| Russian Mathematical Surveys |
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Brief communications Maximum principle and asymptotic properties of Hermite–Padé polynomials S. P. Suetin Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Russian version:
Uspekhi Matematicheskikh Nauk, 2024, Volume 79, Issue 3(477), DOI: https://doi.org/10.4213/rm10176 
Bibliographic databases:
   1.Hermite–Padé polynomials, which are determined by the successive powers of a prescribed function $f$, turn out to be in demand in topical problems related to the Rayleigh–Schrödinger perturbation theory: see [1] and the bibliography there. In the framework of these physical studies one assumes that Katz’s claim [3] holds, namely, the algebraic functions arising can have only square-root branchings. Thus, the class of these algebraic functions is quite a natural object for an analysis from the standpoint of the asymptotic properties of the corresponding Hermite–Padé polynomials. In [9] we proposed a direct proof of Stahl’s theorem for a class of algebraic functions which have only square-root singularities. Our proof was based on the use of the maximum principle alone. In this paper we consider the class of algebraic functions of the fourth order with quadratic branchings that are generated by the inverse Joukowsky function. It is known [7], [5] that the asymptotic properties of Hermite–Padé polynomials are determined by the Nuttall partitioning of the Riemann surface of $f$ into sheets. Here we consider Hermite–Padé polynomials of the first type for the set $[1,f,f^2]$, where $f\in{\mathbb C}(z,w)$, and the function $w$ is defined by (1). As is known [8], for the set $[1,f_1,f_2]$ the construction of the corresponding three-sheeted Riemann surface with Nuttall partitioning is based on the Nuttall condenser, an ordered pair of compact sets $(E,F)$ with certain properties. The problem of the existence of the Nuttall condenser is quite similar to the problem of the existence of the Stahl compact set. In this paper, in connection with the class of functions based on the representation (1), we consider only the geometrically trivial situation, when both plates of the condenser are known in advance and lie on the real line. On the other hand, as the function $f\in{\mathbb C}(z,w)$ is in general complex valued on the real line, the analytic aspect of the problem is no longer standard. Thus, we cannot use the classical Gonchar–Rakhmanov vector method [2], [6], which is real-valued. 2.Let $\varphi(z)=z+(z^2-1)^{1/2}$ be the inverse Joukowsky function (we consider the branch of $(\,\cdot\,)^{1/2}$ such that $(z^2-1)^{1/2}/z\to1$ as $z\to\infty$). Let $m\in{\mathbb N}$, and let $A_j,B_j\in{\mathbb R}$ be real numbers with the following properties: $A_1<B_1<\dots<A_k<B_k<-1$ and $1<A_{k+1}<B_{k+1}<\dots<A_m<B_m$. Set $E:=[-1,1]$ and
$$
\begin{equation}
w(z):=\prod_{j=1}^m \biggl(\frac{A_j-1/\varphi(z)}{B_j-1/\varphi(z)}\biggr)^{1/2},\qquad z\in D:=\widehat{\mathbb C}\setminus{E}.
\end{equation}
\tag{1}
$$
The function $w$ is algebraic of order four. The corresponding Riemann surface ${\mathfrak R}_4(w)$ is four-sheeted. All branchings of $w$ are of square-root type. The corresponding set of branch points is $\Sigma=\Sigma_{w}=\{\pm1,a_1,b_1,\dots,a_m,b_m\}$, where $a_j=(A_j+1/A_j)/2$ and $b_j=(B_j+1/B_j)/2$, $j=1,\dots,m$. For $\varphi(z)$ selected as above there exists an (analytic) element $w_\infty\in{\mathscr H}(\infty)$ of the function $w$ with the following property: $w_\infty(\infty)=\prod_{j=1}^m\sqrt{A_j/B_j}>0$. This element $w_\infty$ extends to the domain $D$ as a holomorphic function. Let $f\in{\mathbb C}(z,w)$, and let $f_\infty\in{\mathscr H}(\infty)$ be the element of $f$ corresponding to $w_\infty$. Set $F:=\bigsqcup_ {j=1}^m[a_j,b_j]$. Then the pair $(E,F)$ forms the Nuttall condenser corresponding to the pair $w$, $w^2$. For $n\in{\mathbb N}$ let $Q_{n,0}$, $Q_{n,1}$, $Q_{n,2}$ denote the Hermite–Padé polynomials of the first type for the multi-index $(n,n,n)$ and the set $[1,f_\infty,f_\infty^2]$, which are defined by the relation
$$
\begin{equation}
(Q_{n,0}+Q_{n,1}f_\infty+Q_{n,2}f_\infty^2)(z)=O(z^{-2n-2}),\qquad z\to\infty.
\end{equation}
\tag{2}
$$
For an arbitrary (positive Borel) measure $\mu$, $\operatorname{supp}\mu\subset{\mathbb C}$, let denote its logarithmic potential at $z\in{\mathbb C}\setminus\operatorname{supp}\mu$. Let $g_F(\zeta,z)$, where $z,\zeta\in \Omega:=\widehat{\mathbb C}\setminus{F}$, be the Green’s function for $\Omega$ with singularity at $\zeta=z$, and let be the corresponding Green’s potential of the measure $\mu$. It is known [8] that there exists a unique probability measure $\lambda_E$ with support on $E$, $\lambda_E\in M_1(E)$, such that $3V^{\lambda_E}(x)+G^{\lambda_E}_F(x)\equiv c_E={\rm const}$ for $x\in E$. Let $\lambda_F\in M_1(F)$ be the balayage of $\lambda_E$ from $\Omega$ to $F=\partial\Omega$. Given a polynomial $Q\in{\mathbb C}[z]\setminus\{0\}$, let $\chi(Q)=\sum_{\zeta:Q(\zeta)=0}\delta_\zeta$ denote its zero-counting measure.The following result holds (cf. [5], [4], and [10]). Theorem 1. Let $f\in{\mathbb C}(z,w)$, and let $f_\infty\in{\mathscr H}(\infty)$ be an element of $f$ satisfying the above condition. Then for the Hermite–Padé polynomials $Q_{n,j}$ the convergence $n^{-1}\chi(Q_{n,j})\xrightarrow{*}\lambda_F$ holds as $n\to\infty$, $j=0,1,2$, and In (3) we let $f(z^{(0)})$ and $f(z^{(1)})$ denote the values of the four-valued function $f$ on the zeroth and first Nuttall sheets of the four-sheeted surface ${\mathfrak R}_4(w)$. The proof of Theorem 1 is direct and based on the maximum principle alone. 
Citation in format AMSBIB
\Bibitem{Sue24} |
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