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This article is cited in 1 scientific paper (total in 1 paper) Solvability of the Nevanlinna-Pick interpolation problem V. I. Buslaev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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     § 1. IntroductionThe following Nevanlinna-Pick interpolation problem is well known in the theory of analytic functions. Let $\Omega $ and $\Upsilon$ be simply connected domains, and let $\{e_p\}_{p\in \mathcal P}$ and $\{h_p\}_{p\in \mathcal P}$ be some set of points lying in $\Omega $ and $\overline{\Upsilon}$, respectively. Find necessary and sufficient conditions for the existence of a holomorphic function $f(z)$ in $\Omega $ that takes values in $\overline{\Upsilon}$ and satisfies $f(e_p )=h_p$ for $p\in \mathcal P$. When the set $\{e_p\}_{p\in \mathcal P}$ has multiple points, the conditions $f(e_p )=h_p$, $p\in \mathcal P$, are modified in the standard way. In particular, if $ e_{p_1}=\dots =e_{p_k}=e$, where $p_j\in \mathcal P$, $j=1,\dots,k$, and the indices $p_1,\dots,p_k$ are pairwise distinct, then the conditions $f(e_{p_j} )=h_{p_j}$, $j=1,\dots,k$, are replaced by the conditions $f^{(j)}(e)=h_e^{(j)}$, which specify the values of the derivatives $f^{(j)}(e)$ of $f(z)$, $j=0,\dots,k-1$, of orders zero through $k-1$ inclusive, at the point $e$ and which are equivalent to the equalities
$$
\begin{equation}
\lim_{z\to e} \frac{f(z)-\sum_{j=0}^{k-1}(h_e^{(j)}/j!)(z-e)^{j}}{(z-e)^{k-1}}=0.
\end{equation}
\tag{1}
$$
When a point $e$ of multiplicity $k$ occurs on the (smooth) boundary of the domain $\Omega$ or, more precisely, when $e_{p_1}=\dots =e_{p_k}=e\in\partial \Omega$, the limit as $z\to e$ in (1) is replaced by $z\,\widehat{\to}\, e$, which means that $z$ tends to $e$ in the nontangential way, always remaining within a fixed angle lying in $\Omega$. Using Riemann’s theorem we can reduce the interpolation problem to the case when $\Omega $ and $\Upsilon$ coincide with prescribed domains $\Omega^*$ and $\Upsilon^*$. In fact, if $\varphi_{\Omega,\Omega^*}(z)$ and $\varphi_{\Upsilon,\Upsilon^*}(z)$ are holomorphic functions mapping $\Omega $ and $\Upsilon$ injectively onto $\Omega^*$ and $\Upsilon^*$, respectively, then the problem reduces to finding condition for the existence of a function $F(t)$ holomorphic in $\Omega^*$ and taking values in $\overline{\Upsilon}^*$ such that $F(d_p )=g_p$, where $d_p =\varphi_{\Omega,\Omega^*}(e_p )$ and $g_p =\varphi_{\Upsilon,\Upsilon^*}(h_p )$, $p\in \mathcal P$. However, we encounter some problems in writing out the solution of the modified Nevanlinna-Pick problem obtained in this way (when there are multiple points), which are related to re-evaluating the values of derivatives. One must make some extra effort to put the solution in an intelligible form in the original terms (for instance, see Remark 1 below). Let $\Omega $ and $\Upsilon$ be simply connected domains, and set
$$
\begin{equation*}
\mathfrak B^{\Omega,\Upsilon}:=\{f(z)\in H(\Omega )\colon f(\Omega )\subset\overline{\Upsilon}\},
\end{equation*}
\notag
$$
where $H(\Omega )$ is the set of holomorphic functions in $\Omega$. Of greatest interest are the sets
$$
\begin{equation}
\mathfrak B^{\mathrm n}:=\mathfrak B^{\mathbb H,\mathbb H}, \qquad \mathfrak B^{\mathrm c}:=\mathfrak B^{\mathbb D,\mathbb K}, \qquad \mathfrak B^{\mathrm s}:=\mathfrak B^{\mathbb D,\mathbb D}\quad\text{and} \quad \mathfrak B^{\mathrm b}:=\mathfrak B^{\mathbb H,\mathbb D},
\end{equation}
\tag{2}
$$
where, here and throughout,
$$
\begin{equation*}
\mathbb H:=\{z\colon \operatorname{Im} z>0\}, \qquad \mathbb K:=\{z\colon \operatorname{Re} z>0\}\quad\text{and} \quad \mathbb D:=\{z\colon |z|<1\}.
\end{equation*}
\notag
$$
The first three sets in (2) are the well-known sets of Nevanlinna, Carathéodory and Schur functions, respectively. The fourth set in (2) was used in [1] in connection with some applied problems. All four sets in (2) have fairly similar definitions and similar properties. In the sets $\mathfrak B^{\zeta}$, $\zeta =\mathrm n, \mathrm c,\mathrm s,\mathrm b$, we distinguish the disjoint subsets $\mathfrak B_N^\zeta$, where $N\in\mathbb Z^\infty_+ :=\mathbb Z_+\cup\{\infty \}$; they are defined by
$$
\begin{equation*}
\mathfrak B_N^{\zeta}:=\mathfrak B^{\zeta}\cap R_N^\zeta \quad \text{for } N\in\mathbb Z_+\quad\text{and} \quad \mathfrak B_\infty^{\zeta}:=\mathfrak B^{\zeta}\setminus \biggl(\bigcup_{N\in\mathbb Z_+}\mathfrak B_N^{\zeta}\biggr),
\end{equation*}
\notag
$$
where $R_N^\zeta$ is the set of rational functions $f(z)$ of degree $N\in\mathbb Z_+$ satisfying the condition
$$
\begin{equation}
f(\partial \mathbb H)= \begin{cases} \partial \mathbb H,&\zeta =\mathrm n, \\ \partial \mathbb D,&\zeta =\mathrm b, \end{cases} \quad\text{or} \quad f(\partial \mathbb D)= \begin{cases} \partial \mathbb K,&\zeta =\mathrm c, \\ \partial \mathbb D,&\zeta =\mathrm s. \end{cases}
\end{equation}
\tag{3}
$$
Explicit representations for functions in $\mathfrak B_N^{\mathrm c}$ and $\mathfrak B_N^{\mathrm s}$, $N\in\mathbb Z_+$, were presented by Carathéodory and Schur in [2] and [3]. In particular, $\mathfrak B_0^{\mathrm s}$ consists of the constants with modulus 1, and $\mathfrak B_N^{\mathrm s}$, where $N\in\mathbb N$, consists of the Blaschke products $\gamma\prod_{k=1}^N((z-e_k)/(1-z\overline{e}_k))$, where $|\gamma |=1$ and $e_k\in\mathbb D$, $k=1,\dots,N$. Letting $(f\circ g)(z):=f(g(z))$ denote the composition of the functions $f(z)$ and $g(z)$ we obtain the following result. Proposition 1. Let $t(z)=(z-i)/(z+i)$ and $T(z)=(1-z)/(1+z)$ be linear fractional maps taking $\mathbb H$ to $\mathbb D$ and $\mathbb D$ to $\mathbb K$, respectively, and let $N\in\mathbb Z^\infty_+$. Then and We do not describe the relationships between $\mathfrak B^{\mathrm n }$ and $\mathfrak B^{\mathrm s }$, or between $\mathfrak B^{\mathrm c }$ and $\mathfrak B^{\mathrm b}$ in Proposition 1 because we do not refer to them in what follows. In addition, note that we could replace $t(z)$ and $T(z)$ in the proposition by any other linear fractional mappings taking $\mathbb H$ to $\mathbb D$ or $\mathbb D$ to $\mathbb K$, respectively. Carathéodory and Nevanlinna functions have well-known Riesz-Herglotz representations; see [4]. Riesz-Herglotz theorem. A function $f(z)$ is a Nevanlinna function if and only if there exists a measure $\tau$ with support on $(-\infty,\infty)$ such that where $\mu$ and $\nu$ are real constants and $ \mu\geqslant 0$. A function $f(z)$ is a Carathéodory function if and only if there exists a measure $\sigma$ with support on the interval $[0,2\pi]$ such that From the integral representations (4) and (5) we can easily derive necessary conditions for the solvability of the Nevanlinna-Pick problem in the Nevanlinna and Carathéodory classes. In fact, assuming that a function $f(z)$ in $\mathfrak B^{\mathrm n}$ solves the Nevanlinna-Pick problem for given sets of points $\{e_p\}_{p\in \mathcal P}$ (in $\mathbb H$) and $\{h_p\}_{p\in \mathcal P}$ (in $\overline{\mathbb H}$), from (4), for all pairwise distinct $p_1,\dots,p_n$ in the index set $\mathcal P$ and all $(\xi_1,\dots,\xi_n)\in\mathbb C^n$ we have the implication
$$
\begin{equation}
\begin{aligned} \, \notag &f(z)\in\mathfrak B^{\mathrm n} \quad\text{and}\quad f(e_p)=h_p \\ \notag &\Longrightarrow\qquad \sum_{j,k=1}^n\frac{h_{p_j}-\overline{h}_{p_k}}{e_{p_j} -\overline{e}_{p_k}}\xi_{j}\overline{\xi}_{k} =\sum_{j,k=1}^n\frac{f(e_{p_j})-\overline{f(e_{p_k})}}{e_{p_j} -\overline{e}_{p_k}}\xi_{j}\overline{\xi}_{k} \\ \notag &\qquad =\sum_{j,k=1}^n\frac{\displaystyle\mu (e_{p_j}-\overline{e}_{p_k})+\int_{-\infty }^{\infty }\biggl(\frac{1+ue_{p_j}}{u-e_{p_j}}- \frac{1+u\overline{e}_{p_k}}{u-\overline{e}_{p_k}}\biggr)\,d\tau (u)}{e_{p_j}-\overline{e}_{p_k}}\xi_{j} \overline{\xi}_{k} \\ \notag &\qquad =\mu\sum_{j,k=1}^n\xi_{j}\overline{\xi}_{k} +\int_{-\infty }^{\infty }\sum_{j,k=1}^n\frac{(1+u^2)\xi_{j} \overline{\xi}_{k}}{(u-e_{p_j})(u-\overline{e}_{p_k})} \,d\tau (u) \\ &\qquad =\mu \biggl|\sum_{j=1}^n\xi_{j}\biggr|^2+ \int_{-\infty }^{\infty }\biggl|\sum_{j=1}^n\frac{\xi_{j}}{u-e_{p_j}}\biggr|^2(1+u^2)\,d\tau (u) \geqslant 0. \end{aligned}
\end{equation}
\tag{6}
$$
It follows from the necessary conditions $(h_{p}-\overline{h}_{p})/(e_{p}-\overline{e}_{p}) \xi_{1}\overline{\xi}_{1}\geqslant 0$ for $n=1$ and the condition $e_p\in\mathbb H$, which is equivalent to $e_{p}-\overline{e}_{p}=2\operatorname{Im} e_p>0$, that $h_{p}-\overline{h}_{p}=2\operatorname{Im} h_p\geqslant 0$. Hence we need not assume a priori that $h_p\in\overline{\mathbb H}$; and we use this observation in similar situations below. For $\{e_p\}_{p\in \mathcal P}$ in $\mathbb D$ necessary conditions for the solvability of the Nevanlinna-Pick problem in the class of Carathéodory functions can be obtained by analogy with (6), from the integral representation (5). However, to develop a uniform approach to the solvability of this problem in the classes $\mathfrak B^{\zeta}$, $\zeta = \mathrm c,\mathrm s,\mathrm b$, we indicate another way, which is based on the equivalence of the Nevanlinna-Pick problems for different classes of functions. Namely, we use the known implication (6), Proposition 1 and the explicit expression for the linear fractional mapping $t(z)$:
$$
\begin{equation}
\begin{aligned} \, \notag &f(z)\in\mathfrak B^{\mathrm c} \ \text{ and }\ f(e_p)=h_p \quad\Longleftrightarrow\quad F(z)= i(f\circ t)(z)\in\mathfrak B^{\mathrm n } \ \text{ and }\ F(d_p)=g_p, \\ \notag &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ \text{where } e_p=t(d_p) \text{ and } h_p=i^{-1}g_p, \\ &\Longrightarrow\ \,\sum_{j,k=1}^n\frac{h_{p_j}+\overline{h}_{p_k}} {1-e_{p_j}\overline{e}_{p_k}}\xi_{j}\overline{\xi}_{k} =\sum_{j,k=1}^n\frac{g_{p_j}-\overline{g}_{p_k}} {2(d_{p_j}-\overline{d}_{p_k})} \bigl((d_{p_j}+i)\xi_{j}\bigr)\bigl(\overline{(d_{p_k}+i)\xi}_{k}\bigr)\geqslant 0. \end{aligned}
\end{equation}
\tag{7}
$$
In a similar way, from Proposition 1, the explicit form of $T(z)$, the implication (6) and the implication (7) just established, we also deduce necessary conditions for the solvability of the Nevanlinna-Pick problem in the classes $\mathfrak B^{\mathrm s}$ and $\mathfrak B^{\mathrm b}$:
$$
\begin{equation}
\begin{aligned} \, \notag &f(z)\in\mathfrak B^{\mathrm s}\ \text{ and }\ f(e_p)=h_p \quad\Longleftrightarrow\quad F(z)=(T\circ f)(z)\in\mathfrak B^{\mathrm c}\ \text{ and }\ F(e_p)=T(h_p) \\ &\Longrightarrow\ \, \sum_{j,k=1}^n\frac{1-h_{p_j}\overline{h}_{p_k}}{1-e_{p_j}\overline{e}_{p_k}}\xi_j\overline{\xi}_k=\frac 12\sum_{j,k=1}^n\frac{T(h_{p_j})+\overline{T(h_{p_k})}}{1-e_{p_j}\overline{e}_{p_k}} (1+h_{p_j})\xi_j\overline{(1+h_{p_k})\xi_k}\geqslant 0, \end{aligned}
\end{equation}
\tag{8}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag &f(z)\in\mathfrak B^{\mathrm b}\quad\text{and}\quad f(e_p)=h_p \\ &\Longleftrightarrow\ \, F(z)=i(T\circ f)(z)\in\mathfrak B^{\mathrm n} \quad\text{and}\quad F(e_p)=iT(h_p)\notag \\ &\Longrightarrow\ \, \sum_{j,k=1}^ni\frac{1-h_{p_j} \overline{h}_{p_k}}{e_{p_j}-\overline{e}_{p_k}}\xi_j\overline{\xi}_k =\frac 12\sum_{j,k=1}^n\frac{iT(h_{p_j})-\overline{iT(h_{p_k})}} {e_{p_j}-\overline{e}_{p_k}}(1\!+\!h_{p_j})\xi_j\overline{(1\!+\!h_{p_k})\xi_k}\geqslant 0. \end{aligned}
\end{equation}
\tag{9}
$$
Krein and Rekhtman [5] showed that for an arbitrary index set $\mathcal P$ (maybe of the cardinality of the continuum) the necessary conditions (6) for the solvability of the Nevanlinna-Pick problem in the class of Nevanlinna functions are also sufficient. This result was previously established by Pick [6] for a finite index set $\mathcal P$ and by R. Nevanlinna [7] for a countable set $\mathcal P$. Krein-Rekhtman theorem. A Nevanlinna function $f(z)$ satisfying the conditions $f(e_p)=h_p$, $p\in \mathcal P$, for given sets of points $\{e_p\}_{p\in \mathcal P}$ in $\mathbb H$ and $\{h_p\}_{p\in \mathcal P}$ exists if and only if all forms are nonnegative. If some form in (10) is degenerate, then $f(z)$ is unique and is a real rational fraction. By Proposition 1 and the Krein-Rekhtman theorem we can replace ‘$\Longrightarrow $’ in the implications (6), (7) and (9) by the equivalence sign ‘$\Longleftrightarrow$’, after which, taking the stronger version (7) into account we can make the same replacement in (8). This means that the following result holds. Extended version of the Krein-Rekhtman theorem. In order that there exist a function $f(z)\in \mathfrak B^{\zeta}$, $\zeta = \mathrm n,\mathrm b,\mathrm c,\mathrm s$, satisfying the conditions $f(e_p)=h_p$, $p\in \mathcal P$, where $\{e_p\}_{p\in \mathcal P}$ and $\{h_p\}_{p\in \mathcal P}$ are given sets and the points $\{e_p\}_{p\in \mathcal P}$ lie in $\mathbb H$ for $\zeta = \mathrm n,\mathrm b$ and in $\mathbb D$ for $\zeta = \mathrm c,\mathrm s$, it is necessary and sufficient that all forms If some of these forms are degenerate, then $f(z)$ is unique, and it is a rational fraction satisfying (3). Kovalishina [8] and Khudaiberganov [9] extended the Krein-Rekhtman theorem to holomorphic functions of matrices. Abrahamse [10] extended the same theorem to functions in the class $\mathfrak B^{\Omega,\mathbb D} $, where $\Omega$ is a finitely connected domain. Baratchart, Olivi and Seyfert [1] and Sarason [11] considered several questions relating to the Nevanlinna-Pick problem in the case when some points of interpolation occur on the boundary of the domain. This author [12] stated (without proof) an extension of the Krein-Rekhtman theorem to the case when $\mathcal P$ is a countable set, some of the points $\{e_p\}_{p\in \mathcal P}=\{e_1,e_2,\dots\}$ are multiple, and the Nevanlinna-Pick problem is modified by prescribing the values of derivatives of the required function at multiple points. A particular case of the modified Nevanlinna-Pick problem with multiple points in the classes $\mathfrak B^{\mathrm c}$ and $\mathfrak B^{\mathrm s}$ was considered for $e_1=e_2=\dots =0$ by Carathéodory [2] and Schur [3], respectively. This author [13]–[17] extended their results to more general situations. For brevity we formulate the classical criteria due to Carathéodory and Schur as a single statement, where the case $\zeta =\mathrm c$ coincides with the Carathéodory criterion and the case $\zeta =\mathrm s$ is the Schur criterion. Also for brevity, for a sequence $\{M_n\}_{n=1}^\infty$ of real numbers we write below $\{M_n\}_{n=1}^\infty\in \mathscr M_N$, $N\in\mathbb Z^\infty_+$, provided that
$$
\begin{equation*}
M_1>0,\ \dots,\ M_N>0\quad\text{and} \quad M_{N+1}=M_{N+2}=\dots =0
\end{equation*}
\notag
$$
(the inequalities $M_1>0,\ \dots,\ M_N>0$ are absent for $N=0$, and the equalities $M_{N+1}=M_{N+2}=\dots =0$ are absent for $N=\infty$). Carathéodory-Schur criterion. Let $f (z)=\sum_{k=0}^\infty a_kz^k$ be a formal power series, let $N\in\mathbb Z^\infty_+$, let $I_n$ be the $n\times n$ identity matrix, and let Remark 1. In the proof of his criterion Schur introduced an algorithm for expanding a function on $\mathfrak B^{\mathrm s}$ in a continued fraction of special form (a Schur continued fraction, which found subsequently many applications to investigations of many other topics in function theory) and proved his criterion without referring to the (previously established) Carathéodory criterion. Note on the other hand that, as shown in [17], for a formal power series $f (z)=\sum_{k=0}^\infty a_kz^k$ such that $a_0\neq -1$, for each $ n=1,2,\dots$ we have Recall that the trigonometric moment problem consists in finding a probability measure $\sigma$ with support on $[0,2\pi]$ which is distinct from a finite set and such that, given an infinite sequence of complex numbers $a_0=1,a_1,a_2,\dots$, we have the equalities
$$
\begin{equation}
\int_0^{2\pi }e^{-in\theta }\,d\sigma (\theta)=a_n, \qquad n=1,2,\dots\,.
\end{equation}
\tag{14}
$$
By the Riesz-Herglotz theorem (see (5))
$$
\begin{equation*}
\begin{aligned} \, &f(z)\in \mathfrak B^{\mathrm c}, \quad f(0)=1 \\ &\quad\Longleftrightarrow\quad f(z)=\int_0^{2\pi }\frac{e^{i\theta }+z}{e^{i\theta }-z}\,d\sigma (\theta) =1+2\sum_{n=1}^\infty z^n\int_0^{2\pi }e^{-in\theta }\,d\sigma (\theta), \end{aligned}
\end{equation*}
\notag
$$
where $\sigma$ is a probability measure with support on $[0,2\pi]$. This means that, given an infinite sequence of complex numbers $a_0=1,a_1,a_2,\dots$, the existence of a probability measure $\sigma$ satisfying (14) whose support lies on $[0,2\pi]$ and is not a finite set is equivalent to the existence of a Carathéodory function in the class $\mathfrak B^{\mathrm c }_\infty $ whose Taylor series is the (prescribed) series $1+2\sum_{n=1}^\infty a_nz^n$. In other words, the solvability of the trigonometric moment problem for $a_0=1,a_1,a_2,\dots$ is equivalent to the solvability of the (modified) Nevanlinna-Pick problem for $e_1=e_2=\dots =0$, for the series $a_0+2\sum_{n=1}^\infty a_nz^n$ in the class of functions $\mathfrak B^{\mathrm c }_\infty $. In turn, this problem is solvable by Carathéodory’s criterion if and only if
$$
\begin{equation*}
\det \begin{pmatrix} a_0 & a_1 & \dots & a_{n-1} \\ \overline{a}_1 & a_0 & \dots & a_{n-2} \\ \dots & \dots & \dots & \dots \\ \overline{a}_{n-1} & \overline{a}_1 & \dots & a_0 \end{pmatrix}>0, \qquad n=1,2,\dots\,.
\end{equation*}
\notag
$$
Also recall that the Hamburger moment problem consists in finding a probability measure $\sigma$ with support on the real axis which is distinct from a finite set, such that for a prescribed sequence of points $a_0=1,a_1,a_2,\dots$ we have
$$
\begin{equation}
\int_{\mathbb R}t^n\,d\sigma (t)=a_n, \qquad n=1,2,\dots\,.
\end{equation}
\tag{15}
$$
Hamburger [18] and Nevanlinna [19] established a connection between the solvability of the Hamburger moment problem and the solvability of the Nevanlinna-Pick problem in the class of Nevanlinna functions of the form for the initial data $e_1=e_2=\dots =\infty$ lying on the boundary of $\mathbb H$.Hamburger-Nevanlinna theorem. If a measure $\sigma$ with support on the real axis has finite moments (15), then (16) defines a Nevanlinna function such that for each $\delta\in (0,\pi /2)$ the equalities hold uniformly in the angle $\delta\leqslant \arg z\leqslant \pi -\delta$, where $0<\delta <\pi /2$. Conversely, if for a Nevanlinna function $f(z)$ and real $a_0=1,a_1,a_2,\dots$ equalities (17) hold at least for $z=iy$ as $y\to\infty$, then $f(z)$ has a representation (16) in which the measure $\sigma$ has the moments (15). Note that (17) is equivalent to the equalities
$$
\begin{equation*}
\lim_{z\widehat{\to }\infty }z^{n}(f(z)+a_0z^{-1}+a_1z^{-2}+\dots +a_{n-1}z^{-n})=0, \qquad n=1,2,\dots,
\end{equation*}
\notag
$$
the transformation $z\to -z^{-1}$ takes the upper half-plane to itself and the point $\infty$ to $0$, and the moment problems for the sequences $\{a_n\}_{n=0}^\infty$ and $\{(-1)^na_n\}_{n=0}^\infty$ are equivalent (because one is obtained from the other by replacing $\sigma $ by the measure $\sigma^*$ defined by the equality $\sigma^*(K)=\sigma(K^*)$ for each $K\subset\mathbb R$, where $K^*=\{t\colon -t\in K\}$). Therefore, Hamburger’s theorem (see [18]), which solves the moment problem, can also be formulated as a statement solving the modified Nevanlinna-Pick problem in the case when all interpolation points merge into $0\in\partial\mathbb H$. Hamburger’s theorem. Let $\sum_{k=0}^\infty a_kz^{k+1}$ be a formal power series, where $a_0=1,a_1,a_2,\dots$ are real numbers. Then the following are equivalent. $1^\circ$. There exists a function $f(z)\in\mathfrak B_\infty^{\mathrm n}$ such that
$$
\begin{equation*}
\lim_{z\widehat{\to }0}z^{-n}(f(z)-a_0z-\dots -a_{n-1}z^n)=0, \qquad n=1,2,\dots,
\end{equation*}
\notag
$$
uniformly in the angle $\delta\leqslant \arg z\leqslant \pi -\delta$, where $0<\delta <\pi /2$. $2^\circ$. The following inequality holds: § 2. Statement of the resultIn this paper, for the classes of functions $\mathfrak B^\zeta_N$, where $\zeta =\mathrm n, \mathrm b,\mathrm c, \mathrm s$ and $N\in\mathbb Z^\infty_+$, we consider the version of the Nevanlinna-Pick problem in which some of the points $\{e_p\}_{p\in \mathcal P}$ are distinct, some points are multiple and the index set $\mathcal P$ is countable. More precisely, let
$$
\begin{equation*}
\{e_p\}_{p\in \mathcal P}=\{e_1,e_2,\dots\}\subset\Omega^\zeta :=\begin{cases} \mathbb H,&\zeta =\mathrm n, \mathrm b, \\ \mathbb D,&\zeta =\mathrm c, \mathrm s, \end{cases}
\end{equation*}
\notag
$$
and let $\{h_p\}_{p\in \mathcal P}=\{h_1,h_2,\dots\}$ be a set of complex numbers. The version of the Nevanlinna-Pick problem under consideration is to find necessary and sufficient conditions for the existence of a function $f(z)\in\mathfrak B_N^\zeta$, where $\zeta =\mathrm n, \mathrm b, \mathrm c, \mathrm s$ and $N\in\mathbb Z^\infty_+$, such that where $\nu_n$ is the multiplicity of the point $e_n$ in the $n$-point set $E_n=\{e_1,\dots,e_n\}$, $n=1,2,\dots$, and $f^{(k)}(z)$ is the $k$th derivative of $f(z)$, $k=0,1,\dots$ . Apart from the notation introduced in the previous section, we need some further notation to state our result. Let $E_n=\{e_1,\dots,e_n\}$, let $\nu_k$ be the multiplicity of the point $e_k\in\mathbb C$ in the set $E_k:=\{e_1,\dots,e_k\}$, $k=1,\dots,n$; let $f(z)\in H(E_n)$, so that $f(z)$ is a function holomorphic in a neighbourhood of the set $E_n$; and let $\psi_0(z)\equiv 1$ and $\psi_k(z)=z^k$, $k=1,2,\dots$ . Set
$$
\begin{equation}
\begin{gathered} \, \notag f[E_n]:=\{f^{(\nu_1-1)}(e_1),\dots,f^{(\nu_n-1)}(e_n)\}, \\ A_{E_n}^{f[E_n]}:=\begin{pmatrix} (\psi_0f)[E_n] \\ \dots \\ (\psi_{n-1}f)[E_n]\end{pmatrix}=\begin{pmatrix} (\psi_0f)^{(\nu_{1}-1)}(e_{1}) & \dots & (\psi_0f)^{(\nu_{n}-1)}(e_{n}) \\ \dots & \dots & \dots \\ (\psi_{n-1}f)^{(\nu_{1}-1)}(e_{1}) & \dots &(\psi_{n-1}f)^{(\nu_{n}-1)}(e_{n}) \end{pmatrix}, \end{gathered}
\end{equation}
\tag{18}
$$
and
$$
\begin{equation}
A_{E_n}:=\begin{pmatrix} \psi_0[E_n] \\ \dots \\ \psi_{n-1}[E_n]\end{pmatrix}=\begin{pmatrix} \psi_0^{(\nu_{1}-1)}(e_{1}) & \dots & \psi_0^{(\nu_{n}-1)}(e_{n}) \\ \dots & \dots & \dots \\ \psi_{n-1}^{(\nu_{1}-1)}(e_{1}) & \dots & \psi_{n-1}^{(\nu_{n}-1)}(e_{n}) \end{pmatrix}.
\end{equation}
\tag{19}
$$
If $E_n=\{e_1,\dots,e_n\}$ and $ H_n=\{h_1,\dots,h_n\}$ are two $n$-point sets in $\mathbb C$, then letting $\varphi_{E_n}^{H_n}(z)$ denote an (arbitrary) holomorphic function on $E_n$ such that
$$
\begin{equation}
\varphi_{E_n}^{H_n}[E_n]=H_n, \quad \text{that is,}\quad (\varphi_{E_n}^{H_n})^{(\nu_{k}-1)}(e_k)=h_{k}, \qquad k=1,\dots,n,
\end{equation}
\tag{20}
$$
we set Note that the definition of the matrix $A_{E_n}^{H_n}$ is independent of the choice of the interpolation function $\varphi_{E_n}^{H_n}(z)$ and can be given directly in terms of the sets $E_n=\{e_1,\dots,e_n\}$ and $H_n=\{h_1,\dots,h_n\}$, without preparatory calculations of the function $\varphi_{E_n}^{H_n}(z)$, which is only introduced for the simplicity of notation and the greater transparency of the definition. An explicit form of the right-hand side of (21) not involving $\varphi_{E_n}^{H_n}(z)$ was presented in [12]. Let $\overline{A}_{E_n}$ and $\overline{A}_{E_n}^{\,H_n}$ denote the matrices obtained from $A_{E_n}$ and $A_{E_n}^{H_n}$ by replacing each entry by the complex conjugate one. We denote by $\widetilde{A}_{E_n}$ and $\widetilde{A}_{E_n}^{H_n}$ the matrices obtained from $\overline{A}_{E_n}$ and $\overline{A}_{E_n}^{\,H_n}$ by reversing the orders of columns and rows. Set
$$
\begin{equation}
M_{E_n}^{\mathrm n;H_n }: =\det \begin{pmatrix} A_{E_n} & i\,\overline{A}_{E_n} \\ A_{E_n}^{H_n } & i\,\overline{A}_{E_n}^{\,H_n } \end{pmatrix}, \qquad M_{E_n}^{\mathrm b;H_n }: =\det \begin{pmatrix} A_{E_n} & \overline{A}_{E_n}^{\,H_n } \\ A_{E_n}^{H_n } & \overline{A}_{E_n} \end{pmatrix},
\end{equation}
\tag{22}
$$
$$
\begin{equation}
M_{E_n}^{\mathrm c;H_n }: =\det \begin{pmatrix} A_{E_n} & -\widetilde{A }_{E_n} \\ A_{E_n}^{H_n } & \widetilde{A}_{E_n}^{H_n } \end{pmatrix}\quad\text{and} \quad M_{E_n}^{\mathrm s;H_n }: =\det \begin{pmatrix} A_{E_n} & \widetilde{A }_{E_n}^{H_n } \\ A_{E_n}^{H_n } & \widetilde{A}_{E_n} \end{pmatrix}.
\end{equation}
\tag{23}
$$
Note that $\overline{M}_{E_n}^{\,\zeta;H_n}=M_{E_n}^{\zeta;H_n}$, so that all determinants $M_{E_n}^{\zeta;H_n}$, $\zeta =\mathrm n, \mathrm b, \mathrm c, \mathrm s$, are real quantities. Also note that if the points $e_1,\dots,e_n$ in $E_n$ are pairwise distinct, then
$$
\begin{equation}
A_{E_n}=\begin{pmatrix} 1 & \dots & 1 \\ e_1 & \dots & e_n \\ \dots & \dots & \dots \\ e_1^{n-1} & \dots & e_n^{n-1} \end{pmatrix}\quad\text{and} \quad A_{E_n}^{H_n}=\begin{pmatrix} h_1 & \dots & h_n \\ e_1h_1 & \dots & e_nh_n \\ \dots & \dots & \dots \\ e_1^{n-1}h_1 & \dots & e_n^{n-1}h_n \end{pmatrix}.
\end{equation}
\tag{24}
$$
It follows from (24) that for $E_n$ consisting of pairwise distinct points we have
$$
\begin{equation}
M_{E_n}^{\zeta;f[E_n]}=\lim_{\varepsilon_1\to 0}\dots \lim_{\varepsilon_n\to 0}M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_n}]}, \qquad \zeta =\mathrm n,\mathrm b,\mathrm c,\mathrm s,
\end{equation}
\tag{25}
$$
where $f\in H(E_n)$ and $E_{n}^{\varepsilon_1,\dots,\varepsilon_n}:=\{e_1+\varepsilon_1,\dots,e_n+\varepsilon_n \}$. For obvious reasons, if $E_n$ contains multiple points, then (25) fails (generically). However, after modifying slightly the quantities $M_{E_n}^{\zeta;f[E_n]}$ (by multiplying then by the positive factor $|{\det A_{E_n}}|^{-2}$), we state in § 3 an analogue of equality (25) (see (41)), which also holds for $E_n$ containing multiple points. Furthermore, for $N\in\mathbb Z^\infty_+$ and $n\leqslant N+2$ we set For $N\in\mathbb Z_+$, $n\geqslant N+3$ and $\nu_n< N+2$ we set
$$
\begin{equation}
E_{n,N}:=\{e_{j_{n,1}},\dots,e_{j_{n,N+2}}\}\quad\text{and} \quad H_{n,N}:=\{h_{j_{n,1}},\dots,h_{j_{n,N+2}}\},
\end{equation}
\tag{27}
$$
where we fix the increasing indices $1\leqslant j_{n,1}<\dots <j_{n,N+2}=n$ so that the multiplicity of a point $e_{j_{n,k}}$ in the set $\{e_{j_{n,1}},\dots, e_{j_{n,k}}\}$ is $\nu_{j_{n,k}}$, that is, it is equal to its multiplicity in $E_{j_{n,k}}=\{e_{1},e_2,\dots, e_{j_{n,k}}\}$, $k=1,\dots,N+2$. This means that for this choice of $j_{n,1},\dots,j_{n,N+2}$ the set $E_{n,N}$ (for $\nu_n< N+2< n$) consists of $N+2$ points, precisely $\nu_n$ of which coincide with $e_n$ (because $j_{n,N+2}=n$), and we have the equality
$$
\begin{equation*}
\begin{aligned} \, &\varphi_{E_n}^{H_n}[E_{n,N}] =\varphi_{E_n}^{H_n}[\{e_{j_{n,1}},\dots,e_{j_{n,N+2}}\} ] \\ &\qquad =\bigl\{(\varphi_{E_n}^{H_n})^{(\nu_{j_{n,1}}-1)}(e_{j_{n,1}}),\dots, (\varphi_{E_n}^{H_n})^{(\nu_{j_{n,N+2}}-1)}(e_{j_{n,N+2}})\bigr\} \\ &\qquad=\{h_{j_{n,1}},\dots,h_{j_{n,N+2}}\}=H_{n,N}. \end{aligned}
\end{equation*}
\notag
$$
Note that we have some freedom in choosing the $N+2-\nu_n$ points distinct from $e_n$. For $N\in\mathbb Z_+$, $n\geqslant N+3$ and $\nu_n\geqslant N+2$ set
$$
\begin{equation}
E_{n,N}:=\{e_{n}\}^{2\nu_n-N-2}\quad\text{and} \quad H_{n,N}:=\{h_{j_{n,1}},\dots,h_{j_{n,\nu_n}},h_{n,1}^*,\dots,h_{n,\nu_n-N-2}^*\},
\end{equation}
\tag{28}
$$
where $\{e_{n}\}^{2\nu_n-N-2}$ is the set of $2\nu_n-N-2$ points each of which is equal to $e_n$, and the tuple of $\nu_n$ indices $\{j_{n,1},\dots,j_{n,\nu_n}\}$ is uniquely defined by the conditions
$$
\begin{equation*}
1\leqslant j_{n,1}<\dots <j_{n,\nu_n}=n, \quad e_{j_{n,k}}=e_n, \qquad k=1,\dots,\nu_n,
\end{equation*}
\notag
$$
where $h_{n,1}^*,\dots,h_{n,\nu_n-N-2}^*$ are arbitrary complex numbers. As $h_{n,1}^*,\dots,h_{n,\nu_n-N-2}^*$ are arbitrary and the values at $e_n$ of derivatives of order at least $\nu_n$ of the interpolation function $\varphi_{E_n}^{H_n}(z)$ (see (20)) can also be arbitrary, for the convenience of notation we assume in what follows that
$$
\begin{equation}
h_{n,k}^*=(\varphi_{E_n}^{H_n})^{(\nu_n-1+k)}(e_n), \qquad k=1,\dots,\nu_n-N-2.
\end{equation}
\tag{29}
$$
Thus, taking all the above into account, along with (20) we also have the equalities
$$
\begin{equation}
\varphi_{E_n}^{H_n}[E_{n,N}]=H_{n,N}, \qquad n=1,2,\dots, \quad N\in\mathbb Z^\infty_+.
\end{equation}
\tag{30}
$$
The ambiguity in the choice of $j_{n,1},\dots,j_{n,N+2}$ and $\nu_n< N+2$, and the arbitrariness of the values $h_{n,1}^*,\dots,h_{n,\nu_n-N-2}^*$ for $\nu_n\geqslant N+2$, $N\in\mathbb Z_+$, have no effect on the statement below, More precisely, for any way of selecting these quantities we have the following. Theorem 1. Let $\zeta = \mathrm n, \mathrm b,\mathrm c, \mathrm s$ and $N\in \mathbb Z^\infty_+$, let $e_1,e_2,\dots $ be a fixed sequence of points in $\Omega^\zeta =\begin{cases} \mathbb H,&\zeta =\mathrm n, \mathrm b,\\ \mathbb D,&\zeta =\mathrm c, \mathrm s,\end{cases}$ and $h_1,h_2,\dots $ be a fixed sequence of complex numbers, let $E_n=\{e_1,\dots,e_n\}$ and $H_n=\{h_1,\dots,h_n\}$. Then, using the above notation, a function $f(z)\in \mathfrak B_N^\zeta$ satisfying the conditions exists if and only if For $N\in\mathbb Z_+$ the function $f(z)\in \mathfrak B_N^\zeta$ is unique. Note that the conditions $f^{(\nu_n-1)}(e_n)=h_n$, $n=1,2,\dots $, are equivalent to $f[E_n]=H_n$, $n=1,2,\dots $, and taking (26)–(28) into account, the shorthand notation in (31) means that
$$
\begin{equation}
\begin{gathered} \, M_{E_{1}}^{\zeta;H_{1}}>0,\ \dots,\ M_{E_{N}}^{\zeta;H_{N}} >0, \\ M_{E_{N+1}}^{\zeta;H_{N+1}} =M_{E_{N+2}}^{\zeta;H_{N+2}} = M_{E_{N+p,N}}^{\zeta;H_{N+p,N}} =0, \qquad p=3,4,\dots\,. \end{gathered}
\end{equation}
\tag{32}
$$
In this extended form of conditions (31) we indicate explicitly the position at which the parameter $N$ arises, when (for $N<\infty$) from determinants of the form $M_{E_{n}}^{\zeta;H_{n}}$ we go over to $M_{E_{n,N}}^{\zeta;H_{n,N}}$. For $N<\infty$ the first $N+1$ conditions in (32), namely,
$$
\begin{equation*}
M_{E_{1}}^{\zeta;H_{1}}>0,\ \dots,\ M_{E_{N}}^{\zeta;H_{N}} >0\quad\text{and} \quad M_{E_{N+1}}^{\zeta;H_{N+1}} =0,
\end{equation*}
\notag
$$
define a unique function $f(z)\in \mathfrak B_N^\zeta$ such that $f[E_{N+1}]=H_{N+1}$; moreover,
$$
\begin{equation*}
f[E_{N+p}]=H_{N+p} \quad\Longleftrightarrow\quad M_{E_{N+2,N}}^{\zeta;H_{N+2,N}} =\dots =M_{E_{N+p,N}}^{\zeta;H_{N+p,N}}= 0, \qquad p=2,3,\dots\,.
\end{equation*}
\notag
$$
Note that the claim that conditions (32) are necessary in Theorem 1 can be supplemented by a stronger result. Supplement to Theorem 1. Let $f(z)\in\mathfrak B^\zeta$, where $\zeta = \mathrm n, \mathrm b,\mathrm c, \mathrm s$, and let $n\in\mathbb N$, $E_n=\{e_1,\dots, e_n\}\subset \Omega^\zeta $ and $N\in\mathbb Z_+^\infty$. Then Note now that both the Carathéodory-Schur criterion and the Krein-Rekhtman theorem for a countable index set $\mathcal P$ are consequences of the two extreme cases of Theorem 1: in the first case all points $e_1,e_2,\dots $ coincide, while in the second case all points $e_1,e_2,\dots $ are pairwise distinct. We establish this result (a more accurate statement of which is presented below, as a combination of Claims 1 and 2) on the basis of Lemmas 1 and 2. Lemma 1 reveals the relationship between the determinants $M_{n}^{\zeta;f}$ for $\zeta =\mathrm c,\mathrm s $ involved in the statement of the Carathéodory-Schur criterion (see (12)) and the determinants $M_{E_n}^{\zeta;H_n }$, $\zeta\!=\!\mathrm c,\mathrm s $, for the points $e_1,e_2,\dots $ coinciding with zero (see (23)). Lemma 2 reveals the relationship between the determinants of the matrices corresponding to the quadratic forms (10), which are involved in the statement of the Krein-Rekhtman theorem, and the determinants $M_{E_n}^{\mathrm n;H_n }$ for pairwise distinct points $e_1,e_2,\dots $ (see (22)). Lemma 1. Let $e_1=e_2=\dots =0$, let $f (z)=\sum_{k=0}^\infty a_kz^k$ be a formal power series, and let $f_{n}(z)=\sum_{k=0}^{n}a_kz^k$, Claim 1. The Carathéodory-Schur criterion follows from the special case of Theorem 1 when $\zeta = \mathrm c,\mathrm s $ and $e_1=e_2=\dots =0$, supplemented by Lemma 1. Before stating Lemma 2 we set
$$
\begin{equation}
C_{E_n}:=i^{n^2}\prod_{j,k=1}^n(\overline{e}_{k}-e_j),
\end{equation}
\tag{35}
$$
where $E_n=\{e_1,\dots,e_{n}\}$ is an arbitrary set of points in $\mathbb C$; note that if $E_n\subset\mathbb H$, then
$$
\begin{equation*}
C_{E_n}=\biggl(i^{n}\prod_{j=1}^n(\overline{e}_{j}-e_j)\biggr) \prod_{1\leqslant j<k\leqslant n}|\overline{e}_{k}-e_j|^2>0.
\end{equation*}
\notag
$$
Lemma 2. Let $E_n=\{e_1,\dots,e_{n}\}$ be a set of pairwise distinct points in $\mathbb H$ and $H_n=\{h_1,\dots,h_{n}\}$ be an arbitrary set of complex numbers. Then Claim 2. The Krein-Rekhtman theorem for the countable index set $\mathcal P$ follows from the special case of Theorem 1 when $\zeta = \mathrm n$ and the points $e_1,e_2,\dots $ are pairwise distinct, supplemented by Lemma 2. In conclusion of this section note that Hamburger’s theorem stated at the end of § 1 is not a consequence of Theorem 1. Nevertheless, apart from Lemmas 1 and 2, it can be interesting that there is a relationship between the limit values of the determinants $M_{E_n}^{\mathrm n;H_n}$ as all the points of $E_n$, which lie in $\mathbb H$, tend to $0\in\partial \mathbb H$ and the determinant involved in Hamburger’s theorem. More precisely, the following holds. Lemma 3. Let $\varepsilon >0$ and $n\,{\in}\,\mathbb N$, let $e_1,\dots,e_n$ be an arbitrary system of points in $\mathbb H$, and let $\varepsilon E_n=\{\varepsilon e_1,\dots, \varepsilon e_n\}$ and $f_{2n-2}(z)=\sum_{k=0}^{2n-2} a_kz^{k+1}$, where $a_0,\dots,a_{2n-2}$ are some real numbers. Then In the special case of pairwise distinct $e_1,\dots,e_n$ equality (37) takes the following form:
$$
\begin{equation}
\lim_{\varepsilon\to +0} \frac{M_{\varepsilon E_n}^{\mathrm n,f_{2n-2}[\varepsilon E_n]}}{\varepsilon^{n(2n-1)} }= C_{E_n}|V_{E_n}|^2\det \begin{pmatrix} a_0 & \dots & a_{n-1} \\ \dots & \dots &\dots \\ a_{n-1} & \dots & a_{2n-2} \end{pmatrix}, \qquad n=1,2,\dots,
\end{equation}
\tag{38}
$$
where $V_{E_n}$ is the Vandermonde determinant of the points $e_1,\dots,e_n$. § 3. Proof of the theoremProceeding to the proof of Theorem 1, note that we proved it in [16] for $\zeta =\mathrm s $ and in [17] for $\zeta =\mathrm c$, in terms of the quantities
$$
\begin{equation*}
\widehat{M}_{E_n}^{\zeta;H_n}:=\frac{M_{E_n}^{\zeta;H_n}}{W_{E_n}}, \quad \text{where } \zeta =\mathrm s,\mathrm c,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
W_{E_n}:=\det \begin{pmatrix} A_{E_n} & \widetilde{A }_{E_n}^{\psi_n[E_n]} \\ A_{E_n}^{\psi_n[E_n]} & \widetilde{A}_{E_n} \end{pmatrix}, \qquad \psi_n(z)=z^n,
\end{equation*}
\notag
$$
which makes no significant difference for $\zeta =\mathrm s, \mathrm c $, because $W_{E_n}>0$ for $E_n\subset\mathbb D$, so that for $\{e_1,e_2,\dots \}\subset\mathbb D$ we have
$$
\begin{equation*}
\{\widehat{M}_{E_{n,N}}^{\zeta;H_{n,N}}\}_{n=1}^\infty\in \mathscr M_N \quad\Longleftrightarrow\quad \{M_{E_{n,N}}^{\zeta;H_{n,N}}\}_{n=1}^\infty\in \mathscr M_N, \quad \zeta = \mathrm s,\mathrm c, \quad N\in\mathbb Z_+^\infty.
\end{equation*}
\notag
$$
Set
$$
\begin{equation}
\breve{M}_{E_n}^{\zeta;H_n}:=\frac{M_{E_n}^{\zeta;H_n}}{|{\det A_{E_{n}}}|^{2}}, \qquad \zeta = \mathrm n,\mathrm b,\mathrm s,\mathrm c,
\end{equation}
\tag{39}
$$
and note that
$$
\begin{equation}
\{\breve{M}_{E_{n,N}}^{\zeta;H_{n,N}}\}_{n=1}^\infty\in \mathscr M_N \quad\Longleftrightarrow\quad \{M_{E_{n,N}}^{\zeta;H_{n,N}}\}_{n=1}^\infty\in \mathscr M_N, \quad \zeta = \mathrm n,\mathrm b,\mathrm s,\mathrm c, \quad N\in\mathbb Z_+^\infty,
\end{equation}
\tag{40}
$$
because the definition (19) means that $\det A_{E_n}\neq 0$ for all $E_n\subset\mathbb C$. In comparison to the determinants $M_{E_n}^{\zeta;H_n}$, the quantities $\breve{M}_{E_n}^{\zeta;H_n}$ (see (39)), different from them by the positive coefficients $|{\det A_{E_{n}}}|^{-2}$, have slightly more awkward definitions, but satisfy a useful (for the proof of Theorem 1) analogue of (25) for sets $E_n$ with multiple points (as noted in § 2, there is no such analogue for the original determinants $M_{E_n}^{\zeta;H_n}$ in general). More precisely, the following result holds. Lemma 4. Let $E_n=\{e_1,\dots,e_n\}$ and $f\in H(E_n)$. Then where $E_{n}^{\varepsilon_1,\dots,\varepsilon_n}:=\{e_1+\varepsilon_1,\dots,e_n+\varepsilon_n \}$. Proof. First we prove that
$$
\begin{equation}
\begin{gathered} \, \lim_{\varepsilon_n\to 0}\breve{M}_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_n}]}=\breve{M}_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}]}, \\ \text{where }\ \zeta =\mathrm n,\mathrm b,\mathrm c,\mathrm s\quad\text{and} \quad \varepsilon_k\in\mathbb C, \ \ k=1,\dots,n-1. \end{gathered}
\end{equation}
\tag{42}
$$
Since for $\varepsilon_n=0$ the limit in (42) is applied to a quantity equal to the required result, in what follows we assume that $\varepsilon_n\neq 0$ without loss of generality. Hence $e_n+\varepsilon_n\notin\{e_1+\varepsilon_1,\dots,e_{n-1}+\varepsilon_{n-1}\}$ for any fixed $\varepsilon_1,\dots,\varepsilon_{n-1}$ and all sufficiently small (nonzero) $\varepsilon_n$. Therefore, the multiplicity of the point $e_n+\varepsilon_n$ in the set $E_{n}^{\varepsilon_1,\dots,\varepsilon_{n}}$ is equal to one. Since all points in $E_{n}^{\varepsilon_1,\dots,\varepsilon_{n}}$, from the first to $(n-1)$st inclusive, coincide with the corresponding points in $E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}$, all columns of the matrix $A_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}$, from the first to $(n-1)$st inclusive, coincide with the corresponding columns of $A_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}$, and all columns in the determinant $M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_n}]}$, apart from the $n$th and $(2n)$th for $\zeta =\mathrm n,\mathrm b $, and apart from the $n$th and $(n+1)$st for ${\zeta =\mathrm c,\mathrm s}$, coincide with the corresponding columns in $M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}]}$ (see the definitions (18), (19), (22) and (23)). Let $\widetilde{\nu }_n=\nu_{n}(\varepsilon_1,\dots,\varepsilon_{n-1})$ be the multiplicity of the point $e_n$ in the set $E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}$. Denoting by $j_1,\dots,j_{\widetilde{\nu }_n}$ the increasing indices $1\leqslant j_1<\dots <j_{\widetilde{\nu }_n}=n$ such that
$$
\begin{equation*}
e_{j_1}+\varepsilon_1=\dots =e_{j_{\widetilde{\nu }_n-1}}+\varepsilon_{n-1}=e_{j_{\widetilde{\nu }_n}}=e_n,
\end{equation*}
\notag
$$
we add to the $n$th column of $A_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}$ times ${\varepsilon_n^{\widetilde{\nu }_n-1}}/{(\widetilde{\nu }_n-1)!}$ the linear combination of the columns with indices $j_1,\dots,j_{\widetilde{\nu }_n-1}$ times ${\varepsilon_n^0}/{0!},\dots,{\varepsilon_n^{\widetilde{\nu }_n-2}}/{(\widetilde{\nu }_n-2)!}$, respectively (if $\widetilde{\nu }_n=1$, then we make no manipulations with the $n$th column). As a result, we obtain a column coinciding, by Taylor’s formula, with the $n$th column of $A_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}$ to within $o(\varepsilon_n^{\widetilde{\nu }_n-1})$ as $\varepsilon_n\to 0$. Hence
$$
\begin{equation*}
\det A_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}=\frac{\varepsilon_n^{\widetilde{\nu }_n-1}}{(\widetilde{\nu }_n-1)!}(\det A_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}+o(1)), \qquad \varepsilon_n\to 0.
\end{equation*}
\notag
$$
In this paragraph (and only here) we give the proof of (42) for $\zeta =\mathrm n,\mathrm b $ (for $\zeta =\mathrm c,\mathrm s $ the proof remains the same bearing in mind that the columns with indices $n+1,\dots,2n$ have the reverse order in the determinants $M_{E_n}^{\mathrm c;f[E_n]}$ and $M_{E_n}^{\mathrm s;f[E_n]}$). To the $n$th and $ 2n $th columns in $M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}]}$ for $\zeta =\mathrm n,\mathrm b $, multiplied by ${\varepsilon_n^{\widetilde{\nu }_n-1}}/{(\widetilde{\nu }_n-1)!}$ and ${\overline{\varepsilon }_n^{\widetilde{\nu }_n-1}}/{(\widetilde{\nu }_n-1)!}$, respectively, we add the linear combinations of the columns with indices $j_1,\dots,j_{\widetilde{\nu }_n-1}$ times ${\varepsilon_n^0}/{0!},\dots,{\varepsilon_n^{\widetilde{\nu }_n-2}}/{(\widetilde{\nu }_n-2)!}$, respectively, and the linear combinations of the columns with indices $n+j_1,\dots,n+j_{\widetilde{\nu }_n-1}$ times ${\overline{\varepsilon }_n^0}/{0!},\dots,{\overline{\varepsilon }_n^{\widetilde{\nu }_n-2}}/{(\widetilde{\nu }_n-2)!}$, respectively. Then we obtain columns which coincide with the $n$th and $ 2n $th columns, respectively, in the determinant $M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n}}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_{n}}]}$, $\zeta =\mathrm n,\mathrm b $, to within $o(\varepsilon_n^{\widetilde{\nu }_n-1})$ as $\varepsilon_n\to 0$. Thus,
$$
\begin{equation*}
M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_n}]}=\frac{|\varepsilon_n|^{2(\widetilde{\nu }_n-1)}}{((\widetilde{\nu }_n-1)!)^2} \bigl(M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots, \varepsilon_{n-1},0}]}+o(1)\bigr), \qquad \varepsilon_n\to 0.
\end{equation*}
\notag
$$
Hence, taking (39) and the inequality $\det A_{E_{n}}\neq 0$ for all $E_n\subset\mathbb C$ into account, we obtain
$$
\begin{equation*}
\begin{aligned} \, \breve{M}_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots, \varepsilon_n}]} &=\frac{M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots, \varepsilon_n}]}}{|{\det A_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}}|^2} =\frac{\frac{|\varepsilon_n|^{2(\widetilde{\nu }_n-1)}}{((\widetilde{\nu }_n-1)!)^2} \bigl(M_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots, \varepsilon_{n-1},0}]}+o(1)\bigr)}{\bigl|\frac{\varepsilon_n^{\widetilde{\nu }_n-1}} {(\widetilde{\nu }_n-1)!} \bigl(\det A_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}+o(1)\bigr) \bigr|^2} \\ &=\breve{M}_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}]}+o(1), \qquad \varepsilon_n\to 0, \end{aligned}
\end{equation*}
\notag
$$
which is equivalent to (42).
Note now that the quantity $\breve{M}_{E_n}^{\zeta;f[E_n]}$, $ \zeta =\mathrm n,\mathrm b,\mathrm c,\mathrm s$, is invariant under rearrangements of the points in $E_n$. Hence, apart from (42), we also have the equalities
$$
\begin{equation*}
\begin{aligned} \, & \lim_{\varepsilon_{n-1}\to 0}\breve{M}_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-1},0}]}=\breve{M}_{E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-2},0,0}}^{\zeta;f[E_{n}^{\varepsilon_1,\dots,\varepsilon_{n-2},0,0}]}, \\ &\qquad \dots, \quad\lim_{\varepsilon_{1}\to 0}\breve{M}_{E_{n}^{\varepsilon_1,0,\dots,0}}^{\zeta;f[E_{n}^{\varepsilon_1,0,\dots,0}]}=\breve{M}_{E_{n}^{0,\dots,0}}^{\zeta;f[E_{n}^{0,\dots,0}]}. \end{aligned}
\end{equation*}
\notag
$$
Using them sequentially and taking the equality $E_n^{0,\dots,0}=E_n$ into account we obtain (41). Lemma 4 is proved. The proof of Theorem 1 for $\zeta =\mathrm c$ was carried out in [17] by reducing this case to $\zeta =\mathrm s$ (considered previously in [16]) with the help of Proposition 1 and the equality
$$
\begin{equation}
\begin{gathered} \, M_{E_n}^{\mathrm c;\phi [E_n]}=\frac{\prod_{k=1}^n|1+\phi (e_k)|^{2}}{2^n}M_{E_n}^{\mathrm s;\Phi [E_n] }, \\ \text{where}\quad \phi (z)\in H(E_n)\quad\text{and}\quad \Phi (z)=\frac{1-\phi (z)}{1+\phi (z)}, \end{gathered}
\end{equation}
\tag{43}
$$
which holds under the assumption $\phi (z)\neq -1$, $z\in E_n$, and which extends equality (13), stated for $E_n=\{0\}^n$, to arbitrary sets $E_n\subset\mathbb D$. For $\phi (z)=\varphi_{E_n}^{H_n}(z)$ the inequalities $\phi (z)\neq -1$, $z\in E_n$, are, as shown in [17], consequences of conditions (31) for $\zeta =\mathrm c$. In the two cases $\zeta =\mathrm b$ and $\zeta =\mathrm n$ we will obtain the proof of Theorem 1 by reduction, with the help of Proposition 1 and Lemma 5, to the cases $\zeta =\mathrm s $ and $\zeta =\mathrm c $, considered already in [16] and [17], respectively. In the statement of Lemma 5 and in what follows, given a set of points $E_n=\{e_1,\dots,e_n\}\subset\mathbb C$ and a linear fractional transformation $t(z)$, we use the notation $t(E_n)$ for the set $\{t(e_1),\dots,t(e_n)\}$, rather than the notation $t[E_n]:=\{t^{(\nu_1-1)}(e_1), \dots,t^{(\nu_n-1)}(e_n)\}$ used before. For $\nu_1=\dots =\nu_n=1$ these two ways of notation produce the same result. In a similar way, given a set of point $H_n=\{h_1,\dots,h_n\}$ and a linear fractional transformation $T(z)$, we set $T(H_n):=\{T(h_1),\dots,T(h_n)\}$. Lemma 5. Let $t(z)=(z-i)/(z+i)$, $E_n=\{e_1,\dots,e_n\}\subset\mathbb H$ and $\varphi (z)\in H(E_n)$. Then Corollary 1. Let $N\in\mathbb Z_+^\infty$ and $t(z)=(z-i)/(z+i)$, let $e_1,e_2,\dots $ be a sequence of points in $\mathbb H$, $h_1,h_2,\dots $ be a sequence of arbitrary complex numbers, and let $E_n=\{e_1,\dots,e_n\}$, $H_n=\{h_1,\dots,h_n\}$, $D_n=t(E_n)$, and $G_n=(\varphi_{E_n}^{H_n} \circ t^{-1})[t(E_n)]$, $n=1,2,\dots$ . Then the determinants $M_{D_{n}}^{\mathrm s;G_n}$ and $M_{E_{n}}^{\mathrm b;H_n}$ have the same signs or vanish simultaneously; the same holds for the determinants $M_{D_{n}}^{\mathrm c;i^{-1}G_n}$ and $M_{E_{n}}^{\mathrm n;H_n}$. Therefore, and Note that for pairwise distinct points $e_1,\dots,e_n$ the points $t(e_1),\dots,t(e_n)$ are also pairwise distinct and $t^{-1}[t(E_n)]=E_n$, so that $G_n=H_n$. However, in the general case, when there are multiple points in $E_n$, there is no equality $G_n=H_n$. Corollary 2. Let $E_{n}=\{e_1,\dots,e_{n}\}$ be a set of pairwise distinct points in $\mathbb H$, ${H_{n}=\{h_1,\dots,h_{n}\}}$ be a set of points in $\mathbb C\setminus\{-i\}$, and let Then the determinants $M_{E_n}^{\mathrm n;H_n}$ and $M_{t(E_n)}^{\mathrm s;T(i^{-1}H_n)}$ have the same signs or vanish simultaneously. Proof of Lemma 5. First assume that $E_n$ consists of pairwise distinct points and $i\notin E_n$. For brevity set
In this notation, under the above assumptions the set ${t(E_n)=\{d_1,\dots,d_n\}\!=:\!D_n}$ consists of pairwise distinct points in $\mathbb D$ all of which are distinct from $0$, and we have
$$
\begin{equation}
(\varphi \circ t^{-1})[t(E_n)] =\varphi [E_n]=\{\varphi (e_1),\dots,\varphi (e_n)\} =\{h_1,\dots,h_n\}=:H_n
\end{equation}
\tag{47}
$$
and
$$
\begin{equation}
\det A_{E_n}=V_{e_1,\dots,e_n}, \qquad \det A_{D_n}=V_{d_1,\dots,d_n},
\end{equation}
\tag{48}
$$
where
$$
\begin{equation}
V_{z_1,\dots,z_n}=\det \begin{pmatrix} z_1^0 & \dots & z_n^0 \\ \dots & \dots & \dots \\ z_1^{n-1} & \dots & z_n^{n-1} \end{pmatrix} = \prod_{1\leqslant j<k\leqslant n}(z_k-z_j)
\end{equation}
\tag{49}
$$
is the Vandermonde determinant of $z_1,\dots,z_n$.
Also assume that $0\notin H_n$. For $m=1,\dots,n$ set
$$
\begin{equation}
e_{n+m}:=\overline{e}_{m}, \qquad d_{n+m}:=\overline{d}_{m}^{-1}\quad\text{and} \quad h_{n+m}:=\overline{h}_{m}^{-1}
\end{equation}
\tag{50}
$$
and note that
$$
\begin{equation*}
\begin{gathered} \, d_{n+m}\neq\infty, \qquad h_{n+m}\neq\infty, \\ d_{n+m}=\overline{d}_{m}^{-1}=\frac{\overline{e}_m-i}{\overline{e}_m+i} =t(\overline{e}_m)=t(e_{n+m}), \qquad m=1,\dots,n, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
d_{k}-d_{j} =\frac{e_{k}-i}{e_{k}+i}-\frac{e_{j}-i}{e_{j}+i}=(e_{k}-e_{j})\frac{2i}{(e_{k}+i)(e_{j}+i)}, \qquad k,j=1,\dots,2n.
\end{equation}
\tag{51}
$$
It follows from (49) and (51) that for $1\leqslant p_1,\dots,p_n\leqslant 2n$ we have
$$
\begin{equation}
V_{d_{p_1},\dots,d_{p_n}}=\prod_{1\leqslant j <k \leqslant n}(d_{p_k}-d_{p_j})=V_{e_{p_1},\dots,e_{p_n}}\frac{(2i)^{(n-1)n/2}}{\prod_{k=1}^n(e_{p_k}+i)^{n-1}}.
\end{equation}
\tag{52}
$$
In particular, by (48) and (52) (for $p_k=k$, $k=1,\dots,n$) we have the equality
$$
\begin{equation}
\det A_{D_n}=L_{E_n}\det A_{E_n}, \quad \text{where } L_{E_n}:= \frac{(2i)^{(n-1)n/2}}{\prod_{k=1}^n(e_{k}+i)^{n-1}}.
\end{equation}
\tag{53}
$$
Note that, as follows from the definition of $L_{E_n}$ (see (53)) and the first equality in (50),
$$
\begin{equation}
\begin{aligned} \, \notag |L_{E_n}|^2 &= \frac{2^{(n-1)n}}{\prod_{k=1}^n(e_{k}+i)^{n-1}(\overline{e}_{k}-i)^{n-1}} \\ &=\frac{2^{(n-1)n}}{\prod_{k=1}^n(e_{k}+i)^{n-1}(\overline{e}_{k}+i)^{n-1}} \prod_{k=1}^n\biggl(\frac{\overline{e}_{k}+i}{\overline{e}_{k}-i}\biggr)^{n-1} \nonumber \\ &=\frac{2^{(n-1)n}}{\prod_{k=1}^{2n}(e_{k}+i)^{n-1}} \prod_{k=1}^n\overline{d}_k^{\,n-1}. \end{aligned}
\end{equation}
\tag{54}
$$
Given some indices $p_1,\dots,p_n$ such that $1\leqslant p_1<\dots <p_n\leqslant 2n$, let $q_1,\dots,q_n$ denote the indices complementary to $\{p_1,\dots,p_n\}$ in the set $\{1,\dots,2n\}$ and put in the increasing order: $1\leqslant q_1<\dots <q_n\leqslant 2n$. In view of the definitions (47) of the set $H_n$ and (23) of the quantity $M_{E_n}^{\mathrm s;H_n}$ we have the chain of equalities
$$
\begin{equation*}
\begin{aligned} \, &M_{t(E_n)}^{\mathrm s;(\varphi \circ t^{-1})[t(E_n)]}=M_{D_n}^{\mathrm s;H_n}=\det \begin{pmatrix} A_{D_n} & \widetilde{A }_{D_n}^{H_n} \\ A_{D_n}^{H_n } & \widetilde{A}_{D_n} \end{pmatrix} \\ &\qquad =(-1)^{(n-1)n/2} \begin{vmatrix} d_1^0 & \dots & d_{n}^0 & \overline{d_1^{n-1}h_1} & \dots & \overline{d_n^{n-1}h_n} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ d_1^{n-1} & \dots & d_{n}^{n-1} & \overline{d_1^{0}h_1} & \dots & \overline{d_n^{0}h_n} \\ d_1^0h_{1} & \dots & d_{n}^0h_{n} & \overline{d_1^{n-1}} & \dots & \overline{d_n^{n-1}} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ d_1^{n-1}h_{1} & \dots & d_{n}^{n-1}h_{n} & \overline{d_1^{0}} & \dots & \overline{d_n^{0}} \end{vmatrix} \\ &\qquad =i^{(n-1)n}\prod_{k=1}^n\overline{h_kd_k^{n-1}} \begin{vmatrix} d_1^{0} & \dots & d_{n}^{0} & d_{n+1}^{0} & \dots & d_{2n}^{0} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ d_1^{n-1} & \dots & d_{n}^{n-1} & d_{n+1}^{n-1} & \dots & d_{2n}^{n-1} \\ d_1^0h_{1} & \dots & d_{n}^0h_{n} & d_{n+1}^0h_{n+1} & \dots & d_{2n}^0h_{2n} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ d_1^{n-1}h_{1} & \dots & d_{n}^{n-1}h_{n} & d_{n+1}^{n-1}h_{n+1} & \dots & d_{2n}^{n-1}h_{2n} \end{vmatrix} \\ &\qquad =i^{(n-1)n}\prod_{k=1}^n\overline{h_kd_k^{n-1}}\sum_{1\leqslant p_1<\dots <p_n\leqslant 2n}\frac{V_{d_{p_1},\dots,d_{p_n}}h_{q_1}\dotsb h_{q_n}V_{d_{q_1},\dots,d_{q_n}}}{(-1)^{(1+p_1)+\dots +(n+p_n)}} \\ &\qquad = \frac{2^{(n-1)n}\prod_{k=1}^n\overline{h_kd_k^{n-1}}}{\prod_{k=1}^{2n}(e_{k}+i)^{n-1}}\sum_{1\leqslant p_1<\dots <p_n\leqslant 2n}\frac{V_{e_{p_1},\dots,e_{p_n}}h_{q_1}\dotsb h_{q_n}V_{e_{q_1},\dots,e_{q_n}}}{(-1)^{(1+p_1)+\dots +(n+p_n)}} \\ &\qquad = \overline{h}_1\dotsb \overline{h}_n |L_{E_n}|^2 \begin{vmatrix} e_1^0 & \dots & e_{n}^0 & e_{n+1}^0 & \dots & e_{2n}^0\\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1} & \dots & e_{n}^{n-1} & e_{n+1}^{n-1} & \dots & e_{2n}^{n-1}\\ e_1^0h_{1} & \dots & e_{n}^0h_{n} & e_{n+1}^0h_{n+1} & \dots & e_{2n}^0h_{2n}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1}h_{1} & \dots & e_{n}^{n-1}h_{n} & e_{n+1}^{n-1}h_{n+1} & \dots & e_{2n}^{n-1}h_{2n} \end{vmatrix} \\ &\qquad = |L_{E_n}|^2 {\begin{vmatrix} e_1^0 & \dots & e_{n}^0 & \overline{e_{1}^0h_1} & \dots & \overline{e_{n}^0h_n}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1} & \dots & e_{n}^{n-1} & \overline{e_{1}^{n-1}h_1} & \dots & \overline{e_{n}^{n-1}h_n}\\ e_1^0h_{1} & \dots & e_{n}^0h_{n} & \overline{e}_{1}^0 & \dots & \overline{e}_{n}^0\\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1}h_{1} & \dots & e_{n}^{n-1}h_{n} & \overline{e}_{1}^{n-1} & \dots & \overline{e}_{n}^{n-1} \end{vmatrix}} \\ &\qquad = |L_{E_n}|^2 \det \begin{pmatrix} A_{E_n} & \overline{A}_{E_n}^{\,H_n } \\ A_{E_n}^{H_n } & \overline{A}_{E_n}\end{pmatrix}= |L_{E_n}|^2M_{E_n}^{\mathrm b;H_n}= |L_{E_n}|^2M_{E_n}^{\mathrm b;\varphi [E_n]}. \end{aligned}
\end{equation*}
\notag
$$
Some comments are here in order. In the third equality, taking the definitions of $\widetilde{A }_{D_n}$ and $\widetilde{A }_{D_n}^{H_n}$ into account, we reversed the order of the last $n$ columns; in the fourth equality we took the coefficient $\overline{d_{m}^{n-1}h_{m}}$ out of the $(n+m)$th column in the determinant for $m=1,\dots,n$ and used the notation $d_{n+m}=\overline{d}_m^{-1}$ and $h_{n+m}=\overline{h}_m^{-1}$ (see (50)); in the fifth equality we used Laplace’s formula to expand a determinant of order $(2n)$ with respect to $n$th-order minors in the first $n$ rows and the definition of the indices $q_1,\dots,q_n$; in the sixth equality we used (52) and the equality $\{p_1,\dots,p_n\}\sqcup \{q_1,\dots,q_n\}=\{1,\dots,2n\}$; in the seventh, (54) and Laplace’s formula; in the eighth we introduced the coefficient $\overline{h}_{m}$ into the $(n+m)$th column in the determinant for $m=1,\dots,n$ and used the notation $e_{n+m}=\overline{e}_m$ and $h_{n+m}=\overline{h}_m^{\,-1}$ (see (50)); in the ninth, tenth, and eleventh equalities we used the definitions of the matrices $A_{E_n}$ and $A_{E_n}^{H_n}$, the definition (22) of $M_{E_n}^{\mathrm b;H_n}$ and the definition (47) of the set $H_n$, respectively.
Hence, from the definition (39) of the quantities $\breve{M}_{E_n}^{\zeta;H_n}$ for $\zeta =\mathrm s,\mathrm b$ and (53) we obtain
$$
\begin{equation*}
\breve{M}_{t(E_n)}^{\mathrm s;(\varphi \circ t^{-1})[t(E_n)]}=\frac{M_{t(E_n)}^{\mathrm s;(\varphi \circ t^{-1})[t(E_n)]}}{|{\det A_{t(E_n)}}|^2}= \frac{|L_{E_n}|^2M_{E_n}^{\mathrm b;\varphi [E_n]}}{|L_{E_n}\det A_{E_n}|^2}=\breve{M}_{E_n}^{\mathrm b;\varphi [E_n]},
\end{equation*}
\notag
$$
which is the first equality in (44).
Setting $h_{n+m}:=\overline{h}_{m}$ in place of the notation $h_{n+m}:=\overline{h}_{m}^{-1}$ used in (50), in a similar way we obtain the chain of equalities
$$
\begin{equation*}
\begin{aligned} \, &M_{t(E_n)}^{\mathrm c;i^{-1}(\varphi \circ t^{-1})[t(E_n)]}=M_{D_n}^{\mathrm c;i^{-1}H_n}=\det \begin{pmatrix} A_{D_n} & -\widetilde{A }_{D_n} \\ A_{D_n}^{i^{-1}H_n } & \widetilde{A}_{D_n}^{i^{-1}H_n }\end{pmatrix} \\ &\qquad =(-1)^{(n-1)n/2}\begin{vmatrix} d_1^0 & \dots & d_{n}^0 & -\overline{d_1^{n-1}} & \dots & -\overline{d_n^{n-1}}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ d_1^{n-1} & \dots & d_{n}^{n-1} & -\overline{d_1^{0}} & \dots & -\overline{d_n^{0}}\\ d_1^0i^{-1}h_{1} & \dots & d_{n}^0i^{-1}h_{n} & \overline{d_1^{n-1}i^{-1}h_1} & \dots & \overline{d_n^{n-1}i^{-1}h_n}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ d_1^{n-1}i^{-1}h_{1} & \dots & d_{n}^{n-1}i^{-1}h_{n} & \overline{d_1^{0}i^{-1}h_1} & \dots & \overline{d_n^{0}i^{-1}h_n} \end{vmatrix} \\ &\qquad =i^{(n-1)n}i^n\prod_{k=1}^n\overline{d}_k^{\,n-1} \begin{vmatrix} d_1^{0} & \dots & d_{n}^{0} & d_{n+1}^{0} & \dots & d_{2n}^{0}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ d_1^{n-1} & \dots & d_{n}^{n-1} & d_{n+1}^{n-1} & \dots & d_{2n}^{n-1}\\ d_1^0h_{1} & \dots & d_{n}^0h_{n} & d_{n+1}^0h_{n+1} & \dots & d_{2n}^0h_{2n}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ d_1^{n-1}h_{1} & \dots & d_{n}^{n-1}h_{n} & d_{n+1}^{n-1}h_{n+1} & \dots & d_{2n}^{n-1}h_{2n} \end{vmatrix} \\ &\qquad =i^{(n-1)n}i^n\prod_{k=1}^n\overline{d}_k^{\,n-1} \sum_{1\leqslant p_1<\dots <p_n\leqslant 2n}\frac{V_{d_{p_1},\dots,d_{p_n}}h_{q_1} \dotsb h_{q_n}V_{d_{q_1},\dots,d_{q_n}}}{(-1)^{(1+p_1)+\dots +(n+p_n)}} \\ &\qquad = \frac{i^{n}2^{(n-1)n}\prod_{k=1}^n\overline{d}_k^{\,n-1}}{\prod_{k=1}^{2n}(e_{k}+i)^{n-1}}\sum_{1\leqslant p_1<\dots <p_n\leqslant 2n} \frac{V_{e_{p_1},\dots,e_{p_n}}h_{q_1} \dotsb h_{q_n}V_{e_{q_1},\dots,e_{q_n}}}{(-1)^{(1+p_1)+\dots +(n+p_n)}} \\ &\qquad = i^{n}|L_{E_n}|^2 \begin{vmatrix} e_1^0 & \dots & e_{n}^0 & e_{n+1}^0 & \dots & e_{2n}^0\\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1} & \dots & e_{n}^{n-1} & e_{n+1}^{n-1} & \dots & e_{2n}^{n-1}\\ e_1^0h_{1} & \dots & e_{n}^0h_{n} & e_{n+1}^0h_{n+1} & \dots & e_{2n}^0h_{2n}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1}h_{1} & \dots & e_{n}^{n-1}h_{n} & e_{n+1}^{n-1}h_{n+1} & \dots & e_{2n}^{n-1}h_{2n} \end{vmatrix} \\ &\qquad =i^n |L_{E_n}|^2 \begin{vmatrix} e_1^0 & \dots & e_{n}^0 & \overline{e_{1}^0} & \dots & \overline{e_{n}^0}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1} & \dots & e_{n}^{n-1} & \overline{e_{1}^{n-1}} & \dots & \overline{e_{n}^{n-1}}\\ e_1^0h_{1} & \dots & e_{n}^0h_{n} & \overline{e_{1}^0h_1} & \dots & \overline{e_{n}^{0}h_n}\\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1}h_{1} & \dots & e_{n}^{n-1}h_{n} & \overline{e_{1}^{n-1}h_1} & \dots & \overline{e_{n}^{n-1}h_n} \end{vmatrix} \\ &\qquad = |L_{E_n}|^2 \det \begin{pmatrix} A_{E_n} & i\qquad \overline{A}_{E_n} \\ A_{E_n}^{H_n } & i\qquad \overline{A}_{E_n}^{\,H_n }\end{pmatrix}= |L_{E_n}|^2M_{E_n}^{\mathrm n;H_n}= |L_{E_n}|^2M_{E_n}^{\mathrm n;\varphi [E_n]}, \end{aligned}
\end{equation*}
\notag
$$
which, in combination with (53), yields the second equality in (44). Thus we have established both equalities in (44) under the above assumptions.
We can drop the assumptions $i\notin E_n$ and $0\notin H_n$ by making an obvious limit transition in (44) if necessary. In the case when $E_n$ contains multiple points we consider the sets $E_{n}^{\varepsilon_1,\dots,\varepsilon_n}:=\{e_1+\varepsilon_1,\dots,e_n+\varepsilon_n \}$, where arbitrarily small $\varepsilon_1,\dots,\varepsilon_n$ are such that the sets $E_{n}^{\varepsilon_1,\dots,\varepsilon_n}$ consist of pairwise distinct points. Set
$$
\begin{equation*}
t(E_{n}^{\varepsilon_1,\dots,\varepsilon_n})= \{t(e_1+\varepsilon_1),\dots,t(e_n+\varepsilon_n)\} =\{d_1+\delta_1,\dots,d_n+\delta_n \}=:D_{n}^{\delta_1,\dots,\delta_n},
\end{equation*}
\notag
$$
where $\delta_k=t(e_k+\varepsilon_k)-t(e_k)$ and therefore $\lim_{\varepsilon_k\to 0}\delta_k=0$ for $k=1,\dots,n$. Then, using (44) and Lemma 4 (see (41)) for the sets $E_{n}^{\varepsilon_1,\dots,\varepsilon_n}$ and $D_{n}^{\delta_1,\dots,\delta_n}$, for $\zeta =\mathrm b$ and $\zeta =\mathrm s$, respectively, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\breve{M}_{E_n}^{\mathrm b;\varphi [E_n]}=\lim_{\varepsilon_1\to 0}\dots \lim_{\varepsilon_n\to 0}\breve{M}_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}^{\mathrm b;\varphi [E_{n}^{\varepsilon_1,\dots,\varepsilon_n}]}=\lim_{\varepsilon_1\to 0}\dots \lim_{\varepsilon_n\to 0}\breve{M}_{t(E_{n}^{\varepsilon_1,\dots,\varepsilon _n})}^{\mathrm s;(\varphi \circ t^{-1})[t(E_{n}^{\varepsilon _1,\dots,\varepsilon _n})]} \\ &\qquad =\lim_{\delta_1\to 0}\dots \lim_{\delta_n\to 0} \breve{M}_{D_{n}^{\delta_1,\dots,\delta_n}}^{\mathrm s;(\varphi \circ t^{-1})[D_{n}^{\delta_1,\dots,\delta_n}]}=\breve{M}_{D_n}^{\mathrm s;(\varphi \circ t^{-1})[D_n]}=\breve{M}_{t(E_n)}^{\mathrm s;(\varphi \circ t^{-1})[t(E_n)]}, \end{aligned}
\end{equation*}
\notag
$$
which is the first equality in (44) in the general case.
In a similar way we obtain the second equality in (44) in the general case:
$$
\begin{equation*}
\begin{aligned} \, \breve{M}_{E_n}^{\mathrm n;\varphi [E_n]} &=\lim_{\varepsilon_1\to 0}\dots \lim_{\varepsilon_n\to 0}\breve{M}_{E_{n}^{\varepsilon_1,\dots,\varepsilon_n}}^{\mathrm n;\varphi [E_{n}^{\varepsilon_1,\dots,\varepsilon_n}]}=\lim_{\varepsilon_1\to 0}\dots \lim_{\varepsilon_n\to 0}\breve{M}_{t(E_{n}^{\varepsilon_1,\dots,\varepsilon _n})}^{\mathrm c;i^{-1}(\varphi \circ t^{-1})[t(E_{n}^{\varepsilon _1,\dots,\varepsilon _n})]} \\ &=\lim_{\delta_1\to 0}\dots \lim_{\delta_n\to 0} \breve{M}_{D_{n}^{\delta_1,\dots,\delta_n}}^{\mathrm c;i^{-1}(\varphi \circ t^{-1})[D_{n}^{\delta_1,\dots,\delta_n}]}=\breve{M}_{D_n}^{\mathrm c;i^{-1}(\varphi \circ t^{-1})[D_n]} \\ &=\breve{M}_{t(E_n)}^{\mathrm c;i^{-1}(\varphi \circ t^{-1})[t(E_n)]}. \end{aligned}
\end{equation*}
\notag
$$
Lemma 5 is proved. To verify Corollary 1 note that it follows from (44) for $\varphi (z)=\varphi_{E_n}^{H_n}(z)$ that, using the notation in Corollary 1 for $N\in\mathbb Z_+^\infty$,
$$
\begin{equation*}
\{\breve{M}_{D_{n}}^{\mathrm s;G_n}\}_{n=1}^\infty\in \mathscr M_N \quad\Longleftrightarrow\quad \{\breve{M}_{E_{n}}^{\mathrm b;H_n}\}_{n=1}^\infty\in \mathscr M_N
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\{\breve{M}_{D_{n}}^{\mathrm c;i^{-1}G_n}\}_{n=1}^\infty\in \mathscr M_N \quad\Longleftrightarrow\quad \{\breve{M}_{E_{n}}^{\mathrm n;H_n}\}_{n=1}^\infty\in \mathscr M_N.
\end{equation*}
\notag
$$
By (39), in the above relations we can replace $\breve{M}$ by $M$, which yields (45) and (46). To verify Corollary 2 recall that, as the points in $E_n=\{e_1,\dots,e_{n}\}$ are pairwise distinct, we have $t^{-1}[t(E_n)]=E_n$. In view of (39), using the second equality in (44) for $\varphi (z)=\varphi_{E_n}^{H_n}(z)$ first and then equality (43) (for $E_n$ replaced by $t(E_n)$) and bearing in mind that for $\phi (z)=i^{-1}(\varphi_{E_n}^{H_n}\circ t^{-1})(z)$ we have
$$
\begin{equation*}
\begin{gathered} \, \phi (t(e_k))=i^{-1}h_k,\qquad k=1,\dots,n, \\ \phi [t(E_n)]=i^{-1}H_n\quad\text{and} \quad (T\circ \phi )[t(E_n)]=T(i^{-1}H_n), \end{gathered}
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation*}
\begin{aligned} \, \frac{M_{E_n}^{\mathrm n;H_n}}{|{\det A_{E_n}}|^2} &=\breve{M}_{E_n}^{\mathrm n;\varphi_{E_n}^{H_n}[E_n]} =\breve{M}_{t(E_n)}^{\mathrm c;i^{-1}(\varphi_{E_n}^{H_n}\circ t^{-1})[t(E_n)]}= \frac{M_{t(E_n)}^{\mathrm c;\phi [t(E_n)]}}{|{\det A_{t(E_n)}}|^2} \\ &=\frac{\prod_{k=1}^n|1+i^{-1}h_k|^2}{2^n|{\det A_{t(E_n)}}|^2}M_{t(E_n)}^{\mathrm s;T(i^{-1}H_n)}. \end{aligned}
\end{equation*}
\notag
$$
This yields the required assertion of Corollary 2. Proof of Theorem 1. We present the proof for $\zeta =\mathrm b,\mathrm n$, as already mentioned, using Proposition 1 and Lemma 5 just proved, by reducing these cases to the cases $\zeta =\mathrm s,\mathrm c$ considered in [16] and [17].
Let $t(z)=(z-i)/(z+i)$. For $n=1,2,\dots$ set
$$
\begin{equation}
\begin{gathered} \, d_n:=t(e_n), \qquad D_n:=\{d_1,\dots,d_n\}=t(E_n), \\ G_{n}=\{g_1,\dots,g_n\}:= (\varphi_{E_n}^{H_n}\circ t^{-1})[D_n] \end{gathered}
\end{equation}
\tag{55}
$$
and note that, first, the set $G_n$ is independent of the choice of the interpolation function $\varphi_{E_n}^{H_n}(z)$ and, second, $(\varphi_{E_n}^{H_n}\circ t^{-1})(z)$ corresponds to the definition (20) of the interpolation function for the sets $D_n$ and $G_n$, so that we can assume that
$$
\begin{equation}
\varphi_{D_n}^{G_n}(z)=(\varphi_{E_n}^{H_n}\circ t^{-1})(z).
\end{equation}
\tag{56}
$$
By Proposition 1
$$
\begin{equation}
f(z)\in\mathfrak B^{\mathrm b}_N\ \ \Longleftrightarrow\ \ (f\circ t^{-1})(z)\in\mathfrak B^{\mathrm s }_N,\quad f(z)\in\mathfrak B^{\mathrm n }_N \ \ \Longleftrightarrow\ \ i^{-1}(f\circ t^{-1})(z)\in\mathfrak B^{\mathrm c }_N.
\end{equation}
\tag{57}
$$
Since, as just mentioned, $G_n$ is independent of the choice of the interpolation function $\varphi_{E_n}^{H_n}(z)$, it follows from the equality $f[E_n]=H_n$ (which means that $f(z)$ is an interpolation function for the sets $E_n$ and $H_n$) and the definition of the set $G_n$ in (55) that for $F(z)=(f\circ t^{-1})(z)$ we have
$$
\begin{equation*}
G_n=(\varphi_{E_n}^{H_n}\circ t^{-1})[D_n]=(f\circ t^{-1})[D_n]=F[D_n].
\end{equation*}
\notag
$$
Note that the converse result also holds: $F[D_n]=G_n$ $\Longrightarrow$ $f[E_n]=H_n$, that is,
$$
\begin{equation}
f[E_n]=H_n \quad\Longleftrightarrow\quad F[D_n]= G_n.
\end{equation}
\tag{58}
$$
In a similar way, for $\Phi (z)=i^{-1}(f\circ t^{-1})(z)$ we have
$$
\begin{equation}
f[E_n]=H_n \quad\Longleftrightarrow\quad \Phi [D_n]=i^{-1}G_n.
\end{equation}
\tag{59}
$$
It follows from (57)–(59) that
$$
\begin{equation}
\begin{aligned} \, \notag &\exists\, f(z)\in\mathfrak B_N^{\mathrm b}, \quad f[E_n]=H_n, \quad n=1,2,\dots \\ &\qquad\Longleftrightarrow\quad \exists\, F(z)\in\mathfrak B_N^{\mathrm s }, \quad F[D_n]=G_n, \quad n=1,2,\dots, \end{aligned}
\end{equation}
\tag{60}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag &\exists\, f(z)\in\mathfrak B^{\mathrm n }_N, \quad f[E_n]=H_n, \quad n=1,2,\dots \\ &\qquad\Longleftrightarrow\quad \exists\, \Phi (z)\in\mathfrak B^{\mathrm c }_N, \quad \Phi [D_n]=i^{-1}G_n, \quad n=1,2,\dots\,. \end{aligned}
\end{equation}
\tag{61}
$$
For the claims on the right-hand sides of (60) and (61) we have Theorem 1, established in [16] and [17] for $\zeta =\mathrm s $ and $\zeta =\mathrm c$, respectively, which shows (see the necessary and sufficient conditions (31)) that (60) and (61) can also be written as follows:
$$
\begin{equation}
\exists\, f(z)\in\mathfrak B_N^{\mathrm b}, \quad f[E_n]=H_n, \quad n=1,2,\dots \quad\Longleftrightarrow\quad \{M_{D_{n,N}}^{\mathrm s;G_{n,N}} \}_{n=1}^\infty\in \mathscr M_N,
\end{equation}
\tag{62}
$$
and
$$
\begin{equation}
\exists\, f(z)\in\mathfrak B^{\mathrm n }_N, \quad f[E_n]=H_n, \quad n=1,2,\dots \quad\Longleftrightarrow\quad \{M_{D_{n,N}}^{\mathrm c;i^{-1}G_{n,N}} \}_{n=1}^\infty\in \mathscr M_N.
\end{equation}
\tag{63}
$$
If $N=\infty$, then for all $n=1,2,\dots$ we have (26) and the analogous equalities $D_{n,\infty }=D_n$ and $G_{n,\infty}=G_n$. Hence for $N=\infty$, by Corollary 1 the right-hand sides of (62) and (63) can be replaced by those of (45) and (46), respectively (for $N=\infty$, $E_n=E_{n,\infty}$ and $H_n=H_{n,\infty}$), that is,
$$
\begin{equation*}
\exists\, f(z)\in\mathfrak B_\infty^{\mathrm b}, \quad f[E_n]=H_n, \quad n=1,2,\dots \quad\Longleftrightarrow\quad \{M_{E_{n,\infty}}^{\mathrm b;H_{n,\infty}} \}_{n=1}^\infty\in \mathscr M_\infty,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\exists\, f(z)\in\mathfrak B^{\mathrm n}_\infty, \quad f[E_n]=H_n, \quad n=1,2,\dots \quad\Longleftrightarrow\quad \{M_{E_{n,\infty}}^{\mathrm n;H_{n,\infty}} \}_{n=1}^\infty\in \mathscr M_\infty.
\end{equation*}
\notag
$$
Thus, we have proved Theorem 1 for $\zeta =\mathrm b,\mathrm n $ in the case when $N=\infty$.
Now consider the case $N\in \mathbb Z_+$. Note that for all $n=1,2,\dots$ the multiplicity $\nu_n$ of the point $e_n$ in $E_n=\{e_1,\dots,e_n\}$ is equal to that of $d_n$ in $D_n=t(E_n)$. Hence if $E_{n,N}$ for $n=1,2,\dots$ and $N\in\mathbb Z$ is the set defined before Theorem 1 by equalities (26)–(28) on the basis of the sequence $e_1,e_2,\dots$, then starting from the sequence $d_1,d_2,\dots$ we can accordingly define a set $D_{n,N}$, namely, for $n\leqslant N+2$ we set $D_{n,N}:=D_n$, and for $n\geqslant N+3$ we set
$$
\begin{equation*}
D_{n,N}:= \begin{cases} \{d_{j_{n,1}},\dots,d_{j_{n,N+2}}\} &\text{for } \nu_n< N+2<n, \\ \{d_n\}^{2\nu_n -N-2} &\text{for }\nu_n\geqslant N+2, \end{cases}
\end{equation*}
\notag
$$
where the indices $ j_{n,1},\dots,j_{n,N+2}$ are taken the same as the ones fixed in (27). This means that, apart from the equalities $D_n=t(E_n)$, we also have the relations
$$
\begin{equation}
D_{n,N}=t(E_{n,N}), \qquad n=1,2,\dots, \quad N\in\mathbb Z_+.
\end{equation}
\tag{64}
$$
In accordance with (27)–(29) and (55), set
$$
\begin{equation*}
G_{n,N}:=\begin{cases} \{g_{j_{n,1}},\dots,g_{j_{n,N+2}}\} &\text{for }\nu_n< N+2<n, \\ \{g_{j_{n,1}},\dots,g_{j_{n,\nu_n}},g^{*}_{n,1},\dots,g^{*}_{n,\nu_n-N-2}\} &\text{for }\nu_n\geqslant N+2, \end{cases}
\end{equation*}
\notag
$$
where $g^{*}_{n,k}=(\varphi_{D_n}^{G_n})^{(\nu_n-1+k)}(d_n)$, $k=1,\dots,\nu_n-N-2$, and note, that along with (30) we also have the equalities $\varphi_{D_n}^{G_n}[D_{n,N}]=G_{n,N}$, so that taking (56) and (64) into account it follows that
$$
\begin{equation}
G_{n,N} =\varphi_{D_n}^{G_n}[D_{n,N}]= (\varphi_{E_n}^{H_n}\circ t^{-1})[t(E_{n,N})], \qquad n=1,2,\dots, \quad N\in\mathbb Z_+.
\end{equation}
\tag{65}
$$
Once we have (40), (64) and (65), using the same arguments as in the derivation of relations (45) and (46), which make up Corollary 1, as consequences of Lemma 5 (as applied for $\varphi (z)=\varphi_{E_n}^{H_n}(z)$ to the sets $E_{n,N}$, rather than to $E_n$) we also obtain (45) and (46) for $E_n$, $H_n$, $D_n$ and $G_n$ replaced by $E_{n,N}$, $H_{n,N}$, $D_{n,N}$ and $G_{n,N}$, respectively. More precisely,
$$
\begin{equation}
\begin{aligned} \, \notag &\{M_{D_{n,N}}^{\mathrm s;G_{n,N}}\}_{n=1}^\infty\in \mathscr M_N\quad\Longleftrightarrow\quad \{M_{t(E_{n,N})}^{\mathrm s; (\varphi_{E_n}^{H_n}\circ t^{-1} )[t(E_{n,N})]}\}_{n=1}^\infty\in \mathscr M_N \\ &\qquad\Longleftrightarrow\quad \{M_{E_{n,N}}^{\mathrm b;\varphi_{E_n}^{H_n}[E_{n,N}]}\}_{n=1}^\infty\in \mathscr M_N \quad\Longleftrightarrow\quad \{M_{E_{n,N}}^{\mathrm b;H_{n,N}}\}_{n=1}^\infty\in \mathscr M_N, \end{aligned}
\end{equation}
\tag{66}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag &\{M_{D_{n,N}}^{\mathrm c;i^{-1}G_{n,N}}\}_{n=1}^\infty\in \mathscr M_N\quad\Longleftrightarrow\quad \{M_{t(E_{n,N})}^{\mathrm c;i^{-1} (\varphi_{E_n}^{H_n}\circ t^{-1} )[t(E_{n,N})]}\}_{n=1}^\infty\in \mathscr M_N \\ &\qquad\Longleftrightarrow\quad \{M_{E_{n,N}}^{\mathrm n;\varphi_{E_n}^{H_n}[E_{n,N}]}\}_{n=1}^\infty\in \mathscr M_N\quad\Longleftrightarrow\quad \{M_{E_{n,N}}^{\mathrm n;H_{n,N}}\}_{n=1}^\infty\in \mathscr M_N. \end{aligned}
\end{equation}
\tag{67}
$$
Hence for $N=\mathbb Z_+$ the right-hand sides of (62) and (63) can be replaced by the right-hand sides of (66) and (67), respectively, so that
$$
\begin{equation*}
\exists\, f(z)\in\mathfrak B_N^{\mathrm b}, \quad f[E_n]=H_n, \quad n=1,2,\dots \quad \Longleftrightarrow\quad \{M_{E_{n,N}}^{\mathrm b;H_{n,N}} \}_{n=1}^\infty\in \mathscr M_N
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\exists\, f(z)\in\mathfrak B^{\mathrm n }_N, \quad f[E_n]=H_n, \quad n=1,2,\dots \quad \Longleftrightarrow\quad \{M_{E_{n,N}}^{\mathrm n;H_{n,N}} \}_{n=1}^\infty\in \mathscr M_N.
\end{equation*}
\notag
$$
That is, we have proved Theorem 1 for $\zeta =\mathrm b $ and $\zeta =\mathrm n $ in the case when $N=\mathbb Z_+$.
Thus, Theorem 1, which we proved previously, in [16] and [17], for $\zeta =\mathrm s,\mathrm c$, is also established for $\zeta =\mathrm b,\mathrm n$. Furthermore, using the same notation and arguments, with the help of Proposition 1, the supplements to Theorem 1 for $\zeta =\mathrm s, \mathrm c$ established in [16] and [17], and Lemma 5 for $\varphi (z)=\varphi_{E_n}^{H_n}(z)$ we obtain the chain of implications
$$
\begin{equation*}
\begin{aligned} \, &f(z)\in\mathfrak B_N^{\mathrm b}, \quad f[E_n]=H_n \\ &\quad\Longrightarrow\quad (f\circ t^{-1})(z)\in\mathfrak B_N^{\mathrm s },\, (f\circ t^{-1})[t(E_n)]= (\varphi_{E_n}^{H_n}\circ t^{-1})[t(E_n)] \\ &\quad\Longrightarrow \quad \begin{cases} M_{t(E_n)}^{\mathrm s;(\varphi_{E_n}^{H_n}\circ t^{-1} )[t(E_n)]}>0, & n\leqslant N, \\ M_{t(E_n)}^{\mathrm s;(\varphi_{E_n}^{H_n}\circ t^{-1} )[t(E_n)]}=0, & n>N \end{cases} \quad\Longrightarrow\quad \begin{cases} M_{E_n}^{\mathrm b;H_n}>0,&n\leqslant N, \\ M_{E_n}^{\mathrm b;H_n}=0,&n>N, \end{cases} \end{aligned}
\end{equation*}
\notag
$$
which proves the supplement to Theorem 1 for $\zeta =\mathrm b$, and the chain of implications
$$
\begin{equation*}
\begin{aligned} \, &f(z)\in\mathfrak B_N^{\mathrm n }, \quad f[E_n]=H_n \\ &\quad\Longrightarrow\quad i^{-1}(f\circ t^{-1})(z)\in\mathfrak B_N^{\mathrm c }, \quad i^{-1}(f\circ t^{-1})[t(E_n)]= i^{-1}(\varphi_{E_n}^{H_n}\circ t^{-1})[t(E_n)] \\ &\quad\Longrightarrow\quad \begin{cases} M_{t(E_n)}^{\mathrm c;i^{-1}(\varphi_{E_n}^{H_n}\circ t^{-1} )[t(E_{n})]}>0,&n\leqslant N, \\ M_{t(E_n)}^{\mathrm c;i^{-1}(\varphi_{E_n}^{H_n}\circ t^{-1} )[t(E_{n})]}=0,& n>N \end{cases} \quad\Longrightarrow\quad \begin{cases} M_{E_n}^{\mathrm n;H_n}>0,&n\leqslant N, \\ M_{E_n}^{\mathrm b;H_n}=0,&n>N, \end{cases} \end{aligned}
\end{equation*}
\notag
$$
which proves the supplement to Theorem 1 for $\zeta =\mathrm n$. Thus we have proved Theorem 1 and the supplement to it in all cases $\zeta =\mathrm n,\mathrm b,\mathrm c,\mathrm s$. § 4. Proofs of Lemmas 1, 2 and 3 and Claims 1 and 2 Proof of Lemma 1. Using the notation from Lemma 1, it follows from its assumptions that for all $n=1,2,\dots$ the multiplicity $\nu_n$ of the point $e_n=0$ in the set $E_n=\{e_1,\dots,e_n\}=\{0\}^n$ is equal to $n$, and as an interpolation function $\varphi_{E_n}^{H_n}(z)$ we can take the partial sum $f_m(z)$ for any $m\geqslant n-1$. Because
$$
\begin{equation*}
(z^kf_{n-1}(z))^{(j)}(0)= \begin{cases} 0,& j<k, \\ j!\,a_{j-k},&j\geqslant k, \end{cases} \qquad j,k=0,\dots,n-1,
\end{equation*}
\notag
$$
and therefore $H_n = f_{n-1}[E_n] = \{0!\,a_0,1!\,a_1, \dots, (n-1)!\,a_{n-1}\}$, $(\psi_1f_{n-1})[E_n]=\{0, 1!\,a_0, \dots, (n-1)!\,a_{n-2}\}$ and so on, from (18) and (19) we obtain
$$
\begin{equation*}
A_{E_n}^{H_n}=\begin{pmatrix} 0!\,a_0 & 1!\,a_1 & \dots & (n\!-\!1)!\,a_{n-1}\\ 0 & 1!\,a_0 & \dots & (n\!-\!1)!\,a_{n-2}\\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & (n\!-\!1)!\,a_{0} \end{pmatrix}\text{ and } A_{E_n}=\begin{pmatrix} 0! & 0 & \dots & 0\\ 0 & 1! & \dots & 0\\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & (n\!-\!1)!\end{pmatrix}.
\end{equation*}
\notag
$$
Hence, from the definition of $C_n:=(0!\,\dotsb (n-1)!)^2$ and the definitions (23) and (11) we obtain the equalities
$$
\begin{equation}
M_{E_n}^{\mathrm c;H_n } =\det \begin{pmatrix} A_{E_n} & -\widetilde{A }_{E_n} \\ A_{E_n}^{H_n } & \widetilde{A}_{E_n}^{H_n } \end{pmatrix} =C_n\det \begin{pmatrix} I_n & -I_n \\ A_{n}^{f} & \widetilde{A}_{n}^{f} \end{pmatrix}
\end{equation}
\tag{68}
$$
and
$$
\begin{equation}
M_{E_n}^{\mathrm s;H_n } =\det \begin{pmatrix} A_{E_n} & \widetilde{A }_{E_n}^{H_n } \\ A_{E_n}^{H_n } & \widetilde{A}_{E_n} \end{pmatrix} =C_n\det \begin{pmatrix} I_n & \widetilde{A}_{n}^{f} \\ A_{n}^{f} & I_n \end{pmatrix}.
\end{equation}
\tag{69}
$$
Using the equality
$$
\begin{equation*}
\det \begin{pmatrix} P & Q \\ R & S\end{pmatrix}=\det (PS-RQ)
\end{equation*}
\notag
$$
(for instance, see [3], § 5), where $P$, $Q$, $R$ and $S$ are $ n\times n $ matrices such that $PR=RP$, from (68), (69) and the definitions (12) we obtain the same equalities
$$
\begin{equation*}
M_{E_n}^{\mathrm c;H_n } =C_n\det (A_{n}^{f} + \widetilde{A}_{n}^{f})=C_nM_{n}^{\mathrm c;f}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
M_{E_n}^{\mathrm s;H_n } =C_n\det (I_n-A_{n}^{f} \widetilde{A}_{n}^{f})=C_nM_{n}^{\mathrm s;f},
\end{equation*}
\notag
$$
as in (34). Lemma 1 is proved. Proof of Claim 1. Using the notation from Lemma 1, it follows from (34), taking (26) into account, that for $N=\infty$
$$
\begin{equation*}
M_{n}^{\zeta;f}>0, \ \ n=1,2,\dots \quad\Longleftrightarrow\quad M_{E_{n,\infty}}^{\zeta;H_{n,\infty}}\,{=}\,M_{E_n}^{\zeta;H_n}\,{=}\,C_nM_{n}^{\zeta;f}\,{>}\,0, \ \ n\,{=}\,1,2,\dots\,.
\end{equation*}
\notag
$$
Hence, from Theorem 1 for $\zeta =\mathrm c,\mathrm s $ and $N=\infty$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &M_{n}^{\zeta;f}>0,\quad n=1,2,\dots \quad\Longleftrightarrow\quad \{M_{E_{n,\infty }}^{\zeta;H_{n,\infty }} \}_{n=1}^\infty\in \mathscr M_\infty \\ &\quad\Longleftrightarrow\quad \exists\, F(z)\in\mathfrak B_\infty^\zeta, \quad F[E_n]=H_n=\{0!\,a_0,\dots,(n-1)!\,a_{n-1}\}, \ \ n=1,2,\dots\,. \end{aligned}
\end{equation*}
\notag
$$
Since $F[E_n]=\{F(0),\dots,F^{(n-1)}(0)\}$, the last result means that the formal power series $f(z)=\sum_{n=0}^\infty a_nz^n$ is the Taylor series of $F(z)\in\mathfrak B_\infty^\zeta$, so that the Carathéodory-Schur criterion holds for $N=\infty$. Now consider the case when $N\in\mathbb Z_+$. For $n\leqslant N+2$, in view of (26) we have the equalities $E_{n,N}=E_n$ and $H_{n,N}=H_n$, while for $n> N+2$, in view of (28) and taking the equality $\nu_n=n$ into account we obtain so that, fixing $m=2n-N-3> n-1$ (that is, setting $\varphi_{E_n}^{H_n}(z)=f_{2n-N-3}(z)$), in view of (30) and (33) we obtain
$$
\begin{equation*}
H_{n,N}=\varphi_{E_n}^{H_n}[E_{n,N}]=f_{2n-N-3}[E_{2n-N-2}] =H_{2n-N-2}.
\end{equation*}
\notag
$$
Therefore, it follows from (34) that
$$
\begin{equation*}
M_{E_{n,N}}^{\zeta;H_{n,N}}= \begin{cases} M_{E_n}^{\zeta;H_n}=C_nM_{n}^{\zeta;f},&n\leqslant N+2, \\ M_{E_{2n-N-2}}^{\zeta;H_{2n-N-2}}=C_{2n-N-2}M_{2n-N-2}^{\zeta;f}, &n>N+2. \end{cases}
\end{equation*}
\notag
$$
Hence by Theorem 1 we have the equivalences
$$
\begin{equation}
\begin{aligned} \, &M_n^{\zeta;f } > 0 \quad \text{for } n=1,\dots,N, \qquad M_{N+1}^{\zeta;f } =M_{N+2}^{\zeta;f } = M_{N+4}^{\zeta;f } = M_{N+6}^{\zeta;f }=\dots = 0 \\ \notag &\Longleftrightarrow\ \ \{M_{E_{n,N}}^{\zeta;H_{n,N}}\}_{n=1}^\infty \in \mathscr M_N \ \ \Longleftrightarrow\ \ \exists\, F(z)\in\mathfrak B_N^\zeta, \quad F[E_n]=H_n, \quad n=1,2,\dots\,. \end{aligned}
\end{equation}
\tag{70}
$$
Because
$$
\begin{equation*}
F[E_n]=\{F(0),\dots,F^{(n-1)}(0)\}\quad\text{and} \quad H_n=\{0!\,a_0,\dots,(n-1)!\,a_{n-1}\},
\end{equation*}
\notag
$$
conditions (70) are necessary and sufficient for the existence of a function in $\mathfrak B_N^\zeta$ with Taylor series equal to the prescribed power series $f(z)=\sum_{n=0}^\infty a_nz^n$.
Note that the sufficient conditions (70) do not contain the relations which are present in the necessary and sufficient conditions $\{M_n^{\zeta;f } \}_{n=1}^\infty\in \mathscr M_N$ of the Carathéodory-Schur criterion.That the conditions $M_{N+3}^{\zeta;f } =M_{N+5}^{\zeta;f } =\dots =0$ are necessary follows from the supplement to Theorem 1 and equalities (34). Thus we have proved Claim 1 and, moreover, have shown that for $N<\infty$ the sufficient conditions in the Carathéodory-Schur criterion can be relaxed slightly. Turning to the proof of Lemma 2, we state and prove Lemma 6, which is a stronger version of Lemma 2. Lemma 6. Let $e_1,\dots,e_{2n}$ be pairwise distinct points and $h_1,\dots,h_{2n}$ be arbitrary complex numbers. Then Proof. Since Lemma 6 for $n=1$ is trivial, we assume below that $n\geqslant 2$. For $2\leqslant k\leqslant 2n$ and pairwise distinct positive integers $p_1,\dots,p_k$ not exceeding $2n$, we set where $V_{z_1,\dots,z_k}$ is the Vandermonde determinant of $z_1,\dots,z_k$. Note that $h_{p_1,\dots,p_k}$ does not change after rearrangements of $(p_1,\dots,p_k)$. We show that
For $k=2$ equality (74) holds in a trivial way because
$$
\begin{equation}
h_{p_1,p_2}:=\det\begin{pmatrix} e_{p_1}^0 & e_{p_2}^0\\ h_{p_1} & h_{p_2}\end{pmatrix}V^{-1}_{e_{p_1},e_{p_2}} =\frac{h_{p_2}-h_{p_1}}{e_{p_2}-e_{p_1}}.
\end{equation}
\tag{75}
$$
Assume that equality (74) holds for all integer indices from 2 to $k-1<2n$ inclusive; we prove it for $k$. For $j=k-1,\dots,2$, from the $j$th row in the determinant in (73) we subtract the $(j-1)$st row multiplied by $e_{p_1}$, and then from the $k$th row we subtract the first multiplied by $h_{p_1}$. We begin a chain of equalities by
$$
\begin{equation*}
h_{p_1,\dots,p_k}=\det\begin{pmatrix} e_{p_{1}}^0 & e_{p_{2}}^0 & \dots & e_{p_{k}}^0 \\ 0 & e_{p_{2}}^0(e_{p_2}- e_{p_1}) & \dots & e_{p_{k}}^0(e_{p_{k}}-e_{p_{1}})\\ \dots & \dots & \dots & \dots \\ 0 & e_{p_2}^{k-3}(e_{p_2}- e_{p_1}) & \dots & e_{p_{k}}^{k-3}(e_{p_{k}}-e_{p_{1}})\\ 0 & h_{p_2}-h_{p_1} & \dots & h_{p_{k}}-h_{p_1}\end{pmatrix}V^{-1}_{e_{p_1},\dots,e_{p_{k}}}.
\end{equation*}
\notag
$$
We expand the determinant on the right with respect to the first column, taking the equalities $h_{p_j}- h_{p_1}=h_{p_1,p_j}(e_{p_j}-e_{p_1})$ into account (see (75)), and move out the factors $ e_{p_j}-e_{p_1} $, $j=2,\dots, k$, bearing in mind formula (49) for Vandermonde determinants. Then we obtain
$$
\begin{equation*}
h_{p_1,\dots,p_k}=\det\begin{pmatrix} e_{p_2}^0 & \dots & e_{p_{k}}^0\\ \dots & \dots & \dots \\ e_{p_2}^{k-3} & \dots & e_{p_{k}}^{k-3}\\ h_{p_1,p_2} & \dots & h_{p_1,p_{k}}\end{pmatrix}V^{-1}_{e_{p_2},\dots,e_{p_{k}}}.
\end{equation*}
\notag
$$
We continue in a similar way, taking account of the induction assumption for the required equality (74); then we obtain
$$
\begin{equation*}
\begin{aligned} \, h_{p_1,\dots,p_k} &=\frac{\det \begin{pmatrix} e_{p_{k-1}}^0 & e_{p_{k}}^0\\ h_{p_1,\dots,p_{k-2},p_{k-1}} & h_{p_1,\dots,p_{k-2},p_{k}} \end{pmatrix}}{V_{e_{p_{k-1}},e_{p_{k}}}} \\ &=\frac{h_{p_1,\dots,p_{k-2},p_{k}}-h_{p_1,\dots,p_{k-2},p_{k-1}}}{e_{p_{k}}-e_{p_{k-1}}}. \end{aligned}
\end{equation*}
\notag
$$
Thus, (74) is proved.
We prove (71). For $j=2,\dots,n,n+2,\dots,2n$, from the $j$th row in the determinant (72) we subtract the $(j-1)$st row times $e_1$, and then from the $(n+1)$st row we subtract the first times $h_1$. Then we can begin a chain of equalities by
$$
\begin{equation*}
\Delta_{e_1,\dots,e_{2n}}^{h_1,\dots,h_{2n}}=\det\begin{pmatrix} e_{1}^0 & e_{2}^0 & \dots & e_{2n}^0\\ 0 & e_2^0(e_2-e_1) & \dots & e_{2n}^0(e_{2n}-e_1)\\ \dots & \dots & \dots & \dots \\ 0 & e_2^{n-2}(e_{2}-e_1) & \dots & e_{2n}^{n-2}(e_{2n}-e_1) \\ 0 & h_{2}-h_{1} & \dots & h_{2n}-h_1\\ 0 & e_2^{0}h_2(e_{2}-e_1) &\dots & e_{2n}^{0}h_{2n}(e_{2n}-e_1)\\ \dots & \dots & \dots & \dots \\ 0 & e_2^{n-2}h_2(e_{2}-e_1) &\dots & e_{2n}^{n-2}h_{2n}(e_{2n}-e_1)\end{pmatrix}.
\end{equation*}
\notag
$$
We expand the determinant with respect to the first column taking the equalities $h_j-h_1=h_{j,1}(e_j- e_1)$ into account (see (75)) and move out the factors $e_j-e_1$, $j=2,\dots,2n$. Then we obtain
$$
\begin{equation*}
\Delta_{e_1,\dots,e_{2n}}^{h_1,\dots,h_{2n}}=\det\begin{pmatrix} e_2^0 & \dots & e_{2n}^0\\ \dots & \dots & \dots \\ e_2^{n-2} & \dots & e_{2n}^{n-2} \\ h_{2,1} & \dots & h_{2n,1}\\ e_2^{0}h_2 &\dots & e_{2n}^{0}h_{2n}\\ \dots & \dots & \dots \\ e_2^{n-2}h_2 &\dots & e_{2n}^{n-2}h_{2n}\end{pmatrix}\prod_{j=2}^{2n}(e_j-e_1).
\end{equation*}
\notag
$$
For $j=2,\dots,n-1,n+2,\dots,2n-1$, from the $j$th row we subtract the $(j-1)$st row multiplied by $e_2$, and then from the $n$th and $(n+1)$st ones we subtract the first row times $h_{2,1}$ and $h_2$, respectively. After this we use (49):
$$
\begin{equation*}
\Delta_{e_1,\dots,e_{2n}}^{h_1,\dots,h_{2n}}=\det\begin{pmatrix} e_{2}^0 & e_{3}^0 & \dots & e_{2n}^0 \\ 0 & e_3^0(e_3-e_2) & \dots & e_{2n}^0(e_{2n}-e_2)\\ \dots & \dots & \dots & \dots \\ 0 & e_3^{n-3}(e_{3}-e_2) & \dots & e_{2n}^{n-3}(e_{2n}-e_2) \\ 0 & h_{3,1}-h_{2,1} & \dots & h_{2n,1}-h_{2,1}\\ 0 & h_{3}-h_{2} & \dots & h_{2n}-h_{2}\\ 0 & e_3^{0}h_3(e_{3}-e_2) &\dots & e_{2n}^{0}h_{2n}(e_{2n}-e_2)\\ \dots & \dots & \dots & \dots \\ 0 & e_3^{n-3}h_3(e_{3}-e_2) &\dots & e_{2n}^{n-3}h_{2n}(e_{2n}-e_2)\end{pmatrix}\frac{V_{e_1,\dots,e_{2n}}} {V_{e_2,\dots,e_{2n}}}.
\end{equation*}
\notag
$$
We expand the determinant with respect to the first column and for $j=3,\dots,2n$ take the factors $e_j-e_2$ out of the suitable columns. Taking (74) into account we obtain
$$
\begin{equation*}
\Delta_{e_1,\dots,e_{2n}}^{h_1,\dots,h_{2n}}=\det\begin{pmatrix} e_3^0 & \dots & e_{2n}^0\\ \dots & \dots & \dots \\ e_3^{n-3} & \dots & e_{2n}^{n-3} \\ h_{3,2,1} & \dots & h_{2n,2,1}\\ h_{3,2} & \dots & h_{2n,2}\\ e_3^{0}h_3 &\dots & e_{2n}^{0}h_{2n}\\ \dots & \dots & \dots \\ e_3^{n-3}h_3 &\dots & e_{2n}^{n-3}h_{2n}\end{pmatrix}\frac{V_{e_1,\dots,e_{2n}}}{V_{e_3,\dots,e_{2n}}}.
\end{equation*}
\notag
$$
Taking (74) into account we continue in a similar way and obtain
$$
\begin{equation*}
\Delta_{e_1,\dots,e_{2n}}^{h_1,\dots,h_{2n}}=\det\begin{pmatrix} h_{n+1,n,\dots,1} & \dots & h_{2n,n,\dots,1}\\ h_{n+1,n,\dots,2} & \dots & h_{2n,n,\dots,2}\\ \dots & \dots & \dots \\ h_{n+1,n} & \dots & h_{2n,n}\end{pmatrix}\frac{V_{e_1,\dots,e_{2n}}}{V_{e_{n+1},\dots,e_{2n}}}.
\end{equation*}
\notag
$$
From the first row in the determinant obtained we subtract the second row multiplied by $(e_2-e_1)^{-1}$, take the equalities
$$
\begin{equation*}
h_{k,n,\dots,3,2,1}-\frac{h_{k,n,\dots,3,2}}{e_2-e_1}=-\frac{h_{k,n,\dots,3,1}}{e_2-e_1}, \qquad k=n+1,\dots,2n,
\end{equation*}
\notag
$$
into account, and move the factor $-(e_2-e_1)^{-1}=-V_{e_1,e_2}^{-1}$ out of the first row. Then we obtain
$$
\begin{equation*}
\Delta_{e_1,\dots,e_{2n}}^{h_1,\dots,h_{2n}}=\frac{(-1)}{V_{e_1,e_2}}\det\begin{pmatrix} h_{n+1,n,\dots,4,3,1} & \dots & h_{2n,n,\dots,4,3,1}\\ h_{n+1,n,\dots,4,3,2} & \dots & h_{2n,n,\dots,4,3,2}\\ h_{n+1,n,\dots,4,3} & \dots & h_{2n,n,\dots,4,3}\\ \dots & \dots & \dots \\ h_{n+1,n} & \dots & h_{2n,n}\end{pmatrix}\frac{V_{e_1,\dots,e_{2n}}}{V_{e_{n+1},\dots,e_{2n}}}.
\end{equation*}
\notag
$$
From the first and second rows in the determinant we subtract the third times $(e_3-e_1)^{-1}$ and $(e_3-e_2)^{-1}$, respectively, taking account of the equalities
$$
\begin{equation*}
h_{k,n,\dots,4,3,1}-\frac{h_{k,n,\dots,4,3}}{e_3-e_1} =-\frac{h_{k,n,\dots,4,1}}{e_3-e_1}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
h_{k,n,\dots,4,3,2}-\frac{h_{k,n,\dots,4,3}}{e_3-e_2}=-\frac{h_{k,n,\dots,4,2}}{e_3-e_2}
\end{equation*}
\notag
$$
for $k=n+1,\dots,2n$, and take the factors $-(e_3-e_1)^{-1}$ and $(e_3-e_2)^{-1}$ out of the first and second rows. Then we obtain
$$
\begin{equation*}
\Delta_{e_1,\dots,e_{2n}}^{h_1,\dots,h_{2n}}=\frac{(-1)^{1+2}}{V_{e_1,e_2,e_3}}\det\begin{pmatrix} h_{n+1,n,\dots,5,4,1} & \dots & h_{2n,n,\dots,5,4,1}\\ h_{n+1,n,\dots,5,4,2} & \dots & h_{2n,n,\dots,5,4,2}\\ h_{n+1,n,\dots,5,4,3} & \dots & h_{2n,n,\dots,5,4,3}\\ h_{n+1,n,\dots,5,4} & \dots & h_{2n,n,\dots,5,4}\\ \dots & \dots & \dots \\ h_{n+1,n} & \dots & h_{2n,n}\end{pmatrix}\frac{V_{e_1,\dots,e_{2n}}}{V_{e_{n+1},\dots,e_{2n}}}.
\end{equation*}
\notag
$$
Taking (74) into account we continue this equality as
$$
\begin{equation*}
\Delta_{e_1,\dots,e_{2n}}^{h_1,\dots,h_{2n}}=\frac{(-1)^{1+2+\dots +(n-1)}}{V_{e_1,\dots,e_n}}\det\begin{pmatrix} h_{n+1,1} & \dots & h_{2n,1}\\ h_{n+1,2} & \dots & h_{2n,2}\\ \dots & \dots & \dots \\ h_{n+1,n} & \dots & h_{2n,n}\end{pmatrix}\frac{V_{e_1,\dots,e_{2n}}}{V_{e_{n+1},\dots,e_{2n}}}.
\end{equation*}
\notag
$$
Hence, taking the equalities $h_{n+k,j}=(h_{n+k}-h_j)/(e_{n+k}-e_j)$, $k,j=1,\dots,n$ (see (75)), and formula (49) into account we obtain the required relation (71). Lemma 6 is proved. Proof of Lemma 2. Under the assumptions of Lemma 2 set $e_{n+k} = \overline{e}_k$ and ${h_{n+k}=\overline{h}_k}$, $k=1,\dots,n$. Since the points $e_1,\dots,e_{n}$ are pairwise distinct and lie in $\mathbb H$, the points $e_1,\dots,e_{2n}$ are pairwise distinct. Hence by Lemma 6 we have equality (71), which, taking the definitions (22), (72) and (35) of $M_{E_n}^{H_n}$, $\Delta_{\{e_{1},\dots,e_{n}, \overline{e}_{1},\dots,\overline{e}_{n}\}}^{\{h_{1}, \dots,h_{n},\overline{h}_1,\dots,\overline{h}_n\}}$ and $C_{E_n}$ into account, yields the equality
$$
\begin{equation*}
\begin{aligned} \, M_{E_n}^{\mathrm n;H_n}&=i^n\Delta_{\{e_{1},\dots,e_{n}, \overline{e}_{1},\dots,\overline{e}_{n}\}}^{\{h_{1},\dots,h_{n},\overline{h}_1,\dots,\overline{h}_n\} } =i^{n+(n-1)n}\det \biggl( \frac{\overline{h}_{k}-h_{j}}{\overline{e}_{k}-e_j} \biggr)_{j,k=1,\dots,n}\prod_{j,k=1}^n(\overline{e}_{k}-e_j) \\ &=C_{E_n}\det \biggl( \frac{h_{j}-\overline{h}_{k}}{e_j-\overline{e}_{k}} \biggr)_{j,k=1,\dots,n}, \end{aligned}
\end{equation*}
\notag
$$
which coincides with (36). Lemma 2 is proved. Proof of Claim 2. Let $\{e_p\}_{p\in\mathcal P}$ and $\{h_p\}_{p\in\mathcal P}$ be fixed sets of points in $\mathbb H$ and $\mathbb C$, respectively. It follows from the supplement to Theorem 1 for $\zeta =\mathrm n$ and Lemma 2 that for all pairwise distinct $p_1,\dots,p_n$ in the index set $\mathcal P$, setting $E_n\!=\!\{e_{p_1},\dots,e_{p_n}\}$ and $H_n=\{h_{p_1},\dots,h_{p_n}\}$ we have the implications
$$
\begin{equation*}
\begin{aligned} \, &f(z)\in\mathfrak B^{\mathrm n },\quad f[E_n]=H_n \\ &\quad\Longrightarrow\quad M_{E_n}^{\mathrm n;H_n}\geqslant 0 \quad\Longrightarrow\quad \det \biggl( \frac{h_{j}-\overline{h}_{k}}{e_j-\overline{e}_{k}} \biggr)_{j,k=1,\dots,n}\geqslant 0, \end{aligned}
\end{equation*}
\notag
$$
which show that all the forms in (10) must be nonnegative in order that a Nevanlinna function $f(z)$ satisfying $f(e_p)=h_p$, $p\in \mathcal P$, exist.
Now we show that, for a countable index set $\mathcal P$, that is, for $\{e_p\}_{p\in \mathcal P}\!=\!\{e_1,e_2,\dots\}$, if all forms in (10) are nonnegative, then there exists a Nevanlinna function $f(z)$ such that $f(e_p)=h_p$, $p=1,2,\dots$ . First assume that all forms in (10) are positive. In particular, then for $n=1,2,\dots$ we have the strict inequalities
$$
\begin{equation*}
\det\biggl( \frac{h_{j}-\overline{h}_{k}}{e_{j}-\overline{e}_{k}} \biggr)_{j,k=1,\dots,n}>0, \qquad n=1,2,\dots,
\end{equation*}
\notag
$$
which are equivalent by Lemma 2 to the strict inequalities
$$
\begin{equation*}
M_{E_n}^{\mathrm n;H_n} >0, \quad \text{where }\ E_n=\{e_1,\dots,e_n\}\ \text{ and }\ H_n=\{h_1,\dots,h_n\}, \quad n=1,2,\dots\,.
\end{equation*}
\notag
$$
The latter, taking (26) into account, imply by Theorem 1 for $\zeta =\mathrm n$ and $N=\infty$ that there exists a function $f(z)\in \mathfrak B_\infty^{\mathrm n}$ such that $f(e_n)=h_n$, $n=1,2,\dots$ .
Now consider the case when some nonnegative forms in (10) are degenerate. Then there exists $N\in\mathbb Z_+$ such that all forms of order at most $N$ in (10) are positive (this condition is void for $N=0$) and all forms of orders $N+1$ and $N+2$ in (10) are nonnegative and at least one form of order $N+1$ in (10) is degenerate. Hence, taking Lemma 2 into account, after rearranging the index set $\mathcal P$ appropriately we have
$$
\begin{equation}
\begin{gathered} \, M_{E_n}^{\mathrm n;H_n}>0,\quad n=1,\dots,N,\qquad M_{E_{N+1}}^{\mathrm n;H_{N+1}}=0, \\ M_{\{E_{N+1} \cup e_{N+p}\} }^{\mathrm n;\{H_{N+1}\cup h_{N+p}\} }\geqslant 0, \qquad p=2,3,\dots \end{gathered}
\end{equation}
\tag{76}
$$
(there are no inequalities of the form $M_{E_n}^{\mathrm n;H_n}>0$ for $N=0$).
We claim that (76) yields the stronger conditions
$$
\begin{equation}
\begin{gathered} \, M_{E_{n}}^{\mathrm n;H_{n}}>0,\qquad n=1,\dots,N, \\ M_{E_{N+1}}^{\mathrm n;H_{N+1}}=M_{E_{N+1} \cup \{e_{N+p}\} }^{\mathrm n;H_{N+1} \cup \{h_{N+p}\} }= 0,\qquad p=2,3,\dots\,. \end{gathered}
\end{equation}
\tag{77}
$$
Since the points $e_1,\dots,e_n$ are pairwise distinct and $h_k\neq -i$, $k=1,2,\dots$ (because the forms in (10) for $n=1$ are nonnegative), by Corollary 2 it follows from the first $N+1$ conditions in (76) that
$$
\begin{equation}
M_{D_n}^{\mathrm s;G_n}>0 \quad \text{for } n=1,\dots,N \quad\text{and} \quad M_{E_{N+1}}^{\mathrm s;G_{N+1}}=0,
\end{equation}
\tag{78}
$$
where
$$
\begin{equation*}
D_n=t(E_n),\quad G_n=T(i^{-1}H_n),\quad t(z)=\frac{z-i}{z+i}\quad\text{and}\quad T(z)=\frac{1-z}{1+z}.
\end{equation*}
\notag
$$
It was shown in [16], Lemma 4 (without stating this explicitly) that conditions (78) imply the inequalities
$$
\begin{equation*}
M_{D_{N+1}\cup \{d\}}^{\mathrm s;G_{N+1}\cup \{g\}}\leqslant 0 \quad\text{for all }\ d\in\mathbb D\setminus D_{N+1}\ \text{ and }\ g\in\mathbb C.
\end{equation*}
\notag
$$
By Corollary 2 this means that $M_{E_{N+1} \cup \{e\} }^{\mathrm n;H_{N+1}\cup \{h\} }\leqslant 0$ for all $e\in\mathbb H\setminus E_{N+1}$ and $h\in\mathbb C$. In particular,
$$
\begin{equation*}
M_{E_{N+1} \cup \{e_{N+p}\} }^{\mathrm n; H_{N+1}\cup \{h_{N+p}\} }\leqslant 0 \quad\text{for all } p=2,3,\dots\,.
\end{equation*}
\notag
$$
Hence conditions (76) and (77) are equivalent.
The points $e_1,e_2,\dots $ being pairwise distinct means, in particular, that $\nu_n={1<N+2}$ for $n=1,2,\dots$ . Hence by (27), as the $(N+2)$-point sets $E_{n,N}$ and $H_{n,N}$ for $n=N+3,N+4,\dots$ we can take
$$
\begin{equation*}
E_{N+p,N}=E_{N+1} \cup \{e_{N+p}\}\quad\text{and} \quad H_{N+p,N}=H_{N+1} \cup \{h_{N+p}\}, \qquad p=3,4,\dots\,.
\end{equation*}
\notag
$$
Hence, taking (26) into account, conditions (77) coincide with conditions (31) for $\zeta =\mathrm n$. By Theorem 1 for $\zeta =\mathrm n$, they ensure the existence and uniqueness of a Nevanlinna function $f(z)\in \mathfrak B_N^{\mathrm n}$ satisfying $f(e_n)=h_n$, $n=1,2,\dots$ . Thus we have proved Claim 2. Proof of Lemma 3. For the convenience of notation below, for $\varepsilon >0$ set
For arbitrary complex numbers $z_1,\dots,z_p$ we introduce the notation
$$
\begin{equation*}
S^0_{z_{1},\dots,z_{p}}:= 1\quad\text{and} \quad S^k_{z_{1},\dots,z_{p}}:=\sum z_1^{t_1}\dotsb z_p^{t_p}, \qquad p,k=1,2,\dots,
\end{equation*}
\notag
$$
where the sum is taken over all $(t_1,\dots,t_p)\in\mathbb Z_+^p$ such that $t_1+\dots +t_p=k$.
Note that $S^k_{\varepsilon_1,\dots,\varepsilon_p}$ is independent of rearrangements of $\varepsilon_1,\dots,\varepsilon_p$. For $m=0,1$ and $j=p+2,\dots,2n$, where $p=1,\dots,2n-2$, we have
$$
\begin{equation}
\begin{aligned} \, \notag &\varepsilon_j^mS^k_{\varepsilon_1,\dots,\varepsilon_p,\varepsilon_j} -\varepsilon_{p+1}^mS^k_{\varepsilon_1,\dots,\varepsilon_p,\varepsilon_{p+1}} =\sum_{l=0}^kS^{k-l}_{\varepsilon_1,\dots,\varepsilon_p}(\varepsilon_j^{l+m} -\varepsilon_{p+1}^{l+m}) \\ &\qquad=(\varepsilon_j-\varepsilon_{p+1})\sum_{l=1-m}^kS^{k-l}_{\varepsilon_1,\dots, \varepsilon_p}S^{l+m-1}_{\varepsilon_{p+1},\varepsilon_j} =(\varepsilon_j-\varepsilon_{p+1})S^{k+m-1}_{\varepsilon_1,\dots,\varepsilon_p\varepsilon_{p+1},\varepsilon_j}. \end{aligned}
\end{equation}
\tag{80}
$$
Assuming first that the points $e_1,\dots,e_n$ are pairwise distinct, we observe that then (using the notation from Lemma 3)
$$
\begin{equation*}
f_{2n-2}[\varepsilon E_n]=\biggl\{\sum_{k=0}^{2n-2}a_{k}\varepsilon_1^{k+1},\dots,\sum_{k=0}^{2n-2}a_{k}\varepsilon_n^{k+1}\biggr\}.
\end{equation*}
\notag
$$
Bearing in mind the definition (24) of the matrices $A_{\varepsilon E_{n}}$ and $A_{\varepsilon E_{n}}^{f_{2n-2}[\varepsilon E_n]}$, the fact that $a_0,a_1,\dots$ are real and the notation (79) we obtain the chain of equalities
$$
\begin{equation}
\begin{aligned} \, &\det \begin{pmatrix} A_{\varepsilon E_{n}} & \overline{A}_{\varepsilon E_{n}} \\ A_{\varepsilon E_{n}}^{f_{2n-2}[\varepsilon E_n]} & \overline{A}_{\varepsilon E_{n}}^{f_{2n-2}[\varepsilon E_n]} \end{pmatrix} \nonumber \\ &=\det\begin{pmatrix} \varepsilon_1^0 & \dots & \varepsilon_{2n}^0 \\ \dots & \dots & \dots \\ \varepsilon_1^{n-1} & \dots & \varepsilon_{2n}^{n-1}\\ \varepsilon_1^{0} \sum_{k=0}^{2n-2}a_{k}\varepsilon_1^{k+1} & \dots & \varepsilon_{2n}^{0} \sum_{k=0}^{2n-2}a_{k}\varepsilon_{2n}^{k+1} \\ \dots & \dots & \dots \\ \varepsilon_1^{n-1} \sum_{k=0}^{2n-2}a_{k}\varepsilon_1^{k+1} & \dots & \varepsilon_{2n}^{n-1}\sum_{k=0}^{2n-2}a_{k}\varepsilon_{2n}^{k+1} \end{pmatrix} \nonumber \\ &= \det \begin{pmatrix} \varepsilon_1^0 & \dots & \varepsilon_{2n}^0\\ \dots & \dots & \dots \\ \varepsilon_1^{n-1} & \dots & \varepsilon_{2n}^{n-1}\\ \varepsilon_1^{n}\sum_{k=0}^{n-1}a_{k+n-1}\varepsilon_{1}^{k} & \dots & \varepsilon_{2n}^{n}\sum_{k=0}^{n-1}a_{k+n-1}\varepsilon_{2n}^{k}\\ \dots & \dots & \dots \\ \varepsilon_1^{n}\sum_{k=0}^{2n-2}a_{k}\varepsilon_{1}^{k} & \dots & \varepsilon_{2n}^{n}\sum_{k=0}^{2n-2}a_{k}\varepsilon_{2n}^{k} \end{pmatrix} \nonumber \\ &=\det \begin{pmatrix} \varepsilon_2^0(\varepsilon_2-\varepsilon_1) & \dots & \varepsilon_{2n}^0(\varepsilon_{2n}-\varepsilon_1)\\ \dots & \dots & \dots \\ \varepsilon_2^{n-2}(\varepsilon_2-\varepsilon_1) & \dots & \varepsilon_{2n}^{n-2}(\varepsilon_{2n}-\varepsilon_1) \\ \varepsilon_2^{n-1}\sum_{k=0}^{n-1}a_{k+n-1} (\varepsilon_2^{k+1}-\varepsilon_1^{k+1}) & \dots & \varepsilon_{2n}^{n-1}\sum_{k=0}^{n-1}a_{k+n-1} (\varepsilon_{2n}^{k+1}-\varepsilon_1^{k+1}) \\ \dots & \dots & \dots \\ \varepsilon_2^{n-1}\sum_{k=0}^{2n-2}a_{k} (\varepsilon_2^{k+1}-\varepsilon_1^{k+1}) & \dots & \varepsilon_{2n}^{n-1}\sum_{k=0}^{2n-2}a_{k} (\varepsilon_{2n}^{k+1}-\varepsilon_1^{k+1}) \end{pmatrix} \nonumber \\ &=\frac{V_{\varepsilon_1,\dots,\varepsilon_{2n}}}{V_{\varepsilon_{2},\dots,\varepsilon_{2n}}}\det \begin{pmatrix} \varepsilon_2^0 & \dots & \varepsilon_{2n}^0\\ \dots & \dots & \dots \\ \varepsilon_2^{n-2} & \dots & \varepsilon_{2n}^{n-2}\\ \varepsilon_2^{n-1}\sum_{k=0}^{n-1}a_{k+n-1}S^k_{\varepsilon_1,\varepsilon_2} & \dots & \varepsilon_{2n}^{n-1}\sum_{k=0}^{n-1}a_{k+n-1}S^k_{\varepsilon_1,\varepsilon_{2n}} \\ \dots & \dots & \dots \\ \varepsilon_2^{n-1}\sum_{k=0}^{2n-2}a_{k}S^{k}_{\varepsilon_1,\varepsilon_2} & \dots & \varepsilon_{2n}^{n-1}\sum_{k=0}^{2n-2}a_{k}S^{k}_{\varepsilon_1,\varepsilon_{2n}} \end{pmatrix} \nonumber \\ &=\frac{V_{\varepsilon_1,\dots,\varepsilon_{2n}}}{V_{\varepsilon_{3},\dots,\varepsilon_{2n}}}\det \begin{pmatrix} \varepsilon_3^0 & \dots & \varepsilon_{2n}^0\\ \dots & \dots & \dots \\ \varepsilon_3^{n-3} & \dots & \varepsilon_{2n}^{n-3}\\ \varepsilon_3^{n-2}\sum_{k=0}^{n-1}a_{k+n-1}S^k_{\varepsilon_1,\varepsilon_2,\varepsilon_3} & \dots & \varepsilon_{2n}^{n-2}\sum_{k=0}^{n-1}a_{k+n-1}S^k_{\varepsilon_1,\varepsilon_2,\varepsilon_{2n}} \\ \dots & \dots & \dots \\ \varepsilon_3^{n-2}\sum_{k=0}^{2n-2}a_{k}S^{k}_{\varepsilon_1,\varepsilon_2,\varepsilon_3} & \dots & \varepsilon_{2n}^{n-2}\sum_{k=0}^{2n-2}a_{k}S^{k}_{\varepsilon_1,\varepsilon_2,\varepsilon_{2n}} \end{pmatrix} \nonumber \\ \nonumber &=\frac{V_{\varepsilon_1,\dots,\varepsilon_{2n}}}{V_{\varepsilon_{n+1},\dots,\varepsilon_{2n}}}\det \begin{pmatrix} \sum_{k=0}^{n-1}a_{k+n-1}S^k_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{n+1}} & \dots & \sum_{k=0}^{n-1}a_{k+n-1}S^k_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{2n}} \\ \dots & \dots & \dots \\ \sum_{k=0}^{2n-2}a_{k}S^k_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{n+1}} & \dots & \sum_{k=0}^{2n-2}a_{k}S^k_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{2n}} \end{pmatrix} \\ &=\frac{V_{\varepsilon_1,\dots,\varepsilon_{2n}}}{V_{\varepsilon_{n+1},\dots,\varepsilon_{2n}}}\det ( {\mathscr A}_n\times {\mathscr L}_{\varepsilon,n}), \end{aligned}
\end{equation}
\tag{81}
$$
where the $ n\times (2n-1) $ matrix ${\mathscr A}_n$ and the $ (2n-1)\times n $ matrix ${\mathscr L}_{\varepsilon,n}$ are defined by
$$
\begin{equation*}
{\mathscr A}_n:=\begin{pmatrix} a_{n-1} & \dots & a_{2n-2} & 0 & \dots & 0\\ a_{n-2} & \dots & a_{2n-3} & a_{2n-2} & \dots & 0\\ \dots & \dots & \dots & \dots & \dots & \dots \\ a_{0} & \dots & a_{n-1} & a_{n} & \dots & a_{2n-2}\end{pmatrix}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
{\mathscr L}_{\varepsilon,n}:=\begin{pmatrix} S^0_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{n+1}} & \dots & S^0_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{2n}}\\ \dots & \dots & \dots \\ S_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{n+1}}^{2n-2} & \dots & S_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{2n}}^{2n-2} \end{pmatrix},
\end{equation*}
\notag
$$
respectively. Some comments are here in order. In the second equality, from the $(n+j)$th row in the determinant, where $j=1,\dots,n-1$, we subtracted linear combinations of the first $n$ rows with appropriate coefficients and then re-labelled the summation indices. In the third equality, from the $(n+j)$th row, $j=1,\dots,n$, we subtracted the $n$th row times $\varepsilon_1\sum_{k=0}^{n-2+j}a_{k+n-j}\varepsilon_{1}^{k} $, then for $j=n,\dots,2$, from the $j$th row we subtracted the $(j-1)$st row times $\varepsilon_1$, and then we expanded the resulting determinant with respect to the first column (all entries in which, apart from the first, are zeros). In the fourth equality, taking the relation $\varepsilon_j^{k+1}- \varepsilon_1^{k+1} =(\varepsilon_j-\varepsilon_1)S_{\varepsilon_1,\varepsilon_j}^{k}$ into account, we took the factor $(\varepsilon_j-\varepsilon_1)$ out of the $(j-1)$st column for each $j=2,\dots,2n$, and then used (49). In the fifth equality, from the $({n-1+j})$th row in the determinant, $j=1,\dots,n$, we subtracted the $(n-1)$st row times $\varepsilon_{2}\sum_{k=0}^{n-2+j}a_{k+n-j}S_{\varepsilon_1,\varepsilon_2}^{k} $, from the $j$th row, $j=n-1,\dots,2$, we subtracted the $(j-1)$st times $\varepsilon_2$, expanded the resulting determinant with respect to the first column and, taking the equality $\varepsilon_jS_{\varepsilon_1,\varepsilon_j}^{k}-\varepsilon_2S_{\varepsilon_1,\varepsilon_2}^{k}=(\varepsilon_j-\varepsilon_2)S_{\varepsilon_1,\varepsilon_2,\varepsilon_j}^k$ into account (see (80) for $m=1$, $p=1$ and $j=3,\dots,2n$), moved out the factors $(\varepsilon_j-\varepsilon_2)$, $j=3,\dots,2n$. Next, repeating the above arguments with the help of relation (80) for $m=1$, $p=2,\dots,n-1$ and $j=p+2,\dots,2n$, we continued the chain of equalities.
Using elementary transformations of columns (standard for the calculation of Vandermonde determinants) we transform ${\mathscr L}_{\varepsilon,n}$ into a matrix $\widetilde{{\mathscr L}}_{\varepsilon,n}$ as follows. For $j=2,\dots,n$, from the $j$th column of the matrix ${\mathscr L}_{\varepsilon,n}$ we subtract the first column and take the equalities $S^0_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{n+j}}-S^0_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{n+1}}=0$ and
$$
\begin{equation*}
S^k_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{n+j}}-S^k_{\varepsilon_1,\dots,\varepsilon_n,\varepsilon_{n+1}}=(\varepsilon_{n+j}-\varepsilon_{n+1})S_{\varepsilon_1,\dots,\varepsilon_{n+1},\varepsilon_{n+j}}^{k-1}, \qquad k=1,2,\dots,
\end{equation*}
\notag
$$
into account (see (80) for $m=0$, $p=n$ and $j=n+2,\dots,2n$). Then for $j=3,\dots,n$, from the $j$th column of the transformed matrix we subtract the second column times $(\varepsilon_{n+j}-\varepsilon_{n+1})/(\varepsilon_{n+2}-\varepsilon_{n+1})$ and take (80) for $m=0$, $p=n+1$ and $j=n+3,\dots,2n$ into account. Proceeding in a similar way we end up with the matrix
$$
\begin{equation*}
\widetilde{{\mathscr L}}_{\varepsilon,n}=\begin{pmatrix} S^{0}_{\varepsilon_1,\dots,\varepsilon_{n+1}} & 0 & 0 & \dots & 0\\ S^{1}_{\varepsilon_1,\dots,\varepsilon_{n+1}} & {\mathscr V}_{n,2}S^{0}_{\varepsilon_1,\dots,\varepsilon_{n+2}} & 0 & \dots & 0\\ \dots & \dots & \dots & \dots & \dots \\ S^{n-1}_{\varepsilon_1,\dots,\varepsilon_{n+1}} & {\mathscr V}_{n,2}S^{n-2}_{\varepsilon_1,\dots,\varepsilon_{n+2}} & {\mathscr V}_{n,3}S^{n-3}_{\varepsilon_1,\dots,\varepsilon_{n+3}} & \dots & {\mathscr V}_{n,n}S^{0}_{\varepsilon_1,\dots,\varepsilon_{2n}}\\ \dots & \dots & \dots & \dots & \dots \\ S^{2n-2}_{\varepsilon_1,\dots,\varepsilon_{n+1}} & {\mathscr V}_{n,2}S^{2n-3}_{\varepsilon_1,\dots,\varepsilon_{n+2}} & {\mathscr V}_{n,3}S^{2n-4}_{\varepsilon_1,\dots,\varepsilon_{n+3}} & \dots & {\mathscr V}_{n,n}S^{n-1}_{\varepsilon_1,\dots,\varepsilon_{2n}}\end{pmatrix},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
{\mathscr V}_{n,j}:=\prod_{k=1}^{j-1}(\varepsilon_{n+j}-\varepsilon_{n+k}), \qquad j=2,\dots,n.
\end{equation*}
\notag
$$
Since $S^{0}_{\varepsilon_1,\dots,\varepsilon_{n+j}}=1$ and
$$
\begin{equation}
S^{k}_{\varepsilon_1,\dots,\varepsilon_{n+j}}=\varepsilon^kS^{k}_{e_1,\dots,e_{n+j}}=O(\varepsilon^k) \quad \text{as } \varepsilon\to 0 \quad\text{for } j=1,\dots,n, \quad k=1,2,\dots,
\end{equation}
\tag{82}
$$
it follows that
$$
\begin{equation*}
\widetilde{{\mathscr L}}_{\varepsilon,n}=\begin{pmatrix} 1 & 0 & 0 & \dots & 0\\ O(\varepsilon ) & {\mathscr V}_{n,2} & 0 & \dots & 0\\ \dots & \dots & \dots & \dots & \dots \\ O(\varepsilon^{n-1}) & {\mathscr V}_{n,2}O(\varepsilon^{n-2}) & {\mathscr V}_{n,3}O(\varepsilon^{n-3}) & \dots & {\mathscr V}_{n,n}\\ \dots & \dots & \dots & \dots & \dots \\ O(\varepsilon^{2n-2}) & {\mathscr V}_{n,2}O(\varepsilon^{2n-3}) & {\mathscr V}_{n,3}O(\varepsilon^{2n-4}) & \dots & {\mathscr V}_{n,n}O(\varepsilon^{n-1}) \end{pmatrix}.
\end{equation*}
\notag
$$
Bearing in mind that the determinant of ${\mathscr A}_n\times {\mathscr L}_{\varepsilon,n}$ does not change under the elementary transformations of ${\mathscr L}_{\varepsilon,n}$ mentioned above and $\prod_{j=2}^n{\mathscr V}_{n,j}= V_{\varepsilon_{n+1},\dots,\varepsilon_{2n}}$ by (49), from the explicit form of ${\mathscr A}_n$ and $\widetilde{{\mathscr L}}_{\varepsilon,n}$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\det ( {\mathscr A}_n\times {\mathscr L}_{\varepsilon,n})=\det ( {\mathscr A}_n\times \widetilde{{\mathscr L}}_{\varepsilon,n}) =\bigl(\det (a_{n-1+j-k})_{j,k=0,\dots,n-1} +O(\varepsilon ) \bigr)\prod_{j=2}^n{\mathscr V}_{n,j} \\ &\qquad =\bigl((-1)^{(n-1)n/2}\det (a_{j+k})_{j,k=0,\dots,n-1} +O(\varepsilon )\bigr) V_{\varepsilon_{n+1},\dots,\varepsilon_{2n}} . \end{aligned}
\end{equation}
\tag{83}
$$
As below, we consider the general case when $E_n$ contains multiple points, noting that, as we can see from the first occurrence of $O(\varepsilon)$ in (82) and the calculations that follows, the quantity $O(\varepsilon)$ on the right-hand side of (83) has modulus at most $C_1\varepsilon$, where the positive constant $C_1=C_1(e_1,\dots,e_n;a_0,\dots,a_{2n-2})$ depends on the points $e_1,\dots,e_n$ and the coefficients $a_0,\dots,a_{2n-2}$ and satisfies
$$
\begin{equation}
C_2=\sup_{\{\delta_1\leqslant |e_1|/2,\dots,\delta_n\leqslant |e_n|/2\} }C_1(e_{1}+\delta_1,\dots,e_{n}+\delta_n;a_0,\dots,a_{2n-2})<\infty.
\end{equation}
\tag{84}
$$
Taking account of the definition (22) of $M_{\varepsilon E_n}^{\mathrm n;f_{2n-2}[\varepsilon E_{n}]}$ (for the left-hand side of (81)) and equality (83) (for the right-hand side of (81) we can write (81) as
$$
\begin{equation}
i^{-n}M_{\varepsilon E_n}^{\mathrm n;f_{2n-2}[\varepsilon E_{n} ]}=i^{(n-1)n}V_{\varepsilon_1,\dots,\varepsilon_{2n}}\bigl(\det (a_{j+k})_{j,k=0,\dots,n-1} +O(\varepsilon)\bigr).
\end{equation}
\tag{85}
$$
It follows from the notation (79), formula (49) and the definition (35) of $C_{E_n}$ that
$$
\begin{equation*}
V_{\varepsilon_1,\dots,\varepsilon_{2n}}=\varepsilon^{n(2n-1)}V_{e_1,\dots,e_n,\overline{e}_1,\dots,\overline{e}_n} =\varepsilon^{n(2n-1)}i^{-n^2}C_{E_n}|V_{E_n}|^2.
\end{equation*}
\notag
$$
Hence (85) coincides with the equality
$$
\begin{equation}
M_{\varepsilon E_n}^{\mathrm n;f_{2n-2}[\varepsilon E_{n} ]}=\varepsilon^{n(2n-1)}C_{E_n}|V_{E_n}|^2\bigl(\det (a_{j+k})_{j,k=0,\dots,n-1}+O(\varepsilon)\bigr),
\end{equation}
\tag{86}
$$
which is equivalent to (38).
In the general case, when $E_n$ contains multiple points, consider the sets
$$
\begin{equation*}
E_{n}^{\delta_1,\dots,\delta_n}:=\{e_1+\delta_1,\dots,e_n+\delta_n \}\quad\text{and}\quad \varepsilon E_{n}^{\delta_1,\dots,\delta_n}=\{\varepsilon (e_1+\delta_1),\dots,\varepsilon (e_1+\delta_1)\},
\end{equation*}
\notag
$$
where $\delta_1,\dots,\delta_n$ are arbitrarily small and such that the sets $E_n^{\delta_1,\dots,\delta_n}$ contain pairwise distinct points in $\mathbb H$. Taking (84) into account we can write equality (86) for the sets $E_{n}^{\delta_1,\dots,\delta_n}$ as the inequality
$$
\begin{equation}
\biggl|\frac{M_{\varepsilon E_{n}^{\delta_1,\dots,\delta_n}}^{\mathrm n,f_{2n-2}[\varepsilon E_{n}^{\delta_1,\dots,\delta_n}]}}{\varepsilon^{n(2n-1)}C_{E_{n}^{\delta_1,\dots,\delta_n}}|V_{E_{n}^{\delta_1,\dots,\delta_n}}|^2}-\det (a_{j+k})_{j,k=0,\dots,n-1} \biggr|\leqslant C_2\varepsilon,
\end{equation}
\tag{87}
$$
where for sufficiently small $\delta_1,\dots,\delta_n$ the finite constant $C_2$ is independent of $\delta_1,\dots,\delta_n$.
From the equality ${\det A_{\varepsilon E_{n}^{\delta_1,\dots,\delta_n}}}|^2=\varepsilon^{(n-1)n}|V_{E_{n}^{\delta_1,\dots,\delta_n}}|^2$ (see (24)) and the definition (39) we obtain
$$
\begin{equation*}
\frac{M_{\varepsilon E_{n}^{\delta_1,\dots,\delta_n}}^{\mathrm n,f_{2n-2}[\varepsilon E_{n}^{\delta_1,\dots,\delta_n}]}}{\varepsilon^{(n-1)n}|V_{E_{n}^{\delta_1,\dots,\delta_n}}|^2} =\frac{M_{\varepsilon E_{n}^{\delta_1,\dots,\delta_n}}^{\mathrm n,f_{2n-2}[\varepsilon E_{n}^{\delta_1,\dots,\delta_n}]}}{|{\det A_{\varepsilon E_{n}^{\delta_1,\dots,\delta_n}}}|^2}= \breve {M}_{\varepsilon E_{n}^{\delta_1,\dots,\delta_n}}^{\mathrm n,f_{2n-2}[\varepsilon E_{n}^{\delta_1,\dots,\delta_n}]}.
\end{equation*}
\notag
$$
Hence by Lemma 4, taking (39) into account we have
$$
\begin{equation}
\lim_{\delta_1\to 0}\dots \lim_{\delta_n\to 0}\frac{M_{\varepsilon E_{n}^{\delta_1,\dots,\delta_n}}^{\mathrm n,f_{2n-2}[\varepsilon E_{n}^{\delta_1,\dots,\delta_n}]}}{\varepsilon^{(n-1)n}|V_{E_{n}^{\delta_1,\dots,\delta_n}}|^2}=\breve {M}_{\varepsilon E_{n}}^{\mathrm n,f_{2n-2}[\varepsilon E_{n}]}=\frac{M_{\varepsilon E_{n}}^{\mathrm n,f_{2n-2}[\varepsilon E_{n}]}}{|{\det A_{\varepsilon E_{n}}}|^2} .
\end{equation}
\tag{88}
$$
Furthermore, it follows directly from (35) that
$$
\begin{equation}
\lim_{\delta_1\to 0}\dots \lim_{\delta_n\to 0}C_{E_{n}^{\delta_1,\dots,\delta_n}}=C_{E_n}.
\end{equation}
\tag{89}
$$
Taking the limit as $\delta_k\to 0$, $k=1,\dots,n$, in (87) and bearing in mind (88) and (89) we obtain
$$
\begin{equation*}
\biggl|\frac{M_{\varepsilon E_{n}}^{\mathrm n,f_{2n-2}[\varepsilon E_{n}]}}{\varepsilon^{n^2}C_{E_{n}}|{\det A_{\varepsilon E_{n}}}|^2}-\det (a_{j+k})_{j,k=0,\dots,n-1} \biggr|\leqslant C_2\varepsilon,
\end{equation*}
\notag
$$
which yields (37) in the general case. Now Theorem 3 is proved. The author is grateful to the referee, who read the manuscript and made valuable comments. 
Citation in format AMSBIB
\Bibitem{Bus23} |
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