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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
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     § 1. The classical Carathéodory and Schur criteriaRecall that a holomorphic function F(z) in the disc D:={|z|<1} is called a Carathéodory function if ReF(z)⩾0 for z∈D, and it is called a Schur function if |F(z)|⩽1 for z∈D. We denote the sets of Carathéodory and Schur functions by BC and BS, respectively. In the sets BC and BS of Carathéodory and Schur functions authors usually distinguish the disjoint subsets BCN and BSN, N∈Z∞+:={∞,0,1,2,…}, where
BC0:={F(z)∈BC:F(z)≡λ,Reλ=0},BS0:={F(z)∈BS:F(z)≡γ,|γ|=1},BCN:={F(z)∈BC:F(z)=λ0+N∑k=1λktk−ztk+z,Reλ0=0,λk>0,|tk|=1,k=1,…,N,t1,…,tN are pairwise distinct},BSN:={F(z)∈BS:F(z)=γN∏k=1z−ek1−z¯ek,|γ|=1,ek∈D,k=1,…,N}
(a bar over a symbol denotes complex conjugation), N=1,2,…, Let T(z) denote the linear fractional map T(z)=(1−z)/(1+z) taking the unit disc D to the right-hand half-plane {Rez>0} and taking this half-plane to the unit disc. It was shown in [1], pp. 229–230, that (the function F(z)≡−1 in BS0, for which (T∘F)(z)≡∞, is an exception here).Set Since (T∘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
\begin{equation*}
f(z)\lessdot \mathfrak B_N^{\zeta} \quad\Longleftrightarrow\quad f(z)\in{\mathscr A}_N^{\zeta} \quad\Longleftrightarrow\quad f(z)\in\hat{\mathscr{A}}_N^{\zeta} \quad\Longleftrightarrow\quad f(z)\in\breve{{\mathscr A}}_N^{\zeta},
\end{equation*}
\notag
where N\in\mathbb Z_+, \zeta ={\mathtt{C}}, {\mathtt{S}},
\begin{equation}
\hat{\mathscr{A}}_N^{\zeta} :=\bigl\{f(z)\colon M_p^{\zeta;f}>0,\, p=1,\dots,N, \,M_{N+1}^{\zeta;f}= M_{N+2p}^{\zeta;f}=0,\, p=1,2,\dots \bigr\}
\end{equation}
\tag{9}
and
\begin{equation}
\breve{{\mathscr A}}^{\zeta}_N :=\bigl\{f(z)\colon M_p^{\zeta;f}>0,\, p=1,\dots,N, \, M_{N+1}^{\zeta;f}=M_{N+2p}^{\zeta;f_{N+p+1}}=0,\, p=1,2,\dots \bigr\}
\end{equation}
\tag{10}
(there are no inequalities of the form M_p^{\zeta;f}>0 for N=0), where f_{n}(z) is the (n-1)st partial sum of the series f(z), n=1,2,\dotsc . 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 kth (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 kth 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
\begin{equation}
(\mathscr F_N^\zeta)^{(\nu_n-1)}(e_n)=F^{(\nu_n-1)}(e_n) \quad \text{for all } n=1,2,\dots,
\end{equation}
\tag{18}
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:
\begin{equation*}
\begin{gathered} \, 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, \\ F(z)\lessdot \mathfrak B_N^{\mathtt{S}}, \qquad F(z)\not\equiv -1 \quad\Longrightarrow\quad (T\circ F)(z)\lessdot \mathfrak B_N^{\mathtt{C}}. \end{gathered}
\end{equation*}
\notag
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
\begin{equation*}
E_{n,N}:=\begin{cases} \{e_n\} ^{2\nu_n-N-2}&\text{for } \nu_n\geqslant N+2, \\ E^{j_1,\dots,j_{N+2}}&\text{for }\nu_n\leqslant N+2, \end{cases}
\end{equation*}
\notag
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
\begin{equation*}
{F}_{n,N}(z) \text{ be the restriction of }F_n(z)\text{ to } \begin{cases} E_{n,N}\subseteq E_n\quad &\text{if }\nu_n\leqslant N+2\leqslant n, \\ \{e_n\} ^{\nu_n}\subseteq E_n \quad &\text{if } N+2\leqslant \nu_n\leqslant n; \end{cases}
\end{equation*}
\notag
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:
\begin{equation}
F(z)\lessdot \mathfrak B_N^{\zeta}\quad \Longrightarrow\quad \begin{cases} M_{E^{j_1,\dots,j_m}}^{\zeta;F^{j_1,\dots,j_m}}>0,&m\leqslant N, \\ M_{E^{j_1,\dots,j_m}}^{\zeta;F^{j_1,\dots,j_m}}=0,&m\geqslant N+1, \\ M_{E_{n,N}}^{\zeta;{F}_{n,N}} = 0,&n\geqslant N+1, \end{cases} \quad\Longrightarrow \quad F(z)\in\Phi_N^{\zeta}.
\end{equation}
\tag{21}
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
\begin{equation*}
\mathbf{F}_{k+1}(z)-\mathbf{F}_{k}(z)= a_k(z-e_1)\dotsb (z-e_k)
\end{equation*}
\notag
for some a_k\in\mathbb C. This means that we can assign to F(z) the Newton series
\begin{equation*}
f(z)=\mathbf{F}_{1}(z)+\sum_{k=1}^{\infty}(\mathbf{F}_{k+1}(z)-\mathbf{F}_{k}(z)) =a_0+\sum_{k=1}^{\infty} a_k(z-e_1)\dotsb (z-e_k)
\end{equation*}
\notag
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
\begin{equation*}
E_{n,N}=E^{1,\dots,N+1,n}=\{e_1,\dots,e_{N+1}, e_n\}, \qquad n\geqslant N+2.
\end{equation*}
\notag
Hence the definition (20) has the following form:
\begin{equation}
\begin{aligned} \, \notag \Phi ^{\zeta}_N &:=\bigl\{F(z)\colon M_{E_p}^{\zeta;F_p}>0,\, p=1,\dots,N, \\ &\qquad M_{E_{N+1}}^{\zeta;{F}_{N+1}}= M_{\{e_1,\dots,e_{N+1}, e_{N+p}\}}^{\zeta;F^{1,\dots,N+1, N+p}} =0,\, p = 2,3,\dots \bigr\}, \end{aligned}
\end{equation}
\tag{23}
where F^{1,\dots,N+1, n}(z) for n\geqslant N+2 is the restriction of F_n(z) to the set {E^{1,\dots,N+1,n} \subseteq E_n}. 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
\begin{equation}
M_{E_1}^{\zeta;F}>0,\quad \dots,\quad M_{E_{n}}^{\zeta;F}>0.
\end{equation}
\tag{26}
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
\begin{equation}
\frac{\mathscr F_{n}^\zeta (z)-F(z)}{\omega_{E_n}(z)}\in H({E_{n}}).
\end{equation}
\tag{27}
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
\begin{equation}
M_{E_1}^{\zeta;F}>0,\quad \dots,\quad M_{E_{N}}^{\zeta;F}>0\quad\textit{and} \quad M_{E_{N+1,N}}^{\zeta;F_{N+1,N}}=\dots = M_{E_{N+p,N}}^{\zeta;F_{N+p,N}}=0,
\end{equation}
\tag{28}
where the sets E_{N+j,N}(z) and polynomials F_{N+j,N}(z) were defined for j=1,\dots,p before Theorem 3. Then there exists a function \mathscr F_{N}^\zeta (z)\in \mathfrak B_N^\zeta such that
\begin{equation}
\frac{\mathscr F_{N}^\zeta (z)-F(z)}{\omega_{E_{N+p}}(z)}\in H({E_{N+p}}).
\end{equation}
\tag{29}
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 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. 
Citation in format AMSBIB
\Bibitem{Bus22}
This publication is cited in the following 2 articles:
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