Effective H^{\infty} interpolation constrained by Hardy and Bergman weighted norms
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Journal articles
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 $\left\Vert g\right\Vert _{Y}$ among all functions $g$ such that $g_{\vert\sigma}=f_{\vert\sigma}$. Generally speaking, the interpolation constant considered is $c\left(\sigma,\, X,\, Y\right)=\mbox{sup}{}_{f\in X,\,\parallel f\parallel_{X}\leq1}\mbox{inf}\left\{ \left\Vert g\right\Vert _{Y}:\, g_{\vert\sigma}=f_{\vert\sigma}\right\} \,.$ 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 $c\left(\sigma,\, X,\, H^{\infty}\right)$. }{\small \par} {\small If $X$ is a Hilbert space belonging to the scale of Hardy and Bergman weighted spaces, we show that $c\left(\sigma,\, X,\, H^{\infty}\right)\leq a\varphi_{X}\left(1-\frac{1-r}{n}\right)$ where $n=\#\sigma$, $r=\mbox{max}{}_{\lambda\in\sigma}\left|\lambda\right|$ 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 $r$. }{\small \par} {\small 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 $c\left(\sigma,\, X,\, H^{\infty}\right)$ (sometimes for special sets $\sigma$) but with some gaps between these bounds.}{\small \par} {\small This constrained interpolation is motivated by some applications in matrix analysis and in operator theory.
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