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This article is cited in 2 scientific papers (total in 2 papers) Necessary and sufficient conditions for extending a function to a Carathéodory function V. I. Buslaev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 11, DOI: https://doi.org/10.4213/sm9611 
Bibliographic databases:
    § 1. The classical Carathéodory and Schur criteriaRecall that a holomorphic function $\mathscr F (z)$ in the disc $\mathbb D:=\{|z|<1\}$ is called a Carathéodory function if $\operatorname{Re} \mathscr F (z)\geqslant 0$ for $z\in\mathbb D$, and it is called a Schur function if $|\mathscr F (z)|\leqslant 1$ for $z\in\mathbb D$. We denote the sets of Carathéodory and Schur functions by $\mathfrak B ^{\mathtt{C}}$ and $\mathfrak B ^{\mathtt{S}}$, respectively. In the sets $\mathfrak B ^{\mathtt{C}}$ and $\mathfrak B ^{\mathtt{S}}$ of Carathéodory and Schur functions authors usually distinguish the disjoint subsets $\mathfrak B_N^{\mathtt{C}}$ and $\mathfrak B_N^{\mathtt{S}}$, $N\in\mathbb Z_+^\infty :=\{\infty,0,1,2,\dots \}$, where
$$
\begin{equation*}
\begin{aligned} \, \mathfrak B_0^{\mathtt{C}} &:=\bigl\{\mathscr F (z)\in\mathfrak B ^{\mathtt{C}}\colon \mathscr F (z)\equiv \lambda,\, \operatorname{Re} \lambda =0\bigr\}, \\ \mathfrak B_0^{\mathtt{S}} &:=\bigl\{\mathscr F (z)\in \mathfrak B ^{\mathtt{S}}\colon \mathscr F (z)\equiv \gamma,\, |\gamma |=1\bigr\}, \\ \mathfrak B_N^{\mathtt{C}} &:=\biggl\{\mathscr F (z)\in\mathfrak B ^{\mathtt{C}}\colon \mathscr F(z)=\lambda_0+\sum_{k=1}^N\lambda_k\dfrac{t_k-z}{t_k+z},\, \operatorname{Re} \lambda_0=0,\, \lambda_k>0,\, |t_k|=1, \\ &\qquad k=1,\dots,N,\, t_1,\dots,t_N\text{ are pairwise distinct}\biggr\}, \\ \mathfrak B_N^{\mathtt{S}} &:=\biggl\{\mathscr F (z)\in \mathfrak B ^{\mathtt{S}}\colon \mathscr F (z)=\gamma \prod_{k=1}^N\dfrac{z-e_k}{1-z\overline{e}_k},\,|\gamma |=1,\, e_k\in\mathbb D,\, k=1,\dots,N\biggr\} \end{aligned}
\end{equation*}
\notag
$$
(a bar over a symbol denotes complex conjugation), $N=1,2,\dots$,
$$
\begin{equation*}
\mathfrak B_\infty ^{\mathtt{C}}:=\mathfrak B ^{\mathtt{C}}\setminus \biggl(\bigcup_{N\in\mathbb Z_+} \mathfrak B_N^{\mathtt{C}}\biggr)\quad\text{and}\quad \mathfrak B_\infty^{\mathtt{S}}:=\mathfrak B ^{\mathtt{S}}\setminus \biggl(\bigcup_{N\in\mathbb Z_+} \mathfrak B_N^{\mathtt{S}}\biggr).
\end{equation*}
\notag
$$
Let $T(z)$ denote the linear fractional map $T(z)=(1-z)/(1+z)$ taking the unit disc $\mathbb D$ to the right-hand half-plane $\{\operatorname{Re} z>0\}$ and taking this half-plane to the unit disc. It was shown in [1], pp. 229–230, that
$$
\begin{equation}
\mathfrak B_N^{\mathtt{C}} =\bigl\{(T\circ \mathscr F )(z)\colon \mathscr F (z)\in\mathfrak B_N^{\mathtt{S}}\bigr\}, \quad\text{for } N\in\mathbb Z_+^\infty, \quad\text{where } (T\circ \mathscr F ) (z):=T(\mathscr F (z))
\end{equation}
\tag{1}
$$
(the function $\mathscr F (z)\equiv -1$ in $\mathfrak B_0^{\mathtt{S}}$, for which $(T\circ \mathscr F )(z)\equiv \infty$, is an exception here). Set
$$
\begin{equation}
t(\zeta ):= \begin{cases} {\mathtt{S}}&\text{if }\zeta = {\mathtt{C}}, \\ {\mathtt{C}}&\text{if }\zeta = {\mathtt{S}}. \end{cases}
\end{equation}
\tag{2}
$$
Since $(T\circ T)(z)=z$, it follows from (1) (for $\mathscr F (z)\not\equiv -1$) that
$$
\begin{equation}
\mathscr F (z)\in \mathfrak B_N^{\zeta} \quad\Longleftrightarrow\quad (T\circ \mathscr F )(z)\in \mathfrak B_N^{t(\zeta )} \quad\text{for } N\in\mathbb Z_+^\infty\quad\text{and} \quad \zeta =\mathtt{C},\mathtt{S}.
\end{equation}
\tag{3}
$$
Let $f (z)=\sum_{k=0}^\infty a_kz^k$ be a formal power series and $I_n$ be the $n\times n $ identity matrix. Set
$$
\begin{equation}
A_n^{f}: =\begin{pmatrix} a_0 & a_1 & \dots & a_{n-1}\\ 0 & a_0 & \dots & a_{n-2}\\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & a_0 \end{pmatrix}, \qquad \widetilde{A}_n^{f}: =\begin{pmatrix} \overline{a}_0 & 0 & \dots & 0\\ \overline{a}_1 & \overline{a}_0 & \dots & 0\\ \dots & \dots & \dots & \dots \\ \overline{a}_{n-1} & \overline{a}_{n-2} & \dots & \overline{a}_0 \end{pmatrix}
\end{equation}
\tag{4}
$$
and
$$
\begin{equation}
M_n^{\zeta;f} := \begin{cases} \det (A_n^{f}+\widetilde{A}_n^{f})&\text{for }\zeta = {\mathtt{C}}, \\ \det(I_n-A_n^{f}\widetilde{A}_n^{f}) &\text{for }\zeta = {\mathtt{S}}, \end{cases} \qquad n=1,2,\dots\,.
\end{equation}
\tag{5}
$$
Note that the matrices $A_n^{f}$ and $\widetilde{A}_n^{f}$ and the determinants $M_n^{\zeta;f}$ depend only on the first $n$ coefficients $a_0,\dots,a_{n-1}$ of the series $f (z)$. In particular,
$$
\begin{equation*}
\begin{gathered} \, A_n^{f}=A_n^{f_m}, \quad\text{where } m\geqslant n\quad\text{and} \\ f_m(z)=\sum_{k=0}^{m-1} a_kz^k \text{ is the $(m-1)$st partial sum of the series } f(z). \end{gathered}
\end{equation*}
\notag
$$
In the set of formal power series we distinguish the following subsets, where $N\in\mathbb Z_+^\infty$ and $\zeta =\mathtt{C}, \mathtt{S}$:
$$
\begin{equation}
{\mathscr A} ^{\zeta}_N :=\bigl\{f(z)\colon M_p^{\zeta;f}>0, \, p=1,\dots,N,\, M_{N+p}^{\zeta;f}=0,\, p=1,2,\dots \bigr\}
\end{equation}
\tag{6}
$$
(if $N=0$, then there are no inequalities of the form $M_p^{\zeta;f} > 0$ for $p=1,\dots,N$, and if $N=\infty$, then there are no equalities of the form $M_{N+p}^{\zeta;f} = 0$ for $p=1,2,\dots$). If a formal power series $f(z)$ is the Taylor series of a Carathéodory or a Schur function, then we write $f(z)\lessdot \mathfrak B ^{\mathtt{C}}$ or $f(z)\lessdot \mathfrak B ^{\mathtt{S}}$, respectively. The expressions $f(z)\lessdot \mathfrak B_N^{\mathtt{C}}$ and $f(z)\lessdot \mathfrak B_N^{\mathtt{S}}$, $N\in\mathbb Z_+^\infty $, are interpreted similarly. In [2] and [3] necessary and sufficient conditions were found which ensure that a given formal power series is the Taylor series of a Carathéodory function. A similar result for Schur functions was obtained in [1]. For brevity we formulate Carathéodory’s and Schur’s criteria as one statement, in which $\zeta =\mathtt{C}$ corresponds to Carathéodory’s criterion and $\zeta =\mathtt{S}$ to Schur’s criterion. Carathéodory-Schur Criterion. Let $f (z)\!=\!\sum_{k=0}^\infty a_kz^k$ be a formal power series. Then using the notation (4) and (5), In § 2 we establish the following result. Theorem 1. Let $f (z)=\sum_{k=0}^\infty a_kz^k$ be a formal power series such that $a_0\neq -1$. Then using the notation (4) and (5), the following equalities hold: From (7) we obtain
$$
\begin{equation*}
M_n^{t(\zeta );T\circ f}=0\ \Longleftrightarrow\ M_n^{\zeta;f}=0 \quad\text{and} \quad M_n^{t(\zeta );T\circ f}>0\ \Longleftrightarrow\ M_n^{\zeta;f}>0.
\end{equation*}
\notag
$$
Hence for $a_0\neq -1$,
$$
\begin{equation}
f(z)\in{\mathscr A}_N^{\zeta}\quad\Longleftrightarrow\quad(T\circ f)(z)\in{\mathscr A}_N^{t(\zeta )}.
\end{equation}
\tag{8}
$$
Taking (3) and (8) into account Theorem 1 means that, with the exception of the case when $a_0=-1$, which can easily be examined separately, Carathéodory’s and Schur’s criteria are equivalent in the following sense: Schur’s criterion follows directly from Carathéodory’s criterion and Theorem 1 and, conversely, Carathéodory’s criterion is a direct consequence of Schur’s criterion and Theorem 1. In § 3 we state and in § 4 we prove an analogue of Carathéodory’s and Schur’s criteria for functions defined (with multiplicities) at points $e_1,e_2,\dots$ in $\mathbb D$. Its particular case in the situation of functions defined by the values of their derivatives at zero (that is, for $e_1=e_2=\dots =0$) reads as follows. Refinement of the Carathéodory-Schur Criterion for $N\in {\mathbb Z}_+$. Let $f (z)=\sum_{k=0}^\infty a_kz^k$ be a formal power series. Then Note that the definition (9) of the sets $\hat{\mathscr{A}}_N^{\zeta}$ involves no equalities of the form
$$
\begin{equation*}
M_{N+2p+1}^{\zeta;f} = 0, \qquad p=1,2,\dots, \quad N\in\mathbb Z_+,
\end{equation*}
\notag
$$
which are present in the definition (6) of the sets ${\mathscr A} ^{\zeta}_N$. This means that for $N\in\mathbb Z_+$ the hypotheses in the implications
$$
\begin{equation*}
f(z)\lessdot \hat{\mathscr{A}}_N^{\zeta} \quad\Longrightarrow\quad f(z)\in \mathfrak B_N^{\zeta}
\end{equation*}
\notag
$$
form subsets of the hypotheses in the implications
$$
\begin{equation*}
f(z)\lessdot {\mathscr A}_N^{\zeta} \quad\Longrightarrow\quad f(z)\in \mathfrak B_N^{\zeta},
\end{equation*}
\notag
$$
which are parts of Carathéodory’s and Schur’s criteria. Also note that from the standpoint of computations, the determinants $M_{{N+2p}}^{\zeta;f_{N+p+1}}$ in the definition (10) of the sets $\breve{{\mathscr A}}^{\zeta}_N$ are slightly simpler than the determinants $M_{{N+2p}}^{\zeta;f}$ in the definition (9) of the sets $\hat{\mathscr{A}}_N^{\zeta}$, because in the calculation of $M_{{N+2p}}^{\zeta;f_{N+p+1}}$ the coefficients $a_{N+p+1},\dots, a_{N+2p-1}$ involved in the determinants $M_{{N+2p}}^{\zeta;f}$, $p=2,3,\dots$, can be replaced by zeros (or any other numbers as shown below). In [4] we proposed a proof of Schur’s criterion in terms of two-point Hankel determinants of the power series $f(z)=\sum_{k=0}^\infty a_kz^k$ ($a_0\neq 0$) with centre $z=0$ and the associated series
$$
\begin{equation*}
f^* (z):=\bigl(\overline{f(\overline{z^{-1}})}\bigr)^{-1}=\sum_{k=0}^\infty a_k^* z^{-k}\,
\end{equation*}
\notag
$$
with centre $z=\infty$; the latter comes into play because of the well-known relation between convergents with even and odd indices of the Schur continued fraction corresponding to $f(z)$. It turns out that two-point Hankel determinants of order $n$ for $f(z)$ and $f^*(z)$ coincide with the Schur determinants $M_n^{\mathtt{S};f}$ up to a factor of $(-1)^n\overline{a}_0^{\,n}$. Using this fact and a two-point analogue of Polya’s theorem (see [5] and [6]) on an upper estimate for the capacity of the singularities of a meromorphic function, in [7] we investigated the convergence and boundary behaviour of a limit periodic Schur continued fraction. It is known (for instance, see [8]) that Schur’s classical algorithm described in [1] has a continuous analogue, which enables one to interpolate a function $\mathscr F (z)\in\mathfrak B ^{\mathtt{S}}$ by convergents of a multipoint Schur continued fraction at a prescribed sequence of points in the unit disc, rather than at the single point $z=0$ (with multiplicity). In [9], for $E_n:=\{e_1,\dots,e_n\}\subset \mathbb D$ we discovered quantities $M_{E_n}^{\mathtt{S};f}$ (coinciding with the Schur determinants $M_n^{\mathtt{S};f}$ for $e_1=\dots =e_n=0$) in terms of which we could state a multipoint analogue of Schur’s criterion. The proof of this multipoint analogue of Schur’s criterion stated in [9] is based on a multipoint version of Schur’s algorithm; it was presented in [10]. In this paper we define certain quantities $M_{E_n}^{\mathtt{C};f}$ (coinciding with the Carathéodory determinants $M_n^{\mathtt{C};f}$ for $e_1=\dots =e_n=0$), and we state and prove a multipoint version of Carathéodory’s criterion in terms of these quantities. As in the classical case, the multipoint Carathéodory criterion and the multipoint Schur criterion are equivalent in the following sense: each of them is a direct consequence of the other and relations between the quantities $M_{E_n}^{\mathtt{C};T\circ f}$ and $M_{E_n}^{\mathtt{S};f}$ (see Theorem 2 in § 2, of which Theorem 1 is a special case). Bearing in mind the proof of the multipoint Schur criterion in [10], the proof of the multipoint Carathéodory criterion in § 4 is based on the equivalence of the two criteria that we have revealed. Apart from the proof of Theorem 2, it reduces to verifying the conditions $f(z)\neq -1$ for $z\in E_n$, $n=1,2,\dots$, from the hypotheses of Theorem 2. § 2. Multipoint analogues of Carathéodory and Schur determinantsLet $F(z)$ be a function defined (with multiplicities taken into account) at the points in a set $E_n:=\{e_1,\dots,e_n\}$ so that if $\nu_j$ is the multiplicity of $e_j$ in the set $\{e_1,\dots,e_j\}$, $j=1,\dots,n$, then the $(\nu_j-1)$st derivative $F ^{(\nu_j-1)}(e_{j})$ of $F(z)$ at the point $e_{j}$ is defined. Recall that if $F(z)$ and $G(z)$ are functions defined (with multiplicities) at the points in $E_n$, then the functions $(F\pm G)(z)$, $(FG)(z)$ and $(F/G)(z)$ are also defined there with multiplicities (provided that $G(z)\neq 0$ for $z\in E_n$ in the last case). In particular, if $F(z)\neq -1$ for $z\in E_n$, then the function $(T\circ F)(z)=(1-F(z))/ (1+F(z))$ is also defined with multiplicities. It is easy to see that any function $\mathbf{F}(z)\in H(E_n)$ (that is, any function holomorphic in a neighbourhood of $E_n=\{e_1,\dots,e_n\}$) is defined with multiplicities at the points in any set $E^{j_1,\dots,j_p}:=\{e_{j_1},\dots,e_{j_p}\}$, where $e_{j_q}\in E_n$, $q=1,\dots,p$, $p\in\mathbb N$, and for any function $F(z)$ defined with multiplicities at the points in $E_n$ there exists a function $\mathbf{F}(z)\in H(E_n)$ that is equal to $F(z)$ on $E_n$ with multiplicities taken into account. In particular, there exists a polynomial $\mathbf{F}_n(z)$ of degree at most $n-1$ (the Lagrange interpolation polynomial) such that
$$
\begin{equation*}
\mathbf{F}_n^{(\nu_j-1)}(e_j)=F^{(\nu_j-1)}(e_j), \qquad j=1,\dots,n,
\end{equation*}
\notag
$$
where $\nu_j$ is the multiplicity of $e_j$ in $E_{n}$. For the convenience of references we state the following obvious result. Proposition 1. Let $F(z)$ be a function defined at the points in $E_n$ with multiplicities taken into account and such that $F(z)\neq -1$, $z\in E_n$. Let $\mathbf{F}(z)\in H(E_n)$ interpolate $F(z)$ with multiplicities at the points in $E_n$. Then the function $(T\circ \mathbf{F})(z)\in H(E_n)$ interpolates $(T\circ F)(z)$ with multiplicities at the points in $E_n$. Let
$$
\begin{equation*}
E_n=\{e_1,\dots,e_n\}\quad\text{and} \quad E^{j_1,\dots,j_p}:=\{e_{j_1},\dots,e_{j_p}\} \quad\text{for } 1\leqslant j_1<\dots <j_p\leqslant n.
\end{equation*}
\notag
$$
By the restriction of a function $F(z)$, defined at the points in $E_n$ with multiplicities taken into account, to the set $E^{j_1,\dots,j_p}$ we mean the function defined withe multiplicities at the points in $E^{j_1,\dots,j_p}$ by means of the Lagrange interpolation polynomial $\mathbf{F}_n(z)$. For the Lagrange interpolation polynomial $\mathbf{F}^{j_1,\dots,j_p}(z)$ of this restriction of $F(z)$ to $E^{j_1,\dots,j_p}$ we have
$$
\begin{equation*}
\frac{\mathbf{F}^{j_1,\dots,j_p}(z)-\mathbf{F}_n(z)}{(z- e_{j_1})\dotsb(z-e_{j_p})}\in H(E^{j_1,\dots,j_p}).
\end{equation*}
\notag
$$
Note that we do not need to calculate the Lagrange polynomial $\mathbf{F}_n(z)$ to define the restriction; we can do this in terms of $F(z)$ itself without using this polynomial. We only mention the Lagrange polynomial to avoid explaining some subtleties related to the fact that the points $e_{j_q}$, $q=1,\dots,p$, can have different multiplicities in $E_n$ and $E^{j_1,\dots,j_p}$. The multipoint analogue of the Carathéodory-Schur criterion that we establish here will be stated in terms of certain quantities $M_{E_n}^{\mathtt{C};F}$ and $M_{E_n}^{\mathtt{S};F}$. To define these we introduce some further notation. Notation 1. Assume that an $n$-point set $E_n=\{e_1,\dots,e_n\}$ is rearranged in the form $E_n:=\bigl\{\{e_1\} ^{r_1},\dots,\{e_k\} ^{r_k}\bigr\}$, where $e_1,\dots,e_k$ are pairwise distinct, ${r_1+\dots +r_k}=n$ and $\{e\}^{r}:=\{\underbrace{e,\dots,e}_{r}\}$. Let $F(z)$ be a function defined at the points in $E_n$ with multiplicities taken into account, and let $\varphi_p (z):=z^p$, $p=0,1,\dots$ (we set $z^0:=1$ even for $z=0$). Bearing in mind that each $(\varphi_pF)(z)$ is a function defined with multiplicities at the points in $E_n$, we let $A_{E_n}^F$ and $\widetilde{A}_{E_n}^F$ denote the matrices
$$
\begin{equation}
{ \begin{pmatrix} \dfrac{(\varphi_0F)(e_1)}{0!} &\dots & \dfrac{(\varphi_0F)^{(r_1-1)}(e_1)}{(r_1{-}\,1)!} &\dots & \dfrac{(\varphi_0F)(e_k)}{0!} &\dots & \dfrac{(\varphi_0F)^{(r_k-1)}(e_k)}{(r_k{-}\,1)!} \\ \dots &\dots &\dots &\dots &\dots &\dots &\dots \\ \dfrac{(\varphi_{n-1}F)(e_1)}{0!} &\dots & \dfrac{(\varphi_{n-1}F)^{(r_1-1)}(e_1)}{(r_1{-}\,1)!} &\dots & \dfrac{(\varphi_{n-1}F)(e_k)}{0!} &\dots & \dfrac{(\varphi_{n-1}F)^{(r_k-1)}(e_k)}{(r_k{-}\,1)!} \end{pmatrix}, }
\end{equation}
\tag{11}
$$
and
$$
\begin{equation}
{ \begin{pmatrix} \dfrac{\overline{(\varphi_{n-1}F)^{(r_k-1)}(e_k)}}{(r_k{-}\,1)!} &\dots & \dfrac{\overline{(\varphi_{n-1}F)(e_k)}}{0!} &\dots & \dfrac{\overline{(\varphi_{n-1}F)^{(r_1-1)}(e_1)}}{(r_1{-}\,1)!} &\dots & \dfrac{\overline{(\varphi_{n-1}F)(e_1)}}{0!} \\ \dots &\dots &\dots &\dots &\dots &\dots &\dots \\ \dfrac{\overline{(\varphi_{0}F)^{(r_k-1)}(e_k)}}{(r_k{-}\,1)!} &\dots & \dfrac{\overline{(\varphi_{0}F)(e_k)}}{0!} &\dots & \dfrac{\overline{(\varphi_{0}F)^{(r_1-1)}(e_1)}}{(r_1{-}\,1)!} &\dots & \dfrac{\overline{(\varphi_{0}F)(e_1)}}{0!} \end{pmatrix}, }
\end{equation}
\tag{12}
$$
respectively, and we set
$$
\begin{equation}
W_{E_n}:=\det \begin{pmatrix} A_{E_n}^{\varphi_0} & \widetilde{A}_{E_n}^{\varphi_n} \\ A_{E_n}^{\varphi_n} & \widetilde{A}_{E_n}^{\varphi_0} \end{pmatrix},
\end{equation}
\tag{13}
$$
$$
\begin{equation}
M_{E_n}^{\mathtt{C};F}: =\dfrac{\det \begin{pmatrix} A_{E_n}^{\varphi_0} & \widetilde{A}_{E_n}^{\varphi_0} \\ -A_{E_n}^{F} & \widetilde{A}_{E_n}^{F} \end{pmatrix}}{W_{E_n}} \quad\text{and}\quad M_{E_n}^{\mathtt{S};F}: =\dfrac{\det \begin{pmatrix} A_{E_n}^{\varphi_0} & \widetilde{A}_{E_n}^{F} \\ A_{E_n}^{F} & \widetilde{A}_{E_n}^{\varphi_0} \end{pmatrix}}{W_{E_n}}.
\end{equation}
\tag{14}
$$
Note the following: we can obtain $\widetilde{A}_{E_n}^F$ from $A_{E_n}^F$ by performing complex conjugation and reverting the order of both columns and rows (if $A_{E_n}^F \,{=}\,(a_{k,j})_{k,j=1,\dots,n}$, then $\widetilde{A}_{E_n}^F =(\overline{a}_{n+1-k,n+1-j})_{k,j=1,\dots,n}$); $W_{E_n}\neq 0$ (provided that $E_n\subset\mathbb D$); each quantity $M_{E_{n}}^{\mathtt{C};F}$ and $M_{E_{n}}^{\mathtt{S};F}$ is real and invariant under rearrangements of the points in $E_n=\{e_1,\dots,e_n\}$. Also note that we can replace $F(z)$ in (11) and (12) by any function equal to $F (z)$ on $E_n$ with multiplicities taken into account (for instance, by the Lagrange interpolation polynomial $\mathbf{F}_n(z)$), and the constants $0!, \dots,(r_j-1)!$ can be replaced by arbitrary nonzero constants (for instance, by ones as in [10], in the definition of $M_{E_{n}}^{\mathtt{S};F}$), because these constants are multiplied out of determinants and occur in a similar way in the numerators and denominators of the quantities $M_{E_n}^{\mathtt{C};F}$ and $M_{E_n}^{\mathtt{S};F}$ as defined in (14). Our choice of $0!, \dots,(r_j-1)!$ in the definitions (11) and (12) is explained by the convenience of comparison of $M_{E_n}^{\mathtt{C};F}$ and $M_{E_n}^{\mathtt{S};F}$, in the special case when $E_n=\{0\} ^n$ (so that all points in the $n$-point set $E_n$ coincide with zero) and $F (z)=\sum_{k=0}^{n-1} a_kz^k$, with the quantities $M_{n}^{\mathtt{C};F}$ and $M_{n}^{\mathtt{S};F}$ introduced before the Carathódory-Schur criterion (see (5)) in terms of the matrices $A_n^F$ and $\widetilde{A}_n^F$ (see (4)). Namely, the following holds. Proposition 2. Let $n\in\mathbb N$, $E_n=\{0\} ^n$, $F (z)=\sum_{k=0}^{n-1} a_kz^k$ and $\zeta =\mathtt{C}, \mathtt{S} $. Then $M_{E_n}^{\zeta;F}=M_{n}^{\zeta;F}$, where the quantities $M_{E_n}^{\zeta;F}$ are defined by (11)–(14) and the $M_{n}^{\zeta;F}$ are defined by (4) and (5). In fact, if $E_n=\{0\} ^n$ and $F (z)=\sum_{k=0}^{n-1} a_kz^k$, then from (11)–(14) and (4), (5) we obtain
$$
\begin{equation*}
\begin{gathered} \, \frac{(\varphi_pF)^{(r)}(0)}{r!}= \begin{cases} 0 \quad &\text{for }0\leqslant r<p\leqslant n-1, \\ a_{r-p}\quad &\text{for } 0\leqslant p\leqslant r\leqslant n-1, \end{cases} \\ A_{E_n}^{F}=\begin{pmatrix} a_0 & a_1 & \dots & a_{n-1} \\ 0 & a_0 & \dots & a_{n-2} \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & a_0 \end{pmatrix} {=}\,A_n^F, \qquad \widetilde{A}_{E_n}^{F} = \begin{pmatrix} \overline{a}_0 & 0 & \dots & 0 \\ \overline{a}_1 & \overline{a}_0 & \dots & 0 \\ \dots & \dots & \dots & \dots \\ \overline{a}_{n-1} & \overline{a}_{n-2} & \dots & \overline{a}_0 \end{pmatrix} {=}\,\widetilde{A}_{n}^{F}, \\ A_{E_n}^{\varphi_0}=\begin{pmatrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & 1 \end{pmatrix} {=}\,\widetilde{A}_{E_n}^{\varphi_0}, \qquad A_{E_n}^{\varphi_n}=\begin{pmatrix} 0 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & 0 \end{pmatrix} {=}\,\widetilde{A}_{E_n}^{\varphi_n}, \\ W_{E_n}=\det \begin{pmatrix} I_n & O_n \\ O_n & I_n \end{pmatrix} =1, \end{gathered}
\end{equation*}
\notag
$$
where $I_n$ and $O_n$ are the $ n\times n $ identity and zero matrices,
$$
\begin{equation}
M_{E_n}^{\mathtt{C};F}: =\det \begin{pmatrix} I_n & I_n \\ -A_n^F & \widetilde{A}_n^F \end{pmatrix}=\det(A_n^F +\widetilde{A}_n^F)=:M_{n}^{\mathtt{C};F}
\end{equation}
\tag{15}
$$
and
$$
\begin{equation}
M_{E_n}^{\mathtt{S};F}: =\det \begin{pmatrix} I_n & \widetilde{A}_n^F \\ A_n^F & I_n \end{pmatrix}=\det(I_n-A_n^F \widetilde{A}_n^F)=:M_{n}^{\mathtt{S};F}.
\end{equation}
\tag{16}
$$
In (15) and (16) we have used the well-known equality (see, for instance, [1], § 5)
$$
\begin{equation*}
\det\begin{pmatrix} P & Q \\ R & S \end{pmatrix} =\det(PS-RQ)
\end{equation*}
\notag
$$
for $ n\times n $-matrices $P,Q,R$ and $S$ such that $PR=RP$. A very useful auxiliary result is as follows. Proposition 3. For $n\in\mathbb N$ let $E_n=\{e_1,\dots,e_n\}\subset\mathbb D$, let $F(z)$ be a function defined at the points in $E_n$ with multiplicities taken into account, $\mathbf{F}(z)\in H(E_n)$ be a function interpolating $F(z)$ with multiplicities at the points in $E_n$, and let $\zeta =\mathtt{C},\mathtt{S}$. Then It was shown in [10], Proposition 5, that for $\zeta =\mathtt{S}$ Proposition 3 is a simple consequence of Taylor’s formula. Moreover, the arguments in [10] can also be used for $\zeta =\mathtt{C} $ almost word for word. The next theorem extends Theorem 1 to the multipoint case. Theorem 2. For $n\in\mathbb N$ let $E_n=\{e_1,\dots,e_n\}\subset\mathbb D$, and let $F(z)$ be a function defined at the points in $E_n$ with multiplicities taken into account and such that $F(z)\neq-1$ for $z\in E_n$. Then, using the notation (11)–(14), the following equalities hold for the function $(T\circ F)(z)$ defined with multiplicities at the points in $E_n$: In particular, Proof. First assume that the points $e_1,\dots,e_n$ in $E_n$ are pairwise distinct, and set for brevity
Then the following chain of equalities holds:
$$
\begin{equation*}
\begin{aligned} \, &W_{E_n}M_{E_{n}}^{\mathtt{C};T\circ F} =\det \begin{pmatrix} e_1^0 & \dots & e_n^0 & \overline{e_n^{n-1}} & \dots & \overline{e_1^{n-1}} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1} & \dots & e_n^{n-1} & \overline{e_n^{0}} & \dots & \overline{e_1^{0}} \\ -e_1^0G_1 & \dots & -e_n^0G_n & \overline{e_n^{n-1}G_n} & \dots & \overline{e_1^{n-1}G_1} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ -e_1^{n-1}G_1 & \dots & -e_n^{n-1}G_n & \overline{e_n^{0}G_n} & \dots & \overline{e_1^{0}G_1} \end{pmatrix} \\ &=\frac{\det { \begin{pmatrix} e_1^0(1+F_1) & \dots & e_n^0(1+F_n) & \overline{e_n^{n-1}(1+F_n)} & \dots & \overline{e_1^{n-1}(1+F_1)} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1}(1+F_1) & \dots & e_n^{n-1}(1+F_n) & \overline{e_n^{0}(1+F_n)} & \dots & \overline{e_1^{0}(1+F_1)} \\ -e_1^0(1-F_1) & \dots & -e_n^0(1-F_n) & \overline{e_n^{n-1}(1-F_n)} & \dots & \overline{e_1^{n-1}(1-F_1)} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ -e_1^{n-1}(1-F_1) & \dots & -e_n^{n-1}(1-F_n) & \overline{e_n^{0}(1-F_n)} & \dots & \overline{e_1^{0}(1-F_1)} \end{pmatrix}}}{\prod_{k=1}^n|1+F_k|^{2}} \\ &=\frac{\det { \begin{pmatrix} 2e_1^0 & \dots & 2e_n^0 & 2\overline{e_n^{n-1}F_n} & \dots & 2\overline{e_1^{n-1} F_1} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 2e_1^{n-1} & \dots & 2e_n^{n-1} & 2\overline{e_n^{0}F_n} & \dots & 2\overline{e_1^{0} F_1} \\ -e_1^0(1-F_1) & \dots & -e_n^0(1-F_n) & \overline{e_n^{n-1}(1-F_n)} & \dots & \overline{e_1^{n-1}(1-F_1)} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ -e_1^{n-1}(1-F_1) & \dots & -e_n^{n-1}(1-F_n) & \overline{e_n^{0}(1-F_n)} & \dots & \overline{e_1^{0}(1-F_1)} \end{pmatrix}}}{\prod_{k=1}^n|1+F_k|^{2}} \\ &=C_{E_n}^F\det \begin{pmatrix} e_1^0 & \dots & e_n^0 & \overline{e_n^{n-1}F_n} & \dots & \overline{e_1^{n-1}F_1} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1} & \dots & e_n^{n-1} & \overline{e_n^{0}F_n} & \dots & \overline{e_1^{0}F_1} \\ e_1^0F_1 & \dots & e_n^0F_n & \overline{e_n^{n-1}} & \dots & \overline{e_1^{n-1}} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ e_1^{n-1}F_1 & \dots & e_n^{n-1}F_n & \overline{e_n^{0}} & \dots & \overline{e_1^{0}} \end{pmatrix} =C_{E_n}^FW_{E_n}M_{E_{n}}^{\mathtt{S};F}, \end{aligned}
\end{equation*}
\notag
$$
where $C_{E_n}^F:=2^n/\prod_{k=1}^n|1+F(e_k)|^{2}$.
Some comments regarding this chain of equalities are in order here. The first equality follows from the definition (11)–(14) of $M_{E_n}^{\mathtt{C};T\circ F}$; the second (taking the inequality $F_k\neq -1$ and the equalities $G_k=(1-F_k)/(1+F_k)$, $k=1,\dots,n$, into account) is deduced by multiplying and dividing by $\prod_{k=1}^n|1+F_k|^{2}$; the third is obtained by subtracting the $(n+k)$th row from the $k$th ($k=1,\dots,n$) in the determinant and the fourth by multiplying out the constant coefficient 2 from the first $n$ rows in the determinant, introducing the constant $C_{E_n}^F$ and adding the $k$th row of the transformed determinant to the ${(n+k)}$th; the fifth equality follows from the definition (11)–(14) of $M_{E_n}^{\mathtt{S};F}$. Since $W_{E_n}\neq 0$ for $E_n\subset\mathbb D$, we have thus completed the proof of (17) for $\zeta =\mathtt{S} $ and pairwise distinct points $e_1,\dots,e_n$. In the general case let $\mathbf{F}(z)\in H(E_n)$ denote the function interpolating $F(z)$ with multiplicities at the points in $E_n$. Using Propositions 1 and 3, where the infinitesimals $\varepsilon_1,\dots,\varepsilon_n$ are selected so that the set $E_{n;\varepsilon_1,\dots,\varepsilon_n}$ consists of pairwise distinct points, from equality (17) for $\zeta =\mathtt{S} $ and pairwise distinct points, which we have established, we obtain
$$
\begin{equation*}
\begin{aligned} \, M_{E_n}^{\mathtt{C};T\circ F} &=\lim_{\varepsilon_1\to 0}\dotsb \lim_{\varepsilon_n\to 0}M_{E_{n;\varepsilon_1,\dots,\varepsilon_n}}^{\mathtt{C};T\circ \mathbf{F}} \\ &=\lim_{\varepsilon_1\to 0}\dotsb \lim_{\varepsilon_n\to 0} C_{E_{n;\varepsilon_1,\dots,\varepsilon_n}}^{\mathbf{F}} M_{E_{n;\varepsilon_1,\dots,\varepsilon_n}}^{\mathtt{S};\mathbf{F}} =C_{E_n}^FM_{E_n}^{\mathtt{S};F}. \end{aligned}
\end{equation*}
\notag
$$
This is the same as (17) for $\zeta =\mathtt{S} $ in the general case.
Replacing $F(z)$ by $(T\circ F)(z)$ in equality (17) for $\zeta =\mathtt{S} $, which we have already proved, and bearing in mind that
$$
\begin{equation*}
(T\circ F)(z)\neq -1, \quad z\in E_n, \qquad (T\circ T\circ F)(z)=F(z)\quad\text{and} \quad C_{E_n}^{T\circ F}=(C_{E_n}^{F})^{-1},
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation*}
M_{E_n}^{\mathtt{C};F}=C_{E_n}^{T\circ F}M_{E_n}^{\mathtt{S};T\circ F}=(C_{E_n}^{F})^{-1}M_{E_n}^{\mathtt{S};T\circ F},
\end{equation*}
\notag
$$
which is (17) for $\zeta =\mathtt{C} $.
Theorem 2 is proved. By Proposition 2, Theorem 1 is the special case of Theorem 2 for $E_n=\{0\} ^n$. § 3. Multipoint Carathéodory-Schur criterionLet $e_1,e_2,\dots $ be an infinite sequence of points in the disc $\mathbb D$ and $F(z)$ be a function defined with multiplicities at these points. We can define the restrictions $F_n(z)$ of $F$ to the sets $E_{n}:=\{e_1,\dots,e_n\}$, $n=1,2,\dots $, in the natural way; in their turn these latter define for (sufficiently large) suitable $n$ the restrictions to any sets of the form $\{e_{j_1},\dots,e_{j_p}\}$, $1\leqslant j_1<\dots <j_p<\infty$ (see § 2). Definition. We say that a function $F(z)$ defined at an infinite sequence of points $e_1,e_2,\dots $ in $\mathbb D$ with multiplicities taken into account admits an extension to a function in $\mathfrak B_N^\zeta$ ($N\in\mathbb Z_+^\infty$ and $\zeta =\mathtt{C}, \mathtt{S}$), and we write $F(z)\lessdot \mathfrak B_N^\zeta$ if there exists a function $\mathscr F_N^\zeta (z)\in \mathfrak B_N^\zeta$ such that where $\nu_n$ is the multiplicity of $e_n$ in the set $E_{n}=\{e_1,\dots,e_n\}$. If $F(z)$ is such that $F(e_n)\neq -1$, $n=1,2,\dots $, then we can also define the function $(T\circ F)(z)$ at the points $e_1,e_2,\dots$ . Furthermore, by Proposition 1 equalities (18) are equivalent to
$$
\begin{equation}
(T\circ \mathscr F_N^\zeta )^{(\nu_n-1)}(e_n)=(T\circ F)^{(\nu_n-1)}(e_n) \quad \text{for all } n=1,2,\dotsc\,.
\end{equation}
\tag{19}
$$
Hence, as
$$
\begin{equation*}
F(z)\lessdot \mathfrak B_N^{\zeta}\quad\text{for } N\in\mathbb Z_+^\infty\quad\text{and} \quad \zeta =\mathtt{C},\mathtt{S} \, \quad\Longrightarrow\quad F(e_n)\neq -1, \quad n=1,2,\dots
\end{equation*}
\notag
$$
(except when $\zeta ={\mathtt{S}}$ and $F(z)\equiv -1$), taking (19) and (3) into account we obtain the following. Proposition 4. Let $F(z)$ be a function defined with multiplicities at points $e_1, e_2,\dots$ in the disc $\mathbb D$. Then the following implications hold for $N\in\mathbb Z_+^\infty$: Apart from the quantities $M_{E_n}^{\mathtt{C};F}$ and $M_{E_n}^{\mathtt{S};F}$ introduced in § 2, to state the multipoint analogue of Schur’s criterion for functions defined with multiplicities at points $e_1,e_2,\dots $ in $\mathbb D$, which was established in [10] (and which we prove in our § 4), we need the following definitions. Notation 2. Let $n\in\mathbb N$ and $N\in\mathbb Z_+$, let $n\geqslant N+2$, and let $\nu_n$ be the multiplicity of $e_n$ in $E_n :=\{e_1,\dots,e_n\}$. Set where the indices $j_1,\dots,j_{N+2}$ satisfy $1\leqslant j_1<\dots <j_{N+2}=n$ and the point $e_{j_{N+2}}=e_n$ has multiplicity $\nu_n$ in $E^{j_1,\dots,j_{N+2}}$ (so that for $ \nu_n\leqslant N+2\leqslant n$ the set $E_{n,N}$ is obtained from $E_n$ by removing arbitrary $n-N-2$ points distinct from $e_n$). It is easy to see that for $n=N+2$ we have $E_{N+2,N}=E_{N+2}$. Let $F(z)$ be a function defined with multiplicities at points $e_1,e_2,\dots$ in $\mathbb D$, let ${m\in\mathbb N}$, and let $j_1,\dots,j_m$ be a set of indices such that $1\leqslant j_1<\dots <j_m<\infty$. Let $F_n(z)$ be the restriction of $F(z)$ to $E_n :=\{e_1,\dots,e_n\}$, $n=1,2,\dots$, $F^{j_1,\dots,j_m}(z)$ be the restriction of $F(z)$ to $E^{j_1,\dots,j_m}:=\{e_{j_1},\dots,e_{j_m}\}$, and here for $ N+2\leqslant \nu_n\leqslant n$ the function ${F}_{n,N}(z)$ is arbitrarily extended to a function defined at the points in $E_{n,N}=\{e_n\} ^{2\nu_n-N-2}$ with multiplicities taken into account (that is, the restriction of $F_n(z)$ to the set $\{e_n\} ^{\nu_n}$ defined by the values of derivatives $F_n^{(0)}(e_n),\dots,F_n^{(\nu_n-1)}(e_n)$ is extended by selecting arbitrary values of derivatives $F_n^{(\nu_n)}(e_n),\dots,F_n^{(2\nu_n-N-3)}(e_n)$).For the brevity of the statements that follow set
$$
\begin{equation*}
E_{N+1,N}:=E_{N+1}\quad\text{and} \quad F_{N+1,N}(z):=F_{N+1}(z).
\end{equation*}
\notag
$$
In the set of functions defined with multiplicities at $e_1,e_2,\dots $, for $N\in\mathbb Z_+^\infty$ and $\zeta =\mathtt{C},\mathtt{S}$ we distinguish the subsets
$$
\begin{equation}
\Phi ^{\zeta}_N :=\bigl\{F(z)\colon M_{E_p}^{\zeta;F_p}>0, \, p=1,\dots,N, \,M_{E_{N+p,N}}^{\zeta;F_{N+p,N}}=0, \, p=1,2,\dots \bigr\}
\end{equation}
\tag{20}
$$
(as above, if $N=0$, then there are no inequalities of the form $M_{E_p}^{\zeta;F_p}>0$ for $p=1,\dots,N$, while for $N=\infty$ there are no equalities of the form $M_{E_{N+p,N}}^{\zeta;F_{N+p,N}}=0$ for $p=1,2,\dots $). For functions defined with multiplicities at some points in $\mathbb D$, in § 4 we prove a multipoint analogue of Carathéodory’s and Schur’s criteria. Theorem 3. Let $F(z)$ be a function defined at points $e_1,e_2,\dots$ in $\mathbb D$ with multiplicities taken into account. Then in the notation introduced above We make some comments on Theorem 3. Remark 1. The necessary conditions for the relation $F(z)\lessdot \mathfrak B_N^{\zeta}$ in Theorem 3 can also be expressed in a more elaborate form: Remark 2. For $\zeta ={\mathtt{S}}$ Theorem 3 and the implication (21) were proved in [10] (Theorem 1 and an addendum to it). Remark 3. Theorem 3 is independent of the choice of the values of the derivatives $F_n^{(\nu_n)}(e_n),\dots,F_n^{(2\nu_n-N-3)}(e_n)$ of ${F}_{n,N}(z)$ for $\nu_n\geqslant N+2$, which can be arbitrary. Remark 4. Theorem 3 can also be stated in terms of formal Newton series. In fact, let $F(z)$ be the function defined at the points $e_1,e_2,\dots$ in $\mathbb D$ with multiplicities taken into account, and let $\mathbf{F}_{k}(z)$ be the Lagrange interpolation polynomial for the restriction of $F(z)$ to $E_k:=\{e_1,\dots,e_k\}$, $k=1,2,\dots$ . It is easy to see that $\mathbf{F}_{k+1}(z)-\mathbf{F}_k(z)$ is a polynomial of degree at most $k$ which vanishes with multiplicities at the points in $E_{k}$, so that for some $a_k\in\mathbb C$. This means that we can assign to $F(z)$ the Newton series such that its $(n-1)$st partial sum coincides with the Lagrange interpolation polynomial for $F(z)$ on $E_n$. Conversely, each Newton series $f(z)$ with nodes at points $e_1,e_2,\dots$ in $\mathbb D$ produces a function $F(z)$ defined at the points $e_1,e_2,\dots$ by the equalities
$$
\begin{equation*}
F^{(\nu_n -1)}(e_n)=f_n^{(\nu_n -1)}(e_n), \qquad n=1,2,\dots,
\end{equation*}
\notag
$$
with multiplicities taken into account. Here $\nu_n$ is the multiplicity of $e_n$ in the set $E_n=\{e_1,\dots,e_n\}$ and $f_n(z)$ is the $(n-1)$st partial sum of the Newton series $f(z)$. If a Newton series $f(z)$ with nodes at points $e_1,e_2,\dots$ in $\mathbb D$ corresponds to a function $F(z)$ defined with multiplicities at $e_1,e_2,\dots$ and if $F(e_n)\neq -1$ for $n=1,2,\dots$, then we let $(T\circ f)(z)$ denote the Newton series corresponding to the function $(T\circ F)(z)$. In the case when
$$
\begin{equation*}
e_n=e, \quad n=1,2,\dots,\quad\text{and} \quad f(z)=\sum_{k=0}^{\infty} a_k(z-e)^k, \quad a_0\neq -1,
\end{equation*}
\notag
$$
the series $(T\circ f)(z)$ coincides with the series obtained by dividing the power series $1-f(z)$ (about $e$) formally by the series $1+f(z)$. Given a Newton series $f(z)$, the expression $f(z)\lessdot \mathfrak B_N^\zeta$ means that there exists a function $\mathscr F_N^\zeta (z)\in \mathfrak B_N^\zeta$ such that the following inclusions hold:
$$
\begin{equation*}
\frac{\mathscr F_N^\zeta (z)-f_n(z)}{(z-e_1)\dotsb (z-e_n)}\in H(\mathbb D) \quad \text{for all } n=1,2,\dots,
\end{equation*}
\notag
$$
where $f_n(z)$ is the $(n-1)$st partial sum of the Newton series. Note that
$$
\begin{equation*}
f(z)\lessdot \mathfrak B_N^\zeta \quad \Longleftrightarrow \quad F(z)\lessdot \mathfrak B_N^\zeta, \qquad N\in\mathbb Z_+^\infty, \quad \zeta =\mathtt{C},\mathtt{S},
\end{equation*}
\notag
$$
where $F(z)$ is the function corresponding to the series $f(z)$. To formulate Theorem 3 in terms of formal Newton series we replace the notation $F$ by $f$ throughout the statement of Theorem 3 and the definition (20) of the sets $\Phi ^{\zeta}_N$ and treat $f(z)$ as a formal Newton series with nodes $e_1,e_2,\dots$ in $\mathbb D$, $f_n(z)$, $n=1,2,\dots$, as its $(n-1)$st partial sum, $f^{j_1,\dots,j_m}(z)$ as an arbitrary fixed polynomial such that
$$
\begin{equation*}
\frac{f^{j_1,\dots,j_m}(z)-f_{n}(z)}{(z-e_{j_1})\dotsb (z-e_{j_m})}\in H(E^{j_1,\dots,j_m}), \qquad 1\leqslant j_1<\dots <j_m\leqslant n,
\end{equation*}
\notag
$$
and ${f}_{n,N}(z)$ as an (arbitrary fixed) polynomial such that
$$
\begin{equation}
\frac{f_{n,N}(z)-f_n(z)}{(z-e_{n})^{\nu_n}}\in H(e_{n}) \quad \text{for } n\geqslant \nu_n\geqslant N+2
\end{equation}
\tag{22}
$$
and
$$
\begin{equation*}
\frac{f_{n,N}(z)-f_n(z)}{\prod_{e\in E_{n,N}}(z-e)}\in H(E_{n,N}) \quad \text{for } \nu_n\leqslant N+2\leqslant n
\end{equation*}
\notag
$$
($\nu_n$ and $E_{n,N}$ were defined before Theorem 3). Remark 5. The definitions of the sets $E_{n,N}$ for $N+2\leqslant n$ are considerably different in the cases when $e_1,e_2,\dots$ are pairwise distinct and when $e_1,e_2,\dots$ coincide with the same point $e\in\mathbb D$. In the first case, for $n\geqslant N+2$ we have $E_{n,N}=E^{j_1,\dots,j_{N+2}}$, where $1\leqslant j_1<\dots <j_{N+1}<n$ are arbitrary fixed indices and $j_{N+2}=n$. In particular, for $j_k=k$, $k=1,\dots,N+1$, we obtain In the second case
$$
\begin{equation*}
E_n=\{e\} ^n, \quad \nu_n=n, \qquad E_{n,N}=\{e\} ^{2n-N-2},
\end{equation*}
\notag
$$
and it is natural to take either $f_{2n-N-2}(z)$ or $f_{n}(z)$ as a polynomial $f_{n,N}(z)$ (${n\geqslant N+2}$) satisfying (22). In particular, for $e=0$, bearing in mind that ${M_{E_n}^{\zeta;f_n}=M_n^{\zeta;f}}$ by Proposition 2 we see that for $N\in\mathbb Z_+$ the sets $\Phi_N^\zeta$ (see (20) for $F$ replaced by $f$) either coincide with the $\hat{\mathscr{A}}_N^\zeta$ (see (9)) or with the $\breve{{\mathscr A}}_N^\zeta$ (see (10)). This means that Theorem 3, as reformulated in terms of the formal power series $f(z)=\sum_{k=0}^\infty a_kz^k$, yields the refinement of Carathéodory’s and Schur’s criteria for $N\in\mathbb Z_+$ that we stated in § 1. Remark 6. It was shown in [10] that for a function $F(z)$ defined by the equalities $F(e_n)=\gamma$ for $n\in\mathbb N\setminus\{k\}$ at pairwise distinct points $e_1,e_2,\dots$ in $\mathbb D$, where $|\gamma |=1$ and $k\geqslant 3$, we have $M_{E_{n}}^{\mathtt{S};F_n}=0$ for all $n=1,2,\dots $ and any value of $F(e_k)$ (we also obtain a similar example for $\zeta =\mathtt{C}$, provided that we replace the condition $|\gamma |=1$ by $\operatorname{Re} \gamma =0$). This means that in contrast to the conditions in (23) for $N=0$, the conditions are not sufficient to claim that $F(z)\lessdot \mathfrak B_0^{\mathtt{S}}$ in the case when $F(e_k)\neq\gamma$. § 4. Proof of the multipoint Carathéodory-Schur criterionTheorem 3 supplemented by implication (21) covers two different cases, ${\zeta =\mathtt{C}}$ and ${\zeta =\mathtt{S}}$. Throughout this section, for brevity we call the case $\zeta = \mathtt{C} $ of Theorem 3, which corresponds to the multipoint version of Carathéodory’s theorem, Theorem 3$^{\mathtt{C}}$, and we call the case $\zeta =\mathtt{S}$, corresponding to the multipoint version of Schur’s theorem, Theorem 3$^{\mathtt{S}}$. In view of Remark 2 after the statement of Theorem 3, to establish Theorem 3 it is sufficient to prove Theorem 3$^{\mathtt{C}}$ alone. We obtain the proof of Theorem 3$^{\mathtt{C}}$ by reducing it to Theorem 3$^{\mathtt{S}}$ (established in [10]) with the help of Theorem 2, proved in § 2. Turning to the proof of Theorem 3$^{\mathtt{C}}$ with implication (21) note that we have the chain of implications
$$
\begin{equation}
\begin{aligned} \, \notag F(z)\lessdot \mathfrak B_N^{\mathtt{C}} &\quad\Longrightarrow\quad (T\circ F)(z)\lessdot \mathfrak B_N^{\mathtt{S}}, \qquad (T\circ F)(e_n)\neq -1, \quad n=1,2,\dots, \\ \notag &\quad\Longrightarrow \quad \begin{cases} M_{E^{j_1,\dots,j_m}}^{\mathtt{S};T\circ F^{j_1,\dots,j_m}}>0,& m\leqslant N, \\ M_{E^{j_1,\dots,j_m}}^{\mathtt{S};T\circ F^{j_1,\dots,j_m}}=0,& m\geqslant N+1, \\ M_{E_{n,N}}^{\mathtt{S};T\circ F_{n,N}}=0,& n\geqslant N+1, \end{cases} \\ & \quad\Longrightarrow \quad \begin{cases} M_{E^{j_1,\dots,j_m}}^{\mathtt{C};F^{j_1,\dots,j_m}}>0,&m\leqslant N, \\ M_{E^{j_1,\dots,j_m}}^{\mathtt{C};F^{j_1,\dots,j_m}}=0,&m\geqslant N+1, \\ M_{E_{n,N}}^{\mathtt{C};F_{n,N}}=0,& n\geqslant N+1, \end{cases} \quad\Longrightarrow \quad F(z)\in\Phi_N^{\mathtt{C}}, \end{aligned}
\end{equation}
\tag{24}
$$
the first of which follows from Proposition 4, the second from Theorem 3$^{\mathtt{S}}$, proved in [10], as applied to the function $(T\circ F)(z)$, the third follows from Theorem 2 and the inequalities $(T\circ F)(e_n)\neq -1$, $n=1,2,\dots$, and the fourth from the definition (20) of the sets $\Phi_N^{\mathtt{C}}$. Thus, to complete the proof of Theorem 3$^{\mathtt{C}}$ it suffices to show that
$$
\begin{equation}
F(z)\in\Phi_N^{\mathtt{C}} \quad\Longrightarrow\quad F(z)\lessdot \mathfrak B_N^{\mathtt{C}}.
\end{equation}
\tag{25}
$$
We prove (25) on the basis of Lemmas 1$^{\mathtt{C}}$ and 2$^{\mathtt{C}}$, which are of independent interest. They are analogues of Lemmas 1$^{\mathtt{S}}$ and 2$^{\mathtt{S}}$, which we proved in [10] (Lemmas 3–5). We combine their assertions into Lemmas 1 and 2, by adding some notation. Namely, for $E_n=\{e_1,\dots,e_n\}$ we set Lemma 1. Let $F(z)$ be a polynomial, let $n\in\mathbb N$, $E_{n}=\{e_1,\dots,e_{n}\}\subset\mathbb D$, $\zeta =\mathtt{C},\mathtt{S}$, and let Then there exists a function $\mathscr F_{n}^\zeta (z)\in \mathfrak B ^\zeta \setminus (\bigcup_{k=0}^{n-1} \mathfrak B_k^\zeta )$ such that In particular, $F(z)\neq -1$, $z\in E_n$. Lemma 2. Let $F(z)$ be a polynomial, let $N\in \mathbb Z_+$, $p \in \mathbb N$, $E_{N+p} = \{e_1,\dots,e_{N+p}\} \subset \mathbb D$, $\zeta =\mathtt{C},\mathtt{S}$, and let Then there exists a function $\mathscr F_{N}^\zeta (z)\in \mathfrak B_N^\zeta $ such that In particular, $F(z)\neq -1$, $z\in E_{N+p}$ (except when $N=0$, $\zeta ={\mathtt{S}}$ and $\mathscr F_{N}^{\mathtt{S}} (z)\equiv -1$). Proof of Lemma 1. Lemma 1$^{\mathtt{S}}$ was proved in [10] (Lemma 3). Now note that under the additional assumption that $F(z)\neq -1$ for $z\in E_{n}$, Lemma 1$^{\mathtt{C}}$ reduces to Lemma 1$^{\mathtt{S}}$ by use of Theorem 2. In fact, if conditions (26) are satisfied for $\zeta =\mathtt{C}$ and a function $F(z)\neq -1$ for $z\in E_{n}$, then by Theorem 2 the same conditions for $\zeta =\mathtt{S}$ hold for the function $(T\circ F)(z)$. Hence by Lemma 1$^{\mathtt{S}}$ there exists a function $\mathscr F_{n}^{\mathtt{S}} (z)\in \mathfrak B ^{\mathtt{S}} \setminus (\bigcup_{k=0}^{n-1} \mathfrak B_k^{\mathtt{S}} )$ such that
$$
\begin{equation*}
\frac{\mathscr F_{n}^{\mathtt{S}} (z)-(T\circ F)(z)}{\omega_{E_n}(z)}\in H({E_{n}}).
\end{equation*}
\notag
$$
By Proposition 1 this yields
$$
\begin{equation*}
\frac{(T\circ \mathscr F_{n}^{\mathtt{S}}) (z)-F(z)}{\omega_{E_n}(z)}\in H({E_{n}}).
\end{equation*}
\notag
$$
Therefore, setting $\mathscr F_{n}^{\mathtt{C}} (z)=(T\circ \mathscr F_{n}^{\mathtt{S}}) (z)$ and observing that $\mathscr F_{n}^{\mathtt{C}} (z)\in \mathfrak B ^{\mathtt{C}} \setminus (\bigcup_{k=0}^{n-1} \mathfrak B_k^{\mathtt{C}} )$ by Proposition 4 we obtain the required inclusion (27) for $\zeta =\mathtt{C}$.
Thus, to complete the proof of Lemma 1 it suffices to show that inequalities (26) for $\zeta =\mathtt{C}$ imply the inequalities $F(z)\neq -1$ for $z\in E_{n}$ in the hypotheses of Theorem 2. Assume the converse: let $F(z)= -1$ for some $z\in E_{n}$. Taking the relation $F(e_1)\neq -1$, which follows from the first inequality in (26) (namely, $0<M_{E_1}^{\mathtt{C};F}=2\operatorname{Re} F(e_1)$), into account, this assumption means that $n\geqslant 2$ and there exists ${k\in\{2,\dots,n\}}$ such that
$$
\begin{equation}
F(z)\neq -1, \quad z\in E_{k-1}:=\{e_1,\dots,e_{k-1}\}, \qquad F(e_{k})=-1.
\end{equation}
\tag{30}
$$
In particular, it follows from (30) that $e_k$ is distinct from all points in $E_{k-1}$ (we write $e_{k}\notin E_{k-1}$ for brevity).
Consider the polynomial $G_{k,\varepsilon}(z)$ that is equal to $F(z)$ for $z\in E_{k-1}$ and to $F(e_k)+\varepsilon$ for $z=e_{k}$. We see directly from this definition and (30) that
$$
\begin{equation*}
G_{k,\varepsilon}\neq -1 \quad \text{for } z\in E_{k}, \quad \varepsilon\neq 0.
\end{equation*}
\notag
$$
Since by the definitions (11)–(14) we have
$$
\begin{equation*}
M_{E_{1}}^{\mathtt{C};G_{k,\varepsilon}}=M_{E_{1}}^{\mathtt{C};F}, \quad \dots,\quad M_{E_{k-1}}^{\mathtt{C};G_{k,\varepsilon}}=M_{E_{k-1}}^{\mathtt{C};F}\quad\text{and} \quad \lim_{\varepsilon\to 0}M_{E_{k}}^{\mathtt{C};G_{k,\varepsilon}}=M_{E_{k}}^{\mathtt{C};F},
\end{equation*}
\notag
$$
it follows from (26) for $\zeta =\mathtt{C}$ that for all sufficiently small $\varepsilon$ we have $M_{E_{j}}^{\mathtt{C};G_{k,\varepsilon}}>0$, $j=1,\dots,k$. Thus, $G_{k,\varepsilon}(z)$ for $\varepsilon\neq 0$ fulfils all assumptions under which Lemma 1$^{\mathtt{C}}$ (for $k$ in place of $n$) was proved. Hence for all sufficiently small $\varepsilon\neq 0$ there exist functions $\mathscr F_{k,\varepsilon}^{\mathtt{C}}(z)\in \mathfrak B ^{\mathtt{C}}$ such that
$$
\begin{equation}
\frac{\mathscr F_{k,\varepsilon}^{\mathtt{C}}(z)-G_{k,\varepsilon}(z)}{\omega_{E_{k}}(z)}\in H(E_{k}).
\end{equation}
\tag{31}
$$
It follows from (31) that
$$
\begin{equation*}
\lim_{\varepsilon \to 0}\mathscr F_{k,\varepsilon}(e_{k})=\lim_{\varepsilon \to 0}G_{k,\varepsilon}(e_{k})=\lim_{\varepsilon \to 0}(F(e_{k})+\varepsilon )=F(e_{k})=-1,
\end{equation*}
\notag
$$
which is impossible because $\mathscr F_{k,\varepsilon}^{\mathtt{C}}\in\mathfrak B ^{\mathtt{C}}$, and therefore $\operatorname{Re} \mathscr F_{k,\varepsilon}^{\mathtt{C}} (e_{k})\geqslant 0$. This means that the assumption $F(e_{k})=-1$ leads to a contradiction. Lemma 1$^{\mathtt{C}}$, and therefore also Lemma 1, are proved. Proof of Lemma 2. Note that Lemma 2$^{\mathtt{S}}$ was proved in [10], explicitly for $p=1,2$ (see [10], Lemmas 3 and 4) and implicitly for $p=3,4,\dots$ (see Lemma 5 and the subsequent proof of Theorem 1 in [10]). Since Lemma 2$^{\mathtt{C}}$, under the additional assumption that $F(z)\neq -1$ for $z\in E_{N+p}$, reduces to Lemma 2$^{\mathtt{S}}$ by use of Theorem 2, to prove Lemma 2 it suffices to show that the hypotheses of Lemma 2$^{\mathtt{C}}$ imply the relations
For $N=0$ and $p=1$ the inequality $F(e_1)\neq -1$ clearly follows from the equality $0=M_{E_1}^{\mathtt{C};F}=2\operatorname{Re} F(e_1)$ in (28). We prove (32) for $N\in\mathbb N$ and $p=1$. Then we must slightly modify the arguments used in the proof of Lemma 1 and rely on the valid equality $M_{E_{N+1}}^{\mathtt{C};F}=0$ in place of the strict inequality $M_{E_{N+1}}^{\mathtt{C};F}>0$ used in Lemma 1. Fix a function $\mathscr F ^{\mathtt{C}}(z)\in \mathfrak B ^{\mathtt{C}}$ such that
$$
\begin{equation}
\frac{\mathscr F ^{\mathtt{C}} (z)-F(z)}{\omega_{E_N}(z)}\in H({E_{N}}).
\end{equation}
\tag{33}
$$
It exists by Lemma 1 for $\zeta =\mathtt{C}$ (and $N$ in place of $n$), which we have already established. In particular, this inclusion yields the inequalities $F(z)\neq -1$, $z\in E_{N}$. Thus, to prove (32) for $p=1$ it suffices to show that conditions (28) (for $\zeta =\mathtt{C}$, $N\in\mathbb N$ and $p=1$) yield the inequality $F(e_{N+1})\neq -1$.
Assuming the converse we obtain
$$
\begin{equation*}
e_{N+1}\notin E_N\quad\text{and} \quad G_{N+1,\varepsilon}\neq -1 \quad \text{for } z\in E_{N+1}\quad\text{and} \quad \varepsilon\neq 0,
\end{equation*}
\notag
$$
where $G_{N+1,\varepsilon}(z)$ is the polynomial equal to $F(z)$ for $z\in E_N$ and to $F(e_{N+1})+\varepsilon$ for $z=e_{N+1}$.
We claim that there exists a null sequence of nonzero complex numbers $\{\varepsilon_l\}_{l=1}^\infty$ such that
$$
\begin{equation*}
M_{E_{N+1}}^{\mathtt{C};G_{N+1,\varepsilon_l}}\geqslant 0 \quad \text{for } l=1,2,\dots\,.
\end{equation*}
\notag
$$
Since $\mathscr F ^{\mathtt{C}}(z)\in \mathfrak B ^{\mathtt{C}}$, we have $M_{E_{N+1}}^{\mathtt{C};\mathscr F ^{\mathtt{C}}}\geqslant 0$ by (24). Hence from the definition of $G_{N+1,\varepsilon} (z)$ and(33) we obtain
$$
\begin{equation}
M_{E_{N+1}}^{\mathtt{C};G_{N+1,\varepsilon_0}}=M_{E_{N+1}}^{\mathtt{C};\mathscr F ^{\mathtt{C}}}\geqslant 0, \quad \text{where } \varepsilon_0:=\mathscr F ^{\mathtt{C}}(e_{N+1})-F(e_{N+1}).
\end{equation}
\tag{34}
$$
Note that $\varepsilon_0\neq 0$, because $\mathscr F ^{\mathtt{C}}(e_{N+1})\neq -1$ and $F(e_{N+1})=-1$ by assumption.
Since $e_{N+1}\notin E_N$, it follows from the definitions of $G_{N+1,\varepsilon} (z)$ and the quantities $M_{E_{N+1}}^{\mathtt{C};G_{N+1,\varepsilon}}$ (see (11)–(14)) that
$$
\begin{equation}
M_{E_{N+1}}^{\mathtt{C};G_{N+1,\varepsilon}}=A|\varepsilon |^2+B\varepsilon +\overline{B\varepsilon}+C,
\end{equation}
\tag{35}
$$
where $A$, $B$ and $C $ are some coefficients depending on $N$ and $F$ and, furthermore,
$$
\begin{equation*}
A \in\mathbb R\quad\text{and} \quad C =M_{E_{N+1}}^{\mathtt{C};G_{N+1,0}}=M_{E_{N+1}}^{\mathtt{C};F}=0.
\end{equation*}
\notag
$$
If $B=0$, then $M_{E_{N+1}}^{\mathtt{C};G_{N+1,\varepsilon}}\geqslant 0$ for all $\varepsilon\in\mathbb C$ because
$$
\begin{equation*}
M_{E_{N+1}}^{\mathtt{C};G_{N+1,\varepsilon}}=A|\varepsilon|^2 = \frac{M_{E_{N+1}}^{\mathtt{C};G_{N+1,\varepsilon_0}}|\varepsilon|^2}{|\varepsilon_0|^2}\geqslant 0
\end{equation*}
\notag
$$
by (35) and (34). If $B\neq 0$, then the equalities
$$
\begin{equation*}
M_{E_{N+1}}^{\mathtt{C};G_{N+1,\varepsilon_l}}=A|\varepsilon_l|^2 +2\operatorname{Re} (B\varepsilon_l) = 0
\end{equation*}
\notag
$$
hold for each null sequence of nonzero complex numbers $\varepsilon_l=|\varepsilon_l|e^{i\theta_l}$, where
$$
\begin{equation*}
0<|\varepsilon_l|<2\biggl|\frac BA\biggr|\quad\text{and} \quad \cos (\arg B+\theta_l)=-\frac{A|\varepsilon_l|}{2|B|}.
\end{equation*}
\notag
$$
Thus, the polynomials $G_{N+1,\varepsilon_l}$, $l=1,2,\dots$, fulfill the assumptions of either Lemma 1$^{\mathtt{C}}$ for $n=N+1$ or of Lemma 2$^{\mathtt{C}}$, which we proved above. Hence for all $l=1,2,\dots$ there exist functions $\mathscr F ^{\mathtt{C},l}(z)\in \mathfrak B ^{\mathtt{C}}$ such that
$$
\begin{equation}
\frac{\mathscr F ^{\mathtt{C},l}(z)-G_{N+1,\varepsilon_l}(z)}{\omega_{E_{N+1}}(z)}\in H(E_{N+1}), \qquad l=1,2,\dotsc\,.
\end{equation}
\tag{36}
$$
It follows from (36) that
$$
\begin{equation*}
\begin{aligned} \, \lim_{l\to \infty}\mathscr F ^{\mathtt{C},l}(e_{N+1}) &=\lim_{l\to \infty}G_{N+1,\varepsilon_l}(e_{N+1}) \\ &=\lim_{l\to \infty}(F(e_{N+1})+\varepsilon_l )=F(e_{N+1})=-1, \end{aligned}
\end{equation*}
\notag
$$
which is impossible because $\mathscr F ^{\mathtt{C},l}\in\mathfrak B ^{\mathtt{C}}$, $l=1,2,\dots$ . This means that the assumption $F(e_{N+1})=-1$ leads to a contradiction. Lemma 2$^{\mathtt{C}}$ is proved for $N\in \mathbb Z_+$ and $p=1$.
Let $p\geqslant 2$. We make the inductive assumption that (32) has been proved for all $N\in\mathbb Z_+$ and all positive indices up to $p-1$ inclusive. Now we prove these inequalities for the index $p$. By our inductive assumption there exists a function $\mathscr F ^{\mathtt{C}}(z)\in \mathfrak B ^{\mathtt{C}}$ such that (29) holds (for $\zeta =\mathtt{C}$ and $p$ replaced by $p-1$). In particular, this yields the inequalities $F(z)\neq -1$ for $z\in E_{N+p-1}$. Thus, to prove (32) is its sufficient to show that conditions (28) for $\zeta =\mathtt{C}$ and $N\in\mathbb N$ imply that $F(e_{N+p})\neq -1$. Assuming the converse we arrive at a contradiction by means of the arguments analogous to the ones used for $p=1$ (except that $N$ must now be replaced by $N+p-1$). This completes the proof of Lemma 2$^{\mathtt{C}}$, and therefore of Lemma 2. To complete the proof of Theorem 3 we note that, given Lemmas 1 and 2 and Theorem 2, implication (25) follows from the chain of implications
$$
\begin{equation*}
F(z)\in\Phi_N^{\mathtt{C}} \quad\Longrightarrow\quad (T\circ F)(z)\in\Phi_N^{\mathtt{S}} \quad\Longrightarrow\quad (T\circ F)(z)\lessdot \mathfrak B_N^{\mathtt{S}} \quad\Longrightarrow\quad F(z)\lessdot \mathfrak B_N^{\mathtt{C}},
\end{equation*}
\notag
$$
the first of which follows from the definition (20) of the sets $\Phi_N^{\zeta}$ for $\zeta = \mathtt{C},\mathtt{S}$ and Theorem 2, taking the inequalities
$$
\begin{equation*}
F(z)\neq -1 \quad\text{for } z\in E_n, \quad n=1,2,\dots,
\end{equation*}
\notag
$$
which follow from Lemmas 1 and 2, into account; the second implication follows from Theorem 3 for $\zeta =\mathtt{S}$, which was proved in [10], and the third follows from Proposition 4. The proof of (25) and therefore of Theorem 3 is complete. In conclusion we note that the arguments in this section show that Theorem 3$^{\mathtt{C}}$ is a consequence of Theorems 2 and 3$^{\mathtt{S}}$. It is easy to see that, in a similar way, Theorem 3$^{\mathtt{S}}$ is a consequence of Theorems 2 and 3$^{\mathtt{C}}$. In other words, Theorem 2 reveals the equivalence of the multipoint analogues of Carathéodory’s and Schur’s criteria, just as Theorem 1 reveals the equivalence of the classical Carathéodory and Schur criteria, with the single reservation that the verification of the inequalities $F(e_n)\neq -1$, $n=1,2,\dots$, in the hypotheses of Theorem 2 is not as trivial as in the classical case. 
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\Bibitem{Bus22} |
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