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\begin{document}

\begin{center}{\large\bf SCHWARZ LEMMA AND OPTIMAL RECOVERY OF
FUNCTIONS IN $H^2$}
\end{center}

\vspace{3mm}
\begin{center}
{\bf Konstantin Yu. Osipenko,$^1$, Michael I. Stessin$^2$}
\end{center}

\begin{center}\textit{$^1$ "MATI" - Russian State Technological University, Orshanskaya Str. 3, Moscow, 121552, Russia}
\end{center}
\begin{center}
\textit{$^2$ Department of Mathematics and Statistics, University at Albany,
Albany, NY 12222, USA\\
e-mail: $^{1}$kosipenko@yahoo.com, $^2$stessin@math.albany.edu
}\end{center}
\begin{abstract}

This paper is devoted to optimal method of the reconstruction of the value of the function $f$ whose values on some sets are known.

\end{abstract}


{\it AMS Subject Classification: 30H05, 30E10}

\medskip
{\bf Keywords. Extremal problems, optimal recovery, Hardy spaces, Shwartz lemma}

\bigskip


Let $D \subset C^k$ be a domain, $\nu $ be a probability measure on $\overline{D}$ and $X$ be
a closed subspace of $L^2(\nu).$ Consider $D_0,\dots ,D_n \subset D$ and probability
measures $\mu_0,\dots , \mu_n$ on $D_0,\dots ,D_n$ respectively. We suppose that $X \subset L^2(\mu_j), j =
0, 1,\dots , n.$ We allow one of $D_j$ to coincide with $D.$ In this case we
assume that $\mu_j$ coincides with $\nu.$

Write ${\cal D} = (D_0,\dots,D_n)$, $\mu = (\mu_0,\dots , \mu_n)$, $\mu = (\mu_1,\dots , \mu_n)$, $y =(y_1,\dots , y_n)$.

\section{Optimal recovery problem}

Given $y_1,\dots , y_n$ defined on $D_1,\dots ,D_n$ such that
$$
\|f_j -  y_j\|_{L^2(\mu j )} \le  \delta_j,\quad  j = 1,\dots , n,
$$
we are to reconstruct $f$. Here $f_j$ is the restriction of $f$ to $D_j$ and $\delta_j \ge 0$, $j = 1,\dots , n$ are accuracy levels. In particular, $\delta_j = 0$ means that $f$ is
known precisely on $D_j$.

A recovery algorithm (method, procedure, etc.) is an operator
$$
A\colon L^2(\mu_1)\times \dots \times L^2(\mu_n) \to L^2(\mu_0).
$$
We consider $A(y)$, $y = (y_1,\dots , y_n),$ to be the recovered value of $f$ on
$D_0.$ At this point we impose no conditions on $A.$

The maximal possible error of a method $A$ is
$$e(X,{\cal D},\mu,\delta,A)=\sup_{\substack{f\in X,\  y\in L^2(\mu_1)\times\dots\times L^2(\mu_n)\\\|f_j-y_j\|_{L^2(\mu_j)}\le \delta_j, j=1,\dots,n}}\|f_0 - A(y)\|_{L^2(\mu_0)}$$

The optimal recovery error is
$$
E(X,{\cal D}, \mu, \delta) =
\inf_{A\colon L^2(\mu_1)\times \dots \times L^2(\mu_n)\to L^2(\mu_0)}e(X,{\cal D}, \mu, \delta, A).
$$
A method ${\hat A}$ such that
$$
E(X,{\cal D}, \mu, \delta) =
e(X,{\cal D}, \mu, \delta, {\hat A})
$$
is called an {\it optimal recovery method.}

The problem of finding an optimal recovery method (and sometimes
an extremal function at which the optimal recovery error is attained)
is usually referred to as {\it optimal recovery problem.}

\section{Extremal problem}

The optimal recovery problem is closely related to the following extremal problem. Find
\begin{equation}\label{e1}
\|f_0\|_{L^2(\mu 0)}\to\max,\quad \|f_j\|^2_{L^2(\mu_j)}\le\delta_j^2,\ j = 1,\dots , n,\quad f\in X.
\end{equation}

A special case of this extremal problem is when $D$ is the unit disk $\mathbb{D}$,
$\mu_0$ and $\mu_1$ are point masses and $\mu_2$ is the normalized Lebesgue measure
on the unit circle. Here the problem turns into
$$
\max\{|f(a_0)|: |f(a_1)|\le \delta_1,\,  \|f\|_{H^2}\le \delta_2\},
$$
which might be viewed as a version of the classical Schwarz lemma.
Here we consider another variant of Scwarz Lemma. Let $a \in  \mathbb{D}$ and
$\Gamma$ be a circle inside of the unit disk, $\mu$ be the normalized Lebesgue
measure on $\Gamma$, and $\mu > 0.$ Find
\begin{equation}\label{e2}
\sup \left \{\int_{\Gamma} |f|^2d\mu: f\in H^2,\, \|f\|^2_{H^2}\le 1,\, |f(a)|\le \delta \right \}.
\end{equation}


We will consider the case when the circle $\Gamma$ passes through the origin
and its center lies on the real axis, so that
$$
\Gamma = \{ z\in \mathbb{C}: |z -\rho|=\rho\},\quad0<\rho <1/2.$$

The corresponding optimal recovery problem is:
{\it Reconstruct a Hardy function $f$  on the circle $\Gamma$ from its value at $a$ given with some tolerance.}

There are several papers where similar problems were considered for Hardy and Bergman spaces in connection with optimal recovery in both one and several dimensional cases (see, for example, \cite{OsS1}--\cite{OsS3}).


\section{Euler equation for the general problem}

Let $K(z,w)$ be the reproducing kernel of $X.$ Write
$$
\tilde{\mu} = -\mu_0 + \sum_{j=1}^n\lambda_j\mu_j.
$$
Then $\tilde{\mu}$ is a regular measure on $D$ and every function from $X$ is square-
integrable with respect to $\tilde{\mu}$. For $w \in D$ we introduce
$$
d\tilde{\mu}_w(z) = K(z,w)d \tilde{\mu}(z).
$$
Obviously every function from $X$ is $\tilde{\mu}_w$-integrable.

We further define
$$
\tau_w^{\lambda}(z) = \int_D K(z \tau ) d\tilde{\mu}_w(\tau ).
$$

\begin{theorem}
If $\tilde{f} \in X$ is a solution of the general extremal problem
above, then there exists a non-negative vector $\widehat{\lambda} = (\widehat{\lambda}_1,\dots ,\widehat{\lambda}_n)$ such that
$$
\widehat{f} = ( \spann \{\tau_{w}^{\widehat{\lambda}}, w\in D\})^{\perp},
$$
and
$$
\widehat{\lambda}_j(\|f\|_{L_2(\mu_j)}-\delta_j) = 0,\,  j = 1,\dots , n.
$$
\end{theorem}

We say that a non-negative vector $ \lambda = (\lambda_1,\dots , \lambda_n)$ belongs to the
spectrum of the problem, if there exists an admissible for this problem
function $f \in X$ such that
\begin{align*}
1.\ & \lambda_j(\|f\|_{L^2(\mu_j )}-\mu_j) = 0.\\
2.\ & f \in (\spann\{\tau_w^{\lambda}: w \in D\})^{\perp}.
\end{align*}
In this case we call $f$ a {\it spectral function.}

\begin{theorem}
Let $\Lambda$ be the spectrum of the problem. Then
\begin{equation}\label{e3}
 \sup_{\substack{f\in X\\ \|f\|_{L_2(\mu_j)}\le \delta_j,\ j=1,\dots, n}}
= \sup_{\lambda \in \Lambda}\sum_{j=1}^n\lambda_j\delta_j^2.
\end{equation}
\end{theorem}

We call a spectral point $(\widehat{\lambda_1},\dots , \widehat{\lambda_n})$ {\it extremal,} if the maximum of the
right-hand side of \eqref{e3} is attained at $(\widehat{\lambda_1},\dots , \widehat{\lambda_n}).$

\section{Spectrum of the Schwarz Lemma}

Here we have.
$$
\tau_w^{\lambda} = -{1\over \pi}
\int_{\Gamma}{1\over 1-z\overline{\tau}}\cdot {1\over 1-\tau\overline{w}}\cdot {|d\tau |\over |\tau - \rho|} +
$$
$$
\lambda_1{1\over 1-z\overline{a}}\cdot {1\over 1-a\overline{w}} + {\lambda_2\over 2\pi}\int_{|\tau|=1}{1\over 1-z\overline{\tau}}\cdot {1\over 1-\tau\overline{w}}|d\tau |=
$$
$$
-{1\over 1-z\rho -\rho \overline{w}}+{\lambda_1\over (1-z\overline{a})(1-a\overline{w})}+{\lambda_2\over 1-z\overline{w}}.
$$

By Theorem 1 every extremal function satisfies the following equation
$$
{1\over 1-\rho w}f\left( {\rho\over 1-\rho w}\right) = \lambda_1{f(a)\over 1-\overline{a}w}
$$
for some $\lambda_1, \lambda_2 \ge 0$ and all $w\in \mathbb{D}.$
Let
$$
b = {1-\sqrt{1-4\rho^2}\over 2\rho }.
$$
Then $b$ is the Denjoy-Wolff point of the following self-mapping of $\mathbb{D}$
$$
z \to {\rho \over 1-\rho z},
$$
and the disk bounded by the circle $\Gamma$ is a hyperbolic neighborhood of
$b.$

Consider the following functions
$$
\varphi_j(z) = {\sqrt{1-b^2}\over 1-bz} \left ( b-z\over 1-bz\right )^j,\quad j=0,1,\dots\,\,.
$$

These functions form an orthonormal basis of $H^2,$ and they are eigenfunctions of the operator
$$
Tf(z) = {1\over 1-\rho z}f \left ({\rho \over 1-\rho z} \right ),
$$
and the corresponding eigenvalues are
%\begin{equation}\label{e4}
$$\alpha_j = {b^{2j}\over 1-\rho b}.$$
%\end{equation}

\begin{theorem}
Let $a\neq b$.

$1$. If
$$
\left | a- {\rho \over 1-\rho^2}\right | \ge {\rho^2\over 1-\rho^2},
$$
or
$$
\delta > {\sqrt {|a|^2\rho^2 - |\rho-a|^2}\over a\rho +\overline{a}\rho - |a|^2},
$$
then the spectrum of Schwarz Lemma extremal problem consists of two
parts $\Lambda = \Lambda_1\cup \Lambda_2$, where
$$
\Lambda_1 = \{(0, \alpha_j) : |\varphi_j(a)| \le \delta\},
$$
$$
\Lambda_2 = \{(\lambda_1, \lambda_2) : \lambda_1, \lambda_2 > 0,\
F(\lambda_2) = \delta^{-2},\ \lambda_1 = h(\lambda_2)\},
$$
where
$$
F(\lambda) = \sum_{j=0}^{\infty}{|\varphi_j(a)|^2\over (a_j-\lambda)^2}h^2(\lambda),\quad
h(\lambda) = \left (\sum_{j=0}^{\infty}{|\varphi_j(a)|^2\over a_j-\lambda}\right )^{-1}.
$$
$2$. If
$$
\left | a- {\rho \over 1-\rho^2}\right | < {\rho^2\over 1-\rho^2},
$$
and
$$
\delta \le {\sqrt {|a|^2\rho^2 - |\rho-a|^2}\over a\rho +\overline{a}\rho - |a|^2},
$$
then the spectrum includes in addition the point
$$
\Lambda_3 =
\left \{ \left ({a\rho +\overline{a}\rho - |a|^2\over \rho^2},0 \right )\right \}.
$$
\end{theorem}

\begin{theorem}
Let $a = b$,
$$
\Lambda_1 = \{(0, \alpha_j):\, j = 1, 2,\dots , \},
$$
$$
\Lambda_2 = \{((1-b^2)(\alpha_0-\alpha_j), \alpha_j):\, j = 1, 2,\dots , \}.
$$
Then the spectrum of problem is $\Lambda = \Lambda_1\cup \Lambda_2,$  if $\delta <\dfrac1{\sqrt{1-b^2}}$,  and
$\Lambda = \Lambda_1\cup \Lambda_2 \cup\{(0,\alpha_0)\},$  if $\delta \ge \dfrac1{\sqrt{1-b^2}}.$
\end{theorem}

It turns out that $\Lambda_2$ is the most important part of the spectrum.

\begin{prop}
If $a$ lies outside $\Gamma$, then $F(\lambda) \to \infty$  as $\lambda \to 0.$
\end{prop}

This Proposition implies that if $a$ lies outside $\Gamma$, then $\Lambda_2$ contains
only finite number of points.

Now we will use Theorem 2 to describe the extremal points of the
spectrum.

\begin{prop}
If $\delta \ge |\varphi_0(a)|,$ then   $(0, \alpha_0)$ is the extremal point of the spectrum.
\end{prop}

\begin{prop}
If $a = b$ and $\delta < 1/\sqrt {1-b^2}$, then the extremal spectral
point  is
$$
(\widehat{\lambda}_1, \widehat{\lambda}_2) = ((1- b^2)(\alpha_0 - \alpha_1), \alpha_1).
$$
\end{prop}

\begin{prop}
If $\delta < |\varphi_0(a)|$, then $\Lambda_1$ does not contain extremal
spectral points.
\end{prop}

Note that the function
%\begin{equation}\label{e7}
$$ g(\lambda) = \sum_{j=0}^{\infty}{|\varphi_j(a)|^2 \over \alpha_j-\lambda}$$
%\end{equation}
is monotone and increases from $-\infty$ to $+\infty$ when $\lambda \in (\alpha_{j+1}, \alpha_j).$ Let
$\zeta_j$ be the only zero of $g$ on the interval $(\alpha_{j+1}, \alpha_j).$

\begin{prop}
Let $a\neq b.$ If $\delta \le |\varphi_1(a)|,$ then the extremal spectral point $(\widehat{\lambda}_1, \widehat{\lambda}_2)$ is unique, belongs to $\Lambda_2$ and is determined by the
condition $\zeta_0 < \widehat{\lambda}_2 < \alpha_0.$
\end{prop}

\begin{prop}
Assume that $|\varphi_1(a)| <\delta < |\varphi_0(a)|$ and
$$
\gamma = \left |{b-a\over 1-ab} \right | \ge b^{2/3},
$$
then the conclusion of Proposition $5$ is valid, that is, the extremal spectral point $(\widehat{\lambda}_1, \widehat{\lambda}_2)$ is unique, belongs to $\Lambda_2$ and is determined by the
condition $\zeta_0 < \widehat{\lambda}_2 < \alpha_0.$
\end{prop}

\section{Optimal Recovery Method}

To construct optimal recovery methods we need the following result (several results of this type may be found in \cite{MM}, \cite{MO2}, \cite{Os}).

\begin{theorem}
Assume that there exist $\widehat{\lambda}_j\ge 0,\, j = 1,\dots , n,$ such that
the value of the extremal problem
$$\|f_0\|^2_{L_2(\mu_0)}\to \max,\quad\sum_{j=1}^{\infty}\widehat{\lambda}_j\|f_j\|^2_{L_2(0,\mu_j)} \le\sum_{j=1}^{\infty}\widehat{\lambda}_j\delta_j^2,\quad f\in X,
$$
is the same as in \eqref{e1}. Moreover, assume that for every $\tilde{y} =
(\tilde{y}_1,\dots , \tilde{y}_n) \in Y_1 \times \cdots \times Y_n,$ where $Y_j$ are dense in $L^2(\mu_j),$ there exists $f_{\tilde{y}}$ which is a solution of the extremal problem
$$
\sum_{j=1}^{\infty}\widehat{\lambda}_j\|f_j-\tilde{y}_j\|^2_{L_2(0,\mu_j)} \to \min,\quad  f\in X.
$$
Moreover, let $\widehat{A}\colon L^2(\mu_1)\times \cdots \times L^2(\mu_n)\to L^2(\mu_0)$ be a linear continuous
operator, where the norm in $L^2(\mu_1)\times \cdots \times L^2(\mu_n)$ is defined as
$$
\|y\|=\left (\sum_{j=1}^{n}\|y_j\|^2_{L^2(\mu_j)}\right )^{1/2},
$$
such that for all $\tilde{y} =
(\tilde{y}_1,\dots , \tilde{y}_n) \in Y_1 \times \cdots \times Y_n,$
$$
\widehat{A}(\tilde{y})=(f_{\tilde{y}})_0.
$$
Then
$$
E(X,{\cal D}, \mu, \delta) = \sup_{\substack{f\in X\\ \|f_j\|_{L^2(\mu_j)}\le \delta_j,\ j=1,\dots, n}}\|f_0\|_{L^2(\mu_0)}
$$
and the method $\widehat{A}(y)$ is optimal.
\end{theorem}


We will apply Theorem 5 to the construction of optimal recovery
method for the Schwarz Lemma type problem considered above.

Consider the extremal problem
\begin{equation}\label{e9}
\int_{\Gamma}|f|^2d\mu \to \max,\quad \widehat{\lambda}_1|f(a)|^2 + \widehat{\lambda}_2\|f\|^2_{H^2} \le \widehat{\lambda}_1\delta^2+\widehat{\lambda}_2,\quad
f\in H^2,
\end{equation}
where as before $\mu $ is the normalized
Lebesgue measure on $\Gamma$ and $(\widehat{\lambda}_1,\widehat{\lambda}_2)$ is an extremal spectral point for problem \eqref{e2}.

\begin{prop}
Suppose that either

\noindent $1$. $a \neq b$ and $\delta \le |\varphi_1(a)|,$ or $|\varphi_1(a)| <\delta <  |\varphi_0(a)|$ and
$
\gamma = \left |\dfrac{b-a}{1-ab}\right | \ge b^{2/3},
$

\noindent or

\noindent $2$. $a = b$ and $\delta <  \varphi (b) = 1/\sqrt{1-b^2}.$

Then the values of extremal problems
\eqref{e2} and \eqref{e9} are the same.
\end{prop}

\begin{theorem}
Suppose that one of the following conditions is satisfied

$1$. $\delta \ge |\varphi_0(a)|,$

$2$. $\delta \le |\varphi_1(a)|,$

$3$. $ |\varphi_1(a)|<\delta < |\varphi_0(a)|,\quad \gamma \ge b^{2/3},$

$4$. $a = b,$

\noindent and $(\widehat{\lambda}_1, \widehat{\lambda}_2)$ is the corresponding extremal spectral point. Then the error
of optimal recovery is given by
$$
\sqrt{\widehat{\lambda}_1\delta^2+\widehat{\lambda}_2}$$
and the method
\begin{equation}\label{e10}
\widehat{A}(y)(z) = {\widehat{\lambda}_1y \over \widehat{\lambda}_1+\widehat{\lambda}_2(1-|a|^2)}\cdot{1-|a|^2\over 1-\overline{a}z}
\end{equation}
is optimal.
\end{theorem}

Note that for $a = b$ the optimal method of recovery \eqref{e10} does not
depend on $\delta$ and has the form
$$
\widehat{A}(y)(z) = {1-|b|^2\over 1-bz}.
$$


\section{Open problems}


1. It would be desirable to identify the extremal spectral point in all
possible cases. We have shown that in a number of cases the extremal
spectral point is the only point in $\Lambda_2$ such that $\zeta_0<\widehat{\lambda}_2 < \alpha_0.$ Our
attempts to find a nontrivial-case when this point is not extremal failed.
Thus, we are tempted to conjecture that the point of $\Lambda_2$ with the biggest
$\lambda_2$ is always extremal.

{\bf Conjecture.} {\it If $a \neq b$ and $\delta < |\varphi_0(a)|$, the point in $\Lambda_2$ such that
$\zeta_0<\widehat{\lambda}_2 < \alpha_0$ is always the spectral extremal point for problem \eqref{e2}.}

\medskip

2. It is natural to ask which choice of $a$ minimizes the value of problem \eqref{e2} (of course, this choice of $a$ leads to the least optimal recovery error). It follows from above discussion that the point $b$ plays a special role.

{\bf Problem.} {\it  Does the choice $a = b$ always lead to the least mean square optimal recovery error?}

\medskip

3. Finally, if in problem \eqref{e2} we replace the constraint $|\varphi(a)|\le  \delta$ with
$$
{1\over 2\pi r}\int_{|z-a|=r}|f(z)|^2|d(z-a)|\le \delta, \quad 0<r< 1-|a|,
$$
then the problem becomes even more difficult. The reason is that in
the right hand side of Euler's equation the term $\lambda_1\dfrac{f(a)}{1-\overline{a}z}$
is replaced
with
$$
\lambda_1 f\left (a - {r^2z\over 1-\overline{a}z} \right )
$$
and the equation turns into
$$
{1\over 1-\rho w} f \left( {\rho \over 1-\rho w}\right ) =
{\lambda_1\over 1-\overline{a}z}f\left (a - {r^2z\over 1-\overline{a}z} \right )+ \lambda_2f(w).
$$
Thus, finding the spectrum in this case is reduced to finding eigenvalues of an operator which is a linear combination of two compact
non-commuting operators.
It would be very interesting to find the eigenbasis which corresponds
to this problem and to find the solution.

\begin{thebibliography}{99}
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prilozh.} {\bf37} (2003), 51--64; English transl. in {\it Funct. Anal and
Its Appl.}, {\bf37} (2003).

\bibitem{MM} A.~A.~Melkman and C.~A.~Micchelli, ``Optimal estimation of
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\bibitem{Os}K.~Yu.~Osipenko, ``The Hardy--Littlewood--P\'olya inequality for analytic functions from Hardy--Sobolev spaces",  {\it Mat. Sb.} {\bf197} (2006), 15--34; English transl. in {\it Sbornic: Mathematics} {\bf197} (2006), 315--334.

\bibitem{OsS1} K.~Yu.~Osipenko, M.~I.~Stessin, ``On problems of recovery in Hardy and Bergman spaces", {\it Mat.
Zametki} {\bf49} (1991), 95--104; English transl. in {\it Math. Notes} {\bf
49} (1991), 395--401.

\bibitem{OsS2} K.~Yu.~Osipenko, M.~I.~Stessin, ``On optimal recovery of a holomorphic function in the unit ball of
$\mathbb C^n$", {\it Constr. Approx.} {\bf8} (1992), 141--159.

\bibitem{OsS3} K.~Yu.~Osipenko, M.~I.~Stessin, ``On some problems of optimal recovery of analytic and harmonic
functions from inaccurate data", {\it J. Approx. Theory} {\bf70} (1992),
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\end{thebibliography}

\end{document} 