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\begin{document}
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\title{Minimal Blashke Products and Optimal Quadratures in $H^\infty$}
\author{K. Yu. Osipenko}

\maketitle

We consider the problem of minimization Blashke products with real nodes $x_j$ of multiplicities $\nu_j$ in integral metric
\begin{equation}\label{1}
\int_a^b\biggl|\prod_{j=1}^n\left(\frac{x-x_j}{1-x_jx}\right)^{\nu_j}\biggr|^qs(x)\,dx\to\inf,
\quad-1<x_1<\ldots<x_n<1,
\end{equation}
where $-1\le a<b\le1$, $1\le q<\infty$, and $s(x)$ is nonnegative weight function which is continuous in $(a,b)$. In \cite{1} it was proved that the solution of the problem \eqref{1} existed but it was not unique in general case (for $s(x)=1$ the existence was also proved in \cite{2}).

The problem \eqref{1} can be transformed to the follows:
$$\int_{-1}^1\biggl|\prod_{j=1}^n\left(\frac{t-t_j}{1-t_jt}\right)^{\nu_j}\biggr|^qp(t)\,dt
\to\inf,\quad-1<t_1<\ldots<t_n<1.$$
This problem has a unique solution for all $k$ sufficiently small. We find it for some weight functions.

\begin{theorem}
Let $\nu_1=\ldots=\nu_n=1$, $q>1$. Then for the weight functions
$$p_1(t)=\frac1{\sqrt{(1-t^2)(1-k^2t^2)}},\quad p_2(t)=p_1(t)\left(\frac{1-t^2}{1-k^2t^2}\right)^{q/2},$$
and all $k$ sufficiently small the unique system of optimal nodes are
\begin{align*}
\ov t_1&=\left\{\sn\left[\left(\frac{2j-1}n-1\right)K,k\right]\right\}_{j=1}^n,\\
\ov t_2&=\left\{\sn\left[\left(\frac{2j}{n+1}-1\right)K,k\right]\right\}_{j=1}^n,
\end{align*}
correspondingly; here
$$K=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.$$
\end{theorem}

The received results are applied to the problem of finding optimal quadrature formulae in $H^\infty(G)$, the set of bounded analytic functions in $G$ with the norm
$$\|f\|_\infty=\sup_{z\in G}|f(z)|$$
(cf.\ \cite{3}). Set
\begin{multline*}
R(\mu,p,G)\\
=\infp_{a\le x_1<\ldots<x_n\le b}\,\infp_{a_{jm}}\,\sup_{\|f\|_\infty\le1}\biggl|
\int_a^bf(x)p(x)\,dx-\sum_{j=1}^n\sum_{m=0}^{\mu_j-1}a_{jm}f^{(m)}(x_j)\biggr|,
\end{multline*}
$[a,b]\subset G$. Let $\mbox{Э}_c$, $c>1$, be the interior of the ellipse given by the equations $2x=(c+c^{-1})\cos\theta$, $2y=(c-c^{-1})\sin\theta$, $0\le\theta\le2\pi$.

\begin{corollary}
Let $[a,b]=[-1,1]$ and $q$ is an even number. Then for $q-1\le\mu_j\le q$, $j=1,\ldots,n$, and for all $c$ sufficiently large the asymptotic equation
$$R\left(\mu,\frac1{\sqrt{1-x^2}},\mbox{Э}_c\right)=\frac{\sqrt\pi\Gamma\left(\dfrac{q+1}2\right)
2^q}{\Gamma\left(\dfrac q2+1\right)}c^{-nq}+O\left(c^{-q(n+4)}\right)$$
holds. The only system of optimal nodes is
$$x_j=\cos\frac{2j-1}{2n}\pi,\quad j=1,\ldots,n.$$
\end{corollary}

\begin{thebibliography}{99}
\selectlanguage{russian}
\bibitem{1} Осипенко К.Ю. О произведениях Бляшке, наименее уклоняющихся от нуля. Матем. заметки, to appear.
\selectlanguage{english}
\bibitem{2} Uluchev R. An extremal problem in the set of Blashke products with fixed multiplicities of the zeros // Serdica Bulg. math. publ., 1988, v.~14, N~1. p.~98--101.
\selectlanguage{russian}
\bibitem{3} Осипенко К.Ю. О наилучших и оптимальных квадратурных формулах на классах ограниченных аналитических функций. Изв. АН СССР. Сер. матем., 1988, 52, \No1. с.~79--99.

\end{thebibliography}
\end{document}

