Interpolation avec contraintes sur des ensembles finis du disque

Abstract

Given a finite set \sigma of the unit disc \mathbb{D}=\{z\in\mathbb{C}:,\,| z|<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 ||g||_{Y} among all functions g such that g_{|\sigma}=f_{|\sigma}. For Y=H^{\infty}, and for the corresponding interpolation constant c(\sigma,\, X,\, H^{\infty}), we show that c(\sigma,\, X,\, H^{\infty})\leq a\phi_{X}(1-\frac{1-r}{n}) where n=#\sigma, r=max_{\lambda\in\sigma}|\lambda| and where \phi_{X}(t) stands for the norm of the evaluation functional f\mapsto f(\lambda) on the space X. The upper bound is sharp over sets \sigma with given n and r.