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 HAL: hal-00381191, version 2
 arXiv: 0905.0573
 Effective H^(\infty) interpolation constrained by Hardy and Bergman normsZarouf R.http://hal.archives-ouvertes.fr/hal-00381191
 Available versions: v1 (2009-05-05) v2 (2010-05-31) v3 (2010-07-06) v4 (2010-11-03)
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Mathematics/Functional Analysis
Effective H^{\infty} interpolation constrained by Hardy and Bergman norms
Rachid Zarouf () 1
 1: Institut de Mathématiques de Bordeaux (IMB) http://www.math.u-bordeaux.fr/IMB/ CNRS : FR2254 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II 351 cours de la Libération 33405 TALENCE CEDEX France
Given a finite set \sigma of the unit disc \mathbb{D}=\{z\in\mathbb{C}:,\,\vert z\vert<1\} and a holomorphic function f in \mathbb{D} which belongs to a class X, we are looking for a function g in another class Y (smaller than X) which minimizes the norm \left\Vert g\right\Vert _{Y} among all functions g such that g_{\vert\sigma}=f_{\vert\sigma}. For Y=H^{\infty}, X=H^{p} (the Hardy space) or X=L_{a}^{2} (the Bergman space), and for the corresponding interpolation constant c\left(\sigma,\, X,\, H^{\infty}\right), we show that c\left(\sigma,\, X,\, H^{\infty}\right)\leq a\varphi_{X}\left(1-\frac{1-r}{n}\right) where n=\#\sigma, r=max_{\lambda\in\sigma}\left|\lambda\right| and where \varphi_{X}(t) stands for the norm of the evaluation functional f\mapsto f(t) on the space X. The upper bound is sharp over sets \sigma with given n and r.
English
2008

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 hal-00381191, version 2 http://hal.archives-ouvertes.fr/hal-00381191 oai:hal.archives-ouvertes.fr:hal-00381191 From: Rachid Zarouf <> Submitted on: Sunday, 14 March 2010 11:27:20 Updated on: Monday, 31 May 2010 11:20:18