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

Rachid Zarouf (1)

1 IMB - Institut de Mathématiques de Bordeaux

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.

Domaines: Analyse fonctionnelle [math.FA]

Rachid Zarouf. Effective H^{\infty} interpolation constrained by Hardy and Bergman norms. 2010. ⟨hal-00381191v3⟩