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

Abstract

Given a finite set $\sigma$ of the unit disc $\mathbb{D}$ and a holomorphic function $f$ in $\mathbb{D}$ which belongs to a class $X$ we are looking for a function $g$ in another class $Y$ which minimizes the norm $|g|_{Y}$ among all functions $g$ such that $g_{|\sigma}=f_{|\sigma}$. Generally speaking, the interpolation constant considered is $c(\sigma,\, X,\, Y)={sup}{}_{f\in X,\,\parallel f\parallel_{X}\leq1}{inf}\{|g|_{Y}:\, g_{|\sigma}=f_{|\sigma}\} \,.$ When $Y=H^{\infty}$, our interpolation problem includes those of Nevanlinna-Pick (1916), Caratheodory-Schur (1908). Moreover, Carleson's free interpolation (1958) has also an interpretation in terms of our constant $c(\sigma,\, X,\, H^{\infty})$.} If $X$ is a Hilbert space belonging to the scale of Hardy and Bergman weighted spaces, 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 $f\mapsto f(t)$ on the space $X$. The upper bound is sharp over sets $\sigma$ with given $n$ and $r$.} If $X$ is a general Hardy-Sobolev space or a general weighted Bergman space (not necessarily of Hilbert type), we also found upper and lower bounds for $c(\sigma,\, X,\, H^{\infty})$ (sometimes for special sets $\sigma$) but with some gaps between these bounds.} This constrained interpolation is motivated by some applications in matrix analysis and in operator theory.