| Russian Mathematical Surveys |
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Brief communications Asymptotic properties of polynomials defined by shifted orthogonality conditions S. P. Suetin Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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    1.Let $\rho$ be a positive Borel measure with support $E$ consisting of several intervals of the real line, $E=\bigsqcup_{j=1}^p E_j$, and such that $\rho'(x):=d\rho/dx>0$ almost everywhere on $E$. Let $f$ be a complex-valued function in $L_\rho^2(E)$, and let the polynomials $Q_n(x)=Q_n(x;f)$, $\deg Q_n\leqslant n$, be defined by the following ‘shifted’ orthogonality conditions:
$$
\begin{equation}
\int_E Q_n(x)T_k(x;\rho)f(x)\,d\rho(x)=0,\qquad k=n+1,\dots,2n,
\end{equation}
\tag{1}
$$
where the $T_k(x;\rho)=x^k+\dotsb$, $k=0,1,2,\dots$, are the system of orthogonal polynomials on $E$ with respect to $\rho$. With regard to the asymptotic properties of the $Q_n$, we discuss here two cases: 1) $f=\widehat\sigma$ is a Markov function (2), where $\sigma$ is a positive Borel mesure with support on a real compact set $F=\bigsqcup_{s=1}^q F_s$ such that $\sigma'>0$ almost everywhere on $F$, and we have $\operatorname{conv}(E)\cap\operatorname{conv}(F)=\varnothing$ ($\operatorname{conv}(\,\cdot\,)$ denotes the convex hull of the set under consideration); 2) $f$ is a holomorphic function on $E$ extending from $E$ to a domain $\widehat{\mathbb{C}}\setminus\Sigma$ as a multivalued analytic function, where $\Sigma=\Sigma(f)$, $\#\Sigma<\infty$; we denote the class of such functions by $\mathscr A_E(\Sigma)$. The following two well-known problems can be reduced to (1). The first arises in connection with the investigation of the convergence of so-called linear Padé approximants of orthogonal expansions [3], and the second in connection with the asymptotic properties of the Hermite–Padé polynomials of the first type for a set of functions $[f_0,f_1,f_2]$, where $f_0 \equiv 1$, and the pair $f_1$, $f_2$ forms a Nikishin system [7], [5]. It was shown in [9] that the two problems are closely connected. As usual, the problem of the asymptotic properties of polynomials defined by orthogonality conditions has two components, a geometric and an analytic one. In the framework of this approach, first we must find a relevant $S$-compact set corresponding to the problem under consideration [8], [6]. After that the problem of the asymptotic behaviour of orthogonal polynomials itself is solved by the Gonchar–Rakhmanov–Stahl method ($\operatorname{GRS}$-method) [10], [2], on the basis of the properties of the $S$-compact set we have found. 2.In the first case, when
$$
\begin{equation}
f(z)=\widehat{\sigma}(z):=\int_F\frac{d\sigma(x)}{z-x}\,,\qquad z\in\widehat{\mathbb{C}}\setminus F,
\end{equation}
\tag{2}
$$
where $\operatorname{supp}\sigma=F$ (we call this the real case), the $S$-compact set is know a priori: it is equal to the real set $F$. Hence the question of the limiting distribution of the zeros of $Q_n$ is solved in the framework of the standard (real-valued) potential-theoretic equilibrium problems [1], [7]. Given an arbitrary (positive Borel) measure $\nu$, $\operatorname{supp}{\nu}\subset\mathbb{C}$, let
$$
\begin{equation*}
V^\nu(z)=-\displaystyle\int\log|z-\zeta|\,d\nu(\zeta),
\end{equation*}
\notag
$$
$z\in\mathbb{C}\setminus\operatorname{supp}{\nu}$, denote the logarithmic potential of $\nu$. Let $g_E(\zeta,z)$, $z,\zeta\in D:=\widehat{\mathbb{C}}\setminus{E}$, be the Green’s function for the domain $D$ with singularity at $\zeta=z$, and
$$
\begin{equation*}
G_E^\nu(z):=\displaystyle\int g_E(\zeta,z)\,d\nu(\zeta)
\end{equation*}
\notag
$$
be the corresponding Green’s potential of $\nu$. There exists [4] a unique probability measure $\lambda_F$ with support on $F$, $\lambda_F\in M_1(F)$, such that $3V^{\lambda_F}(x)+G^{\lambda_F}_E(x)+ 3g_E(x,\infty)\equiv c_F=\operatorname{const}$, $x\in F$; then $\operatorname{supp}\lambda_F=F$. For a polynomial $Q\in\mathbb{C}[z]\setminus\{0\}$ let $\chi(Q)=\sum_{\zeta:Q(\zeta)=0}\delta_\zeta$ denote the zero-counting measure of $Q$ (taking account of multiplicities). Then the following result holds. Theorem 1. Let $f=\widehat\sigma$, where $\sigma'>0$ almost everywhere on $F$, and let $\operatorname{conv}(E)\cap \operatorname{conv}(F)=\varnothing$. Then the polynomial $Q_n(z;\widehat\sigma)$ has degree precisely $n$ and is uniquely determined by the normalization $Q_n(z)=z^n+\cdots$, all of its zeros are real and lie in the compact set $\operatorname{conv}(F)$. In addition, As usual, here we mean by ‘$\xrightarrow{*}$’ weak-$*$ convergence in the space of measures. Note that when the orthogonality conditions in (1) are standard, rather than shifted, all zeros of the corresponding polynomials lie in the compact set $\operatorname{conv}(E)$, and their limiting distribution is described by the Robin measure for $E$. 3.Let $E=[-1,1]$. Then the following result holds. Theorem 2. Let $f\in\mathscr A_E(\Sigma)$. (1) Then there exists a pair of compact sets $E^*$, $F^*$ such that: (a) $E^*$ is an analytic arc joining the points $\pm1$, $F^*\cap E=\varnothing$, $E^*\cap F^*=\varnothing$, and $F^*$ is formed by a finite system of analytic arcs; (b) the set $D^*:=\widehat{\mathbb{C}}\setminus{F^*}$ is a domain; (c) $f$ extends to $D^*$ as a single-valued meromorphic function; (d) the set $F^*$ and the corresponding equilibrium measure $\lambda_{F^*}$ satisfying the conditions
$$
\begin{equation*}
3V^{\lambda_{F^*}}(z)+G^{\lambda_{F^*}}_{E^*}(z)+3g_{E^*}(z,\infty)\equiv c_{F^*}=\operatorname{const},\ \ z\in F^*,\ \ \textit{and}\ \ \operatorname{supp}\lambda_{F^*}=F^{*},
\end{equation*}
\notag
$$
have the following $S$-property:
$$
\begin{equation}
\begin{aligned} \, &\frac{\partial (3V^{\lambda_{F^*}}+G^{\lambda_{F^*}}_{E^*}+3g_{E^*}(\,\cdot\,,\infty))}{\partial {\rm n}^{+}}(\zeta) \nonumber \\ &\qquad=\frac{\partial(3V^{\lambda_{F^*}}+G^{\lambda_{F^*}}_{E^*}+3g_{E^*}(\,\cdot\,,\infty))}{\partial {\rm n}^{-}}(\zeta),\quad \zeta\in(F^*)^\circ. \end{aligned}
\end{equation}
\tag{4}
$$
(2) The following convergence holds: In (4) we mean by $(F^*)^\circ$ the union of the open arcs whose closures form $F^*$, and $\partial/\partial {\rm n}^{\pm}$ are the derivatives in the direction of the outward normal to $\zeta\in (F^*)^\circ$, taken from the opposite sides of $F^*$. 
Citation in format AMSBIB
\Bibitem{Sue25} |
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