}_{k}$, and (4.2) holds, concluding the lemma. ∎

Remark. In establishing the above equivalence, we use the interpolation
formula and Parseval’s identity with respect to the zeros of the $A$ function.
De Branges’s result [6, Theorem 22] actually gives these formulas with respect
to the zeros of _any_ of the functions $e^{i\beta}E(z)-e^{-i\beta}E^{*}(z)$
when (C1) holds and $0\leq\beta<\pi$ is such that
$e^{i\beta}E(z)-e^{-i\beta}E^{*}(z)\notin\mathcal{H}(E)$. Of course, we
specialize to the case $\beta=\pi/2$ here, but an analogous version of the
above lemma may be obtained for other values of $\beta$ when the condition
corresponding to (C3) holds. Of particular note is the case $\beta=0$, which
implies we may do the same things for the zeros of the $B$ function when
$B\notin\mathcal{H}(E)$.

###### Proof of Theorem 4.

Since the hypotheses also apply here, we can invoke Lemma 6. We essentially
solve the right-hand side of (4.1), except in a particular case we will single
out below. The coefficients $\\{c_{n}\\}$ and the space $X$ are the same as
before.

Now let $I:=\\{i_{1}<i_{2}<\ldots<i_{k}\\}$ be a set of indices such that
$\xi_{i_{1}}<\xi_{i_{2}}<\ldots<\xi_{i_{k}}$ are a choice of the $k$-th
closest zeros to $\alpha$. That is, for all $i\in I$ and $n\notin I$,
$|\xi_{n}-\alpha|\geq|\xi_{i}-\alpha|$. If $\\{a_{n}\\}\in X$, given the
constraints (4.3), we may write the following system of $k$ equations

$\displaystyle
a_{i_{1}}(\xi_{i_{1}}-\alpha)^{k-1}+\ldots+a_{i_{k}}(\xi_{i_{k}}-\alpha)^{k-1}=-\sum_{n\notin
I}a_{n}(\xi_{n}-\alpha)^{k-1},$ $\displaystyle
a_{i_{1}}(\xi_{i_{1}}-\alpha)^{k-2}+\ldots+a_{i_{k}}(\xi_{i_{k}}-\alpha)^{k-2}=-\sum_{n\notin
I}a_{n}(\xi_{n}-\alpha)^{k-2},$ $\displaystyle\ldots$ $\displaystyle
a_{i_{1}}+\ldots+a_{i_{k}}=-\sum_{n\notin I}a_{n},$

which in matrix notation becomes

$(a_{i_{1}},\ldots,a_{i_{k}})\,\mathcal{T}=(-S_{1},\ldots,-S_{k}),$

where $\mathcal{T}$ is the matrix given by
$\mathcal{T}_{mj}=(\xi_{i_{m}}-\alpha)^{k-j}$ for $1\leq m,j\leq k$ and
$S_{j}=\sum_{n\notin I}a_{n}(\xi_{n}-\alpha)^{k-j}$ for $1\leq j\leq k$
444Here we adopt the convention $0^{0}=1$ whenever it may occur.. In
particular,

$a_{i_{j}}=-\sum_{r=1}^{k}S_{r}(\mathcal{T}^{-1})_{rj}\ \ \text{ for }\ \
1\leq j\leq k.$ (4.4)

If $k\geq 2$, notice that for one $i^{*}\in I$, we have the strict inequality
$|\xi_{i^{*}}-\alpha|<|\xi_{\ell}-\alpha|$ for all $\ell\notin I$, since there
are at most two $\xi_{i}$ such that
$|\xi_{i}-\alpha|=\min_{A(\xi)=0}|\xi-\alpha|$. If $k=1$, suppose for now this
minimum is attained by exactly one of $\xi_{i_{1}}$ and $\xi_{i_{2}}$ ,
($\alpha$ can be the midpoint of two zeros of $A$, but we deal with this case
later,) which we call $\xi_{i^{*}}$ so that we also have
$|\xi_{i^{*}}-\alpha|<|\xi_{\ell}-\alpha|$ for all $\ell\notin I$.

Now for each $\ell\notin I$ construct the following sequence in $X$:

$a_{n}=\begin{cases}1,&{\rm if}\ \ n=\ell;\\\ 0\,,&{\rm if}\ \ n\notin
I\cup\\{\ell\\};\\\ -\sum_{r=1}^{k}S_{r}(\mathcal{T}^{-1})_{rj},&{\rm if}\ \
n=i_{j}\in I.\end{cases}$

implying that, by the remark in the previous paragraph and the definition of
$I$,

$\displaystyle\lambda_{0}^{2k}:=(\mathbb{EP})(E,k,\alpha)\leq\frac{\textstyle\sum_{n\in
I\cup\\{\ell\\}}c_{n}a_{n}^{2}(\xi_{n}-\alpha)^{2k}}{\textstyle\sum_{n\in
I\cup\\{\ell\\}}c_{n}a_{n}^{2}}<\frac{\textstyle\sum_{n\in
I\cup\\{\ell\\}}c_{n}a_{n}^{2}(\xi_{\ell}-\alpha)^{2k}}{\textstyle\sum_{n\in
I\cup\\{\ell\\}}c_{n}a_{n}^{2}}=(\xi_{\ell}-\alpha)^{2k}.$ (4.5)

Now define the functionals $F$ and $G$ on the space of sequences
$\\{a_{n}\\}_{n\notin I}$ satisfying $\sum_{n\notin
I}c_{n}a_{n}^{2}\xi_{n}^{2k}<\infty$ by

$F(\\{a_{n}\\})=\sum_{j=1}^{k}c_{i_{j}}\bigg{(}\sum_{r=1}^{k}S_{r}(\mathcal{T}^{-1})_{rj}\bigg{)}^{2}(\xi_{i_{j}}-\alpha)^{2k}+\sum_{n\notin
I}c_{n}a_{n}^{2}(\xi_{n}-\alpha)^{2k}$

and

$G(\\{a_{n}\\})=\sum_{j=1}^{k}c_{i_{j}}\bigg{(}\sum_{r=1}^{k}S_{r}(\mathcal{T}^{-1})_{rj}\bigg{)}^{2}+\sum_{n\notin
I}c_{n}a_{n}^{2},$

where again $S_{j}=\sum_{n\notin I}a_{n}(\xi_{n}-\alpha)^{k-j}$ for $1\leq
j\leq k$.

From Proposition 5 and Lemma 6, we know there is
$a^{*}=\\{a_{n}^{*}\\}_{n\notin I}$ extremizer to (4.1). If we perturb this
sequence and consider for small $\varepsilon>0$ and for $m\notin I$ the
function

$\varphi_{m}(\varepsilon)=\frac{F(a^{*}+\varepsilon
e^{m})}{G(a^{*}+\varepsilon e^{m})},$

where $e^{m}$ is the sequence with entries

$e^{m}_{n}=\begin{cases}1\text{ if }m=n;\\\ 0\text{ otherwise},\end{cases}$

then $\varphi_{m}$ is differentiable and from extremality we have
$\varphi_{m}^{\prime}(0)=0$, so the sequence must satisfy the Lagrange
multiplier equations

$\frac{\partial F}{\partial
a_{m}}\left(a^{*}\right)=\lambda_{0}^{2k}\frac{\partial G}{\partial
a_{m}}\left(a^{*}\right)\,\text{for }m\notin I,$

or, equivalenty, defining $(a^{*}_{i_{1}},\ldots,a^{*}_{i_{k}})$ according to
(4.4),

$c_{m}a_{m}^{*}\left((\xi_{m}-\alpha)^{2k}-\lambda_{0}^{2k}\right)=\sum_{j=1}^{k}c_{i_{j}}a_{i_{j}}^{*}\bigg{(}\sum_{r=1}^{k}(\xi_{m}-\alpha)^{k-r}(\mathcal{T}^{-1})_{rj}\bigg{)}\left((\xi_{i_{j}}-\alpha)^{2k}-\lambda_{0}^{2k}\right)\,\text{for
}m\notin I.$ (4.6)

By (4.5), we can divide both sides by
$c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda_{0}^{2k}\right)$ to say

$a_{m}^{*}=\frac{1}{c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda_{0}^{2k}\right)}\sum_{j=1}^{k}c_{i_{j}}a_{i_{j}}^{*}\bigg{(}\sum_{r=1}^{k}(\xi_{m}-\alpha)^{k-r}(\mathcal{T}^{-1})_{rj}\bigg{)}\left((\xi_{i_{j}}-\alpha)^{2k}-\lambda_{0}^{2k}\right)\,\text{for
}m\notin I,$ (4.7)

multiply by $(\xi_{m}-\alpha)^{k-\ell}$, with $1\leq\ell\leq k$, and sum over
$m\notin I$ to get

$S^{*}_{\ell}=\sum_{m\notin
I}a_{m}^{*}(\xi_{m}-\alpha)^{k-\ell}=\sum_{j=1}^{k}a^{*}_{i_{j}}\sum_{m\notin
I}\frac{c_{i_{j}}\bigg{(}\sum_{r=1}^{k}(\xi-\alpha)_{m}^{k-r}(\mathcal{T}^{-1})_{rj}\bigg{)}\left((\xi_{i_{j}}-\alpha)^{2k}-\lambda_{0}^{2k}\right)}{c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda_{0}^{2k}\right)}(\xi_{m}-\alpha)^{k-\ell}.$

Recalling that (4.4) tells us

$S^{*}_{\ell}=-\sum_{j=1}^{k}a_{i_{j}}^{*}\mathcal{T}_{j\ell}=-\sum_{j=1}^{k}a_{i_{j}}^{*}(\xi_{i_{j}}-\alpha)^{k-\ell},$

we can subtract the two expressions for $S^{*}_{\ell}$ to obtain the equality

$\sum_{j=1}^{k}a^{*}_{i_{j}}\left[\sum_{m\notin
I}\frac{c_{i_{j}}\bigg{(}\sum_{r=1}^{k}(\xi_{m}-\alpha)^{k-r}(\mathcal{T}^{-1})_{rj}\bigg{)}\left((\xi_{i_{j}}-\alpha)^{2k}-\lambda_{0}^{2k}\right)}{c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda_{0}^{2k}\right)}(\xi_{m}-\alpha)^{k-\ell}+(\xi_{i_{j}}-\alpha)^{k-\ell}\right]=0$
(4.8)

for each $1\leq\ell\leq k$. This can be equivalently stated in terms of matrix
notation as saying that
$(a^{*}_{i_{1}},\ldots,a^{*}_{i_{k}})\in\mathrm{ker}\,\mathcal{W}(\lambda_{0})$
where, for a generic complex $\lambda$, the $k\times k$ matrix
$\mathcal{W}(\lambda)$ is given by

$\displaystyle\big{(}\mathcal{W}(\lambda)\big{)}_{j\ell}$
$\displaystyle=\sum_{m\notin
I}\frac{c_{i_{j}}\bigg{(}\sum_{r=1}^{k}(\xi_{m}-\alpha)^{k-r}(\mathcal{T}^{-1})_{rj}\bigg{)}\left((\xi_{i_{j}}-\alpha)^{2k}-\lambda^{2k}\right)}{c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda^{2k}\right)}(\xi_{m}-\alpha)^{k-\ell}+(\xi_{i_{j}}-\alpha)^{k-\ell}$
(4.9)
$\displaystyle=c_{i_{j}}\left((\xi_{i_{j}}-\alpha)^{2k}-\lambda^{2k}\right)\sum_{\begin{subarray}{c}{m=-\infty}\\\
{m\neq
0}\end{subarray}}^{\infty}\frac{\bigg{(}\sum_{r=1}^{k}(\xi_{m}-\alpha)^{k-r}(\mathcal{T}^{-1})_{rj}\bigg{)}}{c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda^{2k}\right)}(\xi_{m}-\alpha)^{k-\ell}$
$\displaystyle=c_{i_{j}}\left((\xi_{i_{j}}-\alpha)^{2k}-\lambda^{2k}\right)\left((\mathcal{T}^{-1})^{\top}\mathcal{Q}(\lambda)\right)_{j\ell},$

where we have used that

$\sum_{r=1}^{k}(\xi_{i_{n}}-\alpha)^{k-r}(\mathcal{T}^{-1})_{rj}=\begin{cases}1\text{
if }n=j,\\\ 0\text{ if }n\neq j\end{cases}$

for $1\leq n\leq k$, and $\mathcal{Q}(\lambda)$ is the $k\times k$ matrix with
entries

$\big{(}\mathcal{Q}(\lambda)\big{)}_{n\ell}=\sum_{\begin{subarray}{c}{m=-\infty}\\\
{m\neq
0}\end{subarray}}^{\infty}\frac{(\xi_{m}-\alpha)^{2k-n-\ell}}{c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda^{2k}\right)}$
(4.10)

for $1\leq n,\ell\leq k$.

If instead $0<\lambda^{*}<\min_{\ell\notin I}|\xi_{\ell}-\alpha|$ satisfies
$\det\mathcal{W}(\lambda^{*})=0$, pick
$(a^{*}_{i_{1}},\ldots,a^{*}_{i_{k}})\in\mathrm{ker}\,\mathcal{W}(\lambda^{*})\setminus\\{\mathbf{0}\\}$,
and using the formula given by (4.7) we may build a sequence
$\\{a_{n}^{*}\\}_{\mathbb{Z}^{*}}\in X$. In fact, the constraints (4.3) come
from the expression (4.8) and the summability $\sum
c_{n}(a_{n}^{*})^{2}\xi_{n}^{2k}<\infty$ follows from (4.7) and the fact that
$K(0,z)=A(0)B(z)/z$ implies $\sum\frac{1}{c_{n}\xi_{n}^{2}}<\infty$. Undoing
the step from (4.7) to (4.6) and summing over $m\notin I$, we get

$\displaystyle\sum_{m\notin
I}c_{m}(a_{m}^{*})^{2}\big{(}(\xi_{m}-\alpha)^{2k}-$
$\displaystyle(\lambda^{*})^{2k})\big{)}=\sum_{m\notin
I}a_{m}^{*}\sum_{j=1}^{k}c_{i_{j}}a_{i_{j}}^{*}\bigg{(}\sum_{r=1}^{k}(\xi_{m}-\alpha)^{k-r}(\mathcal{T}^{-1})_{rj}\bigg{)}\big{(}(\xi_{i_{j}}-\alpha)^{2k}-(\lambda^{*})^{2k}\big{)}$
$\displaystyle=\sum_{j=1}^{k}c_{i_{j}}a_{i_{j}}^{*}\bigg{(}\sum_{r=1}^{k}S_{r}^{*}(\mathcal{T}^{-1})_{rj}\bigg{)}\big{(}(\xi_{i_{j}}-\alpha)^{2k}-(\lambda^{*})^{2k}\big{)}$
$\displaystyle=-\sum_{j=1}^{k}c_{i_{j}}(a_{i_{j}}^{*})^{2}\big{(}(\xi_{i_{j}}-\alpha)^{2k}-(\lambda^{*})^{2k}\big{)},$

where in the last equality we have used (4.4). Upon rearranging the terms, the
equation above becomes

$\frac{\sum_{n\in\mathbb{Z}^{*}}c_{n}a_{n}^{2}(\xi_{n}-\alpha)^{2k}}{\textstyle\sum_{n\in\mathbb{Z}^{*}}c_{n}a_{n}^{2}}=(\lambda^{*})^{2k},$

whence the conclusion follows that the desired $\lambda_{0}$ is the smallest
such $\lambda^{*}$.

The job now mostly consists in describing $\lambda_{0}$ as the zero of the
simpler function appearing in the theorem statement. To do this we look at
$\mathcal{Q}(\lambda)$, since

$\displaystyle\det\mathcal{W}(\lambda)=\left(\prod_{i\in
I}c_{i}\left((\xi_{i}-\alpha)^{2k}-\lambda^{2k}\right)\right)\left(\det\mathcal{T}^{-1}\right)\det\mathcal{Q}(\lambda),$

which we can further simplify by using the partial fraction decomposition
$\frac{2kx^{2k-s-1}y^{n}}{x^{2k}-y^{2k}}=\sum_{r=0}^{2k-1}\frac{\omega^{-rs}}{x-\omega^{r}y}$,
as in [10, Lemma 26], where $\omega:=e^{i\pi/k}$ and $0\leq s\leq 2k-1$.
Recall also the formula for $C(z)$

$C(z)=\sum_{m=1}^{\infty}\frac{2z}{c_{m}(\xi_{m}^{2}-z^{2})},$ (4.11)

which converges uniformly in compact subsets of $\mathbb{C}$ away from the
zeros of $A$. Let us now look at the partial sums of (4.10). Since by (C2) the
zeros of $A$ are symmetric, i.e. $\xi_{-m}=-\xi_{m}$, we have

$\displaystyle\sum_{0<|m|\leq M}$
$\displaystyle\frac{(\xi_{m}-\alpha)^{2k-n-\ell}}{c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda^{2k}\right)}=\frac{1}{2k\lambda^{\ell+n-1}}\sum_{0<|m|\leq
M}\frac{2k\lambda^{\ell+n-1}(\xi_{m}-\alpha)^{2k-n-\ell}}{c_{m}\left((\xi_{m}-\alpha)^{2k}-\lambda^{2k}\right)}$
$\displaystyle=\frac{1}{2k\lambda^{\ell+n-1}}\sum_{0<|m|\leq
M}\sum_{r=0}^{2k-1}\frac{\omega^{-r(n+\ell-1)}}{c_{m}(\xi_{m}-\alpha-\omega^{r}\lambda)}$
$\displaystyle=\frac{1}{2k\lambda^{\ell+n-1}}\sum_{r=0}^{2k-1}\sum_{m=1}^{M}\frac{\omega^{-r(n+\ell-1)}}{c_{m}(\xi_{m}-\alpha-\omega^{r}\lambda)}+\frac{\omega^{-r(n+\ell-1)}}{c_{m}(-\xi_{m}-\alpha-\omega^{r}\lambda)}$
$\displaystyle=\frac{1}{2k\lambda^{\ell+n-1}}\sum_{r=0}^{2k-1}\omega^{-r(n+\ell-1)}\sum_{m=1}^{M}\frac{2(\alpha+\omega^{r}\lambda)}{c_{m}(\xi_{m}^{2}-(\alpha+\omega^{r}\lambda)^{2})},$

which by formula (4.11) converges to

$\frac{1}{2k\lambda^{\ell+n-1}}\sum_{r=0}^{2k-1}\omega^{-r(n+\ell-1)}C(\alpha+\omega^{r}\lambda)=\frac{\big{(}\mathcal{V}(\lambda)\big{)}_{n\ell}}{2k\lambda^{\ell+n-1}}$

as $M\to\infty$. Thus, we have $\det\mathcal{W}(\lambda)=0$ if and only if

$\left(\prod_{i\in
I}c_{i}\left((\xi_{i}-\alpha)^{2k}-\lambda^{2k}\right)\right)\det\mathcal{V}(\lambda)=0,$

since the exponent of the factors $\lambda^{-(\ell+n-1)}$ in each entry
becomes constant when the determinant is expanded. Observe that from the
expression (4.9) and the inequality (4.5) we can see that the function
$\lambda\mapsto\det\mathcal{W}(\lambda)$ is meromorphic in $\mathbb{C}$ and
continuous in $(0,\min_{\ell\notin I}|\xi_{\ell}-\alpha|)$, and therefore so
is $\lambda\mapsto\left(\prod_{i\in
I}c_{i}\left((\xi_{i}-\alpha)^{2k}-\lambda^{2k}\right)\right)\det\mathcal{V}(\lambda)$.
Now $\lambda$ is a real solution of the above if and only if

$\left(\prod_{i\in
I}\left(\xi_{i}-\alpha-\lambda\right)\left(\xi_{i}-\alpha+\lambda\right)\right)\det\mathcal{V}(\lambda)=0.$
(4.12)

Recall that inequality (4.5) and item (i) of Proposition 5 also established
that $\lambda_{0}$ is within the interval $\left(0,\min_{\ell\notin
I}|\xi_{\ell}-\alpha|\right)$. We know the zeros of $A(\alpha+x)$ in this
interval are precisely the $\xi_{i}-\alpha$ and those of $A(\alpha-x)$
precisely the $-\xi_{i}+\alpha$. So we can replace the product in (4.12) by
these respective functions obtaining $\lambda_{0}$ must be the first positive
solution of

$A(\alpha+\lambda)A(\alpha-\lambda)\det\mathcal{V}(\lambda)=0.$

Now we come back to the assumption we made when $k=1$. We supposed $\alpha$ is
not the midpoint of two consecutive zeros of $A$, but we will now show that
even in this case the first positive zero of (1.12) yields our answer. In
fact, if there exists $i\in\mathbb{Z}^{*}$ such that
$\alpha=\frac{1}{2}(\xi_{i}+\xi_{i+1})$, then by (1.10), we have that

$\displaystyle\|f\|_{\mathcal{H}(E)}^{2}$
$\displaystyle=\sum_{A(\xi)=0}\frac{|f(\xi)|^{2}}{K(\xi,\xi)}=\sum_{A(\xi)=0}\frac{|\xi-\alpha|^{2}}{|\xi-\alpha|^{2}}\frac{|f(\xi)|^{2}}{K(\xi,\xi)}$
$\displaystyle\leq\frac{1}{|\xi_{i}-\alpha|^{2}}\sum_{A(\xi)=0}|\xi-\alpha|^{2}\frac{|f(\xi)|^{2}}{K(\xi,\xi)}=\frac{1}{|\xi_{i}-\alpha|^{2}}\|(z-\alpha)f\|_{\mathcal{H}(E)}^{2},$

so $\lambda_{0}\geq|\xi_{i}-\alpha|$. Now since
$|\xi_{i}-\alpha|=|\xi_{i+1}-\alpha|$, equality above can be obtained only if
$f(\xi)=0$ when $\xi$ is a zero of $A$ neither equal to $\xi_{i}$ nor to
$\xi_{i+1}$. Hence extremizers can only be of the form

$\displaystyle
f(\xi_{i})\frac{K(\xi_{i},x)}{K(\xi_{i},\xi_{i})}+f(\xi_{i+1})\frac{K(\xi_{i+1},x)}{K(\xi_{i+1},\xi_{i+1})}=\bigg{\\{}\frac{f(\xi_{i})}{A^{\prime}(\xi_{i})}\frac{1}{z-\xi_{i}}+\frac{f(\xi_{i+1})}{A^{\prime}(\xi_{i+1})}\frac{1}{z-\xi_{i+1}}\bigg{\\}}A(x),$

which can possibly occur in $\mathcal{X}_{k}$ if we impose the symmetry
condition
$f(\xi_{i})/A^{\prime}(\xi_{i})=-f(\xi_{i+1})/A^{\prime}(\xi_{i+1})$. For this
reason, we actually have the equality
$\lambda_{0}=|\xi_{i}-\alpha|=\frac{1}{2}(\xi_{i+1}-\xi_{i})$.

We only have to show that this is the first zero of

$\displaystyle
A(\alpha+\lambda)A(\alpha-\lambda)\left(C(\alpha+\lambda)-C(\alpha-\lambda)\right)=B(\alpha+\lambda)A(\alpha-\lambda)-A(\alpha+\lambda)B(\alpha-\lambda).$

Plugging in $\frac{1}{2}(\xi_{i+1}-\xi_{i})$ for $\lambda$ yields

$B(\xi_{i+1})A(\xi_{i})-A(\xi_{i+1})B(\xi_{i})=0,$

but we need to rule out the existence of smaller zeros. To do this, recall
that to every $E$ Hermite-Biehler function satisfying condition (C1) we may
associate an analytic phase function $\varphi:\mathcal{W}\to\mathbb{C}$, where
$\mathbb{R}\subseteq\mathcal{W}\subseteq\mathbb{C}$ is an open simply
connected set, satisfying

$E(x)e^{i\varphi(x)}\in\mathbb{R}\ \ \text{ for all}\ \ x\in\mathbb{R}.$
(4.13)

It is immediate from (C1) and the fact that both $A$ and $B$ are real entire
that $\gamma$ is a zero of $A$ if and only if
$\varphi(\gamma)\equiv\frac{\pi}{2}(\bmod\pi)$.

Now, an expression equivalent to (4.13) is

$e^{2i\varphi(x)}=\frac{E^{*}(x)}{E(x)}=\frac{A(x)+iB(x)}{A(x)-iB(x)}$

and from this it follows that $B(\beta)A(\gamma)-A(\beta)B(\gamma)=0$ if and
only if $\varphi(\beta)\equiv\varphi(\gamma)\bmod\pi$.

Suppose by contradiction there is a
$0<\lambda^{*}<\frac{1}{2}(\xi_{i+1}-\xi_{i})$ such that
$B(\alpha+\lambda^{*})A(\alpha-\lambda^{*})-A(\alpha+\lambda^{*})B(\alpha-\lambda^{*})=0$.
Thus, we would need
$\varphi(\alpha+\lambda^{*})\equiv\varphi(\alpha-\lambda^{*})\bmod\pi$ . Since
$\varphi(x)$ is a strictly increasing function of $x\in\mathbb{R}$ [6, Problem
48] it follows that there exists a
$\gamma\in[\alpha-\lambda^{*},\alpha+\lambda^{*}]$ such that
$\varphi(\gamma)\equiv\frac{\pi}{2}\bmod\pi$, i.e., $A(\gamma)=0$. But
$\xi_{i}<\alpha-\lambda^{*}<\alpha+\lambda^{*}<\xi_{i+1}$, while
$\xi_{i}\text{ and }\xi_{i+1}$ are consecutive zeros of $A$, which is a
contradiction. Hence $\lambda_{0}=\frac{1}{2}(\xi_{i+1}-\xi_{i})$ is indeed
the first positive zero, concluding the proof. ∎

Remark. In the particular case $k=1$ and $\alpha$ is a midpoint of two
consecutive zeros $\xi_{i}<\xi_{i+1}$, we saw in the proof above that
extremizers must be complex multiples of the function

$f(z)=\frac{A(z)}{(z-\xi_{i})(z-\xi_{i+1})},$

but we may also classify extremizers in the other cases contemplated in this
theorem. In view of the proof above, we can obtain a real extremizing sequence
$\\{a_{n}^{*}\\}\in X$ to the right-hand side of (4.1) by taking
$(a_{i_{1}}^{*},\ldots,a_{i_{k}}^{*})\in\ker\mathcal{W}(\lambda_{0})\setminus\\{\boldsymbol{0}\\}$
and building the rest of the sequence by (4.7). Applying formula (1.10),
extremizers must then be in the span over $\mathbb{C}$ of the entire functions

$f(z)=\sum_{\begin{subarray}{c}{n=-\infty}\\\ {n\neq
0}\end{subarray}}^{\infty}a_{n}^{*}\frac{A(z)}{z-\xi_{n}}$

that come from sequences $\\{a_{n}^{*}\\}\in X$ constructed in this way, as
the initial reduction in the proof of Lemma 6 guarantees.

###  4.1. Proof of Theorem 2

A few steps are missing in order to conclude Theorem 2, which instead follows
directly from a slight adaptation of the argument. In fact, as Carneiro,
Chirre, and Milinovich establish in [9, Section 5.2], the spaces
$(\mathcal{H}_{G,\pi\Delta},\langle\cdot,\cdot\rangle_{\mathcal{H}_{G,\pi\Delta}})$
are isometrically isomorphic to certain de Branges spaces. They give a
procedure that allows one to obtain a (not necessarily unique) Hermite-Biehler
function $E$ from the expression of the reproducing kernels of
$\mathcal{H}_{G,\pi\Delta}$. It consists of defining

$L(w,z):=2\pi i(\overline{w}-z)K_{G,\pi\Delta}(w,z),$

and then considering the entire function

$E(z):=L(i,i)^{-\frac{1}{2}}L(i,z),$

which is Hermite-Biehler (see [6, Problems 50-53] and [8, Appendix A] for
details). By [6, Theorem 23], $E$ is such that
$\mathcal{H}_{G,\pi\Delta}=\mathcal{H}(E)$ as sets and

$\displaystyle\|F\|_{\mathcal{H}_{G,\pi\Delta}}=\left(\int_{\mathbb{R}}|F(x)|^{2}\,W_{G}(x)\,\text{\rm
d}x\right)^{\frac{1}{2}}=\left(\int_{-\infty}^{\infty}\left|\frac{F(x)}{E(x)}\right|^{2}\text{\rm
d}x\right)^{\frac{1}{2}}=\|F\|_{\mathcal{H}(E)}$

for each $F$ therein. Such an $E$ can be seen to satisfy conditions (C1) and
(C2), so that Theorem 2 is thus a consequence of the following.

######  Corollary 7.

Let $E$ be a Hermite-Biehler function satisfying (C1) and (C2) and
$\mathcal{X}_{1}\neq\\{0\\}$. Setting
$\lambda_{0}=\big{(}(\mathbb{EP})(E,1,\alpha)\big{)}^{1/2}$, we have that
$\lambda_{0}$ is the first positive zero of the equation

$K(\alpha+\lambda,\alpha-\lambda)=0.$

###### Proof.

If $A\notin\mathcal{H}(E)$ then we can apply Theorem 4 when $k=1$, which
states $\lambda_{0}$ is the first zero of

$\displaystyle
A(\alpha+\lambda)A(\alpha-\lambda)\left(C(\alpha+\lambda)-C(\alpha-\lambda)\right)=B(\alpha+\lambda)A(\alpha-\lambda)-A(\alpha+\lambda)B(\alpha-\lambda),$

which by (1.8) is

$2\pi\lambda\,K(\alpha+\lambda,\alpha-\lambda).$

If $B\notin\mathcal{H}(E)$, we may use the interpolation formula and
Parseval’s identity corresponding to the zeros of $B$. That is, as in (1.10)
and (1.11), we have by [6, Theorem 22] that

$f(z)=\sum_{B(\eta)=0}\frac{f(\eta)}{K(\eta,\eta)}K(\eta,z)\ \ \text{ and }\ \
\|f\|_{\mathcal{H}(E)}^{2}=\sum_{B(\eta)=0}\frac{|f(\eta)|^{2}}{K(\eta,\eta)},$

and an analogous version of Lemma 6 holds (see also the remark immediately
following that lemma). By (C2), the zeros of $B$ are also symmetric in the
sense that $B(\eta)=0$ if and only if $B(-\eta)=0$. Its non-negative zeros can
be enumerated as $0=\eta_{0}<\eta_{1}<\ldots$, and so defining
$d_{m}=B^{\prime}(\eta_{m})/A(\eta_{m})$, we likewise have the formula

$\frac{A(z)}{B(z)}=\frac{1}{d_{0}z}+\sum_{m=1}^{\infty}\frac{2z}{d_{m}(z^{2}-\eta_{m}^{2})}$

converging uniformly in compact sets away from the zeros of $B$. The same
procedure in the proof of Theorem 4 can now be repeated, in which case
$\lambda_{0}$ may be seen to be the first positive zero of

$B(\alpha+\lambda)B(\alpha-\lambda)\left(\frac{A(\alpha+\lambda)}{B(\alpha+\lambda)}-\frac{A(\alpha-\lambda)}{B(\alpha-\lambda)}\right)=-2\pi\lambda\,K(\alpha+\lambda,\alpha-\lambda).$

But for any given $E$, at most one of $A$ and $B$ can be in the space
$\mathcal{H}(E)$, since by integrability considerations $E(z)=A(z)-iB(z)$
cannot itself occur in $\mathcal{H}(E)$, and the result holds. ∎

Remark. The condition $\mathcal{X}_{1}\neq\\{0\\}$ is, of course, satisfied in
the cases that interest us. For instance

$f(z)=\frac{\cos^{2}\pi\Delta z/2}{1-4\Delta^{2}z^{2}}$

is such that $zf(z)\in\mathcal{H}_{\pi\Delta}$. In general, we may conclude
$\mathcal{X}_{k}$ is non-trivial if one of the associated functions $A(z)$ or
$B(z)$ has at least $k+1$ zeros. If, say, $\xi_{1},\ldots,\xi_{k+1}$ are such
zeros, then one may obtain
$g(z)=\frac{A(z)}{(z-\xi_{1})\ldots(z-\xi_{k+1})}\in\mathcal{H}(E)$ (or the
analogous for $B$), from an appropriate linear combination of the reproducing
kernels $K(\xi_{i},z)$. Since $|g(x)|\sim|A(x)||x|^{-k-1}$ for $|x|\gg 1$,
this function satisfies $z^{k}g(z)\in\mathcal{H}(E)$.

##  5\. Concluding Remarks

We conclude with an interesting qualitative property of the extremal problem
in de Branges spaces.

######  Proposition 8.

Let $E$ be Hermite-Biehler and $k\in\mathbb{N}$. If
$\mathcal{X}_{k}\neq\\{0\\}$, then the function
$\alpha\in\mathbb{R}\mapsto(\mathbb{EP})(E,k,\alpha)$ is continuous.

###### Proof.

By Proposition 5, for all $\alpha$ there exists an
$f_{\alpha}\in\mathcal{H}(E)$ that attains the corresponding infimum in (1.9).
That is, if we denote by

$\Phi_{\alpha}(f):=\frac{\|(z-\alpha)^{k}f\|_{\mathcal{H}(E)}^{2}}{\|f\|_{\mathcal{H}(E)}^{2}}$

for $0\neq f\in\mathcal{X}_{k}$, we have
$\Phi_{\alpha}(f_{\alpha})=(\mathbb{EP})(E,k,\alpha)$.

Now fix $\alpha$ and let $\\{\alpha_{n}\\}\subseteq\mathbb{R}$ be a sequence
converging to $\alpha$. We can invoke Proposition 5 again to obtain a sequence
of extremizers $\\{f_{\alpha_{n}}\\}\subseteq\mathcal{H}(E)$ each satisfying
$\Phi_{\alpha_{n}}(f_{\alpha_{n}})=(\mathbb{EP})(E,k,\alpha_{n})$. Without
loss of generality, assume $\|f_{\alpha}\|_{\mathcal{H}(E)}=1$ and
$\|f_{\alpha_{n}}\|_{\mathcal{H}(E)}=1$.

On the one hand we have

$\displaystyle(\mathbb{EP})(E,k,\alpha)$
$\displaystyle=\Phi_{\alpha}(f_{\alpha})=\int_{-\infty}^{\infty}\left|\frac{(x-\alpha)^{k}f_{\alpha}(x)}{E(x)}\right|^{2}\text{\rm
d}x$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}\left|\frac{(x-\alpha_{n})^{k}f_{\alpha}(x)}{E(x)}\right|^{2}\text{\rm
d}x=\lim_{n\rightarrow\infty}\Phi_{\alpha_{n}}(f_{\alpha})$
$\displaystyle\geq\limsup_{n\rightarrow\infty}\,(\mathbb{EP})(E,k,\alpha_{n})$
(5.1)

by dominated convergence, so that $\alpha\mapsto(\mathbb{EP})(E,k,\alpha)$ is
upper semicontinuous.

On the other hand, we can also see that

$\displaystyle(\mathbb{EP})(E,k,\alpha)$
$\displaystyle\leq\Phi_{\alpha}(f_{\alpha_{n}})=\Phi_{\alpha_{n}}(f_{\alpha_{n}})+\int_{-\infty}^{\infty}\left\\{(x-\alpha)^{2k}-(x-\alpha_{n})^{2k}\right\\}\left|\frac{f_{\alpha_{n}}(x)}{E(x)}\right|^{2}\text{\rm
d}x$
$\displaystyle=(\mathbb{EP})(E,k,\alpha_{n})+O\big{(}|\alpha-\alpha_{n}|\big{)},$

because

$\displaystyle\bigg{|}\int_{-\infty}^{\infty}\big{\\{}(x-\alpha)^{2k}-$
$\displaystyle(x-\alpha_{n})^{2k}\big{\\}}\left|\frac{f_{\alpha_{n}}(x)}{E(x)}\right|^{2}\text{\rm
d}x\bigg{|}$
$\displaystyle\leq\int_{-\infty}^{\infty}\left|(x-\alpha)^{2k}-(x-\alpha_{n})^{2k}\right|\left|\frac{f_{\alpha_{n}}(x)}{E(x)}\right|^{2}\text{\rm
d}x$
$\displaystyle\ll|\alpha-\alpha_{n}|\int_{-\infty}^{\infty}(\max\\{|\alpha|,|\alpha_{n}|\\}^{2k-1}+|x|^{2k-1})\left|\frac{f_{\alpha_{n}}(x)}{E(x)}\right|^{2}\text{\rm
d}x$
$\displaystyle\ll|\alpha-\alpha_{n}|(\|f_{\alpha_{n}}\|_{\mathcal{H}(E)}^{2}+\|z^{k}f_{\alpha_{n}}\|_{\mathcal{H}(E)}^{2})$
$\displaystyle=|\alpha-\alpha_{n}|(1+(\mathbb{EP})(E,k,\alpha_{n}))$

and $(\mathbb{EP})(E,k,\alpha_{n})$ is bounded in $n$ by (5.1), so that

$(\mathbb{EP})(E,k,\alpha)\leq\liminf_{n\rightarrow\infty}\,(\mathbb{EP})(E,k,\alpha_{n}),$

i.e., it is lower semicontinuous. Hence, it is also continuous. ∎

##  6\. Acknowledgements

I am grateful to Emanuel Carneiro for the constant guidance and support
throughout the writing of this paper. I would also like to thank Tolibjon
Ismoilov for the helpful discussions and Ariel S. Boiardi for his aid in the
elaboration of the graphs of Figure 1.

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