Interpolation avec contraintes sur des ensembles finis du disque

Given a finite set σof 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_{|σ}=f_{|σ}. For Y=H^{\infty}, and for the corresponding interpolation constant c(σ,\, X,\, H^{\infty}), we show that c(σ,\, X,\, H^{\infty})\leq aϕ_{X}(1-\frac{1-r}{n}) where n=#σ, r=max_{λ\inσ}|λ| and where ϕ_{X}(t) stands for the norm of the evaluation functional f\mapsto f(λ) on the space X. The upper bound is sharp over sets σwith given n and r.

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