Constrained Nevanlinna-Pick interpolation in Weighted Hardy and Bergman spaces

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=l_{a}^{p}(w_{k})=\,\left\{ f={\displaystyle \sum_{k\geq0}\hat{f}(k)z^{k}:\,\Vert f\Vert^{p}=\,{\displaystyle \sum_{k\geq0}\vert\hat{f}(k)\vert^{p}w_{k}^{p}<\infty}}\right\} , with a weight w satisfying w_{k}>0 for every k\geq0 and \overline{lim}_{k}(1/w_{k}^{1/k})=\,1, and for the corresponding interpolation constant c\left(\sigma,\, X,\, H^{\infty}\right), we show that if p=2, 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. For X=l_{a}^{p}(w_{k}), p\neq2 and X=L_{a}^{p}\left(\left(1-\left|z\right|^{2}\right)^{\beta}dxdy\right) (the weighted Bergman space), \beta>-1, 1\leq p<2, we also found upper and lower bounds for c\left(\sigma,\, X,\, H^{\infty}\right) (sometimes for special sets \sigma) but with some gaps between these bounds.

Domaines: Analyse fonctionnelle [math.FA]

Rachid Zarouf. Constrained Nevanlinna-Pick interpolation in Weighted Hardy and Bergman spaces. 2008. ⟨hal-00381193v1⟩