## Annals of Functional Analysis

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

Rachid Zarouf

#### 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

#### 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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