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\begin{document}

\title[Minimal Blashke Products]{Minimal Blashke Products and Optimal Quadrature Formulae in $H^\infty$}
\author{K. Yu.\ Osipenko}

\maketitle

\vspace{-20pt}

\begin{centerline}
{\it Moscow Aviation Technology Institute, 103767, Moscow, Russia}
\end{centerline}

\bigskip
\bigskip

We study the extremal problem
\begin{equation}\label{1}
\int_a^bs(x)|B(x,\ov x)|^q\,dx\to\inf,\quad-1<x_1<\ldots<x_n<1,
\end{equation}
on the set of all Blashke products with fixed multiplicities $\nu_j$ of the zeros $x_j$,
$$B(x,\ov x)=\prod_{j=1}^n\left((x-x_j)/(1-x_jx)\right)^{\nu_j},$$
where $-1\le a<b\le1$, $1\le q<\infty$, $\ov x=(x_1,\ldots,x_n)$, and $s(x)\not\equiv0$ is nonnegative weight function, continuous in the interval $(a,b)$.

Similar problems for $q=\infty$ connected with the optimal recovery of bounded analytic functions in the unit disk are considered in \cite{1,2,3,4,5}.

\begin{theorem}[\hspace{-0.5pt}\cite{6}]\label{T1}
For each fixed $1\le q<\infty$ there exists a solution $\ov x=(x_1,\ldots,x_n)$ of the problem \eqref{1}. Moreover, every such system of points satisfies $a<x_1<\ldots<x_n<b$.
\end{theorem}

The particular case $s(x)\equiv1$ was studied before in \cite{10}.

If $-1<a<b<1$ then under the conformal mapping of the unit disk onto the disk of radius $k^{-1/2}$ the interval $[a,b]$ goes to $[-1,1]$ ($k$ is uniquely defined by $a$ and $b$) and the problem \eqref{1} transforms to the following one:
\begin{equation}\label{2}
\int_{-1}^1p(t)|Q(t,\ov t,k)|^q\,dt\to\inf,\quad-1<t_1<\ldots<t_n<1,
\end{equation}
where $\ov t=(t_1,\ldots,t_n)$ and
$$Q(t,\ov t,k)=\prod_{j=1}^n((t-t_j)/(1-kt_jt))^{\nu_j}.$$
Note that in case $k=0$ the problem \eqref{2} is in fact the problem on polynomials of least deviation with fixed multiplicities of the zeros considered in \cite{7}.

The problem \eqref{2}, and hence the problem \eqref{1}, may has not a unique solution.

\begin{theorem}\label{T2}
Let $n=1$, $\nu q>1$. The problem \eqref{2} has a unique solution for each weight function if and only if
\begin{equation}\label{3}
0\le k\le\frac{\nu q-1}{\nu q+1}.
\end{equation}
\end{theorem}

\begin{proof}
We set
$$\varphi(t_1)=\int_{-1}^1p(t)\left|\frac{t-t_1}{1-kt_1t}\right|^{\nu q}\,dt.$$
If $\varphi'(t_1)=0$ then
\begin{multline*}
\varphi''(t_1)=\varphi''(t_1)+\frac{2kt_1}{1+kt_1^2}\varphi'(t_1)\\
=\nu q\int_{-1}^1p(t)\left|\frac{t-t_1}{1-kt_1t}\right|^{\nu q}\frac{1-kt^2}{(t-t_1)^2(1-kt_1t)^2}\biggl[\nu q(1-kt^2)\\
-\frac{1-kt_1^2}{1+kt_1^2}(1+kt^2)\biggr]\,dt\\
\ge[\nu q-1+(\nu q+1)k]\int_{-1}^1p(t)\left|\frac{t-t_1}{1-kt_1t}\right|^{\nu q}\frac{1-kt^2}{(t-t_1)^2(1-kt_1t)^2}\,dt.
\end{multline*}
Therefore, if inequalities \eqref{3} are satisfied, we have $\varphi''(t_1)>0$ for all $t_1$ with $\varphi'(t_1)=0$. This yields uniqueness of the optimal node $t_1$ for each weight function provided \eqref{3} is fulfilled.

Suppose now that $\dfrac{\nu q-1}{\nu q+1}<k\le1$ and set $p(t)=|t|^\alpha$, $\alpha>0$. The function $\varphi(t_1)$ is even in this case. So, if the optimal node $t_1$ is unique then $t_1=0$. It is not difficult to see that for sufficiently large $\alpha$ we obtain $\varphi(0)<0$ and hence the function $\varphi$ has not minimum at $t_1=0$. The theorem is proved.
\end{proof}

Let us denote
$$r=q\min_{1\le j\le n}\nu_j,\quad N=\sum_{j=1}^n\nu_j,\quad\gamma_m(p,q)=\inf_{t_j}\int_{-1}^1p^*(t)
\biggl|\prod_{j=1}^n(t-t_j)\biggr|^q\,dt,$$
where $p^*$ is the normalized weight function
$$p^*(t)=p(t)\bigg/\int_{-1}^1p(t)\,dt.$$

\begin{theorem}[\hspace{-0.5pt}\cite{6}]\label{T3}
If $r>1$ nd
\begin{equation}\label{4}
0\le k\le(r-1)/\left(9r-7+qN2^{qN+1}\gamma^{-1}_N(p,q)\right)
\end{equation}
the problem \eqref{2}has a unique solution.
\end{theorem}

Let us set
\begin{gather*}
p_1(t)=\left((1-t^2)(1-k^2t^2)\right)^{-1/2},\\ p_2(t)=p_1(t)\left((1-t^2)/(1-k^2t^2)\right)^{q/2},\\
\ov u_1=\left\{\sn\left[\left(\frac{2j-1}n-1\right)K,k\right]\right\}_{j=1}^n,\\
\ov u_2=\left\{\sn\left[\left(\frac{2j}{n+1}-1\right)K,k\right]\right\}_{j=1}^n.
\end{gather*}
Denote by $K$ and $\Lambda_m$ the complete elliptic integrals of the first kind with modulus $k$ and $\lambda_m$, respectively.

\begin{theorem}[\hspace{-0.5pt}\cite{6}]\label{T4}
Suppose that $\nu_1=\ldots=\nu_n=1$, $1<q<\infty$, and
$k\in[0,k_i]$, where
$$k_i=\frac{(q-1)\Gamma\left(\dfrac{q+1}2\right)}{(9q-7)\Gamma\left(\dfrac{q+1}2\right)+2\sqrt\pi q\Gamma\left(\dfrac q2+1\right)n2^{q(2n-2+i)}},\quad i=1,2.$$
Then
\begin{multline}\label{5}
\inf_{t_j\in\mathbb C}\int_{-1}^1p_i(t)\biggl|\prod_{j=1}^n(t-t_j)/(1-k\ov t_jt)\biggr|^q\,dt=\int_{-1}^1p_i(t)\left|Q(t,\ov u_i,k)\right|^q\,dt\\
=\frac{2d^q_{n+i-1}(k)K}{\Lambda_{n+i-1}}I_q(\lambda_{n+i-1}),\quad i=1,2,
\end{multline}
\begin{gather*}
d_m(k)=\prod_{j=1}^{[m/2]}\sn^2\left(\frac{2j-1}mK,k\right),\quad\lambda_m=k^md_m^2(k),\\
I_q(\lambda)=\int_0^1x^q(1-x^2)^{-1/2}(1-\lambda^2x^2)^{-1/2}\,dx.
\end{gather*}
The nodes $\ov u_1$ and $\ov u_2$ for which the infima \eqref{5} are attained are unique.
\end{theorem}

For $\nu_1=\ldots=\nu_n=1$, $k=0$  the rational functions $Q(t,\ov u_i,k)$ coincide with the Chebyshev polynomials
\begin{gather*}
Q(t,\ov u_1,k)=2^{1-n}\cos(n\arccos t),\\
Q(t,\ov u_2,k)=2^{-n}(1-t^2)^{-1/2}\sin((n+1)\arccos t).
\end{gather*}

\begin{corollary}\label{C1}
Suppose that $\nu_1=\ldots=\nu_n=1$, $1<q<\infty$. If $k\in[0,k_i)$ then
\begin{multline}\label{6}
\inf_{z_j\in\mathbb C}\int_{-\sqrt k}^{\sqrt k}s_i(t)\biggl|\prod_{j=1}^n(z-z_j)/(1-\ov z_jz)\biggr|^q\,dz=\int_{-\sqrt k}^{\sqrt k}s_i(t)\left|B(z,Z_i)\right|^q\,dz\\
=\frac{2\lambda^{q/2}_{n+i-1}}{k^{(i-1)q/2}\Lambda_{n+i-1}}I_q(\lambda_{n+i-1}),\quad i=1,2,
\end{multline}
where $Z_1=k^{1/2}\ov u_1$, $Z_2=k^{1/2}\ov u_2$, $s_1(z)=\left((k-z^2)(1-kz^2)\right)^{-1/2}$,
$$s_2(z)=s_1(z)\left((k-z^2)(1-kz^2)\right)^{q/2}.$$
The nodes $Z_1$ and $Z_2$ for which the infima \eqref{6} are attained are unique.
\end{corollary}

Note that the nodes $\ov u_1$ and $\ov u_2$ satisfy the necessary extremum conditions for all $k\in[0,1)$. But the problem whether the equalities \eqref{5} (or \eqref{6}) hold for all $k\in[0,1)$ remains still open.

Consider now the problem of optimal quadrature formula in the class $H^\infty(G)$ of functions analytic in the domain $G$ and such that
$$\|f\|_\infty=\sup_{z\in G}|f(z)|<\infty.$$
We define the error of the optimal quadrature formula by 
\begin{multline}\label{7}
R(\mu,p,G)\\
=\infp_{a\le x_1<\ldots<x_n\le b\vphantom{\|f\|_\infty}}\infp_{a_j\vphantom{\|f\|_\infty}}\sup_{\|f\|_\infty\le1}
\biggl|\int_a^bp(x)f(x)\,dx-\sum_{j=1}^n\sum_{m=0}^{\mu_j}a_{jm}f^{(m)}(x_j)\biggr|,
\end{multline}
$\mu=(\mu_1,\ldots,\mu_n)$, $[a,b]\subset G$. The nodes for which the infimum \eqref{7} is attained are called {\it optimal nodes}. It is well known \cite{8} (also \cite{9}), that if $G=D:=\{z\in\mathbb C:|z|<1\}$ then
$$R(\mu,p,D)=\inf_{a\le x_1<\ldots<x_n\le b}\int_a^bp(x)\prod_{j=1}^n\left((x-x_j)/(1-x_jx)\right)^{\nu_j}\,dx,$$
$\nu_j=2[(\mu_j+1)/2]$. It follows from Theorem~\ref{T3} that when $a=-b=k^{1/2}$ the optimal nodes are unique for sufficiently small $k$. Using Corollary~\ref{C1} we find the optimal nodes for the weight functions $s_1(z)$ and $s_2(z)$. 

Mapping conformally the unit disk $D$ onto a domain $G$, so that the interval $[-\sqrt k,\sqrt k]$ goes to $[a, b]\subset G$, we obtain some results on the problem \eqref{7} for the corresponding weight functions. For example, let us map conformally the unit disk onto the ellipse $E_c$ with foci at $\pm1$ and sum of semi-axis $c$, so that the interval $[-\sqrt k,\sqrt k]$ transforms on $[-1,1]$. If we denote by $K'$ the complete elliptic integral of the first kind corresponding to the modulus $(1-k^2)^{1/2}$ we have
$$c=\exp\left(\frac{\pi K'}{4K}\right),\quad\sqrt k=\frac2c\cdot\biggl(\sum_{m=0}^\infty c^{-4m(m+1)}\biggr)\Big/\biggl(1+2\sum_{m=1}^\infty c^{-4m^2}\biggr).$$
Letting $c_i=\left(\dfrac{\pi K_i'}{4K_i}\right)$, $i=1,2$ ($k_1$, $k_2$ are defined as in Theorem~\ref{T4}),
$$p_1(x)=(1-x^2)^{-1/2},\quad p_2(x)=p_1(x)\sn^q\left(\frac{2K}\pi\arccos x,k\right),$$
we get the following assertion. 

\begin{corollary}\label{C2}
Suppose that $q$ is an even number and $q-1\le\mu_j\le q$, $j=1,\ldots,n$. Then
$$R(\mu,p_i,E_c)=\frac{\sqrt\pi\Gamma\left(\dfrac{q+1}2\right)2^q}
{k^{(i-1)q/2}\Gamma\left(\dfrac q2+1\right)}c^{-(n+i-1)q}+O\left(c^{-n(q+4)}\right)$$
for all $c\ge c_i$, $i=1,2$, and
$$x_j=\begin{cases}\cos\dfrac{2j-1}{2n}\pi,&j=1,\ldots,n,\mbox{ if }i=1,\\[10pt]
\cos\dfrac j{n+1}\pi,&j=1,\ldots,n,\mbox{ if }i=2,\end{cases}$$
are the unique optimal nodes.
\end{corollary}

\begin{thebibliography}{99}

\bibitem{1} K.Yu.~Osipenko, Optimal interpolation of analytic functions, {\it Mat.\ Zametki} {\bf12} (1972), 465--476.

\bibitem{2} K.Yu.~Osipenko, Best approximation of analytic functions on the basis of finite number functional values, {\it Mat.\ Zametki} {\bf19} (1976), 29--40.

\bibitem{3} K.Yu.~Osipenko, On optimal extrapolation and interpolation of fuzzy analytic functions, {\it Anal. Math.} {\bf13} (1987), 199--210.

\bibitem{4} B.D.~Bojanov, Comparison theorems in optimal recovery, In ``Optimal Algorithms'' 
(B1.~Sendov ed.), Sofia, 1986, 15--50.

\bibitem{5} B.D.~Bojanov and G.R.~Grozev, A note on the optimal recovery of functions in $H^\infty$, {\it J.\ Approx.\ Theory} {\bf53} (1988), 67--77.

\bibitem{6}	K.Yu.~Osipenko, On Blashke products of least deviation, {\it Mat.\ Zametki} (to appear).

\bibitem{7}	B.D.~Bojanov, Extremal problems in a set of polynomials with fixed miltiplicities of zeros, {\it C.R.\ Acad.\ Bulg.\ Sci.} {\bf31} (1978), 377--380.

\bibitem{8} B.D.~Bojanov, On the existence of optimal quadrature formulae for smooth functions, {\it Calcolo} {\bf16} (1979), 61--70.

\bibitem{9} K.Yu.~Osipenko, On the best and optimal quadrature formulae for classes of bounded analytic functions, {\it Izv.\ Acad.\ Nauk SSSR Ser.\ Mat.} {\bf52} (1988), 79--99.

\bibitem{10} R.K.~Uluchev, An extremal problem in the set of Blashke products with fixed multiplicities of the zeros, {\it SERDICA Bulg.\ Math.\ Publ.} (1988), 98--101.
\end{thebibliography}
\end{document}