* [16] Fidaleo F., Mukhamedov F. Strict weak mixing of some $C^{*}$-dynamical systems based on free shifts, J. Math. Anal. Appl. 336 (2007), 180-187.
  * [17] Fidaleo F., Zsido L. Quantitative BT-Theorem and automatic continuity for standard von Neumann algebras, Adv. Math. 289 (2016), 1236-1260.
  * [18] Furstenberg H. Strict ergodicity and transformation on the torus, Amer. J. math. 83 (1961), 573-601.
  * [19] Haagerup U. The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271-283.
  * [20] R. Longo, C. Peligrad _Noncommutative topological dynamics and compact actions on $C^{*}$-algebras_, J. Funct. Anal. 58 (1984), 157-174.
  * [21] Longo R., Rehren K. H. Nets of Subfactors Rev. Math. Phys. 7 (1995) 567-598.
  * [22] Murray F. J., von Neumann J. On rings of operators, Ann. Math. 37 (1936), 116-229.
  * [23] Niculescu C. P., Ströh A., Zsidó L. Noncommutative extension of classical and multiple recurrence theorems, J. Operator Theory 50 (2003), 3-52.
  * [24] Osaka H., Phillips N. C. _Furstenberg transformations on irrational rotation algebras_ , Ergod. Theory Dyn. Syst. 26 (2006), 1623-1651.
  * [25] Sakai S. _$C^{*}$ -Algebras and $W^{*}$-Algebras_, Springer-Verlag, Berlin, 1971.
  * [26] Strǎtilǎ Ş., L. Zsidó L. Lectures on von Neumann algebras, Abacus Press, Tunbridge Wells, Kent, (1979).
  * [27] Takesaki M. Duality for crossed products and the structure of von Neumann algebras of type $\mathop{\rm{III}}$, Acta Math. 131 (1973), 249-310.
  * [28] Takesaki M. Faithful states on a $C^{*}$-algebra, Pacific J. Math. 52 (1973), 605-610.
  * [29] Takesaki M. Theory of operator algebras I, Springer, Berlin-Heidelberg-New York, 1979.
  * [30] Walters P. An introduction to ergodic theory, Springer, Berlin-Heidelberg-New York, 1982.


# Mate-Nevai-Totik theorem for Krein systems

Pavel Gubkin  Pavel Gubkin<EMAIL_ADDRESS>Leonhard Euler
International Mathematical Institute 14th Line 29B, Vasilyevsky Island, St.
Petersburg, 199178, Russia St. Petersburg State University Universitetskaya
nab. 7-9, St. Petersburg, 199034, Russia

###### Abstract.

We prove the Cesàro boundedness of eigenfunctions of the Dirac operator on the
half-line with a square-summable potential. The proof is based on the theory
of Krein systems and, in particular, on the continuous version of a theorem by
A. Mate, P. Nevai and V. Totik from 1991.

###### Key words and phrases:

Krein system, Dirac operator, orthogonal polynomials, Szegő class

###### 2010 Mathematics Subject Classification:

34L40, 42C05

The work is supported by Ministry of Science and Higher Education of the
Russian Federation, agreement № 075–15–2019–1619.

##  1\. Introduction

###  1.1. Main result

Consider a symmetric zero trace $2\times 2$ potential
$Q=\left(\begin{smallmatrix}-q&p\\\ p&q\end{smallmatrix}\right)$ on
$\mathbb{R}_{+}=[0,+\infty)$ with real-valued entries. We say that $Q\in
L^{p}$ if both functions $p,q\in L^{p}$. Consider the differential equation

(1) $\displaystyle Jf^{\prime}(t)+Qf(t)$ $\displaystyle=\lambda f(t),\quad
t\in\mathbb{R}_{+},\quad f(0)=\begin{pmatrix}1\\\ 0\end{pmatrix},$

where $J=\left(\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\right)$ is a
square root of the identity matrix. This equation can be read as eigenfunction
equation for Dirac operator $\mathcal{D}_{Q}=J\frac{d}{dr}+Q$ on the half-line
$\mathbb{R}_{+}$.

Motivated by scattering theory, M. Christ and A. Kiselev studied [4], [5], [6]
the one-dimensional Shrödinger operator using a method of multilinear
expansion; their results can as well be applied to Dirac-type operators. Among
other things, they proved the existence of wave operators for the Dirac
operator in the case when $Q\in L^{p}$, $1\leqslant p<2$, which implies that
for such Q the absolutely continuous spectrum of $D_{Q}$ is the entire real
line $\mathbb{R}$. Deift and Killip extended [7] the spectrum result for $Q\in
L^{2}$. Using methods of the theory of Krein systems, Denisov showed [9] the
existence of wave operators in the case when $Q\in L^{2}$; recently Bessonov
extended [2] this result for Dirac operators whose spectral measure belongs to
the Szegő class on the real line. Some related results and approaches can be
found in [18], [19], [22].

M. Christ and A. Kiselev proved in Section $6.3$ in [4] that the solution of
(1) is bounded for almost every $\lambda\in\mathbb{R}$ in the case when $Q\in
L^{p}$, $1\leqslant p<2$. Related question was studied by C. Muscalu, T. Tao,
and C. Thiele in [18]; authors posed the following conjecture.

######  Conjecture I.

Assume that $Q\in L^{2}$. Then for almost every $\lambda\in\mathbb{R}$ the
solution $f$ of the differential equation (1) is bounded on $\mathbb{R}_{+}$.

In present paper we prove an ‘‘averaged’’ version of Conjecture I.

######  Theorem 1.

Assume that $Q\in L^{2}$. Then for almost every $\lambda\in\mathbb{R}$ the
solution $f$ of the differential equation (1) is bounded on $\mathbb{R}_{+}$
in $L^{2}$-Cesàro sense, i.e.,

$\displaystyle\sup_{r>0}\left(\frac{1}{r}\int_{0}^{r}|f(t)|^{2}\,dt\right)<\infty.$

Proof of this theorem is based on the theory of Krein systems which plays a
role of a bridge between the theory of orthogonal polynomials on the unit
circle and spectral theory of second order differential self-adjoint
operators. Krein systems were first introduced by M. G. Krein in [15] and
later developed by different authors (see, e.g., [20], [21], [8], [25], [3]).
Detailed account of the theory can be found in [12].

Let us start with a description of basic objects and results in the theory of
orthogonal polynomials and in the theory of the Krein systems.

###  1.2. Orthogonal polynomials on the unit circle

Orthogonal polynomials on the unit circle appear in many areas of mathematics.
An account of the theory can be found in books [24] by G. Szegő and [23] by B.
Simon. Let $\mu$ be a probability measure on the interval $[-\pi,\pi]$ such
that the support of $\mu$ is not a finite set. Then there exists a family of
orthonormal polynomials $\\{\varphi_{n}\\}_{n\geqslant 0}$ in
$L^{2}([\pi,\pi],\mu)$. More precisely, let the family $\varphi_{n}$ be
defined by

(2)
$\displaystyle\deg\varphi_{n}=n,\quad\langle\varphi_{n},\varphi_{m}\rangle_{L^{2}(\mu)}=\int_{-\pi}^{\pi}\varphi_{n}(e^{it})\overline{\varphi_{m}(e^{it})}\,d\mu(t)=\delta_{n,m},$

where $\delta_{n,m}$ is the Kronecker symbol. Define a family of reversed
orthogonal polynomials by

$\displaystyle\varphi_{n}^{*}(z)=z^{n}\overline{\varphi_{n}(1/\overline{z})},\quad
z\in\mathbb{C}.$

It is easy to see that $\varphi_{n}^{*}$ is a polynomial of degree not greater
that $n$. We say that the measure $\mu$ belongs to the Szegő class on the unit
circle if

(3) $\displaystyle\int_{-\pi}^{\pi}\log\mu^{\prime}(\theta)\,d\theta>-\infty,$

where $\mu^{\prime}$ is the density of the absolutely continuous part of $\mu$
with respect to the Lebesgue measure on $[-\pi,\pi]$. The following theorem is
a result of work of different authors, we refer to it as the Szegő theorem.
This statement is a combination of Theorem 2.4.1 and Theorem 2.7.15 from [23].

######  Theorem A (Szegő theorem).

Let $\mu$ be a nontrivial measure on $[-\pi,\pi]$ and let $\varphi_{n}$ be the
associated orthogonal polynomials. Then the following conditions are
equivalent:

  1. $(a)$

the measure $\mu$ belongs to the Szegő class,

  2. $(b)$

for some $z_{0}$ such that $|z_{0}|<1$ we have
$\sum\limits_{n=0}^{\infty}|\varphi_{n}(z_{0})|^{2}<\infty$,

  3. $(c)$

there exists an analytic function $\Pi$ in the unit disc
$\mathbb{D}=\\{z\in\mathbb{C}\colon|z|<1\\}$ such that

$\displaystyle\Pi(z)=\lim_{n\to\infty}\varphi_{n}^{*}(z),\quad
z\in\mathbb{D},$

and the convergence is uniform on compact subsets in $\mathbb{D}$.

If the equivalent conditions of the Szegő theorem hold, then the series in
part $(b)$ converges uniformly on compact subsets of $\mathbb{D}$. Moreover,
the function $\Pi$ is an outer function in the unit disc (see Theorem 2.3.5
[23]) and $|\Pi(e^{it})|^{-2}=\mu^{\prime}(t)$ almost everywhere on
$[-\pi,\pi]$, i.e,

$\displaystyle\Pi(z)$
$\displaystyle=\exp\left[-\frac{1}{4\pi}\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\log\mu^{\prime}(\theta)\,d\theta\right],\quad
z\in\mathbb{D}.$

The function $D=\Pi^{-1}$ is called the Szegő function and often is used
instead of $\Pi$.

###  1.3. Krein systems

Krein systems were first considered by M. G. Krein [15] in 1954. Solutions of
the Krein systems have properties similar to properties of orthogonal
polynomials on the unit circle. Through such similarities it becomes possible
to apply methods and ideas from the theory of orthogonal polynomials on the
unit circle to spectral problems for self-adjoint differential operators.
Detailed account of this approach can be found in the paper [12] by S.
Denisov.

######  Definition.

Let $a$ be a complex-valued function on the half-line $\mathbb{R}_{+}$. The
Krein system with the coefficient $a$ is the following system of differential
equations:

(4) $\displaystyle\begin{cases}\frac{\partial}{\partial
r}P(r,\lambda)=i\lambda
P(r,\lambda)-\overline{a(r)}P_{*}(r,\lambda),&\quad\,\,P(0,\lambda)=1,\\\
\frac{\partial}{\partial r}P_{*}(r,\lambda)=-a(r)P(r,\lambda),&\quad
P_{*}(0,\lambda)=1.\end{cases}$

In present paper we consider the case when $a\in
L^{1}_{\mathop{\mathrm{loc}}\nolimits}(\mathbb{R}_{+})$, that is, $a\in
L^{1}[0,x]$ for every $x\geqslant 0$. Here the complex number $\lambda$ is a
spectral parameter; solutions $P$ and $P_{*}$ are called continuous analogs of
orthogonal polynomials on the unit circle.

Solutions $P(r,\cdot)$ and $P_{*}(r,\cdot)$ are entire functions. The
parameter $r\geqslant 0$ plays the same role as the index $n\geqslant 0$ in
(2): $P(r,\cdot)$ has the exponential type $r$ and $P_{*}(r,\cdot)$ has the
exponential type not greater than $r$, see Chapter 3 in [12]. For any function
$a\in L^{1}_{\mathop{\mathrm{loc}}\nolimits}(\mathbb{R}_{+})$ there exists a
Borel measure $\sigma$ on the real line (see [12]) such that

(5) $\displaystyle\int_{\mathbb{R}}\frac{d\sigma(x)}{1+x^{2}}<\infty,$

and the map

$\displaystyle\mathcal{U}_{\sigma}:f\mapsto\int_{0}^{\infty}f(r)P(r,\lambda)\,dr,$

initially defined on simple measurable functions with compact support can be
continued to an isometry from $L^{2}(\mathbb{R}_{+})$ to
$L^{2}(\mathbb{R},\sigma)$. This measure is called the spectral measure of
system (4). We say that the measure $\sigma$ belongs to the Szegő class on the
real line if it satisfies condition (5) and

(6)
$\displaystyle\int_{\mathbb{R}}\frac{|\log\sigma^{\prime}(x)|}{1+x^{2}}\,dx<\infty,$

where $\sigma^{\prime}$ is the density of the absolutely continuous part of
$\sigma$ with respect to the Lebesgue measure on $\mathbb{R}$. A key result in
the theory of the Krein systems in an analog of the Szegő theorem. It was
originally formulated by Krein in [15] with minor inaccuracies subsequently
corrected by A. Teplyaev in [25]. For a complete proof see Chapter 8 in [12].

######  Theorem B (Krein theorem).

Let a measure $\sigma$ be the spectral measure of some Krein system (4) and
let $P,P_{*}$ be the solutions of that system. Then the following conditions
are equivalent:

  1. $(a)$

the measure $\sigma$ belongs to Szegő class,

  2. $(b)$

for a point
$\lambda_{0}\in\mathbb{C}_{+}=\\{z\in\mathbb{C}\mid\mathop{\mathrm{Im}}\nolimits(z)>0\\}$
we have

$\displaystyle\int_{0}^{\infty}|P(r,\lambda_{0})|^{2}\,dr<\infty,$

  3. $(c)$

there exists analytic in $\mathbb{C}_{+}$ function $\Pi$ and a sequence
$r_{n}\to+\infty$ such that

$\displaystyle\Pi(\lambda)=\lim_{n\to\infty}P_{*}(r_{n},\lambda),$

and the convergence is uniform on compact subsets in $\mathbb{C}_{+}$.

If the equivalent conditions of the Krein theorem hold then the integral in
part $(b)$ converges uniformly on compact subsets. Moreover, $\Pi$ is an outer
function in the upper half-plane and

(7) $\displaystyle|\Pi(x)|^{2}=|2\pi\sigma^{\prime}(x)|^{-1}$

almost everywhere on $\mathbb{R}$ (see Lemma 8.6 in [12]), i.e.,

(8) $\displaystyle\Pi(\lambda)=\frac{1}{\sqrt{2\pi}}\exp\left[\frac{1}{2\pi
i}\int\limits_{-\infty}^{\infty}\left(\frac{1}{s-\lambda}-\frac{s}{s^{2}+1}\right)\log\sigma^{\prime}(s)\,ds\right],\quad\lambda\in\mathbb{C}_{+}.$

In addition, the function $[(\lambda+i)\Pi(\lambda)]^{-1}$ belongs to the
Hardy space $H^{2}$.

###  1.4. Mate-Nevai-Totik theorem

A. Mate, P. Nevai, V. Totik in 1991 proved [17] asymptotics of so-called
Christoffel functions. Their result in the language of orthogonal polynomials
can be considered as a refinement of part (c) of the Szegő theorem for points
on the unit circle $\mathbb{T}$. Let us formulate their result. Let $\mu$ be a
probability Borel measure on $[-\pi,\pi]$. For $z\in\mathbb{C}$, define an
$n$-th Christoffel function $w_{n}$ as follows:

$\displaystyle
w_{n}(\mu,z)=\min\left\\{\frac{1}{2\pi|P(z)|^{2}}\int_{-\pi}^{\pi}|P(e^{i\theta})|^{2}\,d\mu(\theta)\text{
}\bigg{|}\text{ }P\text{ -- polynomial, }\deg P<n,P(z)\neq 0\right\\}.$

######  Theorem C (A. Mate, P. Nevai, V. Totik Theorem 1, [17]).

Assume that the measure $\mu$ belongs to the Szegő class on the unit circle.
Then we have

$\displaystyle\lim_{n\to\infty}nw_{n}(\mu,e^{it})=\mu^{\prime}(t)$

for almost every $t\in[-\pi,\pi]$.

The function $w_{n}$ admits the representation

(9) $\displaystyle 2\pi
w_{n}(\mu,z)=\left(\sum\limits_{k=0}^{n-1}|\varphi_{k}(z)|^{2}\right)^{-1},\quad
z\in\mathbb{C},$

where $\varphi_{n}$ are the orthonormal polynomials on the unit circle
associated with $\mu$, see Theorem 11.3.1 in [24]. This representation allows
us to formulate Theorem C using only the orthogonal polynomials.

######  Theorem D.

Let $\mu$ be a nontrivial probability Borel measure on $\mathbb{T}$ and
$\varphi_{n}$ be the orthogonal polynomials associated with $\mu$. If $\mu$
belongs to the Szegő class, then for almost every $t\in[-\pi,\pi]$

$\displaystyle\frac{1}{n}\sum\limits_{k=0}^{n-1}|\varphi_{k}(e^{it})|^{2}\to\frac{1}{2\pi\mu^{\prime}(t)},\quad
n\to\infty.$

The core result of the present paper is the following theorem. It can be seen
as a refinement of part (c) of the Krein theorem for points on the real line.

######  Theorem 2.

Consider a Krein system with a coefficient in
$L^{1}_{\mathop{\mathrm{loc}}\nolimits}(\mathbb{R}_{+})$ such that its
spectral measure $\sigma$ is in the Szegő class on the real line. Then for
almost every $x\in\mathbb{R}$ there exists the limit

$\displaystyle\frac{1}{r}\int_{0}^{r}|P(s,x)|^{2}ds\to\frac{1}{2\pi\sigma^{\prime}(x)},\quad
r\to\infty.$

###  1.5. Teplyaev’s conjectures

Notice that the convergence in part (c) of the Krein theorem holds only for
some sequence of real numbers $r_{n}$ while there is no such constraint in the
part (c) of the Szegő theorem. A. Teplyaev proved [25] that the convergence
$P_{*}(r,\lambda)\to\Pi(\lambda)$ holds for $a\in L^{2}(\mathbb{R}_{+})$, see
Theorem 1 in [25] or Theorem 11.1 in [12]. Additionally, the author proved
sharpness of this result in the sense that it cannot be extended to any
$L^{p}$ for $p>2$, see Theorem 3 in [25]. Teplyaev conjectured two positive
results concerning the convergence of $P_{*}$ to $\Pi$ for a Krein system with
the real-valued coefficient, see Conjecture 6.5, Conjecture 6.6 in [25]. The
first conjecture was proved by S. Denisov in [12], we state it below:

######  Theorem E (Lemma 8.6 [12]).

Assume that the conditions of the Krein theorem hold, the function $a$ is
real-valued, and a sequence $t_{n}\to\infty$ is such that
$P(t_{n},\lambda_{0})\to 0$ for some $\lambda_{0}\in\mathbb{C}_{+}$. Then
$P_{*}(t_{n},\lambda)\to\Pi(\lambda)$ uniformly on compact subsets in
$\mathbb{C}_{+}$.

The second conjecture concerns a convergence of $P_{*}$ in average (i.e. in
the Cesàro sense). We prove it in the present paper.

######  Theorem 3.

Assume $a$ is a real-valued function and conditions of Krein theorem hold.
Then $\Pi(\lambda)$ is the limit in average of $P_{*}(r,\lambda)$, that is,

$\displaystyle\Pi(\lambda)=\lim_{r\to\infty}\frac{1}{r}\int_{0}^{r}P_{*}(\rho,\lambda)\,d\rho,$

and the convergence is uniform on compact subsets in $\mathbb{C}_{+}$.

Notice that the limit in average of $P_{*}(r,\lambda)$ does not exist in
general. In other words, there exists a function $a\in
L^{1}_{\mathop{\mathrm{loc}}\nolimits}$ such that the convergence in Theorem 3
does not hold, see discussion in the end of Section 6 in [25] .

###  1.6. Open problems

It is unknown to us whether the convergence in Theorem D or in Theorem 2 can
be improved. We formulate two related conjectures.

######  Conjecture II.

Assume that the measure $\mu$ on the unit circle belongs to the Szegő class on
the unit circle. Let $\varphi_{n}$ be the orthogonal polynomials associated
with $\mu$. Then for almost every $t\in[-\pi,\pi]$ we have

$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}\varphi_{k}^{*}(e^{it})\to\frac{1}{\sqrt{2\pi\mu^{\prime}(t)}},\quad
n\to\infty.$

This may be connected with works [1], [11], [10].

######  Conjecture III.

Consider Krein system (4). Let $P,P_{*}$ be its solitions and let $\sigma$ be
its spectral measure. Assume that $\sigma$ is in the Szegő class on the real
line. Then there exist a function $\xi\colon\mathbb{R}_{+}\to\mathbb{C}$ such
that the convergence

$\displaystyle\frac{1}{r}\int_{0}^{r}P(s,x)\xi(s)\,ds\to\frac{1}{\sqrt{2\pi\sigma^{\prime}(x)}},\quad
r\to\infty$

holds for almost every $x\in\mathbb{R}$. Additionally, the function $\xi$ can
be chosen real-valued if the coefficient of the Krein system is real-valued.

The latter conjecture may be related to the results of [2].

###  1.7. Structure of the paper

In Section 3 we provide some preliminaries on Krein systems and prove Theorem
2. Proof of Teplyaev conjecture, Theorem 3 is given in Section 4. In the next
section we use Theorem 2 to prove Theorem 1.

##  2\. Dirac equation. Proof of Theorem 1

Let us show how to derive Theorem 1 from Theorem 2 using an appropriate
transformation of Dirac differential equation (1) to Krein system (4). To do
that, we need some notation. Consider system (4) and let the functions
$\varphi,\psi,p,q$ be as follows:

(10) $\displaystyle\varphi(r,\lambda)$ $\displaystyle=\frac{\exp(-i\lambda
r)}{2}\left[P(2r,\lambda)+P_{*}(2r,\lambda)\right],\quad\varphi(0,\lambda)=1,$
(11) $\displaystyle\psi(r,\lambda)$ $\displaystyle=\frac{\exp(-i\lambda
r)}{2i}\left[P(2r,\lambda)-P_{*}(2r,\lambda)\right],\quad\psi(0,\lambda)=0,$
$\displaystyle p(r)$ $\displaystyle=-2\mathop{\mathrm{Re}}\nolimits
a(2r),\qquad q(r)=2\mathop{\mathrm{Im}}\nolimits a(2r).$

Direct calculations give that functions $\varphi$ and $\psi$ are the solutions
of the differential equation

(12) $\displaystyle\lambda\begin{pmatrix}\varphi\\\
\psi\end{pmatrix}=J\begin{pmatrix}\varphi\\\
\psi\end{pmatrix}^{\prime}+Q\begin{pmatrix}\varphi\\\
\psi\end{pmatrix},\,\varphi(0,\lambda)=1,\,\psi(0,\lambda)=0,$ $\displaystyle
J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\quad Q=\begin{pmatrix}-q&p\\\
p&q\end{pmatrix}.$

Now we proceed to the proof of Theorem 1. Construct a Krein system with the
coefficient $a(r)=-\frac{1}{2}p(r/2)+\frac{i}{2}q(r/2)$, where $p,q$ are the
potentials in equation (1). Uniqueness theorem and (12) give that functions
$\varphi$ and $\psi$ defined by (10) and (11) respectively are the unique
solutions of equation (1). Since potentials $p,q$ are in
$L^{2}(\mathbb{R}_{+})$, coefficient $a$ is also in $L^{2}(\mathbb{R}_{+})$.
Spectral measure of the Krein system with square-summable coefficient belongs
to the Szegő class (see Theorem 11.1 in [12]). Hence, Theorem 2 can be applied
and the convergence

$\displaystyle\frac{1}{r}\int_{0}^{r}|P(s,x)|^{2}\,ds\to(2\pi\sigma^{\prime}(x))^{-1},\quad\text{
as }r\to\infty,$

holds for almost every $x\in\mathbb{R}$. Consequently, for almost every
$x\in\mathbb{R}$

$\displaystyle\sup_{r>0}\frac{1}{r}\int_{0}^{r}|P(s,x)|^{2}\,ds<\infty.$

Finally,

$\displaystyle\sup_{r>0}\frac{1}{r}\int_{0}^{r}|\varphi(s,x)|^{2}ds$
$\displaystyle=\sup_{r>0}\frac{1}{r}\int_{0}^{r}\left|\frac{\exp(-ixs)}{2}\left[P(2s,x)+P_{*}(2s,x)\right]\right|^{2}ds$
$\displaystyle=\sup_{r>0}\frac{1}{r}\int_{0}^{r}\left|\frac{P(2s,x)+P_{*}(2s,x)}{2}\right|^{2}ds$
$\displaystyle\leqslant\sup_{r>0}\frac{1}{r}\int_{0}^{r}\frac{\left|P(2s,x)\right|^{2}+\left|P_{*}(2s,x)\right|^{2}}{2}\,ds$
$\displaystyle=\sup_{r>0}\frac{1}{r}\int_{0}^{r}\left|P(2s,x)\right|^{2}ds<\infty$

The equality on the last line follows from Corollary 1 to Christoffel-Darboux
formula (see Section 3.1 below). Same argument works for $\psi$.

##  3\. Asymptotic behaviour of Krein system solutions on the real line. Proof
of Theorem 2

###  3.1. Preliminaries

Christoffel-Darboux formula is one of the most important identities in the
theory of orthogonal polynomials on the unit circle, for details see Theorem
2.2.7 in [23] or in Theorem 11.4.2 in [24]. We will use its continuous analog.

######  Lemma (Christoffel-Darboux formula).

Let $P$ and $P_{*}$ be solutions of the Krein system (4).The equalities

(13) $\displaystyle
P(r,\lambda)\overline{P(r,\mu)}-P_{*}(r,\lambda)\overline{P_{*}(r,\mu)}=i(\lambda-\overline{\mu})\int_{0}^{r}P(s,\lambda)\overline{P(s,\mu)}\,ds,$
(14)
$\displaystyle\left|P_{*}(r,\lambda)\right|^{2}=\left|P(r,\lambda)\right|^{2}+2\mathop{\mathrm{Im}}\nolimits(\lambda)\int_{0}^{r}\left|P(s,\lambda)\right|^{2}ds$

hold for any two complex numbers $\lambda,\mu$.

###### Proof.

Equation (14) follows from (13) by substituting $\mu=\lambda$; (13) can be
proved by calculation of derivatives in both sides of the equality. ∎

######  Corollary 1.

For any $\lambda\in\mathbb{C}$ the following is true:

  1. (1)

$|P_{*}(r,\lambda)|=|P(r,\lambda)|$ if $\lambda\in\mathbb{R}$

  2. (2)

$|P_{*}(r,\lambda)|>|P(r,\lambda)|$ if
$\mathop{\mathrm{Im}}\nolimits(\lambda)>0$

  3. (3)

$|P_{*}(r,\lambda)|<|P(r,\lambda)|$ if
$\mathop{\mathrm{Im}}\nolimits(\lambda)<0$

###### Proof.

All three statements easily follow from the second Christoffel-Darboux formula
(14). ∎

######  Corollary 2.

$P_{*}(r,\cdot)$ has no zeros in closed half-plane
$\overline{\mathbb{C}_{+}}=\\{z\colon\mathop{\mathrm{Im}}\nolimits z\geqslant
0\\}$.

###### Proof.

From Part 2 of Corollary 1 it follows that there are no roots in an open half-
plane $\mathbb{C}_{+}$. Hence, to prove Corollary 2 we need to show the
absence of zeros on the real line. Assume the converse. Then there exists
$x\in\mathbb{R}$ such that $P_{*}(r,x)=0$. Part 1 of Corollary 1 gives

(15) $\displaystyle P_{*}(r,x)=P(r,x)=0.$

On the other hand, the functions $P_{*}(\cdot,x),P(\cdot,x)$ are the solutions
of linear differential system (4) with non-zero initial values. Hence, (15)
cannot hold. This contradiction concludes the proof. ∎

######  Lemma A (S. Denisov Lemma 8.5, [12]).

If a measure $\sigma$ belongs to the Szegő class and a sequence $r_{n}$ is
such that $P(r_{n},\lambda_{0})\to 0$ for some $\lambda_{0}\in\mathbb{C}_{+}$.
Then the convergence
$\left|P_{*}(r_{n},\lambda)\right|\to\left|\Pi(\lambda)\right|$ holds
uniformly in $\mathbb{C}_{+}$.

######  Corollary 3.

From equation (14) and Lemma A we have

$\displaystyle|\Pi(\lambda)|^{2}=2\mathop{\mathrm{Im}}\nolimits(\lambda)\int_{0}^{\infty}\left|P(s,\lambda)\right|^{2}ds.$

######  Remark.

A similar relation in the theory of the orthogonal polynomials is given in
Chapter 2.4 in [23].

###  3.2. An equivalent form of Theorem 2

In this section we formulate Theorem 2 in a form similar to original Mate-
Nevai-Totik Theorem C. By $PW_{r}$ denote the Paley-Wiener space with a
spectrum in $[0,r]$, that is, a space of entire functions $f$ that can be
represented as

$\displaystyle f(x)=\int_{0}^{r}\varphi(s)e^{ixs}\,ds,\quad\varphi\in
L^{2}[0,r].$

Let a function $m_{r}$ be given by

(16) $\displaystyle m_{r}(\sigma,z)$
$\displaystyle=\inf\left\\{\frac{1}{|f(z)|^{2}}\int_{-\infty}^{\infty}|f(t)|^{2}\,d\sigma(t)\text{
}\bigg{|}\text{ }f\in PW_{r},f(z)\neq 0\right\\},\quad z\in\mathbb{C}.$

Similarly to (9), the function $m_{r}$ can be calculated using the continuous
analogs of the orthogonal polynomials. This is shown in the following lemma.

######  Lemma B (S. Denisov, Lemma 8.2, [12]).

For any $z_{0}\in\mathbb{C}$ the following is true:

$\displaystyle m_{r}(\sigma,z_{0})=K_{r}(z_{0},z_{0})^{-1},\quad\text{ where
}K_{r}(z^{\prime},z)=\int_{0}^{r}\overline{P(s,z^{\prime})}P(s,z)\,ds.$

Infimum in (16) is achieved by function

$\displaystyle f_{r}(z)$
$\displaystyle=\frac{K_{r}(z_{0},z)}{K_{r}(z_{0},z_{0})},$

in particular, function $f_{r}$ itself belongs to $PW_{r}$.

Lemma gives

$\displaystyle m_{r}(\sigma,z)$
$\displaystyle=K_{r}(z,z)^{-1}=\left(\int_{0}^{r}|P(s,z)|^{2}\,ds\right)^{-1},$
$\displaystyle rm_{r}(\sigma,z)$
$\displaystyle=\left(\frac{1}{r}\int_{0}^{r}|P(s,z)|^{2}\,ds\right)^{-1}.$

Substituting the latter equality into Theorem 2 we get its equivalent form.

######  Theorem 4 (Equivalent form of Theorem 2).

Consider a Krein system with a coefficient in
$L^{1}_{\mathop{\mathrm{loc}}\nolimits}(\mathbb{R}_{+})$ such that its
spectral measure $\sigma$ is in the Szegő class on the real line. Then for
almost all $z\in\mathbb{R}$:

$\displaystyle\lim_{r\to\infty}rm_{r}(\sigma,z)=2\pi\sigma^{\prime}(z).$

We prove Theorem 4 by establishing two inequalities

(17) $\displaystyle\limsup rm_{r}(\sigma,z)$ $\displaystyle\leqslant
2\pi\sigma^{\prime}(z),$ (18) $\displaystyle\liminf rm_{r}(\sigma,z)$
$\displaystyle\geqslant 2\pi\sigma^{\prime}(z).$

in Sections 3.3 и 3.5 respectively.

###  3.3. Upper bound

######  Lemma 1.

Assume that the measure $\sigma$ on the real line satisfies condition (5).
Then the inequality

$\displaystyle\limsup rm_{r}(\sigma,z)\leqslant 2\pi\sigma^{\prime}(z)$

holds for almost every $z\in\mathbb{R}$.

######  Remark.

In the lemma we do not require $\sigma$ to belong to the Szegő class. Hence,
the inequality in the lemma is stronger than (17).

###### Proof.

Consider functions

$\displaystyle R(x)$
$\displaystyle=\frac{e^{ix}-1}{ix}=\mathcal{F}\left(\mathbf{1}_{[0,1]}\right)(x),$
$\displaystyle\Phi(x)$ $\displaystyle=|R(x)|^{2}.$

Notice that

$\displaystyle\mathcal{F}^{-1}\left(\Phi\right)=\mathcal{F}^{-1}\left(|R|^{2}\right)=\mathcal{F}^{-1}\left(R\right)*\mathcal{F}^{-1}\left(\overline{R}\right)=\mathbf{1}_{[0,1]}*\mathbf{1}_{[-1,0]}=\begin{cases}0,&|x|>1\\\
1-|x|,&|x|\leqslant 1\end{cases},$

in other words, $\Phi$ is a continuous Fejer kernel. Approximation identity
constructed with function $\Phi$ has powerful convergence properties. We
believe the following result is folklore.

######  Theorem F (Continuous Fejer kernel property).

The convergence

$\displaystyle\left(\Phi_{r}*\sigma\right)(z)\to 2\pi\sigma^{\prime}(z),\quad
r\to\infty,$

where $\Phi_{r}(x)=r\Phi(rx)$, holds for almost every $z\in\mathbb{R}$.

Let us show how the lemma follows from Theorem F. The function
$f_{r,z}(x)=R(r(x-z))$ is in $PW_{r}$ and $f_{r,z}(z)=1$. Hence

$\displaystyle rm_{r}(\sigma,z)\leqslant
r\int_{-\infty}^{\infty}|f_{r,z}(x)|^{2}\,d\sigma(x)=r\int_{-\infty}^{\infty}|R(r(x-z))|^{2}\,d\sigma(x)=(\Phi_{r}*\sigma)(z).$

It is only remains to pass to the limit superior on $r\to\infty$ and apply the
property of the continuous Fejer kernel to conclude the proof. ∎

###  3.4. Decay of functions from Paley-Wiener space

In this section we prove result concerning functions of Paley-Wiener space; it
is independent of the theory of Krein systems but will be useful to us. Let
the elementary Weierstrass factors $E_{n}$ be defined by formula

$\displaystyle E_{n}(z)=\begin{cases}(1-z)&n=0,\\\
(1-z)\exp\left(\frac{z^{1}}{1}+\frac{z^{2}}{2}+\cdots+\frac{z^{n}}{n}\right)&n>0.\end{cases}$

######  Theorem (Hadamard factorization theorem).

Let $f$ be an entire function of finite order $\rho\geqslant 0$ and
$\\{a_{n}\\}_{n=0}^{\infty}$ be its non-zero roots with multiplicity. Then $f$
admits a factorization

$\displaystyle
f(z)=e^{g(z)}z^{m}\prod_{k=0}^{\infty}E_{d}\left(\frac{z}{a_{k}}\right),$

where $g$ is a polynomial with degree not greater than $\rho$ and $d=[\rho]$.

######  Lemma 2.

Let $f$ be a function from $PW_{r}$. Suppose $f(0)\neq 0$ and every zero of
$f$ is a real number. Then the inequality

$\displaystyle|f(t)|\leqslant|f(t+\gamma i)|e^{\gamma r}$

holds for every $t\in\mathbb{R}$ and $\gamma>0$.

###### Proof.

Denote roots of $f$ with multiplicity by $\\{a_{n}\\}_{n=0}^{\infty}$.
Functions from Paley-Wiener space are of order $1$. Hence, by the Hadamard
factorization theorem, it follows that $f$ can be represented as

(19) $\displaystyle
f(z)=e^{c_{1}z+c_{2}}\prod_{k=0}^{\infty}\left(1-\frac{z}{a_{k}}\right)e^{\frac{z}{a_{k}}}.$

If $f(t)=0$ the inequality in lemma is trivial, otherwise we can write

$\displaystyle\left|\frac{f(t+\gamma i)}{f(t)}\right|$
$\displaystyle=\left|e^{c_{1}\cdot\gamma
i}\right|\prod_{k=0}^{\infty}\left|\frac{1-\frac{t+\gamma
i}{a_{k}}}{1-\frac{t}{a_{k}}}e^{\frac{\gamma
i}{a_{k}}}\right|=\left|e^{c_{1}\cdot\gamma
i}\right|\prod_{k=0}^{\infty}\left|\frac{a_{k}-t-\gamma i}{a_{k}-t}\right|$
(20) $\displaystyle=\left|e^{c_{1}\cdot\gamma
i}\right|\prod_{k=0}^{\infty}\left|1-\frac{\gamma
i}{a_{k}-t}\right|=e^{-\mathop{\mathrm{Im}}\nolimits(c_{1})\gamma}\prod_{k=0}^{\infty}\left|1+\frac{\gamma^{2}}{(a_{k}-t)^{2}}\right|^{1/2}\geqslant
e^{-\mathop{\mathrm{Im}}\nolimits(c_{1})\gamma}.$

Choose a large real number $N$ and substitute $-iN$ for $z$ into (19).

$\displaystyle|f(-iN)|$
$\displaystyle=\left|e^{-c_{1}iN+c_{2}}\prod_{k=0}^{\infty}\left(1-\frac{-iN}{a_{k}}\right)e^{\frac{-iN}{a_{k}}}\right|$
$\displaystyle=e^{N\mathop{\mathrm{Im}}\nolimits
c_{1}+\mathop{\mathrm{Re}}\nolimits
c_{2}}\prod_{k=0}^{\infty}\sqrt{1+\frac{N^{2}}{a_{k}^{2}}}\geqslant
e^{N\mathop{\mathrm{Im}}\nolimits c_{1}+\mathop{\mathrm{Re}}\nolimits c_{2}}.$

Since $f$ belongs to $PW_{r}$, its exponential type is not greater than $r$.
So

$\displaystyle
r\geqslant\limsup_{N\to\infty}\frac{\log|f(-iN)|}{N}\geqslant\limsup_{N\to\infty}\frac{N\mathop{\mathrm{Im}}\nolimits
c_{1}+\mathop{\mathrm{Re}}\nolimits c_{2}}{N}=\mathop{\mathrm{Im}}\nolimits
c_{1}.$

The latter inequality together with inequality $\eqref{im(c_1)_ineq}$
completes the proof. ∎

###  3.5. Lower bound

Throughout this section we suppose that the measure $\sigma$ belongs to the
Szegő class on the real line. Consider the Szegő function $D$ defined by the
equality

(21) $\displaystyle D(z)=\left(\sqrt{2\pi}\Pi(z)\right)^{-1}.$

Now if we recall (7), we get that

(22) $\displaystyle|D(z)|^{2}$ $\displaystyle=\sigma^{\prime}(z)$

holds for almost all $z\in\mathbb{R}$. Taking this into account, we see that
inequality (18) is equivalent to

(23) $\displaystyle\liminf rm_{r}(\sigma,z)\geqslant 2\pi|D(z)|^{2}.$

The singular part of $\sigma$ only increases the left side in (23) so it is
suffices to show (23) only for absolutely continuous measures $\sigma$. Hence,
without loss of generality, we may assume that $\sigma$ is absolutely
continuous with respect to the Lebesgue measure and $d\sigma(x)=|D(x)|^{2}dx.$

Next, let us introduce some constraints for the point $z$. First, assume that
$z$ is a Lebesgue point of $D$, i.e.,

(24) $\displaystyle\frac{1}{h}\int\limits_{|x-z|<h}|D(x)-D(z)|\,dx\to 0,\quad
h\to 0.$

Secondly, assume that $D$ has non-tangential boundary values in $z$. Both of
this constraints are satisfied on a set of a full Lebesgue measure; without
loss of generality, we may assume that $z$ satisfies the constraints and
$z=0$. Let us remember that $f_{r}$ is a function for which a value of
$m_{r}(\sigma,0)$ is achieved (see Lemma B). That means

(25)
$\displaystyle\int\limits_{\mathbb{R}}\left|f_{r}(x)\right|^{2}d\sigma(x)=\int\limits_{\mathbb{R}}\left|f_{r}(x)D(x)\right|^{2}dx=m_{r}(0)|f_{r}(0)|^{2}.$

Fix $\varepsilon>0$. To prove (23), we need to show that for a large enough
$r$

(26) $\displaystyle rm_{r}(\sigma,0)>(2\pi-100\varepsilon)|D(0)|^{2}.$

In addition, we may assume that

(27) $\displaystyle rm_{r}(\sigma,0)<2\pi|D(0)|^{2},$

otherwise there is nothing left to prove.

####  3.5.1. Supporting lemmas

######  Lemma 3.

Let $f_{r}$ be a function defined in Lemma B. For almost every
$t\in\mathbb{R}$ and $\gamma>0$

$\displaystyle|f_{r}(t)|\leqslant|f_{r}(t+\gamma i)|e^{\gamma r}.$

###### Proof.

It is enough to show that $f_{r}$ satisfies all of the conditions of Lemma 2.
Recall that by definition

$\displaystyle f_{r}(\lambda)$
$\displaystyle=\frac{K_{r}(0,\lambda)}{K_{r}(0,0)},\text{ where
}K_{r}(\lambda^{\prime},\lambda)=\int_{0}^{r}\overline{P(s,\lambda^{\prime})}P(s,\lambda)\,ds.$

From Lemma B we already know that $f_{r}$ belongs to the Paley-Wiener space
$PW_{r}$; obviously $f_{r}(0)=1$. Hence, the only nontrivial question is the
absence of zeros outside of the real line. Assume the converse. Then there
exists $\lambda\notin\mathbb{R}$ such that $f_{r}(\lambda)=0$. Therefore,

$\displaystyle
K_{r}(0,\lambda)=\int_{0}^{r}\overline{P(s,0)}P(s,\lambda)\,ds=0$

and the Christoffel-Darboux formula (13) gives

$\displaystyle
P(r,\lambda)\overline{P(r,0)}-P_{*}(r,\lambda)\overline{P_{*}(r,0)}=0,$
$\displaystyle\left|P(r,\lambda)P(r,0)\right|=|P_{*}(r,\lambda)P_{*}(r,0)|.$

The latter equality contradicts Corollary 1 of the Christoffel-Darboux
formula. ∎

######  Lemma 4.

For every point $z\in\mathbb{C}_{+}$ the following inequality holds:

$\displaystyle|f_{r}(z)D(z)|\leqslant
K_{r}(0,0)^{-\frac{1}{2}}(4\pi\mathop{\mathrm{Im}}\nolimits(z))^{-\frac{1}{2}},$

in particular, for any $c>0$ the function $f_{r}D$ is bounded in the half-
plane $\\{z\colon\mathop{\mathrm{Im}}\nolimits(z)>c\\}$.

###### Proof.

$\displaystyle|f_{r}(z)|$
$\displaystyle=\left|\frac{K_{r}(0,z)}{K_{r}(0,0)}\right|=\left|\frac{\int_{0}^{r}\overline{P(s,0)}P(s,z)\,ds}{\int_{0}^{r}\left|P(s,0)\right|^{2}ds}\right|\leqslant\frac{\left(\int_{0}^{r}\left|P(s,0)\right|^{2}ds\right)^{\frac{1}{2}}\left(\int_{0}^{r}\left|P(s,z)\right|^{2}ds\right)^{\frac{1}{2}}}{\int_{0}^{r}\left|P(s,0)\right|^{2}ds}\leqslant$
$\displaystyle\leqslant
K_{r}(0,0)^{-\frac{1}{2}}\left(\frac{|\Pi(z)|^{2}}{2\mathop{\mathrm{Im}}\nolimits(z)}\right)^{\frac{1}{2}}\stackrel{{\scriptstyle\eqref{szego_function_definition}}}{{=}}K_{r}(0,0)^{-\frac{1}{2}}\left(\frac{1}{2\mathop{\mathrm{Im}}\nolimits(z)\cdot
2\pi|D(z)|^{2}}\right)^{\frac{1}{2}}$
$\displaystyle=K_{r}(0,0)^{-\frac{1}{2}}\left(4\pi\mathop{\mathrm{Im}}\nolimits(z)\right)^{-\frac{1}{2}}|D(z)|^{-1}.$

The first inequality on the second line follows from Corollary 3. Multiplying
the last inequality by $|D(z)|$, we obtain the required. ∎

######  Lemma 5.

For every $z\in\mathbb{C}_{+}$ the following holds:

(28) $\displaystyle\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{f_{r}(x)D(x)}{x-z}\,dx$ $\displaystyle=f_{r}(z)D(z),$
$\displaystyle\frac{1}{2\pi i}\int_{\mathbb{R}}\frac{f_{r}(x)D(x)}{x+z}\,dx$
$\displaystyle=0.$

######  Remark.

If the function $f_{r}D$ was in the Hardy space $H^{p}$ for some $p\geqslant
1$, the statement of the lemma would be a well-known fact (see p. 116 in
[14]). It is not our case but function $f_{r}D$ satisfies a ‘‘boundedness’’
condition from Lemma 4.

###### Proof.

We have already mentioned that the function
$\frac{c\Pi(z)^{-1}}{z+i}=\frac{D(z)}{z+i}$ is in the Hardy space $H^{2}$.
Hence, for $z\in\mathbb{C}_{+}$ the function $\frac{D(x)}{x+z}$ also belongs
to $H^{2}$. The function $f_{r}$ belongs to the Paley-Wiener space and
consequently to $H^{2}$. Therefore both integrals in the lemma are absolutely
convergent and define analytic functions in $\mathbb{C}_{+}$. Moreover,
$f_{r}(x)\frac{D(x)}{x+z}$ is a multiplication of two functions from $H^{2}$
and consequently it belongs to $H^{1}$. Integral of $H^{1}$ function over real
line equals $0$ (see p. 122 in [14] or Lemma 3.7 in [13]). Thus, the second
equality in lemma is proved.

From Lemma 4 it follows that the function
$G_{\varepsilon}(z)=f_{r}(z+\varepsilon i)D(z+\varepsilon i)$ is bounded in
the upper half-plane $\mathbb{C}_{+}$. Proceeding as above we deduce that for
all $z\in\mathbb{C}_{+}$

$\displaystyle\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{G_{\varepsilon}(x)}{x+z}\,dx$ $\displaystyle=0.$

Next, the following chain of equalities is true:

$\displaystyle\frac{1}{2\pi i}\int_{\mathbb{R}}\frac{f_{r}(x+\varepsilon
i)D(x+\varepsilon i)}{x-z}\,dx$ $\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{G_{\varepsilon}(x)}{x-z}\,dx$ (29)
$\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{G_{\varepsilon}(x)}{x-z}\,dx$
$\displaystyle-\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{G_{\varepsilon}(x)}{x+z}\,dx=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}\,dx.$

The latter integral can be calculated by the calculus of residues. Take a
large positive number $R$ and consider a circuit $C_{R}$ consisting of the
segment $[-R,R]$ and the half-circle $\Gamma_{R}$ connecting points $R$ and
$-R$ in the upper half-plane.

$\displaystyle G_{\varepsilon}(z)$
$\displaystyle=\text{Res}_{z}\left(\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}\right)=\frac{1}{2\pi
i}\int\limits_{C_{R}}\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}dx$ (30)
$\displaystyle=\frac{1}{2\pi
i}\int\limits_{[-R,R]}\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}dx+\frac{1}{2\pi
i}\int\limits_{\Gamma_{R}}\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}dx.$

The first term in the latter sum has a limit as $R\to\infty$.

$\displaystyle\lim_{R\to\infty}\frac{1}{2\pi
i}\int\limits_{[-R,R]}\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}\,dx=\frac{1}{2\pi
i}\int\limits_{\mathbb{R}}\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}\,dx.$

Since the function $G_{\varepsilon}$ is bounded, the second term tends to zero
as $R\to\infty$:

$\displaystyle\left|\frac{1}{2\pi
i}\int_{\Gamma_{R}}\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}\,dx\right|\leqslant\frac{1}{2\pi}\int_{0}^{\pi}\left|\frac{2zG_{\varepsilon}(Re^{i\theta})}{R^{2}e^{2i\theta}-z^{2}}\right|R\,d\theta\leqslant\frac{R|z|}{R^{2}-|z|^{2}}\sup\nolimits_{\theta}|G_{\varepsilon}(Re^{i\theta})|=O\left(\frac{1}{R}\right).$

We can therefore pass to the limit $R\to\infty$ in the equality (3.5.1) and
obtain

$\displaystyle\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{2zG_{\varepsilon}(x)}{x^{2}-z^{2}}\,dx=G_{\varepsilon}(z).$

Substituting this into (3.5.1) we get

$\displaystyle\frac{1}{2\pi i}\int_{\mathbb{R}}\frac{f_{r}(x+\varepsilon
i)D(x+\varepsilon i)}{x-z}\,dx$
$\displaystyle=G_{\varepsilon}(z)=f_{r}(z+i\varepsilon)D(z+i\varepsilon).$

To prove equality (28) it is only remains to pass to the limit as
$\varepsilon$ tends to $0$. The right part of the equality tends to
$f_{r}(z)D(z)$ so we need to evaluate how the left side differs from the
integral in the lemma.

$\displaystyle\left|\int_{\mathbb{R}}\frac{f_{r}(x+\varepsilon
i)D(x+\varepsilon
i)}{x-z}\,dx-\int_{\mathbb{R}}\frac{f_{r}(x)D(x)}{x-z}\,dx\right|=$
$\displaystyle=\left|\int_{\mathbb{R}}f_{r}(x+i\varepsilon)\frac{D(x+\varepsilon
i)-D(x)}{x-z}\,dx+\int_{\mathbb{R}}D(x)\frac{f_{r}(x+i\varepsilon)-f_{r}(x)}{x-z}\,dx\right|\leqslant$
$\displaystyle\leqslant\int_{\mathbb{R}}\left|f_{r}(x+i\varepsilon)\frac{D(x+\varepsilon
i)-D(x)}{x-z}\right|\,dx+\int_{\mathbb{R}}\left|D(x)\frac{f_{r}(x+i\varepsilon)-f_{r}(x)}{x-z}\right|\,dx\leqslant$
$\displaystyle\leqslant\left\|f_{r}(x+\varepsilon
i)\right\|_{2}\cdot\left\|\frac{x+i}{x-z}\right\|_{\infty}\left\|\frac{D(x+\varepsilon
i)-D(x)}{x+i}\right\|_{2}+\left\|f_{r}(x+i\varepsilon)-f_{r}(x)\right\|_{2}\cdot\left\|\frac{x+i}{x-z}\right\|_{\infty}\left\|\frac{D(x)}{x+i}\right\|_{2}$
$\displaystyle\leqslant C\left(\left\|\frac{D(x+\varepsilon
i)-D(x)}{x+i}\right\|_{2}+\left\|f_{r}(x+i\varepsilon)-f_{r}(x)\right\|_{2}\right)\leqslant$
$\displaystyle\leqslant C\left(\left\|\frac{D(x+\varepsilon
i)}{x+i}-\frac{D(x+\varepsilon i)}{x+\varepsilon
i+i}\right\|_{2}+\left\|\frac{D(x+\varepsilon i)}{x+\varepsilon
i+i}-\frac{D(x)}{x+i}\right\|_{2}+\left\|f_{r}(x+i\varepsilon)-f_{r}(x)\right\|_{2}\right)$

The first term can be bounded in the following way:

$\displaystyle\left\|\frac{D(x+\varepsilon i)}{x+i}-\frac{D(x+\varepsilon
i)}{x+\varepsilon i+i}\right\|_{2}=\varepsilon\left\|\frac{D(x+\varepsilon
i)}{(x+\varepsilon
i+i)(x+i)}\right\|_{2}\leqslant\varepsilon\left\|\frac{D(x)}{x+i}\right\|_{2}\to
0.$

From a general property of functions in $H^{2}$ (see Theorem 3.1 in [13] or
Section 19.2 in [16]) it follows that the second and the third terms in the
bracket tend to zero as $\varepsilon\to 0$. ∎

######  Lemma 6.

For a fixed number $b$ an inequality

$\displaystyle|f_{r}(t)|\leqslant 2\sqrt{2}e\pi|f_{r}(0)|<25|f_{r}(0)|$

holds for any large enough $r$ and for all $t\in[-\frac{b}{r},\frac{b}{r}]$.

###### Proof.

Point $t+i/r$ lies in a cone $K=\\{z\colon
b|\mathop{\mathrm{Im}}\nolimits(z)|\geqslant\mathop{\mathrm{Re}}\nolimits(z)\\}$.
By assumption, $D$ has a non-tangential boundary value at the point $0$ and
$D(0)\neq 0$. Hence, $D$ is continuous at the point $0$ inside cone $K$ and
for a large enough $r$ we have

(31) $\displaystyle\left|D\left(t+\frac{i}{r}\right)\right|>\frac{|D(0)|}{2}.$

Lemma 5 gives

$\displaystyle\left|f_{r}\left(t+\frac{i}{r}\right)D\left(t+\frac{i}{r}\right)\right|^{2}$
$\displaystyle=\left|\int_{\mathbb{R}}\frac{f_{r}(x)D(x)}{x-\left(t+\frac{i}{r}\right)}\,dx\right|^{2}\leqslant$
$\displaystyle\leqslant\int_{\mathbb{R}}\left|f_{r}(x)D(x)\right|^{2}\,dx\int_{\mathbb{R}}\frac{1}{x^{2}+\frac{1}{r^{2}}}\,dx=$
$\displaystyle\stackrel{{\scriptstyle\eqref{integral_f_r(x)D(x)}}}{{=}}m_{r}(0)|f_{r}(0)|^{2}\cdot\pi
r\stackrel{{\scriptstyle\eqref{trivial_condition}}}{{\leqslant}}2\pi^{2}|D(0)f_{r}(0)|^{2}.$

Hence,

$\displaystyle|f_{r}(0)|>\frac{1}{\sqrt{2}\pi}\frac{\left|D\left(t+\frac{i}{r}\right)\right|}{|D(0)|}\left|f_{r}\left(t+\frac{i}{r}\right)\right|.$

Applying Lemma 3 with $\gamma=\frac{1}{r}$ to bound
$\left|f_{r}\left(t+\frac{i}{r}\right)\right|$ and inequality (31) we get

$\displaystyle|f_{r}(0)|>\frac{1}{\sqrt{2}\pi}\frac{1}{2}e^{-1}|f_{r}(t)|>\frac{1}{25}|f_{r}(t)|.$

This concludes the proof. ∎

####  3.5.2. End of the proof

Recall that our aim is the proof of (26). Let $H_{r}$ be a function defined by

$\displaystyle H_{r}(x)=\frac{e^{-irx}-1}{x}.$

######  Statement.

For every positive $\delta$ the following holds:

(32) $\displaystyle\int\limits_{-\infty}^{\infty}\left|H_{r}(x-\delta
i)\right|^{2}\,dx<2\pi r.$

###### Proof.

To do the calculations recall the formula of the Fourier transform of
$\frac{1}{x^{2}+1}$.

$\displaystyle\int\limits_{-\infty}^{\infty}\frac{e^{iwx}}{x^{2}+1}\,dx=\pi
e^{-|w|}.$

With that in mind, the integral can be easily evaluated.

$\displaystyle\int\limits_{-\infty}^{\infty}\left|H_{r}(x-\delta
i)\right|^{2}\,dx$
$\displaystyle=\int\limits_{-\infty}^{\infty}\frac{\left|e^{-r\delta}e^{irx}-1\right|^{2}}{x^{2}+\delta^{2}}\,dx=\int\limits_{-\infty}^{\infty}\frac{e^{-2r\delta}+1-e^{-r\delta}(e^{irx}+e^{-irx})}{x^{2}+\delta^{2}}\,dx$
$\displaystyle=\frac{1}{\delta}\left[(e^{-2r\delta}+1)\int\limits_{-\infty}^{\infty}\frac{\,dy}{y^{2}+1}-e^{-r\delta}\int\limits_{-\infty}^{\infty}\frac{e^{ir\delta
y}+e^{-ir\delta y}}{y^{2}+1}\,dy\right]$
$\displaystyle=\frac{1}{\delta}\left[(e^{-2r\delta}+1)\pi-2e^{-r\delta}\pi
e^{-r\delta}\right]=\pi\frac{1-e^{-2r\delta}}{\delta}<2\pi r$

∎

Put $\delta=\frac{\varepsilon}{r}$. The following inequalities hold:

$\displaystyle\left|\int\limits_{-\infty}^{\infty}f_{r}(x)H_{r}(x-\delta
i)D(x)\,dx\right|^{2}$
$\displaystyle\leqslant\int\limits_{-\infty}^{\infty}\left|f_{r}(x)D(x)\right|^{2}\,dx\int\limits_{-\infty}^{\infty}\left|H_{r}(x-\delta
i)\right|^{2}\,dx$
$\displaystyle\stackrel{{\scriptstyle\eqref{integral_f_r(x)D(x)}}}{{=}}m_{r}(0)|f_{r}(0)|^{2}\int\limits_{-\infty}^{\infty}\left|H_{r}(x-\delta
i)\right|^{2}\,dx\leqslant$ $\displaystyle\leqslant m_{r}(0)|f_{r}(0)|^{2}2\pi
r.$

Hence,

(33) $\displaystyle\left|\int\limits_{-\infty}^{\infty}f_{r}(x)H_{r}(x-\delta
i)D(x)\,dx\right|$ $\displaystyle\leqslant|f_{r}(0)|\sqrt{2\pi rm_{r}(0)}$

Consider quantities $I,I_{1},I_{2}$, defined by the equality

$\displaystyle I=\int\limits_{-\infty}^{\infty}f_{r}(x)H_{r}(x-\delta
i)D(x)\,dx$ $\displaystyle=\int\limits_{-\infty}^{\infty}\frac{e^{-ir(x-\delta
i)}f_{r}(x)D(x)}{x-\delta
i}\,dx-\int\limits_{-\infty}^{\infty}\frac{f_{r}(x)D(x)}{x-\delta
i}\,dx=e^{-r\delta}I_{2}-I_{1}.$

Lemma 5 gives

(34) $\displaystyle I_{1}$
$\displaystyle=\int\limits_{-\infty}^{\infty}\frac{f_{r}(x)D(x)}{x-\delta
i}dx=2\pi if_{r}(\delta i)D(\delta i).$

Our further goal will be an estimation

(35) $\displaystyle|I_{2}|<4\varepsilon f_{r}(0)D(0).$

Before proving this let us show how (35) can be used to establish inequality
(26). Notice that

$\displaystyle|I|=|e^{-r\delta}I_{2}-I_{1}|\geqslant|I_{1}|-e^{-r\delta}|I_{2}|\geqslant|I_{1}|-|I_{2}|.$

Substitution of (34), (33), (35) into the last inequality gives

(36) $\displaystyle|f_{r}(0)|\sqrt{2\pi rm_{r}(0)}\geqslant 2\pi|f_{r}(\delta
i)D(\delta i)|-4\varepsilon|f_{r}(0)D(0)|.$

Since the function $D$ has non-tangential limit at the point $0$, for a large
enough $r$

(37)
$\displaystyle|D(i\delta)|=\left|D\left(i\frac{\varepsilon}{r}\right)\right|>(1-\varepsilon)|D(0)|.$

Moreover, from Lemma 3 we know

(38) $\displaystyle|f_{r}(\delta i)|\geqslant
e^{-r\delta}|f_{r}(0)|=e^{-\varepsilon}|f_{r}(0)|\geqslant(1-\varepsilon)|f_{r}(0)|.$

Finally, the substitution of (37) and (38) into inequality (36) finishes the
proof:

$\displaystyle|f_{r}(0)|\sqrt{2\pi rm_{r}(0)}$ $\displaystyle\geqslant
2\pi(1-\varepsilon)(1-\varepsilon)|f_{r}(0)D(0)|-4\varepsilon|f_{r}(0)D(0)|\geqslant$
$\displaystyle\geqslant(2\pi-(4+4\pi)\varepsilon+2\pi\varepsilon^{2})|f_{r}(0)D(0)|>(2\pi-20\varepsilon)|f_{r}(0)D(0)|,$
$\displaystyle rm_{r}(0)$
$\displaystyle\geqslant\frac{(2\pi-20\varepsilon)^{2}}{2\pi}|D(0)|^{2}>(2\pi-100\varepsilon)|D(0)|^{2}.$

Thus, it is only remains to prove estimate (35). Consider one more integral
$\tilde{I_{2}}$.

$\displaystyle\tilde{I_{2}}=\int\limits_{-\infty}^{\infty}\overline{\left(\frac{e^{-irx}f_{r}(x)}{x-\delta
i}\right)}D(x)\,dx=\int\limits_{-\infty}^{\infty}\frac{e^{irx}\overline{f_{r}(x)}}{x+\delta
i}D(x)\,dx.$

The function $f_{r}$ is in $PW_{r}$ so there exists $\varphi\in L^{2}[0,r]$
such that $f_{r}(x)=\int_{0}^{r}\varphi(t)e^{itx}dt$. Notice that

$\displaystyle e^{irx}\overline{f_{r}(x)}$
$\displaystyle=e^{irx}\int_{0}^{r}\overline{\varphi(t)}e^{-itx}dt=\int_{0}^{r}\overline{\varphi(t)}e^{i(r-t)x}dt=\int_{0}^{r}\overline{\varphi(r-t)}e^{itx}dt.$

Hence, the function $e^{irx}\overline{f_{r}(x)}$ also belongs to $PW_{r}$ and
consequently to the Hardy space $H^{2}$ in the upper half-plane. As we have
mentioned before, the function $\frac{D(x)}{x+\delta i}$ also belong to
$H^{2}$. So

$\displaystyle\frac{e^{irx}\overline{f_{r}(x)}D(x)}{x+\delta i}\in H^{1}.$

An integral of a $H^{1}$ function over the real line equals $0$ (see Lemma 3.7
in [13]), therefore, $\tilde{I_{2}}=0$. By definition, put

$\displaystyle
K_{1}=\left\\{x\in\mathbb{R}\colon|x|>\frac{b}{r}\right\\},\quad
K_{2}=\mathbb{R}\setminus K_{1}.$

Split $I_{2}$ into two terms and consider them separately.

$\displaystyle I_{2}=\int_{-\infty}^{\infty}\frac{e^{-irx}f_{r}(x)}{x-\delta
i}D(x)\,dx=\int_{K_{1}}\frac{e^{-irx}f_{r}(x)}{x-\delta
i}D(x)\,dx+\int_{K_{2}}\frac{e^{-irx}f_{r}(x)}{x-\delta
i}D(x)\,dx=I_{21}+I_{22}.$

Similarly, $\tilde{I}_{2}$ is represented as a sum of $\tilde{I}_{21}$ and
$\tilde{I}_{22}$. The following identity is true:

$\displaystyle I_{2}/D(0)=I_{2}/D(0)-\overline{\tilde{I}_{2}/D(0)}$
$\displaystyle=I_{21}/D(0)+\left(I_{22}/D(0)-\int_{K_{2}}\frac{e^{-irx}f_{r}(x)}{x-\delta
i}\,dx\right)+$ (39)
$\displaystyle-\overline{{\tilde{I}_{21}/D(0)}}-\overline{\left(\tilde{I}_{22}/D(0)-\int_{K_{2}}\overline{\left(\frac{e^{-irx}f_{r}(x)}{x-\delta
i}\right)}\,dx\right)}.$

Let us bound every term in the right side.

$\displaystyle|I_{21}|^{2}=\left|\int_{K_{1}}\frac{e^{-irx}f_{r}(x)D(x)}{x-\delta
i}\,dx\right|^{2}$
$\displaystyle\leqslant\int_{K_{1}}\left|f_{r}(x)D(x)\right|^{2}dx\int_{K_{1}}\left|\frac{e^{-irx}}{x-\delta
i}\right|^{2}dx$ $\displaystyle\int_{K_{1}}\left|f_{r}(x)D(x)\right|^{2}dx$
$\displaystyle\leqslant\int_{\mathbb{R}}\left|f_{r}(x)D(x)\right|^{2}dx\stackrel{{\scriptstyle\eqref{integral_f_r(x)D(x)}}}{{=}}m_{r}(0)|f_{r}(0)|^{2}$
$\displaystyle\int_{K_{1}}\left|\frac{e^{-irx}}{x-\delta i}\right|^{2}dx$
$\displaystyle=\int\limits_{|x|>\frac{b}{r}}\frac{1}{x^{2}+\delta^{2}}\,dx\leqslant
2\int_{\frac{b}{r}}^{\infty}\frac{dx}{x^{2}}=2r/b.$

Thus,

$\displaystyle|I_{21}|^{2}\leqslant m_{r}(0)|f_{r}(0)|^{2}\frac{2r}{b}$
$\displaystyle\leqslant\frac{2}{b}|f_{r}(0)|^{2}\left(rm_{r}(0)\right)\stackrel{{\scriptstyle\eqref{trivial_condition}}}{{\leqslant}}\frac{4\pi}{b}|f_{r}(0)D(0)|^{2},$
$\displaystyle\frac{|I_{21}|}{|D(0)|}$
$\displaystyle\leqslant\sqrt{\frac{4\pi}{b}}|f_{r}(0)|.$

Choose $b$ so large that $\frac{4\pi}{b}\leqslant\varepsilon^{2}$ holds.
Taking this into account, we obtain

(40) $\displaystyle\frac{|I_{21}|}{|D(0)|}\leqslant\varepsilon|f_{r}(0)|.$

Next, we bound the second term in (3.5.2).

$\displaystyle\left|\frac{I_{22}}{D(0)}-\int_{K_{2}}\frac{e^{-irx}f_{r}(x)}{x-\delta
i}\,dx\right|$
$\displaystyle=\frac{1}{|D(0)|}\left|\int_{K_{2}}\frac{e^{-irx}f_{r}(x)}{x-\delta
i}D(x)\,dx-\int_{K_{2}}\frac{e^{-irx}f_{r}(x)}{x-\delta
i}D(0)\,dx\right|\leqslant$
$\displaystyle\leqslant\frac{1}{|D(0)|}\sup_{K_{2}}\left|\frac{f_{r}(x)}{x-\delta
i}\right|\int_{K_{2}}|D(x)-D(0)|\,dx.$

From Lemma 6 it follows that $|f_{r}(x)|<25|f_{r}(0)|$ for any $x\in K_{2}$.
Since $0$ is a Lebesgue point of $D$ (inequality (24) holds), we can choose
$r$ so large that

$\displaystyle\int\limits_{K_{2}}|D(x)-D(0)|\,dx=\int\limits_{|x|<\frac{b}{r}}|D(x)-D(0)|\,dx<\frac{b}{r}\frac{\varepsilon^{2}}{25b}|D(0)|.$

From the two latter inequalities we get

(41)
$\displaystyle\left|\frac{I_{22}}{D(0)}-\int_{K_{2}}\frac{e^{-irx}f_{r}(x)}{x-\delta
i}\,dx\right|$
$\displaystyle\leqslant\frac{1}{|D(0)|}\cdot\frac{25|f_{r}(0)|}{\delta}\cdot\frac{b}{r}\frac{\varepsilon^{2}}{25b}|D(0)|=\frac{\varepsilon^{2}}{\delta
r}|f_{r}(0)|=\varepsilon|f_{r}(0)|.$

The two remaining terms in (3.5.2) can be bounded in a similar way.
Substitution of (40), (41) and two similar inequalities for $\tilde{I}_{2}$
into identity (3.5.2) establishes required estimate (35).

##  4\. Proof of Theorem 3

We begin by proving the average convergence for modules of the continuous
orthogonal polynomials.

######  Lemma 7.

Assume that the coefficient $a$ of Krein system (4) belongs to
$L^{1}_{\mathop{\mathrm{loc}}\nolimits}(\mathbb{R}_{+})$ and the associated
spectral measure $\sigma$ belongs to the Szegő class on the real line. Then
for every $\lambda\in\mathbb{C}_{+}$ we have

$\displaystyle\lim_{r\to\infty}\frac{1}{r}\int_{0}^{r}\left|P_{*}(\rho,\lambda)\right|\,d\rho=\left|\Pi(\lambda)\right|.$

###### Proof.

The required equality is equivalent to two inequalities

(42)
$\displaystyle\liminf_{r\to\infty}\frac{1}{r}\int_{0}^{r}\left|P_{*}(\rho,\lambda)\right|\,d\rho\geqslant\left|\Pi(\lambda)\right|,$
(43)
$\displaystyle\limsup_{r\to\infty}\frac{1}{r}\int_{0}^{r}\left|P_{*}(\rho,\lambda)\right|\,d\rho\leqslant\left|\Pi(\lambda)\right|.$

Take the limit inferior for $r$ in Christoffel-Darboux formula (14).

$\displaystyle\liminf_{r\to\infty}\left|P_{*}(r,\lambda)\right|^{2}$
$\displaystyle=\liminf_{r\to\infty}\left(\left|P(r,\lambda)\right|^{2}+2\mathop{\mathrm{Im}}\nolimits(\lambda)\int_{0}^{r}\left|P(s,\lambda)\right|^{2}ds\right)$

Part (b) of the Krein theorem states that the function $P(\cdot,\lambda)$
belongs to $L^{2}(\mathbb{R})$. Hence,

$\displaystyle\liminf_{r\to\infty}\left|P(r,\lambda)\right|^{2}=0.$

The second term tends to the integral from $0$ to $\infty$; by Corollary 3
this integral is equal to $|\Pi(\lambda)|^{2}$. Thus,

$\displaystyle\liminf_{r\to\infty}\left|P_{*}(r,\lambda)\right|^{2}$
$\displaystyle=|\Pi(\lambda)|^{2},$
$\displaystyle\liminf_{r\to\infty}\left|P_{*}(r,\lambda)\right|$
$\displaystyle=|\Pi(\lambda)|.$

Inequality (42) follows from the last equality by the direct integration. To
prove (43), integrate and divide by $r$ Christoffel-Darboux formula (14).

$\displaystyle\frac{1}{r}\int_{0}^{r}\left|P_{*}(\rho,\lambda)\right|^{2}d\rho$
$\displaystyle=\frac{1}{r}\int_{0}^{r}\left(\left|P(\rho,\lambda)\right|^{2}+2\mathop{\mathrm{Im}}\nolimits(\lambda)\int_{0}^{\rho}\left|P(s,\lambda)\right|^{2}ds\right)d\rho$
$\displaystyle=\frac{1}{r}\int_{0}^{r}\left|P(\rho,\lambda)\right|^{2}d\rho+2\mathop{\mathrm{Im}}\nolimits(\lambda)\frac{1}{r}\int_{0}^{r}\int_{0}^{\rho}\left|P(s,\lambda)\right|^{2}ds\,d\rho$
$\displaystyle\leqslant\frac{1}{r}\int_{0}^{\infty}\left|P(\rho,\lambda)\right|^{2}d\rho+2\mathop{\mathrm{Im}}\nolimits(\lambda)\frac{1}{r}\int_{0}^{r}\int_{0}^{\infty}\left|P(s,\lambda)\right|^{2}ds\,d\rho$
$\displaystyle=\left(\frac{1}{2r\mathop{\mathrm{Im}}\nolimits(\lambda)}+1\right)2\mathop{\mathrm{Im}}\nolimits(\lambda)\int_{0}^{\infty}\left|P(s,\lambda)\right|^{2}ds\stackrel{{\scriptstyle\text{Cor.}\ref{cons_module_PI}}}{{=}}\left(\frac{1}{2r\mathop{\mathrm{Im}}\nolimits(\lambda)}+1\right)|\Pi(\lambda)|^{2}$

Now, the Cauchy-Schwarz inequality gives

(44)
$\displaystyle\frac{1}{r}\int_{0}^{r}\left|P_{*}(\rho,\lambda)\right|\,d\rho\leqslant\sqrt{\frac{1}{r}\int_{0}^{r}\left|P_{*}(\rho,\lambda)\right|^{2}d\rho}\leqslant\sqrt{1+\frac{1}{2r\mathop{\mathrm{Im}}\nolimits(\lambda)}}|\Pi(\lambda)|.$

Passing to the limit superior on $r$ concludes the proof. ∎

######  Corollary 4.

Suppose that the spectral measure $\sigma$ belongs to the Szegő class and the
function $a$ is real-valued. Then

$\displaystyle\lim_{r\to\infty}\frac{1}{r}\int_{0}^{r}P_{*}(\rho,i)\,d\rho=\Pi(i).$

###### Proof.

Substituting $i$ for $\lambda$ in (8), we obtain

$\displaystyle\Pi(i)=\frac{1}{\sqrt{2\pi}}\exp\left[\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{(1+si)\ln\sigma^{\prime}(s)}{(i-s)(1+s^{2})}\,ds\right]=\frac{1}{\sqrt{2\pi}}\exp\left[\frac{-1}{2\pi}\int_{-\infty}^{\infty}\frac{\ln\sigma^{\prime}(s)}{(1+s^{2})}\,ds\right].$

Hence, $\Pi(i)$ is a positive real number. System (4) with $\lambda=i$ has
real coefficients, real initial condition and consequently the solution
$P_{*}(r,i)$ is real for every $r$. By Corollary 2, $P_{*}(r,i)$ does not
equal zero for $r>0$; $P_{*}(0,i)>0$. Therefore, $P_{*}(r,i)$ must be positive
for every $r\geqslant 0$ and

$\displaystyle\lim_{r\to\infty}\frac{1}{r}\int_{0}^{r}P_{*}(\rho,i)\,d\rho=\lim_{r\to\infty}\frac{1}{r}\int_{0}^{r}\left|P_{*}(\rho,i)\right|\,d\rho=\left|\Pi(i)\right|=\Pi(i).$

∎

Now we can proceed directly to the proof of Theorem 3.

###### Proof of Theorem 3.

Consider a family $\mathcal{F}$ of the analytic functions

$\displaystyle
F_{r}(\lambda)=\frac{1}{r}\int_{0}^{r}P_{*}(\rho,\lambda)\,d\rho,\quad r>1.$

Notice that

(45)
$\displaystyle|F_{r}(\lambda)|=\left|\frac{1}{r}\int_{0}^{r}P_{*}(\rho,\lambda)\,d\rho\right|\leqslant\frac{1}{r}\int_{0}^{r}\left|P_{*}(\rho,\lambda)\right|\,d\rho\stackrel{{\scriptstyle\eqref{ineq_for_F_r_lemma}}}{{\leqslant}}\sqrt{1+\frac{1}{2r\mathop{\mathrm{Im}}\nolimits(\lambda)}}|\Pi(\lambda)|.$

Consider an arbitrary compact set $D\subset\mathbb{C}_{+}$.
$\mathop{\mathrm{Im}}\nolimits(\lambda)$ is separated from $0$ in $D$ so

$\displaystyle\sup_{\lambda\in D,r}|F_{r}(\lambda)|\leqslant\sup_{\lambda\in
D}c(\lambda)|\Pi(\lambda)|<\infty.$

Hence, the family $\mathcal{F}$ is uniformly bounded on compact subsets and,
by Montel theorem, every subset of $\mathcal{F}$ has a convergent subsequence.
Let a sequence $r_{n}\to\infty$ be such that a sequence $F_{r_{n}}$ is
convergent and $\lim F_{r_{n}}=\tilde{F}$. Inequality (45) gives

$\displaystyle|\tilde{F}(\lambda)|=\lim_{n\to\infty}|F_{r_{n}}(\lambda)|\leqslant|\Pi(\lambda)|$

for every point $\lambda\in\mathbb{C}_{+}$. On the other hand, from corollary
4 it follows that

$\displaystyle\tilde{F}(i)=\lim_{n\to\infty}F_{r_{n}}(i)=\lim_{n\to\infty}\frac{1}{r_{n}}\int_{0}^{r_{n}}P_{*}(\rho,i)\,d\rho=\Pi(i).$

Applying the maximum modulus principle, we conclude that
$\tilde{F}(\lambda)=\Pi(\lambda)$ for every $\lambda\in\mathbb{C}_{+}$. Any
partial limit of $F_{r}$ as $r\to\infty$ coincides with $\Pi$, so $F_{r}$ has
a limit equal to $\Pi$. This completes the proof.

∎

##  5\. Acknowledgements

The author is grateful to Roman Bessonov for helpful discussions and constant
attention to this work.

## References

  * [1] A. I. Aptekarev, S. A. Denisov, and D. N. Tulyakov.  V. A. Steklov’s problem of estimating the growth of orthogonal polynomials.  Proc. Steklov Inst. Math., 289(1):72–95, 2015.  Translation of Tr. Mat. Inst. Steklova 289 (2015), 83–106.
  * [2] R. V. Bessonov.  Szego condition and scattering for one-dimensional Dirac operators.  Constr. Approx., 51(2):273–302, 2020.
  * [3] Roman Bessonov and Sergey Denisov.  A spectral Szego theorem on the real line.  Adv. Math., 359:106851, 41, 2020.
  * [4] M. Christ and A. Kiselev.  Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials.  Geom. Funct. Anal., 12(6):1174–1234, 2002.
  * [5] Michael Christ and Alexander Kiselev.  Maximal functions associated to filtrations.  J. Funct. Anal., 179(2):409–425, 2001.
  * [6] Michael Christ and Alexander Kiselev.  WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials.  J. Funct. Anal., 179(2):426–447, 2001.
  * [7] P. Deift and R. Killip.  On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials.  Comm. Math. Phys., 203(2):341–347, 1999.
  * [8] S. A. Denisov.  To the spectral theory of Kreĭn systems.  Integral Equations Operator Theory, 42(2):166–173, 2002.
  * [9] S. A. Denisov.  On the existence of wave operators for some Dirac operators with square summable potential.  Geom. Funct. Anal., 14(3):529–534, 2004.
  * [10] S. A. Denisov.  On the growth of polynomials orthogonal on the unit circle with a weight $w$ that satisfies $w,w^{-1}\in L^{\infty}(\mathbb{T})$.  Mat. Sb., 209(7):71–105, 2018.
  * [11] Sergey Denisov and Keith Rush.  Orthogonal polynomials on the circle for the weight $w$ satisfying conditions $w,w^{-1}\in{\rm BMO}$.  Constr. Approx., 46(2):285–303, 2017.
  * [12] Sergey A. Denisov.  Continuous analogs of polynomials orthogonal on the unit circle and Kreĭn systems.  IMRS Int. Math. Res. Surv., pages Art. ID 54517, 148, 2006.
  * [13] John B. Garnett.  Bounded analytic functions, volume 96 of Pure and Applied Mathematics.  Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.
  * [14] Paul Koosis.  Introduction to $H_{p}$ spaces, volume 115 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, second edition, 1998.  With two appendices by V. P. Havin [Viktor Petrovich Khavin].
  * [15] M. G. Kreĭn.  Continuous analogues of propositions on polynomials orthogonal on the unit circle.  Dokl. Akad. Nauk SSSR (N.S.), 105:637–640, 1955.
  * [16] B. Ya. Levin.  Lectures on entire functions, volume 150 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1996.  In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko.
  * [17] Attila Máté, Paul Nevai, and Vilmos Totik.  Szego’s extremum problem on the unit circle.  Ann. of Math. (2), 134(2):433–453, 1991.
  * [18] Camil Muscalu, Terence Tao, and Christoph Thiele.  A Carleson theorem for a Cantor group model of the scattering transform.  Nonlinearity, 16(1):219–246, 2003.
  * [19] Camil Muscalu, Terence Tao, and Christoph Thiele.  A counterexample to a multilinear endpoint question of Christ and Kiselev.  Math. Res. Lett., 10(2-3):237–246, 2003.
  * [20] A. M. Rybalko.  On the theory of continual analogues of orthogonal polynomials.  Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp., 3:42–60, 1966.
  * [21] L. A. Sakhnovich.  On the spectral theory of a class of canonical differential systems.  Funktsional. Anal. i Prilozhen., 34(2):50–62, 96, 2000.
  * [22] Barry Simon.  Schrödinger semigroups.  Bull. Amer. Math. Soc. (N.S.), 7(3):447–526, 1982.
  * [23] Barry Simon.  Orthogonal polynomials on the unit circle. Part 1, volume 54 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 2005.  Classical theory.
  * [24] Gábor Szegő.  Orthogonal polynomials.  American Mathematical Society, Providence, R.I., fourth edition, 1975\.  American Mathematical Society, Colloquium Publications, Vol. XXIII.
  * [25] Alexander Teplyaev.  A note on the theorems of M. G. Krein and L. A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle.  J. Funct. Anal., 226(2):257–280, 2005.