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\begin{document}
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\noindent\parbox{2.95cm}{\includegraphics*[keepaspectratio=true,scale=0.125]{AFA.jpg}}
\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
\qquad Ann. Funct. Anal. 2 (2011), no. 2, 59--74\\
{\footnotesize
\qquad \textsc{\textbf{$\mathscr{A}$}nnals of \textbf{$\mathscr{F}$}unctional
\textbf{$\mathscr{A}$}nalysis}\\
\qquad ISSN: 2008-8752 (electronic)\\
\qquad URL: \textcolor[rgb]{0.00,0.00,0.99}{www.emis.de/journals/AFA/}
}\\[.5in]}

\title[$H^{\infty}$ interpolation in
weighted Hardy and Bergman norms]{Effective $H^{\infty}$ interpolation constrained
by weighted Hardy and Bergman norms}

\author[R. Zarouf]{Rachid Zarouf}

\address{CMI-LATP, UMR 6632, Universit\'e de Provence, 39, rue Fr\'ed\'eric Joliot-Curie,
13453 Marseille cedex 13, France}

\email{\textcolor[rgb]{0.00,0.00,0.84}{rzarouf@cmi.univ-mrs.fr}}

\dedicatory{{\rm Communicated by K. Guerlebeck}}


\subjclass[2010]{Primary 30E05; Secondary  32A35, 32A36, 46E20, 46J15.}

\keywords{Nevanlinna--Pick interpolation, Carleson interpolation,
weighted Hardy spaces, weighted Bergman spaces.}

\date{Received: 10 January 2011; Accepted: 10 August 2011.}



\begin{abstract}
Given a finite subset $\sigma$ of the unit disc $\mathbb{D}$ and a
holomorphic function $f$ in $\mathbb{D}$ belonging to a class $X$,
we are looking for a function $g$ in another class $Y$ which
satisfies $g_{\vert\sigma}=f_{\vert\sigma}$ and is of minimal norm
in $Y$. More precisely, we consider the interpolation constant
$c\left(\sigma,\, X,\, Y\right)=\mbox{sup}{}_{f\in X,\,\parallel
f\parallel_{X}\leq1}\mbox{inf}_{g_{\vert\sigma}=f_{\vert\sigma}}\left\Vert
g\right\Vert _{Y}.$ When $Y=H^{\infty}$, our interpolation problem
includes those of Nevanlinna--Pick  and Carath\'eodory--Schur. If
$X$ is a Hilbert space belonging to the families of weighted Hardy
and Bergman spaces, we obtain a sharp upper bound for the constant
$c\left(\sigma,\, X,\, H^{\infty}\right)$ in terms of
$n=\mbox{card}\,\sigma$ and
$r=\mbox{max}{}_{\lambda\in\sigma}\left|\lambda\right|<1$. If $X$ is
a general Hardy--Sobolev space or a general weighted Bergman space
(not necessarily of Hilbert type), we also establish upper and lower
bounds for $c\left(\sigma,\, X,\, H^{\infty}\right)$ but with some
gaps between these bounds. This problem of constrained interpolation
is partially motivated by applications in matrix analysis and in
operator theory.
\end{abstract}
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\end{document}

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