Annals of Functional Analysis

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

Rachid Zarouf

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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 Carathéodory-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 $r=\mbox{max}{}_{\lambda\in\sigma}\left|\lambda\right|\lt 1$. 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.

Article information

Source
Ann. Funct. Anal., Volume 2, Number 2 (2011), 59- 74.

Dates
First available in Project Euclid: 12 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1399900195

Digital Object Identifier
doi:10.15352/afa/1399900195

Mathematical Reviews number (MathSciNet)
MR2855287

Zentralblatt MATH identifier
1254.30036

Subjects
Primary: 30E05: Moment problems, interpolation problems
Secondary: 32A35‎ ‎32A36‎ ‎46E20‎ ‎46J15

Keywords
Nevanlinna-Pick interpolation ‎Carleson interpolation‎ ‎weighted Hardy spaces ‎weighted Bergman space

Citation

Zarouf, Rachid. Effective $H^{\infty}$ interpolation constrained‎ ‎by weighted Hardy and Bergman norms. Ann. Funct. Anal. 2 (2011), no. 2, 59-- 74. doi:10.15352/afa/1399900195. https://projecteuclid.org/euclid.afa/1399900195


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