 |
 |
 |
|
|
|
| HAL: hal-00381191, version 2 |
| arXiv: 0905.0573 |
| Short view |  Export this paper |
 |
| Effective H^(\infty) interpolation constrained by Hardy and Bergman norms Zarouf 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) |
|
|
|
|
| Publication type: | |
Preprint, Working Paper, ... | ||||||||||||||||||
|
||||||||||||||||||||
| Subject: | |
Mathematics/Functional Analysis | ||||||||||||||||||
|
||||||||||||||||||||
| Title: | |
Effective H^{\infty} interpolation constrained by Hardy and Bergman norms | ||||||||||||||||||
|
||||||||||||||||||||
| Author(s): | |
Rachid Zarouf 1 |
||||||||||||||||||
|
||||||||||||||||||||
| Laboratory: | |
|
||||||||||||||||||
|
||||||||||||||||||||
| Abstract: | |
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. | ||||||||||||||||||
|
||||||||||||||||||||
| Fulltext language: | |
English | ||||||||||||||||||
|
||||||||||||||||||||
| Production date: | |
2008 | ||||||||||||||||||
|
||||||||||||||||||||
|
|
| Attached file list to this document: | |||||||||||||||
|
|||||||||||||||
|
|
|
| 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 | |
