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\title{Density of polynomials 
in the de Branges spaces of entire functions}
\author{A. D. Baranov}
\date{}

\begin{document}
\maketitle
\sloppy
\begin{center}
{\bf Introduction\footnote{The work is partially supported 
by RFBR grant No 03-01-00377, by the grant for Leading Scientific Schools
No NSH-2266.2003.1, and by the Ministry of Education grant
No E02-1.0-66.}}
\end{center}

Let $E $ be an entire function satisfying the inequality
\begin{equation}
\label{1}
|E (z) | > |E (\overline {z}) |, \qquad z\in {\mathbb{C}^+},
\end{equation}
where $ {\mathbb{C}^+} = \{z: {\rm Im}\, z > 0 \} $ is the upper half-plane. 
The class of such functions (known as the Hermite-Biehler class) 
we denote by $HB$. With each function $E\in HB$ 
we associate a Hilbert space ${\cal H} (E) $ which consists
of all entire functions $F$ such that $F/E$ and $F^*/E$ belong
to the Hardy class $H^2({\mathbb{C}^+})$ (here and later on
$F^* (z) = \overline {F (\overline z)} $). 
The inner product,
which makes ${\cal H} (E)$ a Hilbert space, 
is defined by the formula
$$
   \langle F, G\rangle_{{\cal H} (E)} = \int\limits_{\mathbb{R}} 
\frac{F (t) \overline {G (t)}} {|E (t) |^2} dt.
$$
\par
The theory of spaces $ {\cal H} (E) $ introduced by L. de Branges
\cite{br} has important applications in mathematical physics.
At the same time, de Branges spaces are of interest from the
point of view of theory of entire functions.
As a special case of de Branges spaces the Paley -- Wiener space
$PW_a $ of entire functions of exponential type at most $a$
square summable on the real axis may be considered
(this space corresponds to the function $E (z) = \exp(-iaz)$, $a > 0$).
\smallskip
\par
De Branges spaces are also closely connected with
the shift-coinvariant subspaces $ {K_\Theta}$ 
of the Hardy class $H^2({\mathbb{C}^+})$ known
also as model subspaces \cite{ko,nk}.
Let $ \Theta $ be an inner function in the upper half-plane.
Put $ K_\Theta = H^2 ({\mathbb{C}^+})\ominus\Theta H^2 ({\mathbb{C}^+}) $.
If $E $ is an entire function satisfying (\ref{1}), then
$ \Theta_E=E^*/E $ is inner and the mapping
$ F\mapsto F/E $ is a unitary operator from
$ {\cal H} (E) $ onto  $ K_{\Theta_E} $.
\medskip
\par 
In the present note we are concerned with the following three
related problems. By ${\cal P}$ we denote the set of all polynomials.
\medskip
\par
{\it
1. For which entire functions $E\in HB$ the function $f\equiv 1$ 
belongs to the space ${\cal H}(E)$?
\smallskip
\par
2. For which $E\in HB$ the inclusion ${\cal P}\subset{\cal H}(E)$ holds?
\smallskip
\par
3. For which $E\in HB$ the polynomials are dense in ${\cal H}(E)$?}
\bigskip
\par
Density of polynomials is a classical problem
for weighted functional spaces. At the same time, there
are rather deep motivations for considering these 
problems for the de Branges spaces particularly. 
The first question is inspired by the recent works
by V.P. Havin and J. Mashreghi \cite{hm1,hm2}
where the admissible majorants
for the model subspaces were studied in the spirit of the 
Beurling -- Malliavin theorem. By an admissible majorant
for the space $K_\Theta$ we mean a nonnegative function $w$ 
on the real line such that $w\ge |f|$ for some nonzero function 
$f\in K_\Theta$. It turns out that the condition $1\in{\cal H}(E)$
is crucial for the existence of the majorant with the fastest
rate of decrease. This problem was also investigated in \cite{vor}.
\smallskip
\par
The third problem is closely connected with the 
Hamburger moment problem on the line. It was shown in
\cite{bs1} that the answer to this question
will lead to the description of all canonical solutions
of the indeterminate Hamburger problem. Moreover, in \cite{bs2}
a criterion for the density of polynomials was obtained,
which is, however, somewhat implicit and not so easy to apply.
At the same time, there exist
very simple geometrical conditions 
on the zeros of $E$ sufficient for the density of polynomials. 
It was mentioned by N.I. Akhiezer \cite{akh}
that if $E\in HB$ is a canonical product of genus $0$ with zeros lying 
in a strip $|{\rm Re}\, z|\le h$ then ${\cal P}\subset{\cal H}(E)$ and
${\rm Clos}_{{\cal H}(E)}{\cal P}={\cal H}(E)$. Another example is due to V.P. Gurarii \cite{gur}:  
if E is an even function with zeros in the angle
$-3\pi/4\le \arg z\le -\pi/4$ then the polynomials are also dense in ${\cal H}(E)$.
\smallskip
\par
It should be noted that problems 2 and 3 lead to different
classes of functions. It is easy to construct an example of the space
${\cal H}(E)$ such that ${\cal P}\subset{\cal H}(E)$, 
but ${\cal P}$ is not dense in ${\cal H}(E)$.
It is known that if the function $B=\frac{E^*-E}{2i}$ belongs 
to ${\cal H}(E)$ then it is orthogonal to the domain of 
the operator of  multiplication by $z$ (see \cite{br}, Theorem 29). 
Thus, $B$ is orthogonal to all polynomials in ${\cal H}(E)$.
For a function $E\in HB$ of zero type one has 
$B\in{\cal H}(E)$ if and only if
the zeros $z_n$ of $E$ satisfy the condition
$\sum_n |{\rm Im}\,z_n|^{-1}<\infty$ \cite{bar1}.
P. Koosis \cite{ko2} has constructed a much more subtle example 
showing that polynomials may be not dense in ${\cal H}(E)$
even in the case when $B\notin{\cal H}(E)$.
\medskip
\\
{\bf Acknowledgements.} The author thanks V.P. Havin, E. Abakumov and 
M. Sodin for posing of the problems and helpful discussions. This
work was partially done while the author was visiting McGill
University, Montreal, in October 2002. 
The author thanks this institution for hospitality.
\bigskip
\\
\begin{center}
{\bf \S 1. Main results}
\end{center}

In the present note we give some simple geometric conditions
generalizing the results of Akhiezer and Gurarii. In particular, we
study the following problem: to describe the subsets 
$\Omega$ of $\mathbb{C^+}$ 
such that for any function $E\in HB$ with zeros in 
the conjugate set $\overline{\Omega}$ 
we have the inclusion ${\cal P}\subset{\cal H}(E)$.
In this case the inclusion ${\cal P}\subset{\cal H}(E)$
holds independently of the distribution of zeros
of $E$ in $\Omega$, and we say 
that the set $\Omega$ is a 
distribution independent (d. i.) set. The result of Akhiezer
states that $\{z=x+iy:\, y>0, |x|\le h\}$ is a d. i. set for any $h>0$.
\par
In what follows we assume that $E$ is a function of the form 
\begin{equation}
\label{2}
E(z)=\prod\limits_{n=1}^\infty \left(1-\frac{z}{\overline z_n}\right),
\end{equation}
where $z_n\in\mathbb{C^+}$, $\sum_n|z_n|^{-1}<\infty$.
For the sake of convenience we denote the zeros of $E$ by
$\overline z_n$ so that $z_n\in\mathbb{C^+}$.
The case when zeros $z_n$ lie in a Stolz angle 
$\Gamma_\gamma=\{z:\,y\ge\gamma |x|\}$, $\gamma>0$,
is of special interest. In this case $E$ belongs 
to the so-called class $A$ discussed in details in 
B.Ya. Levin's monograph \cite{lev}).
\medskip

The following theorem gives a condition on the growth of 
the zeros of $E$ which is sufficient for the density 
\medskip
of polynomials.
\\
{\bf Theorem 1.} {\it  Let $z_n=x_n+iy_n$ and $\inf_n y_n>0$. If 
\begin{equation}
\label{3}
\sum\limits_{|x_n|>R}\frac{|x_n|}{|z_n|^2}\le  AR^{-1},\qquad R>0,
\end{equation}
for some $A>0$, then ${\cal P}\subset{\cal H}(E)$ and ${\rm Clos}\,_{{\cal H}(E)}{\cal P}={\cal H}(E)$.}
\medskip
\\
{\bf Theorem 2.} {\it
Let $E$ be an even entire function of the form (\ref{1}),
$\inf_n y_n>0$ and
\begin{equation}
\label{4}
\sum\limits_{|x_n|^2>y_n^2+R^2}\frac{|x_n|^2}{|z_n|^4}\le AR^{-2},\qquad R>0.
\end{equation}
Then ${\cal P}\subset{\cal H}(E)$ and ${\rm Clos}_{{\cal H}(E)}{\cal P}={\cal H}(E)$.}
\medskip
\par
As a particular case of Theorem 1 we obtain a description
of distribution independent sets generalizing the result of Akhiezer.
\medskip
\\
{\bf Theorem 3.} {\it Let $\Omega\subset \{z=x+iy:\,y > qx^2\}$
for some constant $q>0$. Then
$\Omega $ is a d. i. set. Moreover, this condition is in a sense sharp:
for any $\alpha<2$ there exists a function $E\in HB$ 
such that $z_n\in \{y \ge |x|^\alpha\}$ and 
$1\notin {\cal H}(E)$.}
\medskip
\par
Let zeros $z_n$ lie outside some Stolz angle
$\Gamma_\gamma$. Then $|x_n|\asymp|z_n|$ and
the condition (\ref{3}) is equivalent to 
$\sum_{|z_n|>R}|z_n|^{-1}\le CR^{-1}$. Thus, the sequence $z_n$
tends to infinity very fast (say, as a progression $\rho^n$, 
$\rho>1$). To deal with a more slow growth of $z_n$ 
one should consider more precise estimates. By $n(t)$ we denote
the number of the zeros in the disc $\{|z|\le t\}$.
\medskip
\\
{\bf Theorem 4.} {\it Let zeros $z_n$ lie in a Stolz angle 
$\Gamma_\gamma$. If
\begin{equation}
\label{5}
\sum\limits_{|z_n|>R}\frac{|x_n|}{|z_n|^2}=
o\left(\frac{1}{R}\int\limits_0^R\frac{n(t)}{t}dt+
R^2\int\limits_R^\infty\frac{n(t)}{t^3}dt \right),\qquad R\to\infty,
\end{equation}
then ${\cal P}\subset{\cal H}(E)$.}
\smallskip
\par
Condition (\ref{5}) is applicable to sequences $z_n$ growing "faster than 
any power" (say, $|z_n|\asymp \exp(\log^\alpha n)$, $\alpha>1$), 
but it fails for $|z_n|\asymp n^{\alpha}$, $\alpha>1$.
\medskip
\\
{\bf Theorem 5.} {\it Let $\{z_n\}\subset
\{\gamma_1 x\le y\le\gamma_2x\}$, $\gamma_1,\gamma_2>0$,
and $C_1 n^\alpha\le |z_n| \le C_2 n^\alpha$.
Then there exist exponentials $\alpha_1$ and $\alpha_2$ (depending on 
$\gamma_1,\gamma_2, C_1, C_2$) such that
$E(x)\to \infty$, $|x|\to\infty$, and ${\cal P}\subset{\cal H}(E)$ if $\alpha>\alpha_1$,
whereas $E(x)\to 0$, $|x|\to\infty$, if $\alpha<\alpha_2$.}
\medskip
\par
We construct an example showing that in this case the behavior of $E$
depends essentially on the zeros' distribution. Namely, for any
$\alpha>1$ there exist $z_n$ with $\arg z_n=\pi/4$, 
$|z_n|\asymp n^\alpha$ and $1\notin {\cal H}(E)$. 
We also consider the case when zeros are distributed regularly 
along a single ray:
in this case we obtain an explicit formula for the limit exponential.
\medskip
\\
{\bf Theorem 6.} {\it Let $\gamma>0$ and $z_n=n^\alpha(1+\gamma i)$,
$\alpha>1$. Then $E(x)\to \infty$, $x\to\infty$, if 
$\alpha>2-\frac{2{\rm arctg}\,\gamma}{\pi}$, and
$E(x)\to 0$, $x\to\infty$, if $\alpha<2-\frac{2{\rm arctg}\,\gamma}{\pi}$.}
\medskip
\par
In the last section we discuss the relationship between the 
density of polynomials and the hypercyclicity phenomenon.
Recall that a vector $f$ is said to be hypercyclic for a bounded 
linear operator $T$ in a Fr\'echet space $F$ 
if its orbit $\{T^n f\}_{n=0}^\infty$
is dense in $F$. In this case the operator $T$ is also 
said to be 
\smallskip hypercyclic. 
\par
The first example of a hypercyclic operator was obtained
by G.D. Birkhoff \cite{bir} who showed that
the translation operators $T_w: f\mapsto f(\cdot+w)$, $w\in\mathbb{C}$, 
$w\ne 0$, are hypercyclic in the space of all entire functions
with the topology of uniform convergence on compact 
subsets of the plane. It was shown in \cite{mcl}
that the differentiation operator is also hypercyclic 
in this space,
whereas in \cite{gosh} this result was extended to all
operators commuting with the differentiation.
Thus, spaces of entire functions proved to be 
an important source of hypercyclic operators.
K.C. Chan and J.H. Shapiro \cite{chsh}
studied the hypercyclicity of translations
in the setting of Hilbert spaces of entire functions of
"slow growth". As a corollary of the results of \cite{chsh}
we obtain the following theorem.
\medskip
\\
{\bf Theorem 7.} {\it Let ${\cal H}(E)$ be a de Branges space such that
${\cal P}\subset{\cal H}(E)$ and the differentiation operator
is bounded in ${\cal H}(E)$. Then the translation operators 
$T_w$, $w\ne 0$, are hypercyclic in ${\cal H}(E)$.}
\medskip

Note that the translations are bounded in ${\cal H}(E)$
whenever the differentiation operator ${\cal D}$
is bounded, since $T_w=\exp(w{\cal D})$. The de Branges 
spaces ${\cal H}(E)$ such that the differentiation is bounded on 
${\cal H}(E)$ were described in the author's work \cite{bar2}. 
In particular, it is the case if $E$ is an entire function 
of the form (\ref{2}) with zeros in a Stolz angle.
\bigskip
\\
\begin{center}
{\bf \S 2. Proof of Theorems 1 and 2}
\end{center}

We begin with a condition sufficient  for the density
of polynomials in ${\cal H}(E)$. The following proposition is 
analogous to Theorem 6 in \cite{gur}. 
Recall that the reproducing kernel of the space
${\cal H}(E)$ corresponding to the point $w\in\mathbb{C}$
is of the form
$$
K(z,w)=\frac{i}{2\pi}\cdot\frac{\overline {E(w)}
E(z)-\overline{E^*(w)}E^*(z)}{z-\overline w}.
$$
\smallskip
\\
{\bf Proposition 8.} {\it Assume that there is a 
sequence $P_n$ of polynomials, which converges to the function $E$
uniformly on any compact set, and
\begin{equation}
\label{6}
|P_n(x)|\le C|E(x)|,\qquad x\in\mathbb{R},\quad n\in\mathbb{N}.
\end{equation} 
Then ${\cal P}\subset{\cal H}(E)$ and ${\rm Clos}_{{\cal H}(E)}{\cal P}={\cal H}(E)$.}
\smallskip
\\
{\bf Proof.} The inclusion ${\cal P}\subset{\cal H}(E)$ follows
immediately from (\ref{6}). 
To prove the density we note first that the set of reproducing kernels 
$\{K(\cdot,s)$, $s\in\mathbb{R}\}$ is dense in ${\cal H}(E)$. Thus,
it suffices to show that $K(\cdot,s)\in {\rm Clos}_{{\cal H}(E)}{\cal P}$.
Fix $s\in\mathbb{R}$ and put
$$
K_n(z,s)=\frac{i}{2\pi}\cdot\frac{\overline {P_n(s)}
P_n(z)-P_n(s)P_n^*(z)}{z-s}.
$$
Clearly, $K_n(\cdot,s)$ is a polynomial, and
the sequence $K_n(\cdot,s)$ converges to $K(\cdot,s)$ uniformly
on any compact set. Let us show that $K_n(\cdot,s)$ converge to $K(\cdot,s)$
in the norm of the space ${\cal H}(E)$. Take $A>0$ and split the norm
$\|K_n(\cdot,s)-K(\cdot,s)\|^2_{{\cal H}(E)}$
into two parts:
$$
\int\limits_{|t-s|\le A}\left|\frac{K_n(t,s)-K(t,s)}{2\pi E(t)(t-s)}\right|^2dt+
\int\limits_{|t-s|>A}\left|\frac{K_n(t,s)-K(t,s)}{2\pi E(t)(t-s)}\right|^2dt
=I_1+I_2.
$$
We estimate the integral $I_2$. 
By inequality (\ref{6}) we have
$$
\left|\frac{K_n(t,s)-K(t,s)}{E(t)}\right|\le 
2|P_n(s)|\cdot
\left|\frac{P_n(t)}{E(t)}\right|+2|E(s)|\le (2C^2+2)|E(s)|.
$$
Hence, $I_2\le C_1 |E(s)|^2A^{-1}$ for all
$n\in\mathbb{N}$. Thus, choosing a sufficiently large $A$
we can make the integral $I_2$ as small as we wish 
uniformly with respect to $n$. Now, when $A$ is fixed,
the integral $I_1$ tends to zero when $n$ 
tends to infinity.
$\bigcirc$
\medskip
\par
We introduce certain notations which are used in what 
follows. Note that
$$
|E(x)|^2=\prod\limits_n\frac{(x-x_n)^2+y_n^2}{|z_n|^2}=
\prod\limits_n\left(1-\frac{2xx_n-x^2}{|z_n|^2}\right).
$$
We split this product into two parts $\Pi_+(x)$ and $\Pi_-(x)$,
where 
$$
\Pi_-(x)=\prod\limits_{x_n>x/2}
\left(1-\frac{2xx_n-x^2}{|z_n|^2}\right)
$$
for $x>0$; if $x<0$ we take the product over $n$ such that
$x_n<x/2$. Thus, each factor in the product $\Pi_-$ is smaller
than 1 whereas the factors in the product $\Pi_+$
are greater or equal to 1.
\medskip
\\
{\bf Proof of Theorem 1.} We show that there exist $l\in\mathbb{N}$ and
$C>0$ such that 
\begin{equation}
\label{7}
\Pi_-(x)\ge C(\delta/x)^{2l}
\end{equation}
for sufficiently large $x$;
here $\delta=\inf_n y_n$. Once the inequality (21) is proved the 
rest of the theorem follows immediately. Indeed,
in this case the polynomials 
$$
P_k(z)=
\prod\limits_{|n|\le k} \left(1-\frac{z}{\overline z_n}\right).
$$
satisfy the conditions of the Proposition 8.

Without loss of generality let $x>0$. Denote by $N(x)$
the set of $n$ such that $2xx_n-x^2\ge |z_n|^2/2$ 
and let $|N(x)|$ be the number of elements in $N(x)$.
Then, by (\ref{3}), we have the estimate
$$
\frac{1}{2}|N(x)|\le \sum\limits_{x_n>x/2}
\frac{2xx_n-x^2}{|z_n|^2}\le 2x
\sum\limits_{|x_n|>x/2} \frac{|x_n|}{|z_n|^2}\le 4A,
$$                                                  
which implies that there is $l\in \mathbb{N}$ 
such that $|N(x)|\le l$ for any $x>0$. Note also that
$|z_n|\le 4x$ whenever $n\in N(x)$. Hence,
$$
\prod\limits_{n\in N(x)}\frac{(x-x_n)^2+y_n^2}{|z_n|^2}\ge
\left(\frac{\delta}{4x}\right)^{2l}.
$$
Finally, making use of the elementary estimate 
$\log(1-t)\ge -2t$, $t\in [0,\,1/2]$, we get 
$$
\log \prod\limits_{n\notin N(x),\, x_n>x/2}
\left(1-\frac{2xx_n-x^2}{|z_n|^2}\right) \ge
$$
$$ 
\ge -2 \sum\limits_{n\notin N(x),\, x_n>x/2}
\frac{2xx_n-x^2}{|z_n|^2}\ge -4x
\sum\limits_{x_n>x/2} \frac{|x_n|}{|z_n|^2}\ge 8A.
$$
Combining the last two inequalities we obtain the estimate 
(\ref{7}) and Theorem 1 is proved. $\bigcirc$
\medskip
\\
{\bf Proof of Theorem 2.}
In the case of the even entire functions 
we have the pairs of symmetric zeros
$z_n=x_n+iy_n$ and $\tilde z_n=-x_n+iy_n$.
Hence,
$$
|E(x)|^2=\prod\limits_n
\left(1+\frac{x^4+2x^2(2x_n^2-|z_n|^2)}{|z_n|^2}\right).
$$
Now, one should split this product into two parts
with factors greater than 1 and smaller than 1 respectively.
The inequality (\ref{4}) implies an estimate from below
for the "small" product as in the proof of Theorem 1. We omit the details. 
$\bigcirc$
\bigskip
\\
\begin{center}
{\bf \S 3. Distribution independent sets}
\end{center}
\noindent
{\bf Proof of Theorem 3.} Let $z_n\in \{y > qx^2\}$. 
Then
$$
\sum\limits_{|x_n|>R}\frac{|x_n|}{|z_n|^2}\le
\sum\limits_{|x_n|>R}\frac{|x_n|}{y_n^2}\le
\sum\limits_{|x_n|>R}\frac{1}{qx_n y_n}\le\frac{1}{qR}
\sum\limits_{|x_n|>R}\frac{1}{y_n}.
$$
Since $\sum_n y_n^{-1}<\infty$ we get the estimate (\ref{3}) 
(moreover, here $\sum_{|x_n|>R} |x_n|/|z_n|^2=o(R)$, 
$R\to\infty$).
\smallskip
\par
Now we show that for any $\alpha<2$ there exists a 
function $E\in HB$ such that $z_n$ are in
the set $\{y \ge |x|^\alpha\}$ and $1\notin {\cal H}(E)$.
Without loss of generality let $\alpha>1$.
Put $z_n=x_n+ix_n^\alpha$ and
$$
E(z)=\prod_n\left(1-\frac{z}{\overline z_n}\right)^{k_n}.
$$
We choose sequences $x_n$ and $k_n$ tending to infinity
very fast. Put $x_1=3$, $k_1=1$. Assume that
 $x_1,k_1,\dots, x_{n-1},k_{n-1}$ are already chosen.
Then take any $x_n$ such that $x_n>x_{n-1}^{\alpha}$ and
\begin{equation}
\label{8}
x_n^{1-\frac{\alpha}{2}} > 2^n\log x_n \sum\limits_{j=1}^{n-1} k_j,
\end{equation}
and put $k_n=[x_n^{\frac{3\alpha}{2}-1}]$ (by $[s]$ we denote the 
integer part of the number $s$). It is easy to see that
the sequence $x_n$ increases and tends to infinity.
\smallskip

First of all let us check the condition $\sum_n k_n|z_n|^{-1}<\infty$ 
ensuring that the product for $E$ converges:
$$
\sum\limits_n k_n|z_n|^{-1}\le \sum\limits_n
x_n^{\frac{3\alpha}{2}-1}x_n^{-\alpha}=
\sum\limits_n x_n^{\frac{\alpha}{2}-1}
\le \sum\limits_n 2^{-n}<\infty.
$$

Now we show that $\sup_{[x_n-1,\, x_n+1]}|E(x)|\to 0$, $n\to+\infty$.
Thus, $1/E\notin L^2(\mathbb{R})$. 
Take $x\in [x_n-1,x_n+1]$ and
split the product for $E$ into three parts:
$$
|E(x)|^2=
\left|\frac{x-z_n}{z_n}\right|^{2k_n}
\prod\limits_{l=1}^{n-1}
\left|\frac{x-z_l}{z_l}\right|^{2k_l}
\prod\limits_{l=n+1}^{\infty}
\left|\frac{x-z_l}{z_l}\right|^{2k_l}.
$$
Clearly, the last product does not exceed 1. Since $|z_l|>1$ and
$y_l\le x_n$, $l\le n_1$, we have 
$$
\prod\limits_{l=1}^{n-1}
\left(\frac{(x-x_l)^2+y_l^2}{|z_l|^2}\right)^{k_l}\le
\prod\limits_{l=1}^{n-1}(2x_n^2)^{k_l}
\le \prod\limits_{l=1}^{n-1}(x_n^3)^{k_l}.
$$
Taking into account that $|x-x_n|\le 1$ we get
$$
\log\frac{(x-x_n)^2+y_n^2}{|z_n|^2}\le 
-\frac{x_n^2-1}{x_n^2+x_n^{2\alpha}}\le-
\frac{1}{4x_n^{2\alpha-2}}.
$$
Hence,
$$
\log|E(x)|^2\le 3 \log x_n \sum\limits_{j=1}^{n-1} k_j
-\frac{k_n }{4x_n^{2\alpha-2}},
$$
and, by (\ref{8}), we have
$$
\log|E(x)|^2\le 3 \log x_n \sum\limits_{j=1}^{n-1} k_j-
\frac{[x_n^{\frac{3\alpha}{2}-1}]}
{4x_n^{2\alpha-2}}\le
-\left(\frac{1}{8}- \frac{3}{2^{n}}\right)
x_n^{1-\frac{\alpha}{2}}.
$$
The right-hand side of the last inequality
tends to $-\infty$ when n tends to infinity. 
$\bigcirc$
\medskip
\\
{\bf Remarks.}
1. Though, as Theorem 3 shows, the exponential 2 is sharp,
the condition $\Omega\subset \{y > qx^2\}$ for some $q>0$
is not necessary: there exist d. i. sets $\Omega$
such that 
$$
\limsup_{z\in\,\Omega,\, z\to\infty}\frac{x^2}{y}=\infty.
$$
For example, one can take $\Omega=\{n!+i(n-1)!n!,\, n\in\mathbb{N}\}$. 
At the same time, the arguments analogous to the proof
of sharpness in Theorem 3 show that
if $\Omega$ is a d. i. set then
$$
\limsup_{z\in\,\Omega,\,z\to\infty} \frac{x^2}{y\log|x|}<\infty.
$$
It is not clear whether the last condition is sufficient
for $\Omega$ to be a d. i. set.
\smallskip

2. One can show that for even entire functions the angle
$\{y\ge|x|\}$ is the widest distribution independent angle.
Namely, for each $\gamma<1$ there exists a function $E\in HB$ 
such that $z_n$ are in $\{y\ge\gamma|x|\}$ and $1\notin{\cal H}(E)$.
%
%
\bigskip
\\
\begin{center}
{\bf \S 4. Functions with zeros in an angle}
\end{center}

In this section we find in a sense sharp asymptotic 
of the growth on the real axis for an entire function
with zeros in a Stolz angle. 
\medskip
\\
{\bf Proposition 9.}
{\it Let $z_n\in \Gamma_\gamma$. Then there exist 
functions $A_j(x)>0$, $j=1,\dots,4$, 
such that
$$
0<m_j\le A_j(x)\le M_j, \qquad x\in\mathbb{R},
$$
and for $x\in\mathbb{R}$ we have
$$
\log|E(3x)|^2= A_1\int\limits_0^{|x|}\frac{n(t)}{t}dt+
A_2x^2 \int\limits_{|x|}^{\infty}\frac{n(t)}{t^3}dt+
$$
\begin{equation}
\label{9}
+ A_3|x|\sum\limits_{|z_n|\ge |x|,\, x_nx< 0} 
\frac{|x_n|}{|z_n|^2}
- A_4|x|\sum\limits_{|z_n|\ge |x|,\, x_nx> 0} 
\frac{|x_n|}{|z_n|^2}.
\end{equation}
Here the constants $m_j$ and $M_j$ may depend on $\gamma$.}
\smallskip
\\
{\bf Proof.} 
We use the following elementary estimates:
$-t/\delta \le \log(1-t)\le -t$, $t\in [0,1-\delta]$, and
$t/K\le \log(1+t)\le t$, $t\in[0,K]$.

Without loss of generality $x>0$. Then
$$
|E(x)|^2=
\prod\limits_{|z_n|<x/3}\frac{|x-z_n|^2}{|z_n|^2}
\prod\limits_{|z_n|\ge x/3,\, x_n<x/2}\frac{|x-z_n|^2}{|z_n|^2}
\prod\limits_{x_n\ge x/2}\frac{|x-z_n|^2}{|z_n|^2}=
$$
$$
=\Pi_1(x) \Pi_2(x) \Pi_3(x).
$$
Note that since $z_n\in \Gamma_\gamma$ we have
$\frac{2xx_n-x^2}{|z_n|^2}\le \frac{1}{1+\gamma^2}$
whenever $x_n\ge x/2$.
Hence,
\begin{equation}
\label{10}
-\frac{\gamma^2+1}{\gamma^2}\sum\limits_{x_n\ge x/2} 
\frac{2xx_n-x^2}{|z_n^2|} \le \log\Pi_3(x) \le
-\sum\limits_{x_n\ge x/2} \frac{2xx_n-x^2}{|z_n^2|}.
\end{equation}                 
Analogously,
$$
C_1\sum\limits_{|z_n|\ge x/3,\, x_n<x/2} 
\frac{x^2-2xx_n}{|z_n^2|} \le \log\Pi_2(x) \le 
$$
\begin{equation}
\label{11}
\le C_2 \sum\limits_{|z_n|\ge x/3,\,x_n<x/2} 
\frac{x^2-2xx_n}{|z_n^2|} 
\end{equation}
for some absolute positive constants $C_1$ and $C_2$.

Next we find a rough asymptotic for the product $\Pi_1$.
Since $|z_n|<x/3$ we get $2x/3\le |x-z_n|\le 2x$ and
$$
2^{2n(x/3)} \frac{(x/3)^{2n(x/3)}}
{|z_1|^2|z_2|^2\dots|z_{n(x/3)}|^2}\le
\Pi_1(x)\le
2^{2n(x/3)} 
\frac{(x/3)^{2n(x/3)}}{|z_1|^2|z_2|^2\dots|z_{n(x/3)}|^2}.
$$
Hence,
$$
\log\Pi_1(x) =2\log \frac{(x/3)^{n(x/3)}}{|z_1z_2\dots z_{n(x/3)}|}+
A(x)n(x/3)=
$$
\begin{equation}
\label{12}
=\int\limits_0^{x/3}\frac{n(t)}{t}dt+A(x)n(x/3),
\end{equation}
where $A(x)\asymp 1$. Combining the estimates (\ref{10})-(\ref{12})
and replacing $x$ by $3x$
we get the formula (\ref{9}). We also used the fact that 
$$
n(R)+R^2\sum\limits_{|z_n|>R}\frac{1}{|z_n|^2}=R^2\int\limits_{R}^\infty
\frac{n(t)}{t^3}dt. \qquad\bigcirc
$$

Thus, only the last term in the formula (\ref{9}) is responsible
for the possible smallness of the function $E$ on the real axis.
If this term is asymptotically smaller than the positive 
summands in (\ref{9}), as stated in Theorem 4, then
$E$ tends to infinity along $\mathbb{R}$ and Theorem 4 is proved.
\medskip
\\
{\bf Proof of Theorem 5.} Now let 
$z_n \in \{\gamma_1 x\le y\le\gamma_2x\}$
and $C_1 n^\alpha\le |z_n| \le C_2 n^\alpha$.
In this case $n(t)\asymp t^{1/\alpha}$ and we have the 
following asymptotic for the summands in the formula
(\ref{9}) (all the constants involved depend on $\gamma_i$ and $C_i$,
but do not depend on $R$ and $\alpha$):
$$
\int\limits_0^{R}\frac{n(t)}{t}dt\asymp
\int\limits_0^{R}t^{-1+1/\alpha}dt
\asymp \alpha R^{1/\alpha},\qquad R\to\infty
$$
$$
R^2 \int\limits_{R}^\infty \frac{n(t)}{t^3}dt \asymp 
R^2 \int\limits_{R}^\infty t^{-3+1/\alpha} dt \asymp
\frac{\alpha}{2\alpha-1}R^{1/\alpha}
$$
and, finally,
$$
R\sum\limits_{|z_n|>R}\frac{x_n}{|z_n|^2}\asymp
R\sum\limits_{n=[R^{1/\alpha}]}^\infty \frac{1}
{n^{1/\alpha}}\asymp \frac{R^{1/\alpha}}{\alpha-1}.
$$
Note that $\alpha+\frac{\alpha}{2\alpha-1}=
o\left(\frac{1}{\alpha-1}\right)$, $\alpha\to 1$,
and $\frac{1}{\alpha-1}=o(\alpha)$, $\alpha\to\infty$.
Hence, the coefficient at $R^{1/\alpha}$ 
in the formula for $\log|E(3R)|$ is positive 
when $\alpha$ is sufficiently large and negative when
$\alpha$ is close to 1. $\bigcirc$
\medskip
\par
In the case of regular distribution along a single ray 
we obtain an explicit formula for the limit exponential.
\medskip
\\
{\bf Proof of Theorem 6.} Here we apply the Levin -- Pfluger
theory of entire functions of completely regular growth
(see \cite{lev}).
Recall that now $z_n=n^{\alpha}(1+i\gamma)$, $\gamma>0$.
Then for any $\theta$, $\psi\in (-\pi,\pi]$
there exists the limit
$$
\Delta(\theta,\psi)=\lim\limits_{R\to\infty}\frac
{n(R,\theta,\psi)}{R^{1/\alpha}},
$$
where $n(R,\theta,\psi)$ denotes the number of zeros 
$z_n$ in the sector $\{|z|<R,\,\arg z\in [\theta,\psi)\}$.
Note that $\Delta(\theta,\psi)=0$, ${\rm arctg}\,\gamma\notin
[\theta,\psi)$,
and $\Delta(\theta,\psi)=C(\gamma)$, ${\rm arctg}\,\gamma \in [\theta,\psi)$,
where $C(\gamma)$ is some positive constant.
Hence, $E$ is a function of completely regular growth
and 
$$
\lim\limits_{R\to\infty}
\frac{\log|E(R)|}{R^{1/\alpha}}=H_\alpha=\frac{\pi C(\gamma)}
{\sin \frac{\pi}{\alpha}} \cos\frac{\pi-{\rm arctg}\,\gamma}{\alpha}.
$$
Thus, $H_\alpha>0$ ($H_\alpha<0$) if and only if
$\alpha>2-\frac{2{\rm arctg}\,\gamma}{\pi}$ 
($\alpha<2-\frac{2{\rm arctg}\,\gamma}{\pi}$). $\bigcirc$
\medskip
\par
We conclude this section with an example illustrating 
the subtlety of the growth of a function with power growth
of zeros.
\medskip
\\
{\bf Theorem 10.}
{\it 
For any $\alpha>1$ there exist $z_n$
such that $\arg z_n=\pi/4$, $|z_n|\asymp n^\alpha$ 
and $1\notin {\cal H}(E)$. }
\smallskip
\\
{\bf Proof.}
Take an integer $K>2$ and let $E$ be the entire 
function of the form (\ref{2}) with zeros at the points 
$K^{\alpha l} e^{-i\pi/4}$
with multiplicities equal to $K^l-K^{l-1}$, $l\in\mathbb{N}$.
Now, if $z_1$, $z_2,\dots,z_n\dots$ are zeros of $E$ repeated according
to the multiplicities, then $|z_n|\asymp n^\alpha$ (actually, 
we have replaced the group of zeros $n^\alpha e^{-i\pi/4}$,
$K^{l-1}<n\le K^l$, by a single zero of the corresponding 
multiplicity).

Let $N\in\mathbb{N}$ and put $t_N=K^{\alpha N-\frac{1}{2}}$.
We show that one can choose a sufficiently large $K$
such that
\begin{equation}
\label{13}
\lim\limits_{N\to\infty}|E(3t_N)|=0.
\end{equation}
Then $1/E\notin L^2(\mathbb{R})$ since $E'/E\in L^\infty(\mathbb{R})$
and (\ref{13}) implies that $\log|E|$ is negative
on the intervals $(3t_N-\delta, 3t_N+\delta)$
for some $\delta>0$ and for sufficiently large $N$.
                
By Proposition 9 
$$
\log|E(3t_N)|^2= B_1\int\limits_0^{|t_N|}\frac{n(t)}{t}dt+
B_2n(t_N)+
\qquad\qquad\qquad
$$
\begin{equation}
\label{14}
\qquad\qquad
+B_3 t_N^2
\sum\limits_{|z_n|>t_N}\frac{1}{|z_n|^2}
- B_4|t_N|\sum\limits_{|z_n|\ge t_N} 
\frac{|x_n|}{|z_n|^2},
\end{equation}
where $B_1(t_N)+B_2(t_N)+B_3(t_N)\le C$ 
and $B_4(t_N)\ge c$
for some absolute positive constants $C$ and $c$.
Now we estimate the summands in the formula (\ref{14}).
Note that $K^{\alpha(N-1)}<t_N< K^{\alpha N}$ and
$$
n(t_N)=\sum\limits_{l=1}^{N-1}K^l(K-1)= K^{N}-K.
$$
Hence,
$$
\int\limits_0^{|t_N|}\frac{n(t)}{t}dt
=\log\frac{t_N^{n(t_N)}}{|z_1z_2\dots z_{n(t_N)}|}=
$$
$$
=\left[(K^N-K)(\alpha N-\frac{1}{2})-\alpha \sum_{l=1}^{N-1}lK^l(K-1)
\right]\log K < 2\alpha K^N\log K.
$$
Also, we have 
$$
t_N^2 \sum\limits_{|z_n|>t_N}\frac{1}{|z_n|^2}=
K^{2\alpha N-1}\sum\limits_{l=N}^{\infty}
\frac{K^l(K-1)}{K^{2\alpha l}}< 2 K^N
$$
Now we estimate the negative part of $\log|E(t_N)|^2$:
$$
t_N \sum\limits_{|z_n|\ge t_N} 
\frac{|x_n|}{|z_n|^2}=
K^{\alpha N-\frac{1}{2}}
\sum\limits_{l=N}^{\infty}
\frac{K^l(K-1)}{\sqrt{2} K^{\alpha l}}\ge 
\frac{K^{n+\frac{1}{2}}}{\sqrt{2}}.
$$
Thus,
$$
\log|E(3t_N)|^2 \le C(3K^N+
2\alpha K^N\log K)- \frac{c}{\sqrt{2}} 
K^{N+\frac{1}{2}},
$$
and taking a sufficiently large $K$ we can make
$\log|E(3t_N)|\to-\infty$, $N\to\infty$.
$\bigcirc$
\bigskip
\\
\begin{center}
{\bf \S 5. Hypercyclic operators in the de Branges spaces}
\end{center}

To prove Theorem 7 we need an auxiliary class of spaces introduced
in \cite{chsh}. Let $\gamma(z)=\sum_{n=0}^\infty \gamma_n z^n$
be an entire function such that $\gamma_n>0$ for each $n\ge 0$
and the sequence $n \gamma_{n}/\gamma_{n-1}$
decreases when $n$ tends to infinity (in this case we say that $\gamma$
is an admissible comparison function).
Consider the space $E^2(\gamma)$ of all entire functions $f$
such that
$$
f(z)=\sum\limits_{n=0}^\infty f_nz^n
$$
with
$$
\|f\|_{2,\,\gamma}=\sum\limits_{n=0}^\infty \gamma_n^{-2}|f_n|^2<\infty.
$$ 
Then $E^2(\gamma)$ is a Hilbert space. Moreover, it is easy to see  
that the condition $\sup_n n \gamma_{n}/\gamma_{n-1}<\infty$
implies that the differentiation operator ${\cal D}$ 
is bounded on $E^2(\gamma)$.
K.C. Chan and J.H. Shapiro have showed that translations are hypercyclic
in $E^2(\gamma)$; they also have obtained a much more general 
result. 
\medskip
\\
{\bf Theorem} (K.C. Chan, J.H. Shapiro \cite{chsh}){\bf.}
{\it Suppose that $X$ is a Fr\'echet space of 
entire functions with the 
following properties:
\par
1. ${\cal P}\subset X$ and ${\rm Clos}_X {\cal P}=X$;
\par 
2. the topology of $X$ is stronger than the topology of uniform convergence 
on compact subsets of the plane;
\par
3. $T_w$ is bounded on $X$;
\par
4. $E^2(\gamma)\subset X$ for some admissible comparison 
function $\gamma$.
\\
Then $T_w$, $w\ne 0$, is hypercyclic on $X$. }
\medskip

To apply the Chan -- Shapiro theorem to the de Branges spaces 
satisfying conditions of Theorem 7 
(that is, ${\rm Clos}_{{\cal H}(E)} {\cal P}={\cal H}(E)$
and ${\cal D}$ is bounded in ${\cal H}(E)$) we need to verify only the last 
condition. Let $r_n=\|z^n\|_{{\cal H}(E)}$ and take a sequence 
$\gamma_n>0$ such that the sequence
$n \gamma_{n}/\gamma_{n-1}$ decreases and 
$\{r_n\gamma_n\}\in \ell^2(\mathbb{Z}_+)$. 
We show that
$E^2(\gamma)\subset {\cal H}(E)$. Let $f\in E^2(\gamma)$,
$f(z)=\sum_n f_nz^n$. Then, for any $y>0$, we have
$$
\left\|\frac{f}{E}(t+iy)\right\|_2\le
\sum\limits_n |f_n|\left\|\frac{(t+iy)^n}{E(t+iy)}\right\|_2
\le \sum\limits_n |f_n|r_n
$$
since the functions $z^n/E$ are in the Hardy class $H^2(\mathbb{C^+})$.
Hence,
$$
\sup\limits_{y>0}
\left\|\frac{f}{E}(\cdot+iy)\right\|_2\le \|f\|_{2,\,\gamma}
\|r_n\gamma_n\|_{\ell^2}
$$
and $f/E\in H^2(\mathbb{C^+})$. Analogously, 
$f^*/E\in H^2(\mathbb{C^+})$. Thus,
we get the inclusion $E^2(\gamma)\subset {\cal H}(E)$, and
Theorem 7 is proved.
\medskip
\\
{\bf Remark.} As shown in \cite{chsh}, an interesting feature
of the spaces $E^2(\gamma)$ is that they provide examples of hypercyclic 
operators which are compact or even Schatten class perturbations
of the identity operator $I$. Indeed, choosing the sequence
$\gamma_n$ tending to zero rapidly one can ensure that
the differentiation operator in $E^2(\gamma)$ belongs
to all Schatten -- von Neumann classes
and so does the operator $T_w-I$.

The same is true also for "larger" de Branges spaces.
In \cite{bar3} a number of examples
are constructed showing that the operator ${\cal D}$
in ${\cal H}(E)$ may be compact or belong to all Schatten -- von Neumann 
classes. In particular, ${\cal D}$ is always compact
if the zeros of $E$ lie in a Stolz angle.




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\end{thebibliography}
\bigskip
Address:
Saint Petersburg State University,\\
Department of Mathematics and Mechanics,\\
28, Universitetski pr., St. Petersburg,\\
198504, Russia
\bigskip
\\
E-mail: {antonbaranov@netscape.net}


\end{document}
