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Multipoint Geronimus and Schur parameters of measures on a circle and on a line V. I. Buslaev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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   § 1. Multipoint Schur and Geronimus parametersThroughout, we let $\mathbb C$ denote the set of points in the complex plane and set
$$
\begin{equation*}
\begin{gathered} \, \overline{\mathbb C}:=\mathbb C\sqcup \{\infty\}, \qquad \mathbb C_+:=\{ \operatorname{Im} z>0\}, \qquad\mathbb C_-:=\{ \operatorname{Im} z<0\}, \\ \mathbb K:=\{ \operatorname{Re} z>0\}, \qquad \mathbb R:=\{ \operatorname{Im} z=0\}, \qquad\overline{\mathbb R}:=\mathbb R\sqcup \{\infty\}, \\ \mathbb D:=\{ |z| <1\}, \qquad\overline{\mathbb D}:=\{ |z| \leqslant 1\}, \qquad \mathbb T:=\{ |z|=1\}. \end{gathered}
\end{equation*}
\notag
$$
We denote by $H(\Omega)$ the set of holomorphic functions in $\Omega$ in the case when $\Omega$ is open and the set of functions holomorphic in a neighbourhood of $\Omega$ (depending on the function) in the case when $\Omega$ is closed. Recall that $F(z)$ is called a Carathéodory function if $F(z)\in H(\mathbb D)$ and ${\operatorname{Re} F(z)\geqslant 0}$, $f(z)$ is called a Schur function if $f(z)\in H(\mathbb D)$ and $|f(z)|\leqslant 1$, and $G(z)$ is called a Nevanlinna function if $G(z)\in H(\mathbb C_+)$ and $\operatorname{Im} G(z)\geqslant 0$. We denote the sets of Carathéodory, Schur and Nevanlinna functions by $\mathfrak B^{\mathrm c}$, $\mathfrak B^{\mathrm s}$ and $\mathfrak B^{\mathrm n}$, respectively. For $a\in\mathbb D$ we let denote a linear fractional transformation taking $\mathbb D$ to $\mathbb D$. Here and in what follows a bar over a symbol denotes complex conjugation (with the exception of the sets $\overline{\mathbb C}$, $\overline{\mathbb R}$ and $\overline{\mathbb D}$ defined above). The following simple result holds.Statement 1. Let $a\in\mathbb D$, $f(z)\in H(\mathbb D)$ and $|f(a)|< 1$. Then In addition, Proof. In fact, equality (1) is trivial. The implication ‘$\Rightarrow$’ in (2) follows from the definition (1) of the function and from Schwarz’s lemma; the reverse implication ‘$\Leftarrow$’ follows from the equality
The proof is complete. The multipoint Schur algorithm (at points $a_1,a_2,\dots$ in $\mathbb D$), as applied to a Schur function $f(z)\in\mathfrak B^{\mathrm s}$, returns (finite or infinite) sequences of functions $f_0(z),f_1(z),\dots$ and parameters $\gamma_0,\gamma_1,\dots$ calculated as follows: here $|\gamma_0|\leqslant 1$ (by the definition of the Schur function); if $|\gamma_0|=1$, then (by the maximum modulus principle) the Schur function $f_{0}(z)$ is identically equal to $\gamma_{0}$ and the algorithm terminates; if $|\gamma_0|<1$, then setting
$$
\begin{equation}
f_{n+1}(z):= f_n^{\mathrm{s},a_{n+1}}(z) = \frac{f_n(z)-\gamma_n}{\varphi_{a_{n+1}}(z)(1-f_n(z)\overline{\gamma }_n)},
\end{equation}
\tag{3}
$$
(notice the notation $\overline{\gamma}_n:=\overline{\gamma_n}$ used in (3); we also use it below in similar situations), by Statement 1, for $n=0$ we have the equality
$$
\begin{equation*}
f_0(z)=\gamma_0+\cfrac{(1-|\gamma_0|^2)\varphi_{a_{1}}(z)}{\overline{\gamma }_0\varphi_{a_{1}}(z)+\cfrac{1} {f_1(z)}}
\end{equation*}
\notag
$$
and the implication
$$
\begin{equation*}
f_0(z)\in\mathfrak B^{\mathrm s}, \quad |f_{0}(a_{1})|=|\gamma_0|<1 \quad\Longrightarrow\quad f_{1}(z)\in\mathfrak B^{\mathrm s}.
\end{equation*}
\notag
$$
Hence we can apply the above argument to $f_{1}(z)\in\mathfrak B^{\mathrm s}$, so that if $|\gamma_{1}|=1$, then $f_{1}(z)\equiv \gamma_{1}$ and the algorithm terminates; if $|\gamma_{1}|<1$, then see (3) and (4) for $n=1$ and so on. If the parameters obtained are strictly less than one in modulus for all $n=0,1,\dots$, then the sequences of functions $f_0(z),f_1(z),\dots$ and parameters $\gamma_0,\gamma_1,\dots$ are infinite. Thus, the multipoint Schur algorithm either returns $f_{0}(z)\equiv \gamma_{0}$ for $|\gamma_0|=1$ or the finite (if $|\gamma_0|<1,\dots,|\gamma_{n-1}|<1$ and $|\gamma_n|=1$ for some $n\in\mathbb N$) or infinite continued fraction
$$
\begin{equation}
\gamma_0+\cfrac{(1-|\gamma_0|^2)\varphi_{a_{1}}(z)}{\overline{\gamma }_0\varphi_{a_{1}}(z)+\cfrac{1} {\gamma_1+\cfrac{(1-|\gamma_1|^2)\varphi_{a_{2}}(z)}{\overline{\gamma }_1\varphi_{a_{2}}(z)+\cfrac{1}{\gamma_2+\dotsb }}}}\,
\end{equation}
\tag{5}
$$
with coefficients $\gamma_0,\gamma_1,\dots$ of modulus less than one. The continued fraction (5) is called the Schur multipoint continued fraction (at the points $a_1,a_2,\dots $). In the case when all $a_1,a_2,\dots $ are equal to zero we have the equalities $\varphi_{a_1}(z)=\varphi_{a_2}(z)=\dots =z$, and the multipoint Schur algorithm and Schur continued fraction (5) coincide with the classical ones, described originally in [1]. Apart from Statement 1, we can state another result, which is as easy to verify. Statement 2. Let $a\in\mathbb D$, $F(z)\in H(\mathbb D) $ and $\operatorname{Re} F(a)> 0$. Then In addition, Proof. In fact, letting $\tau_\gamma (z)$ denote the linear fractional map $(z-\gamma)/(z+\overline{\gamma})$, which takes $\mathbb K$ to $\mathbb D$ for $\gamma \in\mathbb K$, we see that the implication ‘$\Rightarrow$’ in (7) follows from the definition (6) of the function and Schwarz’s lemma, while the reverse implication ‘$\Leftarrow$’ follows from the equality
The proof is complete. Let $F(z)\in\mathfrak B^{\mathrm c}$ be a Carathéodory function and $a_0,a_1,\dots$ be a sequence of points in $\mathbb D$. Arguments similar to the above ones but using the equalities
$$
\begin{equation}
\gamma_{-1}:=F(a_0)\quad\text{and} \quad f_0(z):=F^{\mathrm{c},a_0}(z)
\end{equation}
\tag{8}
$$
and Statement 2 at the first step of the algorithm and using equalities (3) and (4) and Statement 1 at the subsequent steps show that either $F(z)\equiv \gamma_{-1}$ if $\operatorname{Re} \gamma_{-1}=0$, or $F(z)$ is a finite fraction (if $\operatorname{Re} \gamma_{-1}>0$, $|\gamma_0|<1,\dots $, $|\gamma_{n-1}|<1$ and $|\gamma_n|=1$ for some $n\in\mathbb Z_+$), or the algorithm, as applied to a function $F(z)\in\mathfrak B^{\mathrm c}$, defines an infinite sequence of functions $F_{-1}(z)=F(z)$, $f_0(z)=F^{\mathrm{c},a_0}(z)$, $f_1(z), \dots$ and produces the continued fraction
$$
\begin{equation}
\gamma_{-1}+\cfrac{(\gamma_{-1}+\overline{\gamma}_{-1})\varphi_{a_0}(z)} {-\varphi_{a_0}(z)+\cfrac{1}{\gamma_0+ \cfrac{(1-|\gamma_0|^2)\varphi_{a_{1}}(z)}{\overline{\gamma }_0\varphi_{a_{1}}(z)+\cfrac{1}{\gamma_1+ \cfrac{(1-|\gamma_1|^2)\varphi_{a_{2}}(z)}{\overline{\gamma }_1\varphi_{a_{2}}(z)+\cfrac{1}{\gamma_2+\dotsb }}}}}}\,
\end{equation}
\tag{9}
$$
with coefficients satisfying $\operatorname{Re} \gamma_{-1}>0$ and $|\gamma_n|<1$, $n=0,1,\dots$ . We call the algorithm applied to the Carathéodory function $F(z)$ and returning a finite or an infinite continued fraction of the form (9) the analogue of Schur’s algorithm, and we call the finite or infinite sequence of coefficients $\gamma_{-1},\gamma_{0},\gamma_1,\dots$ of the continued fraction the sequence of Schur parameters of the Carathéodory function $F(z)$ with respect to the points $a_0,a_1,\dots$ . Carathéodory functions have Riesz–Herglotz integral representations (see [2]). Riesz–Herglotz Theorem. A function $F(z)$ is a Carathéodory function if and only if there exists a nonnegative Borel measure $\sigma $ with support on the circle $\mathbb T$ such that We call the function $F(z)$ defined by (10) the Carathéodory function of the measure $\sigma $. Remark 1. Let $F(z)$ be a Carathéodory function, let $\operatorname{Re} c_1=0$ and $c_2>0$, and let $a_0,a_1,\dots$ be a sequence of points in $\mathbb D$. Then (1) $\widetilde{F}(z)=c_1+c_2F(z)$ is a Carathéodiry function; (2) $\widetilde{F}^{\mathrm{c},a}(z)=F^{\mathrm{c},a}(z)$ (by (6) for all $a\in\mathbb D$), and therefore for the Schur parameters $\gamma_{-1},\gamma_0,\dots$ and $\widetilde{\gamma }_{-1},\widetilde{\gamma }_0,\dots$ of the functions $F(z)$ and $\widetilde{F}(z)$ with respect to the points $a_0,a_1,\dots$ we have Let $\gamma_{-1},\gamma_{0},\gamma_1,\dots$ be the Schur parameters with respect to the points $a_0,a_1,\dots$ of the Carathéodory function $F(z)$ defined by equation (10). We call the parameters $\gamma_{0},\gamma_1,\dots$, which are (by Remark 1) independent of the purely imaginary term $i\operatorname{Im} F(0)$ in (10), the Schur parameters of the measure $\sigma $ with respect to $a_0,a_1,\dots$ . It also follows from Remark 1 that the Schur parameters with respect to $a_0,a_1,\dots$ of the measures $\sigma $ and $c\sigma $, where $c>0$, are equal. We call the case when $a_0=a_1=\dots =0$ the classical case. In this case $\gamma_{0},\gamma_1,\dots$ are called the classical Schur parameters of $\sigma $. Remark 2. The Schur parameters of a measure $\sigma $ with respect to points $a_0,a_1,a_2,\dots$ are different from the parameters of $\sigma $ with respect to points $\widetilde{a}_0\neq a_0,a_1,a_2,\dots $, because in general Before we define the (classical) Geronimus parameters of a measure $\sigma $ with support on $\mathbb T$, recall that the theory of orthogonal polynomials on a circle is based on the fact that if $\sigma $ is a measure on $\mathbb T$ with infinite support, then for the sequence of $\sigma $-orthogonal polynomials
$$
\begin{equation}
\begin{gathered} \, \notag p_n(z)=k_nz^n+\dots +p_n(0), \qquad k_n\neq 0, \qquad n=0,1,\dots , \\ \int_{\mathbb T}p_n(z)\overline{z}^{\,k}\,d\sigma (z)=0, \qquad k=0,\dots,n-1, \end{gathered}
\end{equation}
\tag{12}
$$
for all $n=1,2,\dots$ we have
$$
\begin{equation}
p_{n}(z)=\frac{k_n}{k_{n-1}}zp_{n-1}(z)-\frac{k_n}{\overline{k}_{n-1}}\overline{\alpha }_{n-1}p_{n-1}^*(z), \quad \text{where } p_n^*(z)=z^n\overline{p\biggl(\frac{1}{\overline{z}}\biggr)},
\end{equation}
\tag{13}
$$
$$
\begin{equation}
\alpha_{n-1}=-\frac{\overline{p_{n}(0)}}{p_{n}^*(0)}, \qquad n=1,2,\dots\,.
\end{equation}
\tag{14}
$$
In fact, bearing in mind that $\overline{z}=z^{-1}$ for $z\in\mathbb T$, from the definition (12) of orthogonal polynomials, for $zp_{n-1}(z)$ and $k=1,\dots,n-1$ we obtain
$$
\begin{equation*}
\int_{\mathbb T}zp_{n-1}(z)\overline{z}^{\,k}\,d\sigma (z) =\int_{\mathbb T}p_{n-1}(z)\overline{z}^{\,k-1}\,d\sigma (z)=0,
\end{equation*}
\notag
$$
and for $p_{n-1}^*(z)$ we obtain
$$
\begin{equation*}
\int_{\mathbb T}p_{n-1}^*(z)\overline{z}^{\,k}\,d\sigma (z) =\int_{\mathbb T}z^{n-1}\overline{p_{n-1}\biggl(\frac{1}{\overline{z}}\biggr)}\overline{z}^{\,k}\,d\sigma (z) =\int_{\mathbb T}z^{n-1-k}\overline{p_{n-1}(z)}\,d\sigma (z)=0.
\end{equation*}
\notag
$$
In combination with (12), these equalities mean that each of the polynomials $p_n(z)$, $zp_{n-1}(z)$ and $p_{n-1}^*(z)$ lies in the linear $(n+1)$-dimensional space $\mathscr P_n$ of polynomial of degree at most $n$ and is orthogonal to the $(n-1)$-dimensional space $\{p(z)\colon {p(z)\in\mathscr P_{n-1}}\text{ and } p(0)=0\}$. Thus, $p_n(z)$, $zp_{n-1}(z)$ and $p_{n-1}^*(z)$ lie in a two-dimensional space, and therefore, as the nontrivial polynomials $zp_{n-1}(z)$ and $p_{n-1}^*(z)$ (of degree $n$ and degree at most $n-1$, respectively) are linearly independent, we have In view of the equalities $p_n^*(0)=\overline{k}_n$, $n=0,1,\dots $, from (15), after equating the coefficients of $z^0$ and $z^n$ we obtain (13) and (14). The parameter $\alpha_{n-1}$ defined by $p_n(z)$ via (14) is independent of the normalization of $p_n(z)$ and is called the (classical) $(n-1)$st Geronimus parameter of the measure $\sigma $. The paper [3] contains a remarkable result. Geronimus’s Theorem. Let $\sigma $ be a measure with infinite support on $\mathbb T$. Then where $\alpha_n$ and $\gamma_n$ are the (classical) Geronimus and Schur parameters of $\sigma $. The multipoint Geronimus parameters of a measure $\sigma $ on $\mathbb T$ are defined similarly to the classical parameters, except that orthogonal polynomials must be replaced by orthogonal rational functions with poles outside $\overline{\mathbb D}$. More precisely, let $a_0,a_1,\dots$ be a sequence of points in $\mathbb D$ and $\mathscr L_0(z),\mathscr L_1(z),\dots$ be the sequence of rational functions (defined in terms of Blaschke products) of the special form
$$
\begin{equation}
\mathscr L_0(z):\equiv 1, \quad \mathscr L_1(z):=\varphi_{a_1}(z),\quad\dots,\quad \mathscr L_n(z):=\prod_{k=1}^n\varphi_{a_k}(z),\quad\dots ,
\end{equation}
\tag{16}
$$
let $\sigma $ be a positive Borel measure with infinite support on $\mathbb T$ and $\{ r_n(z)\}_{n=0}^\infty$ be the sequence of $\sigma $-orthogonal rational functions of the form
$$
\begin{equation}
\begin{gathered} \, \notag r_n(z)=\sum_{j=0}^n\varkappa_{n,j}\mathscr L_j(z), \qquad \varkappa_{n,n}\neq 0 , \\ \int_{\mathbb T}r_n(z)\overline{r_k(z)}\,d\sigma (z)=0, \qquad 0\leqslant k<n<\infty, \end{gathered}
\end{equation}
\tag{17}
$$
obtained by the orthogonalization of system (16). Note that $\overline{\mathscr L_n(1/\overline{z})}=\mathscr L_n(z)^{-1}$ and set
$$
\begin{equation}
r^*_n(z):=\mathscr L_n(z)\overline{r_n\biggl(\frac{1}{\overline{z}}\biggr)}=\sum_{j=0}^n\overline{\varkappa }_{n,j}\frac{\mathscr L_n(z)}{\mathscr L_{j}(z)}
\end{equation}
\tag{18}
$$
and
$$
\begin{equation}
\alpha_{n-1}:=-\frac{\overline{r_{n}(a_{n-1})}}{r_{n}^*(a_{n-1})}, \qquad n=1,2,\dots\,.
\end{equation}
\tag{19}
$$
The parameters $\alpha_{0},\alpha_{1},\dots$, defined by (19) are independent of the (nontrivial) leading coefficients $\varkappa_{0,0},\varkappa_{1,1},\dots$ of the expansions of rational functions $r_{0}(z),r_1(z),\dots$; they are called the Geronimus parameter of the measure $\sigma $ with respect to the sequence $a_0,a_1,a_2,\dots$ . Note that for $a_0=a_1=\dots =0$ we have the equalities $\mathscr L_n(z)=z^n$, $n=0,1,\dots$, so that in this case the definition (19) is the same as the classical definition (14). Remark 3. The system of functions (16) and the orthogonal rational functions (17) obtained by its orthogonalization are independent of $a_0$, so the Geronimus parameters $\alpha_1,\alpha_2,\dots $ are independent of $a_0$. Only the Geronimus parameter $\alpha_0=-\overline{r_{1}(a_{0})}/r_{1}^*(a_{0})$ depends on $a_0$. Apart from the relations between $r_n(z)$ and $r_{n-1}(z)$, which we do not present here (although we present their modified analogues in § 2), § 6.4 of [4] contains the following result. Multipoint Version of Geronimus’s Theorem. Let $a_0,a_1,a_2,\dots$ be a sequence of points in $\mathbb D$, let $a_0=0$, and let $\sigma $ be a measure with infinite support on $\mathbb T$. Then where the $\alpha_n$ and $\gamma_n$ are the Geronimus and Schur parameters, respectively, of the measure $\sigma $ with respect to the sequence $a_0,a_1,\dots$ . Note that it is quite important in this result that $a_0=0$. In fact, if we change the point $a_0=0$ without changing $a_1,a_2,\dots$, then the Schur parameters $\gamma_0,\gamma_1,\dots$ change (see Remark 2), but the Geronimus parameters $\alpha_1,\alpha_2,\dots$ remain the same (see Remark 3). A result established by Rakhmanov [5], [6] in the classical case also holds in the multipoint case (covering the classical one). Namely, the following theorem was proved in [4], § 6.4. Multipoint Version of Rakhmanov’s Theorem. Let $a_0,\mkern-1mu a_1,\mkern-1mu a_2,\mkern-1mu \dots$ be a sequence of points with compact closure in $\mathbb D$ and $\sigma $ be a measure on $\mathbb T$ such that $\sigma' >0$ almost everywhere with respect to the Lebesgue measure $m$ on $\mathbb T$, where $\sigma' =d\sigma /dm$ is the derivative of $\sigma$ with respect to $m$. Then where the $\alpha_n$ are the Geronimus parameters of $\sigma $ with respect to $a_0,a_1,\dots$ . Note that for $a_0=0$ it follows from (20) and (21) that $\lim_{n\to\infty}\gamma_{n}=0$. However, as mentioned after the statement of the multipoint version of Geronimus’s theorem, the inequality $a_0\neq 0$ results in the failure of (20). Nonetheless, the limit equality $\lim_{n\to\infty}\gamma_{n}=0$ also holds without the condition $a_0=0$. Namely, in § 4 we prove the following result. Refined Version of Rakhmanov’s Theorem. Let $a_0,a_1,a_2,\dots$ be a sequence of points with compact closure in $\mathbb D$ and $\sigma $ be a measure on $\mathbb T$ such that $\sigma' >0$ almost everywhere with respect to Lebesgue measure $m$ on $\mathbb T$, where $\sigma' =d\sigma /dm$ is the derivative of $\sigma$ with respect to $m$. Then where $\gamma_n$ and $\alpha_n$ are the Geronimus and Schur parameters of $\sigma $ with respect to $a_0,a_1,\dots$ , respectively. Based on Geronimus’s theorem for the classical case on the one hand and, on the other hand, using Fatou’s theorem on nontangential limits (see [7], Ch. I, § 5), which states that, given a measure $\sigma $ on $\mathbb T$, almost everywhere on $\mathbb T$ with respect to the measure $m$ we have where $\sigma' =d\sigma /dm$ and $F(z)$ is the Carathéodory function of $\sigma$, Khrushchev found in [8] a new proof of Rakhmanov’s theorem (for the classical case), in which he used the properties of the Schur classical continued fraction in place of the properties of orthogonal polynomials. This proof is based on a theorem due to Khrushchev ([8], Theorem 1), which is of independent interest and was subsequently extended to the multipoint case ([9], Theorem 3.4).Multipoint Version of Khrushchev’s Theorem. Let $f(z)$ be a Schur function and let the multipoint Schur algorithm at points $a_1,a_2,\dots$ lying on a compact subset of $\mathbb D$, as applied to $f(z)$, return a sequence of functions $f_0(z)=f(z),f_1(z),\dots$ . Then A statement in the opposite direction of Rakhmanov’s theorem to a certain extent was proved in the classical case by Geronimus [10]. Geronimus’s Second Theorem. Let $\gamma_0,\gamma_1,\dots$ be the Schur parameters of a measure $\sigma$ on $\mathbb T$ such that Then $\operatorname{supp} \sigma =\mathbb T$, where $\operatorname{supp} \sigma $ is the support of $\sigma $. In connection with the assumption (25) of Geronimus’s second theorem we observe that for $|\gamma_n|< 1$, $n=0,1,\dots$, we have the implications
$$
\begin{equation}
\lim_{n\to\infty}\gamma_n=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\prod_{k=0}^n (1-|\gamma_k|^2)^{1/n}=1 \quad\Longrightarrow\quad \varlimsup_{n\to\infty}\prod_{k=0}^n (1-|\gamma_k|^2)^{1/n}=1.
\end{equation}
\tag{26}
$$
In this paper we prove a multipoint version of Geronimus’s second theorem . Before stating it we introduce some notation. We let $\xi_a$ denote the Dirac measure with support at $a$, and $\xrightarrow[n\to\infty]{*}$ denote weak-$*$ convergence of measures depending on $n$ as $n\to\infty$. In particular,
$$
\begin{equation*}
\frac{\xi_{a_0}+\dots +\xi_{a_{n-1}}}{n} \xrightarrow[n\to\infty]{*}\eta \quad\Longleftrightarrow\quad \lim_{n\to\infty}\frac{v(a_0)+\dots +v(a_{n-1})}{n}=\int v(z)\,d\eta (z),
\end{equation*}
\notag
$$
where $v(z)$ is an arbitrary continuous function on the Riemann sphere $\overline{\mathbb C}$. Theorem 1. Let $a_0,a_1,\dots$ be points in a compact set $E\subset\mathbb D$, and let the coefficients $\gamma_{-1},\gamma_{0},\gamma_1,\dots$ of the multipoint continued fraction (9) (at $a_0,a_1,\dots$) satisfy $\operatorname{Re}\gamma_{-1}>0$ and $|\gamma_n|<1$, $n=0,1,\dots$ . Then the following results hold. 1. The convergents of (9) with even indices converge locally uniformly in $\mathbb D$ to the Carathéodory function $F(z)$, the convergents with odd indices converge locally uniformly in $\overline{\mathbb C}\setminus\overline{\mathbb D}$ to $F^*(z):=-\overline{F(1/\overline{z})}$, and, as applied to $F(z)$, the analogue of Schur’s algorithm at the points $a_0,a_1,\dots$ returns the original continued fraction (9). 2. If the points $a_0,a_1,\dots$ have a limit distribution such that
$$
\begin{equation}
\frac{\xi_{a_0}+\dots +\xi_{a_{n-1}}}{n} \xrightarrow[n\to\infty]{*}\eta,
\end{equation}
\tag{27}
$$
where $\eta$ is a probability measure with support on $E$, and if the coefficients $\gamma_{-1},\gamma_0,\dots$ of (9) satisfy
$$
\begin{equation}
\varlimsup_{n\to\infty}\prod_{k=0}^n(1-|\gamma_k|)^{1/n}=1,
\end{equation}
\tag{28}
$$
then in the Riesz–Herglotz integral representation (10) for the Carathéodory function $F(z)$ the support of the measure coincides with $\mathbb T$. Supplement to Theorem 1. Assertion 2 of Theorem 1 also holds if we replace (27) and (28) by the two conditions In connection with the hypotheses of Theorem 1 and the supplement to it, note that for $|\gamma_n|< 1$, $n=0,1,\dots$, apart from (26), we have the implications
$$
\begin{equation*}
\lim_{n\to\infty}\gamma_n=0 \quad\Longrightarrow\quad \varlimsup_{n\to\infty}\prod_{k=0}^n (1-|\gamma_k|)^{1/n}=1 \quad\Longrightarrow\quad \varlimsup_{n\to\infty}\prod_{k=0}^n (1-|\gamma_k|^2)^{1/n}=1,
\end{equation*}
\notag
$$
whereas the reverse implications fail in general. This means that the classical second Geronimus theorem, which is not a consequence of Theorem 1, follows from the supplement to Theorem 1 (for $a_0=a_1=\dots =0$). Also note that the following question is still open. Question. Is Theorem 1 also valid in the case when the assumption (27) is dropped (retaining only the condition that $a_0,a_1,\dots $ lie in a compact subset of $\mathbb D$) and (28) is preserved or replaced by the stronger condition $\lim_{n\to\infty}\gamma_n=0$? § 2. Modifications of the multipoint versions of Geronimus’s and Rakhmanov’s theoremsNote that it immediately follows from the definitions of Carathéodory and Nevanlinna functions (see the beginning of § 1) that
$$
\begin{equation}
F(z)\in\mathfrak B^{\mathrm c} \quad\Longleftrightarrow\quad G(z)\in\mathfrak B^{\mathrm n}, \quad\text{where } G(z)=iF (\psi (z)) ,
\end{equation}
\tag{29}
$$
and $\psi (z)$ is a linear fractional transformation taking $\mathbb C_+$ to $\mathbb D$. Hence the class $\mathfrak B^{\mathrm n}$ of Nevanlinna functions has the properties close to the class $\mathfrak B^{\mathrm c}$ of Carathéodory functions and, as is known, the Riesz–Herglotz theorem, stated in § 1 for Carathéodory functions $F(z)$, can be modified for Nevanlinna functions $G(z)$. In fact, let $F(z)$ be the Carathéodory function of a measure $\sigma$ on $\mathbb T$. Fixing in (29) the linear fractional transformation we let $\varsigma$ denote the measure on $\overline{\mathbb R}$ that for $L\subset\overline{\mathbb R}$ is defined by
$$
\begin{equation}
\varsigma (L)=\sigma (\psi (L)), \quad\text{where } \psi (L)=\{ t=\psi (u)\colon u\in L\},
\end{equation}
\tag{30}
$$
and set $\theta :=\sigma (\{ 1\})=\varsigma (\{\infty \})\geqslant 0$. From (29) and (10) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag G(z)&=iF (\psi (z))=i^2\operatorname{Im} F(0) +i\int_{\mathbb T}\frac{t+\psi (z)}{t-\psi (z)}\,d\sigma (t) \\ \notag &=\operatorname{Re} G(i)+i\int_{\overline{\mathbb R}}\frac{\psi (u)+\psi (z)}{\psi (u)-\psi (z)}\,d\varsigma (u)=\operatorname{Re} G(i)+\int_{\overline{\mathbb R}}\frac{1+uz}{u-z}\,d\varsigma (u) \\ &=\operatorname{Re} G(i)+\theta z+\int_{-\infty }^{\infty }\frac{1+uz}{u-z}\,d\varsigma (u). \end{aligned}
\end{equation}
\tag{31}
$$
Note that if two measures $\sigma$ and $\varsigma$ are connected by (30), then $\operatorname{supp}\sigma =\psi (\operatorname{supp}\varsigma)$ and, in particular,
$$
\begin{equation}
\operatorname{supp}\sigma =\mathbb T \quad\Longleftrightarrow\quad \operatorname{supp}\varsigma =\overline{\mathbb R},
\end{equation}
\tag{32}
$$
while for the derivatives $\sigma' ={d\sigma}/{dm}$ and $\varsigma' ={d\varsigma}/{dM}$ with respect to the Lebesgue measures $m$ and $M$ on $\mathbb T$ and $\mathbb R$, respectively, we have
$$
\begin{equation}
\begin{aligned} \, \notag \varsigma' (u) &=\lim_{\Delta u\to 0}\frac{\varsigma ([u,u+\Delta u])}{|\Delta u|} \\ \notag &=\lim_{\Delta u\to 0}\biggl (\frac{\sigma (\psi [u,u+\Delta u])}{m (\psi [u,u+\Delta u])} \,\frac{m (\psi [u,u+\Delta u])}{|\Delta u|}\biggr) \\ &=\sigma' (\psi (u))|\psi' (u)|=\sigma' (\psi (u))\frac{2}{1+u^2}, \qquad u\in\mathbb R. \end{aligned}
\end{equation}
\tag{33}
$$
Let $\widetilde{m}$ be the measure on $\overline{\mathbb R}$ obtained from $m$ on $\mathbb T$ by the formula ${\widetilde{m}(L)=m(\psi(L))}$. Then by (33)
$$
\begin{equation}
\widetilde{m}' (u)=\frac{2}{1+u^2}, \quad\text{where } \widetilde{m}' =\frac{d\widetilde{m}}{dM}, \quad u\in\mathbb R.
\end{equation}
\tag{34}
$$
It follows from (34) that for $|u|\leqslant C<\infty$
$$
\begin{equation*}
\frac{2}{1+C^2}\leqslant\widetilde{m}' (u)\leqslant 2.
\end{equation*}
\notag
$$
Therefore, for each bounded subset $L$ of the line $\mathbb R$
$$
\begin{equation}
M(L)=0 \quad\Longleftrightarrow\quad \widetilde{m}(L)=0 \quad\Longleftrightarrow\quad m (\psi (L))=0;
\end{equation}
\tag{35}
$$
moreover, (35) also holds for arbitrary subsets $L$ of $\mathbb R$ in view of the equality
$$
\begin{equation*}
L=\bigsqcup_{k\in\mathbb Z} L_k, \quad\text{where } L_k:=L\cap [ k,k+1).
\end{equation*}
\notag
$$
Let $G(z)$ be the Nevanlinna function of $\varsigma $ on $\overline{\mathbb R}$, that is, the function defined by the right-hand side of (31), and let $F(z)=i^{-1}G (\psi^{-1}(z))$ be the Carathéodory function of the measure $\sigma $ on $\mathbb T$. It follows from (33), (35) and Fatou’s theorem (see (23)) that almost everywhere with respect to the Lebesgue measure $M$ on $\mathbb R$ we have
$$
\begin{equation*}
\varsigma' (z)=\frac{2}{1+z^2}\sigma' (\psi (z))=\frac{2}{1+z^2}\operatorname{Re} F (\psi (z)) =\frac{2}{1+z^2}\operatorname{Im} G(z), \qquad z\in\mathbb R,
\end{equation*}
\notag
$$
which can be regarded as a modified version of Fatou’s theorem. It also follows from (33) and (35) that
$$
\begin{equation}
\begin{split} &\sigma' >0 \quad \text{a.e. w.r.t. the Lebesgue measure $m$ on} \ \mathbb T \\ &\qquad\Longleftrightarrow\quad \varsigma' >0 \quad \text{a.e. w.r.t. the Lebesgue measure $M$ on} \ \mathbb R. \end{split}
\end{equation}
\tag{36}
$$
In this section we show that the results on Carathéodory functions defined by measures on $\mathbb T$, which were stated in § 1, have analogues for Nevanlinna functions defined by measures on $\overline{\mathbb R}$. First of all, we introduce a (multipoint) analogue of Schur’s algorithm for Nevanlinna functions. To do this, apart from the class $\mathfrak B^{\mathrm n}$ of Nevanlinna functions we will also use the class $\mathfrak B^{\mathrm{b}}$, considered in [11], of holomorphic functions $g(z)$ in $\mathbb C_+$ such that $|g(z)|\leqslant 1$. Because
$$
\begin{equation*}
f(z)\in\mathfrak B^{\mathrm s} \quad\Longleftrightarrow\quad g(z)\in\mathfrak B^{\mathrm{b}}, \quad\text{where } g(z)=f (\psi (z)), \quad \psi (z)=\frac{z-i}{z+i} ,
\end{equation*}
\notag
$$
the class $\mathfrak B^{\mathrm{b}}$ has the properties close to those of of the class $\mathfrak B^{\mathrm s}$ of Schur functions. The multipoint Schur algorithms described in § 1 for functions in $\mathfrak B^{\mathrm s}$ can almost literally be carried over to functions in $\mathfrak B^{\mathrm{b}}$, provided that in place of points of interpolation $a_1,a_2,\dots$ in $\mathbb D$ we use points $d_1,d_2,\dots$ in $\mathbb C_+$; in place of the linear fractional transformations which for $a_n\in\mathbb D$ take $\mathbb D$ to $\mathbb D$, we use the linear fractional transformations which for $d_n\in\mathbb C_+$ take $\mathbb C_+$ to $\mathbb D$, and in place of Statement 1 we use a modified version of it, which reads as follows.Statement 3. Let $d\in\mathbb C_+$, $g(z)\in H(\mathbb C_+)$ and $|g(d)|< 1$. Then Moreover, Equalities (3) and (4), based on Statement 1 and lying at the core of Schur’s algorithm, are replaced in its modified version by the following equalities, based on Statement 3:
$$
\begin{equation}
g_{n+1}(z):= g_n^{\mathrm{b},d_{n+1}}(z) = \frac{g_n(z)-\delta_n}{\psi_{d_{n+1}}(z)(1-g_n(z)\overline{\delta_n})}
\end{equation}
\tag{37}
$$
and in which we denote the emerging parameters by $\delta_0,\delta_1,\dots$ (to stress the distinction from Schur’s algorithm as applied to functions $f(z)\in\mathfrak B^{\mathrm s}$). Thus, as applied to $g(z)\in\mathfrak B^{\mathrm{b}}$, the multipoint analogue of Schur’s algorithm (at points $d_1,d_2,\dots $ in $\mathbb C_+$) returns either $g(z)\equiv \delta_0$ for $|\delta_0|=1$, or the finite (in the case when $|\delta _0|<1,\dots,|\delta _{n-1}|<1$ and $|\delta _n|=1$ for some $n\in\mathbb N$) or infinite continued fraction
$$
\begin{equation}
\delta _0+\cfrac{(1-|\delta _0|^2)\psi_{d_{1}}(z)}{\overline{\delta }_0\psi_{d_{1}}(z)+\cfrac{1}{\delta _1+\cfrac{(1-|\delta _1|^2)\psi_{d_{2}}(z)}{\overline{\delta }_1\psi_{d_{2}}(z)+\cfrac{1}{\delta _2+\dotsb }}}}\,
\end{equation}
\tag{39}
$$
whose coefficients $\delta _0,\delta _1,\dots$ are less than one in modulus. Now observe that the following result holds along with Statements 1, 2 and 3. Statement 4. Let $d\in\mathbb C_+$, $G(z)\in H(\mathbb C_+)$ and $\operatorname{Im} G(d)> 0$. Then Furthermore, As above, in follows from the equalities
$$
\begin{equation}
G_{-1}(z):=G(z), \qquad \delta_{-1}(z):=G(d_0), \qquad g_0(z):=G^{\mathrm{n},d_0}(z)
\end{equation}
\tag{41}
$$
and Statement 4 (at the first step) and from (37), (38) and Statement 3 (at the subsequent steps) that the application of the multipoint analogue of Schur’s algorithm (at points $d_0,d_1,\dots $ in $\mathbb C_+$) to $G(z)\in\mathfrak B^{\mathrm n}$ returns either $G(z)\equiv G(d_0)=:\delta_{-1}$ for $\operatorname{Im}\delta_{-1}=0$, or a finite fraction (in the case when $\operatorname{Im}\delta_{-1}>0$, $|\delta _0|<1$, $\dots$, $|\delta _{n-1}|<1$ and $|\delta _n|=1$ for some $n\in\mathbb Z_+$), or it defines an infinite sequence of functions $G_{-1}(z):=G(z)$, $g_0(z):=G^{\mathrm{n},d_0}(z),g_1(z)$, $\dots$, and produces an infinite continued fraction
$$
\begin{equation}
\delta_{-1}+\cfrac{(\delta_{-1}-\overline{\delta}_{-1})\psi_{d_0}(z)} {-\psi_{d_0}(z)+\cfrac{1}{\delta_0+\cfrac{(1-|\delta_0|^2)\psi_{d_{1}}(z)}{\overline{\delta }_0\psi_{d_{1}}(z)+\cfrac{1} {\delta_1+\cfrac{(1-|\delta_1|^2)\psi_{d_{2}}(z)}{\overline{\delta }_1\psi_{d_{2}}(z)+\cfrac{1}{\delta_2+\dotsb }}}}}}\,
\end{equation}
\tag{42}
$$
with coefficients satisfying $\operatorname{Im} \delta_{-1}>0$ and $|\delta_n|<1$, $n=0,1,\dots$ . We call the coefficients $\delta_0,\delta_1,\dots$ of the fraction (42) obtained by applying the analogue of Schur’s algorithm at the points $d_0,d_1,\dots$ to the Nevanlinna function $G(z)$ of $\varsigma$ on $\overline{\mathbb R}$ the Schur parameters of the measure $\varsigma$ with respect to the points $d_0,d_1,\dots$ . Now, before we define an analogues of the multipoint Geronimus parameters of the measure $\varsigma$ on $\overline{\mathbb R}$, we indicate (by following the scheme proposed in [4] and [9]) relations for orthogonal rational functions with respect to a measure $\varsigma$ with infinite support on $\overline{\mathbb R}$. Let $d_0,d_1,\dots$ be a sequence of points in $\mathbb C_+$, and let $\psi_{d}(z)=(z-d)/(z-\overline{d})$, let
$$
\begin{equation}
\mathscr B_0(z):\equiv 1, \quad \mathscr B_1(z):=\psi_{d_1}(z), \quad\dots, \quad \mathscr B_n(z):=\prod_{k=1}^n\psi_{d_k}(z), \quad\dots ,
\end{equation}
\tag{43}
$$
be a sequence of rational functions (coinciding for $d_1=d_2=\dots =i$ with the sequence $1,\psi (z),\dots,\psi^n(z),\dots$, where $\psi (z)=(z-i)/(z+i)$), $\varsigma$ be a measure with infinite support on $\overline{\mathbb R}$ and $s_0(z),s_1(z),\dots$ be a sequence of orthonormal rational functions with respect to $\varsigma $ of the form
$$
\begin{equation}
s_n(z)=\sum_{j=0}^n\varkappa_{n,j}\mathscr B_j(z), \qquad \varkappa_{n,n}\neq 0 ,
\end{equation}
\tag{44}
$$
$$
\begin{equation}
\int_{\overline{\mathbb R}}s_n(u)\overline{s_k(u)}\,d\varsigma (u)=\delta_{n,k}, \qquad 0\leqslant n,k<\infty,
\end{equation}
\tag{45}
$$
which are obtained by the orthogonalization of the system (43). Here and below $\delta_{n,k}$ is the Kronecker delta equal to $0$ for $n\neq k$ and to $1$ for $n=k$. Let $\mathscr P_n$ be the set of polynomials of degree at most $n$. Since $s_0(z),\dots,s_n(z)$ is an orthonormal basis in the $(n+1)$-dimensional space
$$
\begin{equation*}
L_{n+1}=\biggl \{ \frac{p(z)}{\prod_{j=1}^n(z-\overline{d}_j)}\colon p(z)\in\mathscr P_n \biggr\}
\end{equation*}
\notag
$$
with scalar product defined by the left-hand side of (45), it follows that is the reproducing kernel of the space $L_{n+1}$, that is, for all $s(z)\in L_{n+1}$
$$
\begin{equation*}
\int_{\overline{\mathbb R}}s(u)\overline{h_n(u,w)}\,d\varsigma (u)=s(w).
\end{equation*}
\notag
$$
For $ n=0,1,\dots$ we set
$$
\begin{equation}
\begin{gathered} \, \overline{\psi}_{d_n}(z)=\overline{\psi_{d_n}(\overline{z})},\qquad \overline{\mathscr B}_n(z)=\overline{\mathscr B_n(\overline{z})}, \\ \overline{s}_n(z):=\overline{s_n(\overline{z})} \quad\text{and}\quad s_n^*(z):=\mathscr B_n(z)\overline{s}_n(z) \end{gathered}
\end{equation}
\tag{47}
$$
and note that (taking (43) and (44) into account)
$$
\begin{equation}
\overline{\psi }_{d_n}(z)= (\psi_{d_n}(z))^{-1}, \qquad \overline{\mathscr B }_n(z)= (\mathscr B_n(z))^{-1}\quad\text{and} \quad s_n^*(z)=\sum_{j=0}^n\overline{\varkappa}_{n,j}\frac{\mathscr B_n(z)}{\mathscr B_j(z)}.
\end{equation}
\tag{48}
$$
Since $\psi_{d_n}(d_n)=0$, we have
$$
\begin{equation}
s_n^*(d_n)=\overline{k}_{n}, \quad\text{where } k_{n}:=\varkappa_{n,n} ,
\end{equation}
\tag{49}
$$
$$
\begin{equation}
\overline{\mathscr B_n(d_n)}s_k(\overline{d}_n)=0 \quad\text{for } 0\leqslant k<n \quad\text{and}\quad \overline{\mathscr B_n(d_n)}s_n(\overline{d}_n)=\overline{s_n^*(d_n)}.
\end{equation}
\tag{50}
$$
Note also that all functions in the set
$$
\begin{equation*}
\{ \mathscr B_n(z)\overline{s}_l(z) \}_{l=0}^n =\biggl\{ \sum_{j=0}^l\overline{\varkappa}_{l,j}\frac{\mathscr B_n(z)}{\mathscr B_j(z)}\biggr\}_{l=0}^n
\end{equation*}
\notag
$$
belong to $L_{n+1}$ and form an orthonormal basis there, because for $u\in\overline{\mathbb R}$ we have $|\mathscr B_n(u)|=1$ and $\overline{s}_k(u)=\overline{s_k(u)}$, and therefore for $0\leqslant k,l\leqslant n$, by (47) and (45)
$$
\begin{equation*}
\int_{\overline{\mathbb R}}\mathscr B_n(u)\overline{s}_k(u)\overline{\mathscr B_n(u)\overline{s}_l(u)}\,d\varsigma (u) =\int_{\overline{\mathbb R}}|\mathscr B_n(u)|^2\overline{s_k(u)}s_l(u)\,d\varsigma (u) =\delta_{k,l}.
\end{equation*}
\notag
$$
Hence, as the reproducing kernel is unique (for instance, see [12]), apart from (46), we also have the equality
$$
\begin{equation}
h_n(z,w)=\mathscr B_n(z)\overline{\mathscr B_n(w)}\sum_{j=0}^n\overline{s}_j(z)s_j(\overline{w}).
\end{equation}
\tag{51}
$$
It follows from (51) for $w=d_n$, (50), (47) and (49) that
$$
\begin{equation}
h_n(z,d_n)=\mathscr B_n(z)\overline{\mathscr B_n(d_n)}\overline{s}_n(z)s_n(\overline{d}_n) =s_n^*(z)\overline{s_n^*(d_n)} =s_n^*(z)k _{n} ,
\end{equation}
\tag{52}
$$
From (46), (51) (for the indices $n-1$ and $n$) and (47) we obtain
$$
\begin{equation*}
\begin{aligned} \, s_n(z)\overline{s_n(w)}+h_{n-1}(z,w) &= h_n(z,w) \\ &=\mathscr B_n(z)\overline{\mathscr B_n(w)}\biggl(\overline{s}_n(z)s_n(\overline{w}) +\sum_{k=0}^{n-1}\overline{s}_k(z)s_k(\overline{w})\biggr) \\ &=s_n^*(z)\overline{s_n^*(w)}+\psi_{d_n}(z)\overline{\psi_{d_n}(w)}h_{n-1}(z,w) \end{aligned}
\end{equation*}
\notag
$$
and, as a consequence,
$$
\begin{equation}
h_{n-1}(z,w)=\frac{s_n^*(z)\overline{s_n^*(w)}-s_n(z)\overline{s_n(w)}} {1-\psi_{d_n}(z)\overline{\psi_{d_n}(w)}}.
\end{equation}
\tag{54}
$$
From (52) (with $n$ replaced by $n-1$) and (54) for $w=d_{n-1}$ it follows that
$$
\begin{equation}
k _{n-1} s_{n-1}^*(z)=h_{n-1}(z,d_{n-1})=\frac{s_n^*(z) \overline{s_n^*(d_{n-1})}-s_n(z)\overline{s_n(d_{n-1})}} {1-\psi_{d_n}(z)\overline{\psi_{d_n}(d_{n-1})}}.
\end{equation}
\tag{55}
$$
From (53) (with $n$ replaced by $n-1$) and (55) for $z=d_{n-1}$ it follows that
$$
\begin{equation}
|k _{n-1} |^2=k _{n-1} s_{n-1}^*(d_{n-1})=\frac{|s_n^*(d_{n-1})|^2-|s_n(d_{n-1})|^2}{1-|\psi_{d_n}(d_{n-1})|^2}.
\end{equation}
\tag{56}
$$
Replacing each coefficient in (55) by the complex conjugate one (but keeping the variable $z$) and then multiplying the left-hand side by $\mathscr B_{n-1}(z)$ and the right-hand side by ${\mathscr B_{n}(z)}/{\psi_{d_n}(z)}$, using the facts that
$$
\begin{equation*}
s_{n-1}(z)=\mathscr B_{n-1}(z)\overline{s^*_{n-1}}(z), \qquad s_{n}(z)=\mathscr B_{n}(z)\overline{s^*_{n}}(z)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\psi_{d_n}(z)(1-\overline{\psi}_{d_n}(z)\psi_{d_n}(d_{n-1})) =\psi_{d_n}(z)-\psi_{d_n}(d_{n-1}),
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation}
\overline{k }_{n-1} s_{n-1}(z)=\frac{s_n(z)s_n^*(d_{n-1})-s_n^*(z)s_n(d_{n-1})}{\psi_{d_n}(z)-\psi_{d_n}(d_{n-1})}.
\end{equation}
\tag{57}
$$
Eliminating $s_n^*(z)$ from (57) and (55), namely, adding (57) and (55) after multiplication by
$$
\begin{equation*}
\overline{s_n^*(d_{n-1})} (\psi_{d_n}(z)-\psi_{d_n}(d_{n-1}))\quad\text{and}\quad s_{n}(d_{n-1}) (1-\psi_{d_n}(z)\overline{\psi_{d_n}(d_{n-1})}),
\end{equation*}
\notag
$$
respectively, and taking (56) into account we obtain
$$
\begin{equation*}
\begin{aligned} \, &\overline{k }_{n-1} \overline{s_n^*(d_{n-1})} (\psi_{d_n}(z)-\psi_{d_n}(d_{n-1}))s_{n-1}(z) \\ &\qquad\qquad +k_{n-1} s_n(d_{n-1}) (1-\psi_{d_n}(z)\overline{\psi_{d_n}(d_{n-1})})s_{n-1}^*(z) \\ &\qquad =\bigl(|s_n^*(d_{n-1})|^2-|s_{n}(d_{n-1})|^2\bigr)s_n(z) =|k_{n-1}|^2(1-|\psi_{d_n}(d_{n-1})|^2)s_n(z), \end{aligned}
\end{equation*}
\notag
$$
which is equivalent to the equality
$$
\begin{equation}
\begin{aligned} \, \notag s_n(z) &=\frac{\overline{s_n^*(d_{n-1})}}{1-|\psi_{d_n}(d_{n-1})|^2}\biggl [ (\psi_{d_n}(z)-\psi_{d_n}(d_{n-1}))\frac{s_{n-1}(z)}{k_{n-1}} \\ &\qquad - (1-\psi_{d_n}(z)\overline{\psi_{d_n}(d_{n-1})})\overline{\beta }_{n-1}\frac{s_{n-1}^*(z)}{\overline{k}_{n-1}}\biggr], \end{aligned}
\end{equation}
\tag{58}
$$
where
$$
\begin{equation}
\beta_{n-1}=-\frac{\overline{s_n(d_{n-1})}}{s_n^*(d_{n-1})}, \qquad n=1,2,\dots\,.
\end{equation}
\tag{59}
$$
In the particular case of $d_{n-1}=d_n$, bearing in mind that
$$
\begin{equation*}
\overline{s_n^*(d_{n-1})}=\overline{s_n^*(d_{n})}=k_{n} \quad\text{and}\quad \psi_{d_n}(d_{n-1})=\psi_{d_n}(d_{n})=0,
\end{equation*}
\notag
$$
equalities (58) assume the following form (similar to (13)):
$$
\begin{equation*}
s_n(z)=\frac{k_{n}}{k_{n-1}}\psi_{d_n}(z)s_{n-1}(z) -\frac{k_{n}}{\overline{k}_{n-1}}\overline{\beta }_{n-1}s_{n-1}^*(z).
\end{equation*}
\notag
$$
Remark 4. Equalities (58), obtained under the assumption that system of functions $s_n(z)$ is orthonormal, as well as the coefficients $\beta_{n-1}$, $n=1,2,\dots$, in these equalities, which are defined by (59), are preserved by multiplication of $s_n(z)$ and $s_{n-1}(z)$ by arbitrary nonzero constants $c_n$ and $c_{n-1}$. This means that (58) and (59) hold for each sequence of $\varsigma$-orthogonal rational functions of the form (44). We call the coefficients $\beta_{0},\beta_{1},\dots$, defined by (59) the Geronimus parameters of the measure $\varsigma$ on $\overline{\mathbb R}$ with respect to the points $d_0,d_1,d_2,\dots$ . The following modifications of the theorems stated in § 1 take place. Modification of Geronimus’s Theorem. Let $d_0=i$, $d_1,d_2,\dots$ be a sequence of points in $\mathbb C_+$ and $\varsigma $ be a measure on $\overline{\mathbb R}$ with infinite support. Then where $\beta_n$ and $\delta_n$ are Geronimus and Schur parameters, respectively, of the measure $\varsigma $ with respect to the points $d_0=i$, $d_1,d_2,\dots$ . The condition $d_0=i$ is essential in the above theorem because the Geronimus parameters $\beta_1,\beta_2,\dots$ of $\varsigma $ are independent of $d_0$ but the Schur parameters of $\varsigma $ are significantly different for different $d_0$. Modification of Rakhmanov’s Theorem. Let $d_0,d_1,d_2,\dots$ be a sequence of points with compact support in $\mathbb C_+$ and $\varsigma $ be a measure on $\overline{\mathbb R}$ such that $\varsigma' >0$ almost everywhere with respect to the Lebesgue measure $M$ on $ \mathbb R$, where $\varsigma' =d\varsigma /dM$ is the derivative of $\varsigma$ with respect to $M$ on $\mathbb R$. Then where the $\delta_n$ and $\beta_n$ are the Schur and Geronimus parameters, respectively, of $\varsigma $ with respect to the points $d_0,d_1,\dots$ . Modification of Khrushchev’s Theorem. Let $g(z)\in\mathfrak B^{\mathrm{b}}$, and let the multipoint analogue of Schur’s algorithm at points $d_1,d_2,\dots$ lying in a compact subset of $\mathbb C_+$, as applied to $g(z)$, return a sequence of functions $g_0(z)=g(z),g_1(z),\dots$ . Then Theorem 2 (modification of Theorem 1). Let $d_0,d_1,\dots$ be points in a compact set $E$ in $\mathbb C_+$, and let the parameters $\delta_{-1},\delta_0,\dots$ of the multipoint continued fraction (42) (at $d_0,d_1,\dots$) satisfy $\operatorname{Im}\delta_{-1}>0$ and $|\delta_n|<1$, $n=0,1,\dots$ . Then the following hold. 1. The even convergents of (42) converge locally uniformly in $\mathbb C_+$ to a Nevanlinna function $G(z)$, the odd convergents converge locally uniformly in $\mathbb C_-:=\{ \operatorname{Im} z<0\}$ to the function $\overline{G}(z):=\overline{G(\overline{z})}$, and, as applied to $G(z)$, the analogue of Schur’s algorithm at $d_0,d_1,\dots$ returns the original continued fraction (42). 2. If the points $d_0,d_1,\dots$ have a limit distribution, namely,
$$
\begin{equation}
\frac{\xi_{d_0}+\dots +\xi_{d_{n-1}}}{n} \xrightarrow[n\to\infty]{*}\zeta,
\end{equation}
\tag{63}
$$
where $\zeta$ is a probability measure with support on $E$, and if the coefficients $\delta_{-1},\delta_0,\dots$ of the continued fraction (42) satisfy the condition
$$
\begin{equation}
\varlimsup_{n\to\infty}\prod_{k=0}^n(1-|\delta_k|)^{1/n}=1,
\end{equation}
\tag{64}
$$
then the measure $\varsigma $ in the Riesz–Herglotz integral representation (31) for the Nevanlinna function $G(z)$ has support $\overline{\mathbb R}$. Supplement to Theorem 2. Assertion 2 of Theorem 2 also holds when (63) and (64) are replaced by the two conditions § 3. Lemma 1 and its consequencesThe modifications of theorems from § 1 that were stated in § 1 can be proved by modifying the proofs of the original theorems in § 1. However, a shortcut is to prove Lemma 1 below, which shows that the theorems stated in § 1 are equivalent to their modifications in § 2. Lemma 1. Let $\sigma$ be a measure on $\mathbb T$ with infinite support, $a_0,a_1,\dots$ be points in $\mathbb D$, and let $\gamma_0,\gamma_{1},\dots $ and $\alpha_0,\alpha_{1},\dots $ be the Schur and Geronimus parameters, respectively, of $\sigma$ with respect to the points $a_0,a_1,\dots$ . Let
$$
\begin{equation}
\psi (z)=\frac{z-i}{z+i}\quad\textit{and} \quad t_n=\frac{1-a_n}{1-\overline{a}_n}, \quad n=0,1,\dots,
\end{equation}
\tag{65}
$$
let $\varsigma$ be the measure on $\overline{\mathbb R}$ defined by the equality $\varsigma (L)=\sigma (\psi (L))$ for $L\subset\overline{\mathbb R}$, and let $\delta_0,\delta_{1},\dots $ and $\beta_0,\beta_{1},\dots $ be the Schur and Geronimus parameters, respectively, of $\varsigma$ with respect to the points $d_0,d_1,\dots$, where $d_n=\psi^{-1}(a_n)$, $n=0,1,\dots$ . Then and Proof. From the second equality in (65) and the equalities we obtain
$$
\begin{equation}
\begin{aligned} \, \notag t_{n}\psi_{d_n}(z) &=\frac{1-a_n}{1-\overline{a}_n}\, \frac{z-d_n}{z-\overline{d}_n}=\frac{1-a_n}{1-\overline{a}_n}\, \frac{z-i(1+a_n)/(1-a_n)}{z+i(1+\overline{a}_n)/(1-\overline{a}_n)} \\ \notag &=\frac{z(1-a_n)-i(1+a_n)}{z(1-\overline{a}_n)+i(1+\overline{a}_n)} =\frac{(z-i)-a_n(z+i)}{(z+i)-\overline{a}_n(z-i))} \\ &=\frac{\psi (z)-a_n}{1-\psi (z)\overline{a}_n}=\varphi_{a_n} (\psi (z)), \qquad n=0,1,\dots\,. \end{aligned}
\end{equation}
\tag{68}
$$
Let $r_n(z)$ be the $\sigma$-orthogonal functions defined by (17), and let $r_n^*(z)$ be defined by (18). By (68) and the definitions (16) and (43) of the functions $\mathscr L_n(z)$ and $\mathscr B_n(z)$ we have the equalities $\mathscr L_0 (\psi (z))\equiv 1\equiv \mathscr B_0(z)$ and
$$
\begin{equation}
\begin{aligned} \, \mathscr L_n (\psi (z))&=\prod_{k=1}^n\varphi_{a_k} (\psi (z)) \notag \\ &=\prod_{k=1}^nt_k\psi_{d_k}(z)=t_1\dotsb t_n\mathscr B_n(z), \qquad n=1,2,\dots\,. \end{aligned}
\end{equation}
\tag{69}
$$
It follows from (17) and (69) that the functions $s_0(z):=r_0 (\psi (z))=\varkappa_{0,0}$ and
$$
\begin{equation}
\begin{aligned} \, s_n(z) &:=r_n (\psi (z))=\sum_{j=0}^n\varkappa_{n,j}\mathscr L_j (\psi (z)) \notag \\ &=\sum_{j=0}^n\varkappa_{n,j}t_1\dotsb t_j\mathscr B_j(z), \qquad n=1,2,\dots \end{aligned}
\end{equation}
\tag{70}
$$
(we set the product $t_1\cdots t_j$ equal to 1 for $j=0$), first, have the form (44) and, second, are orthogonal with respect to $\varsigma$ because $\varsigma (L)=\sigma (\psi (L))$, and therefore
$$
\begin{equation*}
\int_{\overline{\mathbb R}}s_n(u)\overline{s_k(u)}\,d\varsigma (u)=\int_{\mathbb T}r_n(t)\overline{r_k(t)}\,d\sigma (t)=0, \qquad 0\leqslant k<n<\infty.
\end{equation*}
\notag
$$
Since $|t_k|=1$, $k=0,\dots,n$, it follows from (70), (48), (69) and (18) that for ${n=1,2,\dots}$
$$
\begin{equation*}
\begin{aligned} \, s^*_n(z) &=\sum_{j=0}^n\overline{\varkappa }_{n,j}\overline{t}_1\dotsb \overline{t}_j\frac{\mathscr B_n(z)}{\mathscr B_{j}(z)} \\ &=\overline{t}_1\dotsb \overline{t}_{n}\sum_{j=0}^n\overline{\varkappa }_{n,j} \frac{\mathscr L_n (\psi (z))}{\mathscr L_{j} (\psi (z))} =\overline{t}_1\dotsb \overline{t}_{n}r^*_n (\psi (z)). \end{aligned}
\end{equation*}
\notag
$$
Hence from the definitions (59) and (19), for $n=1,2,\dots$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \beta_{n-1} &=-\frac{\overline{s_n(d_{n-1})}}{s_n^*(d_{n-1})} =-\frac{\overline{r_n (\psi(d_{n-1}))}}{\overline{t}_1\dotsb \overline{t}_{n}r_n^* (\psi(d_{n-1}))} \\ &=-t_1\dotsb t_n\frac{\overline{r_n(a_{n-1})}}{r_n^*(a_{n-1})}=t_1\dotsb t_n\alpha_{n-1}. \end{aligned}
\end{equation*}
\notag
$$
This yields (66).
Now let $F(z)$ be the Carathéodory function of the measure $\sigma$. As mentioned at the beginning of § 2, $G(z)=iF (\psi (z))$ is the Nevanlinna function of the measure $\varsigma$. Assume that, as applied to $F(z)$, the multipoint analogue of Schur’s algorithm at the points $a_0,a_1,\dots$ produces a sequence of functions
$$
\begin{equation*}
F_{-1}(z)=F(z), \quad f_0(z)=F^{\mathrm{c},a_0}(z), \quad f_1(z), \quad \dots
\end{equation*}
\notag
$$
and returns the continued fraction (9) with coefficients $\gamma_{-1},\gamma_0,\gamma_1,\dots$, while, as applied to $G(z)$, the multipoint analogue of Schur’s algorithm at $d_0,d_1,\dots$ produces a sequence of functions
$$
\begin{equation*}
G_{-1}(z)=G(z), \quad g_0(z)=G^{\mathrm{n},d_0}(z), \quad g_1(z), \quad \dots
\end{equation*}
\notag
$$
and returns (42) with coefficients $\delta_{-1},\delta_0,\delta_1,\dots$ . Let us show that
$$
\begin{equation}
g_n(z)=t_0\cdots t_nf_n (\psi (z)), \qquad n=0,1,\dots,
\end{equation}
\tag{71}
$$
and equalities (67) hold. In fact, from the obvious equalities
$$
\begin{equation*}
\delta_{-1}=G(d_0)=iF (\psi (d_0))=iF(a_0)=i\gamma_{-1},
\end{equation*}
\notag
$$
relations (68) and the definitions of $g_0(z)$ and $f_0 (\psi (z))$ (see (40), (41) and (6), (8)) we obtain
$$
\begin{equation*}
\begin{aligned} \, g_0(z) &=\frac{iF (\psi (z))-i\gamma_{-1}}{\psi_{d_0}(z) (iF (\psi (z))-\overline{i\gamma_{-1}})} \\ &=\frac{F (\psi (z))-\gamma_{-1}}{t_0^{-1}\varphi_{a_0} (\psi (z)) (F (\psi (z))+\overline{\gamma }_{-1})}= t_0f_0 (\psi (z)), \end{aligned}
\end{equation*}
\notag
$$
which coincides with (71) for $n=0$. Hence taking (38) and (4) into account we see that
$$
\begin{equation*}
\delta_0=g_0(d_{1})=t_{0}f_0 (\psi (d_{1}))=t_{0}f_0(a_{1})=t_{0}\gamma_0 .
\end{equation*}
\notag
$$
We make the inductive assumption that (71) and (67) hold for the indices from $0$ to $n-1$ inclusive. Then, bearing in mind that $\overline{t}_kt_k=|t_{k}|^2=1$, $k=0,1,\dots$, from (37), (38), (68) and (3), (4) we obtain
$$
\begin{equation*}
\begin{aligned} \, g_n(z) &=\frac{g_{n-1}(z)-\delta_{n-1}}{\psi_{d_n}(z) (1-\overline{\delta }_{n-1}g_{n-1}(z))} \\ &=\frac{t_{0}\dotsb t_{n-1} (f_{n-1} (\psi (z))-\gamma_{n-1})} {t_{n}^{-1}\varphi_{a_n} (\psi (z)) (1-\overline{\gamma }_{n-1}f_{n-1}(\psi (z)))} =t_{0}\dotsb t_{n}f_n (\psi (z)). \end{aligned}
\end{equation*}
\notag
$$
Hence (71) holds for all $n=0,1,\dots$ . From (71), taking (38) and (4) into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \delta_n=g_n(d_{n+1}) &=t_{0}\dotsb t_{n}f_n (\psi (d_{n+1})) \\ &=t_{0}\dotsb t_{n}f_n(a_{n+1})=t_{0}\dotsb t_{n}\gamma_n, \qquad n=0,1,\dots , \end{aligned}
\end{equation*}
\notag
$$
which proves (67).
Lemma 1 is proved. Now we show that, as a simple consequence of this lemma, the theorems stated in § 1 are equivalent to their modifications in § 2. In fact, let the assumptions of Lemma 1 be fulfilled. Then using the notation from the lemma and its proof we have the following. 1. The equalities
$$
\begin{equation*}
t_n=\frac{1-a_n}{1-\overline{a}_n}=\frac{i-\overline{d}_n}{i+d_n},
\end{equation*}
\notag
$$
and $t_0=1$ for $a_0=0$ (or, equivalently, for $d_0=i$) and relations (66) and (67) established in Lemma 1 yield the equivalences
$$
\begin{equation*}
\begin{aligned} \, \alpha_n=\gamma_n \quad&\Longleftrightarrow\quad t_1\dotsb t_{n+1}\alpha_n=t_0\dotsb t_{n+1}\gamma_n \\ &\Longleftrightarrow\quad \beta_n=t_{n+1}\delta_n,\qquad n=0,1,\dots, \end{aligned}
\end{equation*}
\notag
$$
which show that the multipoint Geronimus theorem (see (20)) is equivalent to its modified version (see (60)). 2. The equivalence (36) and the equalities $|\alpha_n|=|\beta_n|$ and $|\gamma_n|=|\delta_n|$, ${n=0,1,\dots}$ (see (65), (66) and (67)), means that the refined version of Rakhmanov’s theorem (see (22)) is equivalent to its modified version (see (61)). 3. Since in view of (71) and (65) we have
$$
\begin{equation*}
|g_n(z)|=|t_0\dotsb t_nf_n (\psi (z))|=|f_n (\psi (z))|, \qquad n=0,1,\dots,
\end{equation*}
\notag
$$
it follows from (34) and (35) that
$$
\begin{equation*}
\int_{\mathbb T}|f_n(t)|^2\,dm(t)=\int_{\mathbb R}|g_n(u)|^2\,d\widetilde{m}(u)=\int_{\mathbb R}|g_n(u)|^2\frac{2}{1+u^2}\,dM(u)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &|f_0(t)|<1\quad \text{a.e. w.r.t. the Lebesgue measure on}\ \mathbb T \\ &\qquad\Longleftrightarrow\quad |g_0(u)|<1\quad \text{a.e. w.r.t. the Lebesgue measure on } \mathbb R. \end{aligned}
\end{equation*}
\notag
$$
Therefore, the multipoint Khrushchev theorem (see (24)) is equivalent to its modified version (see (62)). 4. Note that it follows from the equalities $a_n=\psi (d_n)$ for $n=0,1,\dots$ and relations (67), supplemented by the equality $\delta_{-1}=i\gamma_{-1}$ for $n=-1$, that the continued fraction (9), after multiplication by $i$ and the substitution of $\psi (z)$ for $z$, becomes in view of (68) a fraction equivalent to (42). Therefore,
$$
\begin{equation}
\pi_{k}^{\mathrm{n}}(z)=i\pi_{k}^{\mathrm{c}} (\psi(z)), \qquad k=0,1,\dots,
\end{equation}
\tag{72}
$$
where $\pi_{k}^{\mathrm{c}}(z)$ and $\pi_{k}^{\mathrm{n}}(z)$ are the $k$-convergents of the continued fractions (9) and (42), respectively. Denoting local uniform convergence in the indicated domains by $\rightrightarrows $, from (72) and the equalities
$$
\begin{equation*}
\frac{1}{\overline{\psi(z)}}=\frac{\overline{z}-i}{\overline{z}+i}=\psi (\overline{z})
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation*}
\begin{aligned} \, &\pi_{2k}^{\mathrm{c}}(z)\rightrightarrows F(z)\in\mathfrak B^{\mathrm c}, \qquad z\in\mathbb D \\ &\ \Longleftrightarrow\quad \pi_{2k}^{\mathrm{n}}(z)=i\pi_{2k}^{\mathrm{c}} (\psi(z))\rightrightarrows iF (\psi(z))=G(z)\in\mathfrak B^{\mathrm n}, \qquad z\in\mathbb C_+, \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\pi_{2k+1}^{\mathrm{c}}(z)\rightrightarrows -\overline{F\biggl(\frac{1}{\overline{z}}\biggr)}, \qquad z\in\overline{\mathbb C}\setminus\overline{\mathbb D} \\ &\ \Longleftrightarrow\quad \pi_{2k+1}^{\mathrm{n}}(z)=i\pi_{2k+1}^{\mathrm{c}} (\psi(z))\rightrightarrows -i\overline{F \biggl(\frac1{\overline{\psi(z)}}\biggr)} =\overline{iF (\psi (\overline{z}))}=\overline{G(\overline{z})}, \qquad z\in\mathbb C_-. \end{aligned}
\end{equation*}
\notag
$$
This means that assertions 1 of Theorems 1 and 2 are equivalent. Now note that it follows from the equalities $a_n=\psi (d_n)$ for $n=0,1,\dots$ and the definition of weak convergence that
$$
\begin{equation*}
\frac{\xi_{d_0}+\dots +\xi_{d_{n-1}}}{n} \xrightarrow[n\to\infty]{*}\zeta \quad\Longleftrightarrow\quad \frac{\xi_{a_0}+\dots +\xi_{a_{n-1}}}{n} \xrightarrow[n\to\infty]{*}\eta,
\end{equation*}
\notag
$$
where the measures $\zeta$ and $\eta$ are connected by (30) (with $\zeta$ in place of $\varsigma$ and $\eta$ in place of $\sigma$). Hence equalities $|\gamma_n|=|\delta_n|$, $n=0,1,\dots$ (see (67)), which imply the equivalence of the assumptions (28) and (64), and the assertion (32) that the properties $\operatorname{supp}\sigma =\mathbb T$ and $\operatorname{supp}\varsigma =\overline{\mathbb R}$ are equivalent mean that the second assertions of Theorems 1 and 2 are equivalent.
§ 4. Proof of refined version of Rakhmanov’s theoremNote that in view of (21) we must only prove the equality $\lim_{n\to\infty}\gamma_n=0$. Our proof is based on the scheme of the proof of Rakhmanov’s theorem (in the classical case) proposed by Khrushchev, in which the multipoint version of Khrushchev’s theorem, stated in § 1 (see (24)) and proved in [9], is used. Let $F(z)$ be the Carathéodory function of the measure $\sigma$, and let
$$
\begin{equation*}
\gamma_{-1}=F(a_0)\quad\text{and} \quad F^{\mathrm{c},a_0}(z)=\frac{F(z)-\gamma_{-1}}{\varphi_{a_0}(z) (F(z)+\overline{\gamma }_{-1})}.
\end{equation*}
\notag
$$
By Proposition 2, $F^{\mathrm{c},a_0}(z)\in\mathfrak B^{\mathrm s}$ and
$$
\begin{equation*}
F(z)=\gamma_{-1}+\cfrac{(\gamma_{-1}+\overline{\gamma }_{-1})\varphi_{a_0}(z)}{-\varphi_{a_0}(z)+\cfrac{1}{F^{\mathrm{c},a_0}(z)}} =\frac{F^{\mathrm{c},a_0}(z)\varphi_{a_0}(z)\overline{\gamma}_{-1}+\gamma_{-1}}{1-F^{\mathrm{c},a_0}(z)\varphi_{a_0}(z)}.
\end{equation*}
\notag
$$
By Fatou’s theorem on nontangential limits (see (23)), almost everywhere with respect to the Lebesgue measure on $\mathbb T$ we have
$$
\begin{equation}
\begin{aligned} \, \sigma' (z) &=\operatorname{Re} F(z)= \operatorname{Re} \frac{F^{\mathrm{c},a_0}(z)\varphi_{a_0}(z)\overline{\gamma}_{-1}+\gamma_{-1}} {1-F^{\mathrm{c},a_0}(z)\varphi_{a_0}(z)} \notag \\ &=\frac{1-|F^{\mathrm{c},a_0}(z)\varphi_{a_0}(z)|^2}{|1-F^{\mathrm{c},a_0}(z) \varphi_{a_0}(z)|^2}\operatorname{Re} \gamma_{-1}. \end{aligned}
\end{equation}
\tag{73}
$$
Since $F(z)$ is a Carathéodory function, we have $\operatorname{Re} \gamma_{-1}\geqslant 0$, and the equality $\operatorname{Re} \gamma_{-1}=0$ is impossible since otherwise $F(z)\equiv \gamma_{-1}$, which leads to the contradiction
$$
\begin{equation*}
0<\int_{\mathbb T}d\sigma (t)=\operatorname{Re} F(0)=\operatorname{Re} F(a_0)=\operatorname{Re} \gamma_{-1}=0.
\end{equation*}
\notag
$$
Furthermore, as $\varphi_{a_0}(z)F^{\mathrm{c},a_0}(z)$ is a Schur function, the equality on a subset of $\mathbb T$ of positive Lebesgue measure yields the identity
$$
\begin{equation*}
\varphi_{a_0}(z)F^{\mathrm{c},a_0}(z)\equiv 1, \qquad z\in\mathbb D
\end{equation*}
\notag
$$
(see [7], Ch. II, Corollary 4.2.), which fails for $ z=a_0$. Thus,
$$
\begin{equation}
\operatorname{Re} \gamma_{-1}>0 , \qquad |1-\varphi_{a_0}(z)F^{\mathrm{c},a_0}(z)|>0 \quad \text{a.e. w.r.t. the Lebesgue measure on } \mathbb T.
\end{equation}
\tag{74}
$$
It follows from (73), (74) and equality $|\varphi_{a_0}(z)|=1$ for $z\in\mathbb T$ that almost everywhere with respect to the Lebesgue measure on $\mathbb T$
$$
\begin{equation}
\sigma' (z)>0 \quad\Longleftrightarrow\quad 1-|F^{\mathrm{c},a_0}(z)\varphi_{a_0}(z)|^2>0 \quad\Longleftrightarrow\quad 1>|F^{\mathrm{c},a_0}(z)|.
\end{equation}
\tag{75}
$$
Let $F_{-1}(z)=F(z)$, $f_0(z)=F^{\mathrm{c},a_0}(z),f_1(z),f_2(z),\dots$ be the functions obtained by the multipoint Schur algorithm (at the points $a_0,a_1,\dots$) as applied to ${F(z)\,{\in}\,\mathfrak B^{\mathrm c}}$. By the hypotheses of the theorem the points $a_0,a_1,\dots$ lie in a compact subset of $\mathbb D$, so from (75) and (24) we obtain the implications
$$
\begin{equation}
\begin{aligned} \, \sigma' (z)>0 \quad \text{a.e. on}\ \mathbb T \quad&\Longrightarrow\quad |F^{\mathrm{c},a_0}(z)|<1 \quad \text{a.e. on}\ \mathbb T \notag \\ &\Longrightarrow\quad \lim_{n\to\infty}\int_0^{2\pi }|f_n(e^{i\theta })|^2\,d\theta =0, \end{aligned}
\end{equation}
\tag{76}
$$
and the inequalities
$$
\begin{equation*}
|e^{i\theta }-a_n|\geqslant \varepsilon >0 \quad\text{for } \theta\in [0,2\pi ], \qquad n=1,2,\dots\,.
\end{equation*}
\notag
$$
Hence using the definition of the Schur parameters of the Carathéodory function $F(z)$, Cauchy’s theorem and the Cauchy–Schwarz–Bunyakovskii inequality we obtain
$$
\begin{equation}
\begin{aligned} \, \notag |\gamma_n| &=|f_n(a_{n+1})| \\ \notag &=\biggl|\frac{1}{2\pi i}\int_{\mathbb T} \frac{f_n(z)}{z-a_{n+1}}\,dz\biggr| =\biggl|\frac{1}{2\pi i}\int_0^{2\pi }f_n(e^{i\theta })\frac{ie^{i\theta }}{e^{i\theta }-a_{n+1}}\,d\theta\biggr| \\ \notag &\leqslant \frac{1}{2\pi } \biggl(\int_0^{2\pi }|f_n(e^{i\theta })|^2\,d\theta\biggr)^{1/2} \biggl(\int_0^{2\pi }\frac{1}{|e^{i\theta }-a_{n+1}|^2}\,d\theta \biggr)^{1/2} \\ &\leqslant \frac{1}{\varepsilon \sqrt{2\pi }}\biggl(\int_0^{2\pi }|f_n(e^{i\theta })|^2\,d\theta\biggr)^{1/2}. \end{aligned}
\end{equation}
\tag{77}
$$
It follows from (76) and (77) that
$$
\begin{equation*}
\sigma' (z)>0\quad \text{a.e. w.r.t. the Lebesgue measure on } \mathbb T \quad\Longrightarrow\quad \lim_{n\to\infty}\gamma_n=0.
\end{equation*}
\notag
$$
This proves the strong version of Rakhmanov’s theorem for any $a_0\in\mathbb D$.
§ 5. Proof of Theorems 1 and 2In view of the equivalence of Theorems 1 and 2, which we mentioned in § 3, it is sufficient to prove just one of them. We prove Theorem 2. We choose it because in the case of Theorem 1, for the proof proposed below, given a Carathéodory function $F(z)$, we need to introduce the associated function $F^*(z):=-\overline{F(1/\overline{z})}$, $z\in\overline{\mathbb C}\setminus\overline{\mathbb D}$, while in the case of Theorem 2, given a Nevanlinna function $G(z)$, we introduce the function $\overline{G}(z):=\overline{G(\overline{z})}$, $z\in\mathbb C_-$, which is more convenient to handle. The proof of Theorem 2 is based on Lemma 2, which has some independent interest. Before stating it, we note that, as follows from the definitions
$$
\begin{equation*}
\psi_{d_n}(z):=\frac{z-d_n}{z-\overline{d}_n}, \qquad n=0,1,\dots,
\end{equation*}
\notag
$$
the continued fraction (42) is equivalent to the fraction
$$
\begin{equation}
\delta_{-1}+\cfrac{(\delta_{-1}-\overline{\delta }_{-1})(z-d_0)}{-(z-d_0) +\cfrac{z-\overline{d}_0}{\delta_0+\cfrac{(1-|\delta_0|^2)(z-d_1)} {\overline{\delta}_0(z-d_1)+\cfrac{z-\overline{d}_1}{\delta_1+\dotsb}}}},
\end{equation}
\tag{78}
$$
with polynomial coefficients, which is more convenient to deal with. The equivalence of the fractions (42) and (78) means that the $n$th numerator and $n$th denominator of the fraction (78) are obtained from the $n$th numerator and $n$th denominator of (42) by multiplication by
$$
\begin{equation*}
\prod_{k=0}^{[(n-1)/2]}(z-\overline{d}_k), \qquad n=1,2,\dots\,.
\end{equation*}
\notag
$$
The well-known three-term equalities for the numerators $P_n (z)$ and denominators $Q_n (z)$ of the convergents $\pi_n(z)=P_n(z)/Q_n(z)$ of (78) have the following form for ${n=1,2,\dots}$:
$$
\begin{equation}
P_{2n} (z)=\delta_{n-1}P_{2n-1} (z)+(z-\overline {d}_{n-1})P_{2n-2} (z),
\end{equation}
\tag{79}
$$
$$
\begin{equation}
Q_{2n} (z)=\delta_{n-1}Q_{2n-1} (z)+(z-\overline {d}_{n-1})Q_{2n-2} (z),
\end{equation}
\tag{80}
$$
$$
\begin{equation}
P_{2n+1} (z)=(z-d_{n}) (\overline{\delta}_{n-1}P_{2n} (z)+(1-|\delta_{n-1}|^2)P_{2n-1} (z)),
\end{equation}
\tag{81}
$$
$$
\begin{equation}
Q_{2n+1} (z)=(z-d_{n}) (\overline{\delta}_{n-1}Q_{2n} (z)+(1-|\delta_{n-1}|^2)Q_{2n-1} (z)),
\end{equation}
\tag{82}
$$
with the initial conditions
$$
\begin{equation}
P_{0}(z)=\delta_{-1}, \qquad P_{1}(z)=-\overline{\delta }_{-1}(z-d_0),
\end{equation}
\tag{83}
$$
We introduce the following notation. Let $d_0,d_1,\dots$ be a sequence of points in $\mathbb C_+$ and $G(z)$ be a holomorphic function on an open set $\Omega\subseteq\mathbb C_+$ containing $d_0,d_1,\dots$ . Set
$$
\begin{equation}
\mathbf G(z):=\begin{cases} G(z),\, z\in\Omega, \\ \overline{G}(z),\, z\in\overline{\Omega}, \end{cases} \quad\text{where } \overline{G}(z):=\overline{G(\overline{z})}\quad\text{and} \quad \overline{\Omega}=\{ z\colon \overline{z}\in \Omega\},
\end{equation}
\tag{85}
$$
$$
\begin{equation}
\nonumber \begin{gathered} \, \mathbf E_n:=E_n\sqcup \overline{E}_n, \\ \text{where } E_n:=\{ d_0,\dots,d_{n-1}\} \quad\text{and}\quad \overline{E}_n:=\{ \overline{d}_0,\dots,\overline{d}_{n-1}\}, \quad n=1,2,\dots, \end{gathered}
\end{equation}
\notag
$$
$$
\begin{equation}
\begin{gathered} \, \mathbf \Psi_n(z):=\Psi_n(z)\overline{\Psi }_n(z), \\ \text{where } \Psi_n(z):=\prod_{k=0}^{n-1}(z-d_k) \quad\text{and} \quad \overline{\Psi }_n(z):=\overline{\Psi_n(\overline{z})}=\prod_{k=0}^{n-1}(z-\overline{d}_k), \end{gathered}
\end{equation}
\tag{86}
$$
and
$$
\begin{equation}
\Delta_{\mathbf E_n}^\mathbf G:=\det \biggl(\frac{1}{2\pi i} \oint_{\mathbf E_n}\frac{\mathbf G(z)}{\mathbf \Psi_n(z)}z^{l+j-2}\,dz\biggr)_{l,j=1,\dots,n}, \qquad n=1,2,\dots\,.
\end{equation}
\tag{87}
$$
Here and in what follows, for each function $\Phi (z)$ defined on $\Omega\subseteq\mathbb C$ we let $\overline{\Phi }(z)$ denote the function on the set $\overline{\Omega}$ defined by $\overline{\Phi }(z)=\overline{\Phi (\overline{z})}$, and for each $m$-point set $E_m$ of $l\leqslant m$ geometrically distinct points $e_1,\dots,e_l$ and each function $\Phi (z)$ holomorphic in a neighbourhood of $E_m$ with punctures at $e_1,\dots,e_l$ we set
$$
\begin{equation*}
\oint_{E_m}\Phi (z)\,dz:=\sum_{j=1}^l\int_{|z-e_j|=\varepsilon_j}\Phi (z)\,dz,
\end{equation*}
\notag
$$
where $\varepsilon_1,\dots,\varepsilon_n$ are sufficiently small so that the sets $\bigcup_{j=1}^l \{0<|z-e_j|<\varepsilon_j\}$ lie in the domain of holomorphy of $\Phi (z)$. Lemma 2. Let the continued fraction (78) satisfy Then for $n=0,1,\dots $
$$
\begin{equation}
\begin{pmatrix} P_{2n+1} (z)\\ Q_{2n+1} (z)\end{pmatrix} =-(z-d_{n}) \begin{pmatrix} \overline{P}_{2n} (z)\\ \overline{Q}_{2n} (z)\end{pmatrix}, \qquad \pi_{2n+1} (z)=\overline{\pi }_{2n} (z),
\end{equation}
\tag{89}
$$
$$
\begin{equation}
(\pi_{2n+1} -\pi_{2n} )(z)=\frac{\rho_{n}\Psi_{n+1}(z)\overline{\Psi }_{n}(z)}{Q_{2n} (z)Q_{2n+1} (z)},
\end{equation}
\tag{90}
$$
$$
\begin{equation}
(\pi_{2n+2} -\pi_{2n} )(z)=\frac{\delta_{n}\rho_{n}\Psi_{n+1}(z)\overline{\Psi }_{n}(z)}{Q_{2n} (z)Q_{2n+2} (z)}
\end{equation}
\tag{91}
$$
and
$$
\begin{equation}
\Delta_{\mathbf E_n}^\mathbf G = \prod_{k=0}^{n-1}\frac{\rho_{k}}{|Q_{2k}(d_{k})|^2(d_{k}-\overline{d}_{k})} , \qquad n=1,2,\dots ,
\end{equation}
\tag{92}
$$
$$
\begin{equation}
\begin{gathered} \, \textit{where } \Psi_0(z)=\overline{\Psi }_0(z)=1, \quad \rho_{0} :=\delta_{-1}-\overline{\delta }_{-1} \\ \textit{and}\quad \rho_k:=\rho_{0} \prod_{j=0}^{k-1}(1-|\delta_j|^2), \qquad k=1,2,\dots\,. \end{gathered}
\end{equation}
\tag{93}
$$
For $z\in\mathbb C_+$
$$
\begin{equation}
0<\prod_{j=0}^{n-1} (1-|\delta_j\psi_{d_j}(z)|)\leqslant \frac{|Q_{2n}(z)|}{\prod_{j=0}^{n-1}|z-\overline{d}_j|}\leqslant \prod_{j=0}^{n-1} (1+|\delta_j\psi_{d_j}(z)|), \qquad n=1,2,\dots,
\end{equation}
\tag{94}
$$
and, as a consequence,
$$
\begin{equation}
0<\prod_{j=0}^{n-1} (1-|\delta_j|)\leqslant \frac{|Q_{2n}(z)|}{\prod_{j=0}^{n-1}|z-\overline{d}_j|} \leqslant \prod_{j=0}^{n-1} (1+|\delta_j |), \qquad z\in\mathbb C_+, \quad n=1,2,\dots,
\end{equation}
\tag{95}
$$
and
$$
\begin{equation}
|Q_{2n}(z)|>0, \qquad z\in\mathbb C_+, \quad n=0,1,2,\dots\,.
\end{equation}
\tag{96}
$$
In addition,
$$
\begin{equation}
\pi_{2n}(z)\in\mathfrak B^{\mathrm n}, \quad n=0,1,\dots, \qquad \pi_{2n}(z)\rightrightarrows G(z)\in\mathfrak B^{\mathrm n}\quad\textit{and}\quad \quad \pi_{2n+1}(z)\rightrightarrows \overline{G}(z)
\end{equation}
\tag{97}
$$
locally uniformly in $\mathbb C_+$ and $\mathbb C_-$, respectively. The Nevanlinna function $G(z)$ coincides with $\pi_{2n}(z)$ at the points $d_0,\dots,d_{n}$ counting multiplicities, that is,
$$
\begin{equation}
\frac{G(z)-\pi_{2n}(z)}{\Psi_{n+1}(z)}\in H(\mathbb C_+).
\end{equation}
\tag{98}
$$
The analogue of Schur’s algorithms, as applied to the Nevanlinna function $G(z)$, returns the continued fraction (42) equivalent to (78). Note that assertions similar to the ones of Lemma 2 can also be stated for polynomial analogues of the continued fractions (5), (9) and (39). In particular, in [13] an analogue of (92) was (implicitly) obtained for a classical Schur continued fraction and in [14] for a multipoint Schur continued fraction (5). Proof of Lemma 2. From the initial conditions (83) and (84) we obtain
$$
\begin{equation*}
\begin{pmatrix}P_{1} (z)\\ Q_{1} (z)\end{pmatrix} =-(z-d_{0}) \begin{pmatrix} \overline{\delta }_{-1}\\ 1\end{pmatrix} =-(z-d_{0}) \begin{pmatrix} \overline{P}_{0} (z)\\ \overline{Q}_{0} (z)\end{pmatrix}\quad\text{and} \quad \pi_{1} (z)=\overline{\pi }_{0} (z),
\end{equation*}
\notag
$$
which coincides with equalities (89) for $n=0$. We make the inductive assumption that (89) holds for the indices from $0$ to $n-1$ inclusive. Then using (81) and (79), then (after collecting like terms) the inductive assumption that (89) holds for the index $n-1$ and then, again, equality (79) with complex conjugate coefficients, for the index $n$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \frac{P_{2n+1}(z)}{z-d_{n}} &=\overline{\delta}_{n-1}P_{2n} (z)+(1-|\delta_{n-1}|^2)P_{2n-1} (z) \\ &=\overline{\delta }_{n-1}\bigl(\delta_{n-1}P_{2n-1} (z)+(z-\overline{d}_{n-1})P_{2n-2} (z)\bigr)+(1-|\delta_{n-1}|^2)P_{2n-1} (z) \\ &=P_{2n-1} (z)+\overline{\delta }_{n-1}(z-\overline{d}_{n-1})P_{2n-2} (z) \\ &=-(z-d_{n-1})\overline{P}_{2n-2} (z)-\overline{\delta }_{n-1}\overline{P}_{2n-1} (z) =-\overline{P}_{2n} (z), \end{aligned}
\end{equation*}
\notag
$$
which proves the first equality in (89). Replacing $P$ by $Q$ in this chain of equalities and using (82) and (80) we obtain the second equality in (89). The third equality is an obvious consequence of the first two.
Now by following a well-known standard scheme, subtracting (82) times $P_{2n}$ from (81) times $Q_{2n}$ we obtain
$$
\begin{equation}
\begin{aligned} \, &(P_{2n+1}Q_{2n}- Q_{2n+1}P_{2n})(z) \notag \\ &\qquad=(z-d_{n})(1-|\delta_{n-1}|^2) ( Q_{2n}P_{2n-1}-P_{2n}Q_{2n-1})(z). \end{aligned}
\end{equation}
\tag{99}
$$
In a similar way, from (79) and (80) it follows that
$$
\begin{equation}
(Q_{2n}P_{2n-1}-P_{2n}Q_{2n-1})(z)=(z-\overline{d}_{n-1}) (P_{2n-1}Q_{2n-2}- Q_{2n-1}P_{2n-2})(z),
\end{equation}
\tag{100}
$$
$$
\begin{equation}
(P_{2n}Q_{2n-2}- Q_{2n}P_{2n-2})(z)=\delta_{n-1} (P_{2n-1}Q_{2n-2}- Q_{2n-1}P_{2n-2})(z).
\end{equation}
\tag{101}
$$
Taking into account the definition (93) of the $\rho_n$, the definition (86) of the $\Psi_{n}(z)$ and $\overline{\Psi }_n(z)$ and the equality
$$
\begin{equation*}
\begin{aligned} \, (P_{1}Q_{0}-Q_{1}P_{0})(z) &=-\overline{\delta}_{-1}(z-d_0)+(z-d_{0})\delta_{-1} \\ &=(\delta_{-1}-\overline{\delta }_{-1})(z-d_0)=\rho_{0}(z-d_0), \end{aligned}
\end{equation*}
\notag
$$
using repeatedly and in turn (99) and (100) leads to the inequalities
$$
\begin{equation}
\begin{aligned} \, \notag &(P_{2n+1}Q_{2n}- Q_{2n+1}P_{2n})(z) \\ \notag &\qquad =(1-|\delta_{n-1}|^2)(z-d_{n})(z-\overline{d}_{n-1}) (P_{2n-1}Q_{2n-2}- Q_{2n-1}P_{2n-2})(z)=\dotsb \\ &\qquad =\biggl(\prod_{j=0}^{n-1}(1-|\delta_j|^2)(z-d_{j+1})(z-\overline{d}_j)\biggr) (P_{1}Q_{0}-Q_{1}P_{0})(z)= \rho_{n}\Psi_{n+1}(z)\overline{\Psi }_{n}(z). \end{aligned}
\end{equation}
\tag{102}
$$
It follows from (101) (for $n$ replaced by $n+1$) and (102) that
$$
\begin{equation}
(P_{2n+2}Q_{2n}- Q_{2n+2}P_{2n})(z)=\delta_{n}\rho _{n}\Psi_{n+1}(z)\overline{\Psi }_{n}(z), \qquad n=0,1,\dots\,.
\end{equation}
\tag{103}
$$
Dividing (102) and (103) by $Q_{2n} (z)Q_{2n+1} (z)$ and $Q_{2n} (z)Q_{2n+2} (z)$, respectively, we obtain equalities (90) and (91). From (80) (for $n$ replaced by $n+1$) and (89) it follows that
$$
\begin{equation}
\begin{aligned} \, \notag \frac{Q_{2n+2}(z)}{z-\overline {d}_{n}} &=\delta_{n}\frac{Q_{2n+1}(z)}{z-\overline {d}_{n}}+Q_{2n}(z) =-\delta_{n}\frac{z-d_{n}}{z-\overline {d}_{n}}\overline{Q}_{2n}(z)+Q_{2n}(z) \\ &=Q_{2n}(z)\biggl(1-\delta_{n}\psi_{d_{n}}(z)\frac{\overline{Q}_{2n}(z)}{Q_{2n}(z)}\biggr). \end{aligned}
\end{equation}
\tag{104}
$$
Since $Q_{0}(z)\equiv 1$, $|\delta_0|<1$, $d_{0}\in \mathbb C_+$ and therefore
$$
\begin{equation*}
\biggl|\delta_0\psi_{d_0}(z)\frac{\overline{Q}_{0}(z)}{Q_{0}(z)} \biggr|<1
\end{equation*}
\notag
$$
for $z\in\mathbb C_+$, it follows from (104) (for $n=0$) that (94) holds for $n=1$.
We make an inductive assumption: inequalities (94) hold for the index $n$ for $z\in\mathbb C_+$. In particular, $|Q_{2n}(z)|> 0$ for $z\in\mathbb C_+$. This means that all zeros of $Q_{2n}(z)$ lie in $\mathbb C\setminus \mathbb C_+$, and therefore
$$
\begin{equation*}
\biggl|\frac{\overline{Q}_{2n}(z)}{Q_{2n}(z)}\biggr|\leqslant 1 \quad\text{for } z\in\mathbb C_+.
\end{equation*}
\notag
$$
Hence we obtain inequalities (94) for the index $n+1$ for $z\in\mathbb C_+$ from the relations $|\delta_{n}|<1$, $|\psi_{d_{n}}(z)|<$ for $z\in\mathbb C_+$, (104) and the inductive assumption (see (94) for the index $n$). Thus we have proved inequalities (94) and their consequences (95) and (96).
Since the continued fractions (42) and (78) are equivalent, turning to (97) we will mean by $\pi_{n}(z)$ the $n$th convergent of (42) (which is incidentally the $n$th convergent of (78)). By the assumption the $\operatorname{Im} \delta_{-1}>0$ in Lemma 2 (see (88)) the inclusion $\pi_{0}(z)=\delta_{-1}\in\mathfrak B^{\mathrm n}$ is trivial. For $n=0,1,\dots$ set $g_{n,n}(z):\equiv \delta_{n-1}$ and for ${0\leqslant k<n<\infty}$ let $g_{k,n}(z)$ denote the finite part of the continued fraction (42) that starts with ‘$\delta_{k-1}+$’ and ends with ‘$+{1}/{\delta_{n-1}}$’. It follows directly from our notation and the form of the continued fraction (42) that
$$
\begin{equation}
g_{0,n}(z)=\delta_{-1}+\cfrac{ (\delta_{-1}-\overline{\delta }_{-1}) \psi_{d_0}(z)}{-\psi_{d_0}(z)+\cfrac{1}{g_{1,n}(z)}}, \qquad n=1,2,\dots,
\end{equation}
\tag{105}
$$
and
$$
\begin{equation}
g_{k,n}(z)=\delta_{k-1}+\cfrac{ (1-|\delta_{k-1}|^2) \psi_{d_k}(z)} {\overline{\delta }_{k-1}\psi_{d_k}(z)+\cfrac{1}{g_{k+1,n}(z)}}, \qquad k=1,\dots,n-1, \qquad n=2,3,\dots\,.
\end{equation}
\tag{106}
$$
Note that by conditions (88), for $n=1,2,\dots$ we have the inclusions $g_{n,n}(z)=\delta_{n-1}\in\mathfrak B^{\mathrm{b}}$ and the chain of equivalences
$$
\begin{equation}
\begin{aligned} \, \notag g_{n,n}(z)\in\mathfrak B^{\mathrm{b}} \quad&\Longleftrightarrow\quad g_{n-1,n}(z)\in\mathfrak B^{\mathrm{b}} \\ &\Longleftrightarrow\quad \cdots \quad\Longleftrightarrow\quad g_{1,n}(z)\in\mathfrak B^{\mathrm{b}} \quad\Longleftrightarrow\quad g_{0,n}(z)\in\mathfrak B^{\mathrm n}, \end{aligned}
\end{equation}
\tag{107}
$$
in which all the assertions, apart from the last one, are consequences of (106) and Statement 3, and the last equivalence follows from (105) and Statement 4. Since $\pi_{2n}(z)=g_{0,n}(z)$, $n=0,1,\dots $, from the chain (107) we obtain the first relation in (97).
Let $L$ be an arbitrary compact subset of $\mathbb C_+$, and let $E$ be a compact subset of $\mathbb C_+$ containing $d_0,d_1,\dots$ (see (88)). We showed in (107) that $g_{1,n}(z)\in\mathfrak B^{\mathrm{b}}$ for ${n\geqslant 1}$. Hence by the definition of the class $\mathfrak B^{\mathrm{b}}$
$$
\begin{equation}
|g_{1,n}(z)-g_{1,m}(z)|\leqslant 2 \quad\text{for all } z\in\mathbb C_+ \quad\text{and all } 1\leqslant n,m<\infty.
\end{equation}
\tag{108}
$$
Expressing $g_{1,n}(z)$ in (105) in terms of $g_{0,n}(z)$ (see (40) in Statement 4) and taking the equality $g_{0,n}(z)=\pi_{2n}(z)$ into account we obtain
$$
\begin{equation*}
g_{1,n}(z)=\frac{g_{0,n}(z)-\delta_{-1}}{\psi_{d_0}(z) (g_{0,n}(z)-\overline{\delta }_{-1})} =\frac{\pi_{2n}(z)-\delta_{-1}}{\psi_{d_0}(z) (\pi_{2n}(z)-\overline{\delta }_{-1})}, \qquad n=1,2,\dots\,.
\end{equation*}
\notag
$$
Hence we see from (91) that for all $ 1\leqslant n< m<\infty$
$$
\begin{equation*}
\begin{aligned} \, &g_{1,m}(z)-g_{1,n}(z)=\frac{\pi_{2m}(z)-\delta_{-1}}{\psi_{d_0}(z) (\pi_{2m}(z)-\overline{\delta }_{-1})}- \frac{\pi_{2n}(z)-\delta_{-1}}{\psi_{d_0}(z) (\pi_{2n}(z)-\overline{\delta }_{-1})} \\ &\qquad =\frac{(\pi_{2m}(z)-\pi_{2n}(z))(\delta_{-1}-\overline{\delta }_{-1})}{\psi_{d_0}(z) (\pi_{2m}(z)-\overline{\delta }_{-1}) (\pi_{2n}(z)-\overline{\delta }_{-1})} \\ &\qquad =\frac{(\delta_{-1}-\overline{\delta }_{-1})}{\psi_{d_0}(z) (\pi_{2m}(z)-\overline{\delta }_{-1}) (\pi_{2n}(z)-\overline{\delta }_{-1})}\sum_{k=n}^{m-1} \frac{\delta_{k}\rho_{k}\Psi_{k+1}(z)\overline{\Psi }_{k}(z)}{Q_{2k} (z)Q_{2k+2} (z)}. \end{aligned}
\end{equation*}
\notag
$$
It follows from the above equality (in view of the definition (86) of the $\Psi_{n}(z)$, inequalities (96) and the inclusions $\pi_{2n}(z)\in\mathfrak B^{\mathrm n}$ (see (97)), which imply that $|\pi_{2n}(z)-\overline{\delta }_{-1}|>0$ for $z\in\mathbb C_+$) that the difference $g_{1,m}(z)-g_{1,n}(z)$ vanishes at the points $d_1,\dots,d_n$ counting multiplicities, and therefore
$$
\begin{equation*}
\frac{g_{1,m}(z)-g_{1,n}(z)}{\psi_{d_1}(z)\dotsb \psi_{d_{n}}(z)}\in H(\mathbb C_+).
\end{equation*}
\notag
$$
Hence setting from (108), by Schwarz’s lemma we obtain
$$
\begin{equation*}
|g_{1,n}(z)-g_{1,m}(z)|\leqslant 2 |\psi_{d_1}(z)\dotsb \psi_{d_{n}}(z)|\leqslant 2q^{n}, \qquad z\in L, \quad 1\leqslant n< m<\infty .
\end{equation*}
\notag
$$
Thus, the sequence $\{ g_{1,n}(z)\}_{n=1}^\infty$ of functions in the class $\mathfrak B^{\mathrm{b}}$ is Cauchy on each compact set $L\subset \mathbb C_+$ so that it has a locally uniform limit $g(z)\in\mathfrak B^{\mathrm{b}}$ in $\mathbb C_+$. Therefore, by (105)
$$
\begin{equation*}
\pi_{2n}(z)=g_{0,n}(z)=\delta_{-1}+\cfrac{ (\delta_{-1}-\overline{\delta }_{-1}) (z-d_0)}{-(z-d_0)+\cfrac{z-\overline{d}_0}{g_{1,n}(z)}} \,\rightrightarrows \, G(z)=\delta_{-1}+\cfrac{ (\delta_{-1}-\overline{\delta }_{-1}) (z-d_0)}{-(z-d_0)+\cfrac{z-\overline{d}_0}{g(z)}}
\end{equation*}
\notag
$$
locally uniformly in $\mathbb C_+$, and we have $G(z)\in\mathfrak B^{\mathrm n}$ by Statement 4. Hence it also follows from (89) that
$$
\begin{equation*}
\pi_{2n+1}(z)=\overline{\pi }_{2n}(z)\rightrightarrows \overline{G}(z)
\end{equation*}
\notag
$$
locally uniformly in $\mathbb C_-$. This proves (97).
Since $\Psi_{n+1}(z)$ divides $\Psi _{k+1}(z)$ for all $k\geqslant n$, it follows from (97), (91) and (96) that for all $n=0,1,\dots$
$$
\begin{equation*}
\begin{aligned} \, \frac{G(z)-\pi_{2n}(z)}{\Psi_{n+1}(z)} &=\sum_{k=n}^\infty \frac{\pi_{2k+2}(z)-\pi_{2k}(z)}{\Psi_{n+1}(z)} \\ &=\sum_{k=n}^\infty \frac{\delta_{k}\rho_{k}\Psi_{k+1}(z)\overline{\Psi }_{k}(z)}{\Psi_{n+1}(z)Q_{2k} (z)Q_{2k+2} (z)} \in H(\mathbb C_+). \end{aligned}
\end{equation*}
\notag
$$
This proves inclusion (98).
Note that by (98) the first $n+1$ coefficients of the continued fractions obtained by applying the multipoint analogue of Schur’s algorithm at $d_0,d_1,\dots$ to the Nevanlinna functions $G(z)$ and $\pi_{2n}(z)$ are the same and equal to $\delta_{-1},\delta_0,\dots,\delta_{n-1}$. As $n$ is arbitrary, the application of the analogue of Schur’s algorithm to $G(z)$ returns the continued fraction (42). Proceeding to (92), note that by (85)–(87), (89) and (98) the determinants $\Delta_{\mathbf E_{n+1}}^\mathbf G$ and $\Delta_{\mathbf E_{n}}^\mathbf G$ can be written as follows:
$$
\begin{equation}
\begin{aligned} \, \notag &\Delta_{\mathbf E_{n+1}}^\mathbf G =\frac{1}{(2\pi i)^{n+1}}\det \biggl(\oint _{E_{n+1}}\frac{G(z)z^{l+j-2}}{\mathbf \Psi_{n+1}(z)}\,dz + \oint _{\overline{E}_{n+1}}\frac{\overline{G}(z)z^{l+j-2}}{\mathbf \Psi_{n+1}(z)}\,dz\biggr)_{l,j=1,\dots,n+1} \\ &\quad=\frac{1}{(2\pi i)^{n+1}}\det \biggl(\oint _{E_{n+1}}\frac{\pi_{2n}(z)z^{l+j-2}}{\mathbf \Psi_{n+1}(z)}\,dz + \oint _{\overline{E}_{n+1}}\frac{\pi_{2n+1}(z)z^{l+j-2}}{\mathbf \Psi_{n+1}(z)}\,dz\biggr)_{l,j=1,\dots,n+1} \end{aligned}
\end{equation}
\tag{109}
$$
and
$$
\begin{equation}
\Delta_{\mathbf E_{n}}^\mathbf G =\frac{1}{(2\pi i)^{n}}\det \biggl(\oint _{E_{n}}\frac{\pi_{2n}(z)z^{l+j-2}}{\mathbf \Psi_{n}(z)}\,dz + \oint _{\overline{E}_{n}}\frac{\pi_{2n+1}(z)z^{l+j-2}}{\mathbf \Psi_{n}(z)}\,dz\biggr)_{l,j=1,\dots,n}.
\end{equation}
\tag{110}
$$
Since
$$
\begin{equation*}
E_1=\{ d_0\}, \qquad \overline{E}_1=\{ \overline{d}_0\}, \qquad \pi_{0}(z)=\delta_{-1}, \qquad \pi_{1}(z)=\overline{\delta }_{-1}
\end{equation*}
\notag
$$
and by Cauchy’s theorem it follows from (109) for $n=0$ that
$$
\begin{equation*}
\begin{aligned} \, \Delta_{\mathbf E_1}^\mathbf G &=\frac{1}{2\pi i} \biggl(\oint _{E_{1}}\frac{\delta_{-1}}{(z-d_0)(z-\overline{d}_0)}\,dz + \oint_{\overline{E}_{1}} \frac{\overline{\delta }_{-1}}{(z-d_0)(z-\overline{d}_0)}\,dz\biggr) \\ &=\frac{\delta_{-1}}{d_0-\overline{d}_0}+\frac{\overline{\delta }_{-1}}{\overline{d}_0-d_0}= \frac{\rho_{0}}{d_0-\overline{d}_0}, \end{aligned}
\end{equation*}
\notag
$$
which is (92) for $n=1$. We make the inductive assumption that (92) holds for the index $n$ and prove (92) for $n+1$.
Since $\deg P_{2n}(z)\leqslant n$ (by the three-term equalities (79) and (81) and the initial conditions (83)) and $\deg \mathbf \Psi_{n+1}(z)=2n+2$, for an arbitrary polynomial $T(z)$ of degree at most $n$, for $R>\max\{ |d_0|,\dots,|d_{n}|\}$ we have
$$
\begin{equation}
\oint_{E_{n+1}}\frac{P_{2n}(z)T(z)\,dz}{\mathbf \Psi_{n+1}(z)}+\oint_{\overline{E}_{n+1}}\frac{P_{2n}(z)T(z)\,dz}{\mathbf \Psi_{n+1}(z)} =\int_{|z|=R}\frac{P_{2n}(z)T(z)\,dz}{\mathbf \Psi_{n+1}(z)}=0.
\end{equation}
\tag{111}
$$
It is immediate from the definition of $\pi_{2n}(z)$ that for all $z\in\mathbb C$ and by (90), (89) and (86) we also have
$$
\begin{equation}
\begin{aligned} \, \notag &Q_{2n}(z)\pi_{2n+1}(z)-P_{2n}(z)=Q_{2n}(z) (\pi_{2n+1}(z)-\pi_{2n}(z)) \\ &\qquad =\frac{\rho_{n} \Psi_{n+1}(z)\overline{\Psi }_n(z)}{Q_{2n+1}(z)} =\frac{\rho_{n} \Psi_{n+1}(z)\overline{\Psi }_n(z)}{-(z-d_{n})\overline{Q}_{2n}(z)} =-\frac{\rho_{n} \mathbf \Psi_{n}(z)}{\overline{Q}_{2n}(z)}. \end{aligned}
\end{equation}
\tag{113}
$$
From (111)–(113), the definition (86) of the polynomials $\mathbf \Psi_{n}(z)$, the inequalities $\overline{Q}_{2n}(z)\neq 0$ for $z\in\overline{E}_{n+1}$ (see (96)) and Cauchy’s theorem, for an arbitrary polynomial $T(z)$ of degree at most $n$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\oint _{E_{n+1}}\frac{Q_{2n}(z)\pi_{2n}(z)}{\mathbf \Psi_{n+1}(z)}T(z)\,dz +\oint _{\overline{E}_{n+1}}\frac{Q_{2n}(z)\pi_{2n+1}(z)}{\mathbf \Psi_{n+1}(z)}T(z)\,dz \\ \notag &\qquad =\oint _{E_{n+1}}\frac{Q_{2n}(z)\pi_{2n}(z)-P_{2n}(z)}{\mathbf \Psi_{n+1}(z)}T(z)\,dz \\ \notag &\qquad\qquad +\oint _{\overline{E}_{n+1}}\frac{Q_{2n}(z)\pi_{2n+1}(z)-P_{2n}(z)}{\mathbf \Psi_{n+1}(z)}T(z)\,dz \\ \notag &\qquad =-\oint _{\overline{E}_{n+1}}\frac{\rho_{n}\mathbf \Psi_{n}(z)}{\mathbf \Psi_{n+1}(z)\overline{Q}_{2n}(z)} T(z)\,dz =\oint _{\overline{E}_{n+1}}\frac{\rho_{n}}{(d_{n}-z)(z-\overline{d}_{n})\overline{Q}_{2n}(z)} T(z)\,dz \\ &\qquad =2\pi i\frac{\rho_n}{(d_{n}-\overline{d}_{n})\overline{Q_{2n}(d_{n})}}T (\overline{d}_{n}). \end{aligned}
\end{equation}
\tag{114}
$$
Making elementary transformations of determinants (namely, subtracting from the $j$th row the $(j-1)$st row times $d_{n}$, $j=n+1,\dots,2$, and then subtracting from the $l$th column the $(l-1)$st column times $\overline{d}_{n}$, $l=n+1,\dots,2$), we reduce the determinant $\Delta_{\mathbf E_{n+1}}^{\mathbf G} $ defined by (109) to the form
$$
\begin{equation}
\begin{aligned} \, \Delta_{\mathbf E_{n+1}}^{\mathbf G} &=\det\biggl(\oint _{E_{n+1}}\frac{\pi_{2n}(z)U_j(z)V_l(z)}{2\pi i\mathbf \Psi_{n+1}(z)}\,dz \notag \\ &\qquad\qquad+\oint _{\overline{E}_{n+1}}\frac{\pi_{2n+1}(z)U_j(z)V_l(z)}{2\pi i\mathbf \Psi_{n+1}(z)}\,dz \biggr)_{j,l=1,\dots,n+1}, \end{aligned}
\end{equation}
\tag{115}
$$
where
$$
\begin{equation*}
\begin{gathered} \, U_1(z)=V_1(z)=1, \\ U_j(z)=z^{j-2}(z-d_{n})\quad\text{and} \quad V_{l}(z)=z^{l-2}(z-\overline{d}_{n}), \qquad j,l=2,\dots,n+1, \end{gathered}
\end{equation*}
\notag
$$
and note that the $n\times n$ matrix in the lower right corner of the $ (n+1)\times (n+1) )$ matrix on the right-hand side of (115) (with the standard numbering of rows and columns) coincides with the matrix
$$
\begin{equation*}
\biggl(\oint _{E_n}\frac{\pi_{2n}(z)z^{l+j-2}}{2\pi i\mathbf \Psi_{n}(z)}\,dz +\oint _{\overline{E}_n}\frac{\pi_{2n+1}(z)z^{l+j-2}}{2\pi i\mathbf \Psi_{n}(z)}\,dz\biggr)_{l,j=1,\dots,n},
\end{equation*}
\notag
$$
whose determinant is equal to $\Delta_{\mathbf E_{n}}^{\mathbf G} $ by (110). Furthermore, bearing in mind that $Q_{2n}(d_{n})\neq 0 $ (see (96)) and $\deg Q_{2n}(z)\leqslant n$ (by (80), (82) and the initial conditions (84)), for some $t_2,\dots,t_{n+1}$ we have
$$
\begin{equation*}
\frac{Q_{2n}(z)}{Q_{2n}(d_{n})}=1+\sum_{j=2}^{n+1}t_jU_j(z).
\end{equation*}
\notag
$$
To the first row in (115) we add the linear combination of the other rows with coefficients $t_2,\dots,t_{n+1}$, respectively. Then the $l$th entry, $l=1,\dots,n+1$, of the first row of this transformation of the matrix (115) is in view of (114) equal to
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{Q_{2n}(d_{n})} \biggl(\oint _{E_{n+1}}\frac{\pi_{2n}(z)Q_{2n}(z)}{2\pi i\mathbf \Psi_{n+1}(z)}V_l(z)\,dz +\oint _{\overline{E}_{n+1}}\frac{\pi_{2n+1}(z)Q_{2n}(z)}{2\pi i\mathbf \Psi_{n+1}(z)}V_l(z)\,dz \biggr) \\ &\qquad =\frac{\rho_n}{Q_{2n}(d_{n})(d_{n}-\overline{d}_{n})\overline{Q_{2n}(d_{n})}}V_l (\overline{d}_{n}). \end{aligned}
\end{equation*}
\notag
$$
Since
$$
\begin{equation*}
V_1(z)\equiv 1, \qquad V_l (\overline{d}_{n})=0 \quad\text{for } l=2,\dots,n+1,
\end{equation*}
\notag
$$
this equality means that all but the first entries in the first row of the transformed matrix (115) (which has preserved its determinant) are zeros, and the first entry is
$$
\begin{equation*}
\frac{\rho_n }{(d_{n}-\overline{d}_{n})|Q_{2n}(d_{n})|^2}.
\end{equation*}
\notag
$$
It follows from the above that
$$
\begin{equation*}
\Delta_{\mathbf E_{n+1}}^{\mathbf G}= \frac{\rho_n }{(d_{n}-\overline{d}_{n})|Q_{2n}(d_{n})|^2}\Delta_{\mathbf E_{n}}^{\mathbf G}.
\end{equation*}
\notag
$$
Hence from the inductive assumption for the index $n$ we obtain that (92) holds for $n+1$. Lemma 2 is proved. Apart from Lemma 2, the proof of Theorem 2 is based on the two results stated below, which we proved before in [15] and [16], respectively. The first supplements the well-known result due to Pólia [17] on estimates for Hankel determinants of a meromorphic function in terms of the standard transfinite diameter (with no external field) of its singular set: namely, in [15] we established an analogue of Polya’s theorem in the presence of an external field. We present here the statement of the result in [15] for the (most interesting) special case when $K$ is a compact subset of the Riemann sphere $\overline{\mathbb C}$, and the external field has the special form ${v(z)=-\mathscr V^\lambda (z)}$, where
$$
\begin{equation}
\mathscr V^\lambda (z):=-\int\log |z-t|\,d\lambda (t)
\end{equation}
\tag{116}
$$
is the logarithmic potential of the unit positive Borel measure $\lambda$ with support away from $K$. Recall (see details in [18]) that the transfinite diameter $\mathbf d_{v}{K}$ of the compact set ${K\subset\overline{\mathbb C}}$ in the external field $v$, where $v$ is a continuous real function on $K$, is the quantity
$$
\begin{equation*}
\mathbf d_{v}K:=\lim_{n\to\infty}\biggl(\max_{z_1,\dots,z_n\subset K}\prod_{1\leqslant q<r\leqslant n}|z_q-z_r|e^{-(v (z_q)+v (z_r))}\biggr)^{2/((n-1)n)}.
\end{equation*}
\notag
$$
Refined Version of Pólia’s Theorem. Let $\mathbf E$ be a compact subset of $\mathbb C$, let $\mathbf G(z)\in H(\mathbf E)$, let $\lambda$ be a unit positive Borel measure with support on $\mathbf E$ and $\{\mathbf \Phi_n(z)\}_{n=1}^\infty$ be a sequence of holomorphic functions in $\overline{\mathbb C}\setminus \mathbf E$ such that locally uniformly in $\overline{\mathbb C}\setminus \mathbf E$. Then for each compact set $\mathbf K\subset\overline{\mathbb C}$ disjoint from $\mathbf E$ and consisting of a finite number of continua and such that $\mathbf G(z)$ has meromorphic extensions to all connected components of $\overline{\mathbb C}\setminus \mathbf K$ intersecting $\mathbf E$ the inequalities hold, where $\mathbf d_{-\mathscr V^\lambda}\mathbf K$ is the transfinite diameter of $\mathbf K$ in the external field $-\mathscr V^\lambda$. The result from [16] is a formula for the calculation of the transfinite diameter of a compact set in an external field $-\mathscr V^\lambda$. Formula for the Calculation of $\mathbf d_{-\mathscr V^\lambda}\mathbf K$. Let $\mathbf K$ be a compact subset of the Riemann sphere $\overline{\mathbb C}$ and $\lambda$ be a unit positive Borel measure with support on a compact set $\mathbf E\subset\mathbb C$ disjoint from $\mathbf K$. Then where $g_\mathbf K(z,t)$ is the Green’s function for the complement $\overline{\mathbb C}\setminus \mathbf K$ with singularity at $z=t$. Note that the logarithmic potential (116) exists only provided that For this reason (118) was stated in [16] in a slightly more general form, in terms of the (well-defined) spherically normalized logarithmic potential
$$
\begin{equation*}
{\cal V}^\lambda (z):=-\int_{|t|\leqslant 1}\log |z-t|\,d\lambda (t) -\int_{|t|>1}\log \biggl|\frac {z-t}{t}\biggr|\,d\lambda (t),
\end{equation*}
\notag
$$
which is different from the standard logarithmic potential (when the latter exists) by the additive constant In [16] this allowed us to cover also the case $\infty\in \mathbf E$ (which we do not need in the proof of Theorem 2). Proof of Theorem 2. Assertion 1 of Theorem 2 on the limit behaviour of the convergents of the continued fraction (42), which is equivalent to (78), was proved already in Lemma 2. We prove assertion 2.
Assume that the Nevanlinna function $G(z)$ (equal to the limit of the even convergents of (42)) has (by the Riesz–Herglotz theorem) the integral representation (31) with the measure $\varsigma $. Arguing by contradiction suppose that $\operatorname{supp} \varsigma \neq \overline{\mathbb R}$. Then for some $x\in \mathbb R$ and $\varepsilon >0$ we have the inclusion
$$
\begin{equation}
\operatorname{supp} \varsigma \subseteq \mathbf K, \quad\text{where } \mathbf K=\overline{\mathbb R}\setminus (x-\varepsilon,x+\varepsilon).
\end{equation}
\tag{119}
$$
Given $G(z)$, we consider the function $\mathbf G(z)$ on $\mathbb C_+\sqcup\mathbb C_-$ defined by (85) and note that by (31) and (119) $\mathbf G(z)$ has a holomorphic extension to $\mathbb C\setminus \mathbf K$ given by
$$
\begin{equation*}
\mathbf G(z)=\operatorname{Re} G(i) +\int_\mathbf K\frac{1+uz}{u-z}\,d\varsigma (u).
\end{equation*}
\notag
$$
Let $E$ be a compact subset of $\mathbb C_+$ containing the points $d_0,d_1,\dots$ (it exists by the hypotheses of Theorem 2). Apart from the notation (85)–(87) before Lemma 2, we also set
$$
\begin{equation}
\begin{gathered} \, \mathbf\Phi_n(z):=\mathbf \Psi_n^{-1}(z), \qquad \mathbf E:=E\sqcup \overline{E}, \\ \zeta_n:=\frac{\sum_{k=0}^{n-1}\xi_{d_k}}{n}, \qquad \overline{\zeta }_n:=\frac{\sum_{k=0}^{n-1}\xi_{\overline{d}_k}}{n}\quad\text{and} \quad \lambda_n:=\frac {\zeta_n+\overline{\zeta}_n}{2}. \end{gathered}
\end{equation}
\tag{120}
$$
Note that it follows from assumption (63) of Theorem 2 that
$$
\begin{equation}
\zeta_n \xrightarrow[n\to\infty]{*}\zeta, \qquad \overline{\zeta }_n \xrightarrow[n\to\infty]{*}\overline{\zeta } \quad\text{and} \quad \lambda_n \xrightarrow[n\to\infty]{*}\lambda :=\frac {\zeta +\overline{\zeta}}{2},
\end{equation}
\tag{121}
$$
where the measure $\overline{\zeta}$ is defined by the equality $\overline{\zeta}(L)=\zeta (\overline{L})$ for all subsets $L$ of $ \mathbb C$.
Also note that
$$
\begin{equation*}
\mathbf E\cap \mathbf K\subseteq \mathbf E\cap \overline{\mathbb R}=\varnothing, \qquad \operatorname{supp}\zeta\subseteq E, \qquad \operatorname{supp}\overline{\zeta}\subseteq \overline{E} \quad\text{and} \quad \operatorname{supp}\lambda\subseteq E\sqcup \overline{E}=\mathbf E,
\end{equation*}
\notag
$$
and it follows from (120), (86) and the weak convergence of the measures $\lambda_n$ mentioned above that
$$
\begin{equation*}
\begin{aligned} \, \frac{1}{2n}\log|\mathbf\Phi_n(z)| &=-\frac{1}{2n}\log|\mathbf \Psi_n(z)| =-\frac{1}{2n}\sum_{k=0}^{n-1} (\log |z-d_k|+\log |z-\overline{d}_k|) \\ &=-\int\log |z-t|\,d\lambda_n(t)\,\underset{n\to\infty}{\rightrightarrows} \, -\int\log |z-t|\,d\lambda (t)=\mathscr V^{\lambda }(z) \end{aligned}
\end{equation*}
\notag
$$
locally uniformly in $\overline{\mathbb C}\setminus \mathbf E$. Hence by the strong version of Pólya’s theorem (see (117)), in the notation (87) and (120) we have
$$
\begin{equation}
\varlimsup_{n\to\infty} |\Delta^\mathbf G_{\mathbf E_n} |^{1/n^2}= \varlimsup_{n\to\infty}\biggl| \det \biggl(\oint_{\mathbf E}\mathbf G(z)\mathbf\Phi_n(z)z^{j+k-2}\,dz\biggr)_{j,k=1,\dots,n}\biggr|^{1/n^2} \leqslant \mathbf d_{-\mathscr V^\lambda }\mathbf K.
\end{equation}
\tag{122}
$$
Using Lemma 2 (see (92)) we estimate the left-hand side of (122) from below. To do this we show that the limit
$$
\begin{equation}
\lim_{n\to\infty}\frac{2}{(n-1)n}\sum_{k=1}^{n-1}\sum_{j=0}^{k-1}\log |d_{k}-\overline{d}_j| =\iint\log|z-t|\,d\overline{\zeta }(t)\,d\zeta (z)
\end{equation}
\tag{123}
$$
exists. We write the expression after the limit sign on the left in (123) as
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{2}{(n-1)n}\sum_{k=1}^{n-1}k \int\log |d_k-t|\,d\overline{\zeta}_k(t) \\ &\qquad=\frac{2}{(n-1)n}\sum_{k=1}^{n-1}k \int\log |d_k-t|\,d(\overline{\zeta}_k-\overline{\zeta})(t)+A_n, \end{aligned}
\end{equation}
\tag{124}
$$
where
$$
\begin{equation}
\begin{gathered} \, \notag A_n=\frac{2}{(n-1)n}\sum_{k=1}^{n-1}k \int\log |d_k-t|\,d\overline{\zeta}(t) =\iint\log |z-t|\,d\overline{\zeta}(t)\,d\breve{\zeta }_n(z), \\ \begin{split} \breve{\zeta }_n &:=\frac{2\sum_{k=1}^{n-1}k\xi_{d_{k}}}{(n-1)n}= \frac{2}{n}\biggl(\sum_{k=0}^{n-1}\xi_{d_{k}} -\frac{\sum_{k=0}^{n-2}\sum_{j=0}^k\xi_{d_{j}}}{n-1}\biggr) \\ &= 2\zeta_n-\frac{2\sum_{k=0}^{n-2}(k+1)\zeta_{k+1}}{(n-1)n}. \end{split} \end{gathered}
\end{equation}
\tag{125}
$$
Since $E\cap\overline{E}=\varnothing$, the function $\log |z-t|$ is bounded and uniformly continuous on $\{ (z,t)\in (E\times \overline{E})\}$. Therefore, the first term on the right-hand side of (124) tends to zero as $n\to\infty$ by (121). The second term on the right in (124) tends to the right-hand side of (123) because by (125) and (121) we have
$$
\begin{equation*}
\breve{\zeta}_n \xrightarrow[n\to\infty]{*} 2\zeta -\zeta =\zeta .
\end{equation*}
\notag
$$
Thus we have proved (123). Now it follows from (92)–(94) and (123) that
$$
\begin{equation}
\begin{aligned} \, \notag &\varlimsup_{n\to\infty} |\Delta^\mathbf G_{\mathbf E_n} |^{1/n^2} =\varlimsup_{n\to\infty} \biggl|\prod_{k=0}^{n-1}\frac{\rho_{k}}{(d_{k}-\overline{d}_{k})|Q_{2k}(d_{k})|^2}\biggr|^{1/n^2} \\ \notag &\qquad \geqslant \varlimsup_{n\to\infty}\biggl|\prod_{k=1}^{n-1}\frac{\rho_0}{d_{k}-\overline{d}_{k}} \prod_{j=0}^{k-1} \frac{(1-|\delta_j|^2)}{(1+|\delta_j\psi_{d_j}(d_k)|)^2|d_{k}-\overline{d}_{j}|^2}\biggr|^{1/n^2} \\ &\qquad =\exp \biggl\{ -\iint\log|z-t|\,d\overline{\zeta }(t)\,d\zeta (z)\biggr\} \varlimsup_{n\to\infty}\biggl|\prod_{k=1}^{n-1} \prod_{j=0}^{k-1} \frac{(1-|\delta_j|^2)}{(1+|\delta_j\psi_{d_j}(d_k)|)^2}\biggr|^{1/n^2}. \end{aligned}
\end{equation}
\tag{126}
$$
Since $|\psi_{d_j}(d_k)|<1$, $j,k=0,1,\dots$, and therefore
$$
\begin{equation*}
\begin{aligned} \, &\varlimsup_{n\to\infty}\biggl|\prod_{k=1}^{n-1} \prod_{j=0}^{k-1} \frac{(1-|\delta_j|^2)}{(1+|\delta_j\psi_{d_j}(d_k)|)^2}\biggr|^{1/n^2}\geqslant \varlimsup_{n\to\infty}\prod_{k=1}^{n-1} \prod_{j=0}^{k-1} (1-|\delta_j|)^{2/n^2} \\ &\qquad =\varlimsup_{n\to\infty}\prod_{j=0}^{n-2}(1-|\delta_j|)^{2(n-1-j)/n^2}\geqslant \varlimsup_{n\to\infty}\prod_{j=0}^{n-2}(1-|\delta_j|)^{2/n}, \end{aligned}
\end{equation*}
\notag
$$
from (126) and condition (64) in Theorem 2 we obtain
$$
\begin{equation}
\varlimsup_{n\to\infty} |\Delta^\mathbf G_{\mathbf E_n} |^{1/n^2}\geqslant \exp \biggl\{ -\iint\log|z-t|\,d\overline{\zeta }(t)\,d\zeta (z)\biggr\}.
\end{equation}
\tag{127}
$$
In connection with the supplement to Theorem 2 note that for $j,k=0,1,\dots$
$$
\begin{equation*}
|\delta_j\psi_{d_j}(d_k)|<\biggl|\frac{d_k-d_j}{d_k-\overline{d}_j}\biggr|\leqslant C(E)|d_k-d_j|, \quad \text{where } C(E)=\max_{d\in E,\widetilde{d}\in\overline{E}}\frac{1}{|d-\widetilde{d}|}<\infty,
\end{equation*}
\notag
$$
and if the limit $\lim_{n\to\infty}d_n=d\in\mathbb C_+$ exists, then
$$
\begin{equation*}
\lim_{n\to\infty}\prod_{k=1}^{n-1}\prod_{j=0}^{k-1} (1+|\delta_j\psi_{d_j}(d_k)|)^{2/n^2}=1 \quad\text{and} \quad \zeta_n=\frac{1}{n}\sum_{k=0}^{n-1}\xi_{d_k}\,\underset{n\to\infty}{\overset{*}\longrightarrow }\zeta =\xi_d .
\end{equation*}
\notag
$$
Hence in this case (127) is a consequence of (126) and the condition
$$
\begin{equation*}
\varlimsup_{n\to\infty}\prod_{j=0}^{n} (1-|\delta_j|^2)^{1/n}=1
\end{equation*}
\notag
$$
(which is weaker than (64) in Theorem 2).
Now we estimate the right-hand side of (122) from above. It follows from the definition (119) of $\mathbf K$ that $g_\mathbf K(z,t)>g_{\overline{\mathbb R}}(z,t)$ for $z,t\in \mathbb C\setminus\overline{\mathbb R}$. Hence for all $z,t\in \mathbf E$ we have $g_\mathbf K(z,t)>g_{\overline{\mathbb R}}(z,t)+\epsilon$, where $\epsilon >0$. This and (118) yield the strict inequality
$$
\begin{equation}
\mathbf d_{-\mathscr V^\lambda }\mathbf K <\exp \biggl\{\iint (-\log |z-t|-g_{\overline{\mathbb R}}(z,t))\,d\lambda (z)\,d\lambda (t)\biggr \}.
\end{equation}
\tag{128}
$$
Thus, by inequality (122) and the lower and upper bounds (127) and (128) we obtain the strict inequality
$$
\begin{equation}
\begin{aligned} \, &\exp \biggl\{ -\iint\log|z-t|\,d\overline{\zeta }(t)\,d\zeta (z)\biggr\} \notag \\ &\qquad<\exp \biggl\{\iint (-\log |z-t|-g_{\overline{\mathbb R}}(z,t))\,d\lambda (z)\,d\lambda (t)\biggr\}. \end{aligned}
\end{equation}
\tag{129}
$$
By the definition of the Green’s function
$$
\begin{equation*}
g_{\overline{\mathbb R}}(z,t)= \begin{cases} \log \biggl|\dfrac{z-\overline{t}}{z-t}\biggr | &\text{if both } z \text{ and } t \text{ lie in } \mathbb C_+ \text{ or } \mathbb C_- , \\ 0 &\text{if } z \text{ and } t \text{ lie in different components of } \mathbb C\setminus\overline{\mathbb R}. \end{cases}
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\log |z-t|+g_{\overline{\mathbb R}}(z,t) \\ &\qquad =\begin{cases} \log |z-\overline{t}| &\text{if both}\ z\ \text{and}\ t\ \text{lie in}\ \mathbb C_+\ \text{or}\ \mathbb C_- , \\ \log |z-t| &\text{if}\ z\ \text{and}\ t\ \text{lie in different components of}\ \mathbb C\setminus\overline{\mathbb R}. \end{cases} \end{aligned}
\end{equation*}
\notag
$$
Then the exponent on the right-hand side of (129) is equal to
$$
\begin{equation*}
\begin{aligned} \, &-\int_{\mathbb C_+}\int_{\mathbb C_+}\log |z-\overline{t}|\,d\lambda (z)\,d\lambda (t) -2\int_{\mathbb C_+}\int_{\mathbb C_-}\log |z-t|\,d\lambda (z)\,d\lambda (t) \\ &\qquad\qquad - \int_{\mathbb C_-}\int_{\mathbb C_-}\log |z-\overline{t}|\,d\lambda (z)\,d\lambda (t) \\ &\qquad=-\iint \log |z-\overline{t}|\frac {d\zeta (z)\,d\zeta (t)}{4} -\iint \log |z-t|\frac {d\overline{\zeta }(t)\,d\zeta (z)}{2} \\ &\qquad\qquad -\iint \log |z-\overline{t}|\frac {d\overline{\zeta }(z)\,d\overline{\zeta }(t)}{4} \\ &\qquad=-\iint \log |z-t|\,d\overline{\zeta }(t)\,d\zeta (z) \end{aligned}
\end{equation*}
\notag
$$
and coincides with the exponent on the left-hand side of the strict inequality (129). This contradiction means that our assumption $\operatorname{supp}\varsigma \neq\overline{\mathbb R}$ is impossible. Hence $\operatorname{supp}\varsigma =\overline{\mathbb R}$.
Theorem 2 and our supplement to it are proved. . As noted in the beginning of this section, Theorem 1 needs no separate proof because it is equivalent to Theorem 2. 
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