Interpolation avec contraintes sur des ensembles finis du disque

TL;DR

This paper investigates constrained interpolation problems in the unit disk, providing sharp bounds on the interpolation constant for holomorphic functions in various classes, with implications for approximation theory.

Contribution

It establishes a sharp upper bound for the interpolation constant in the unit disk for functions in class X interpolating on finite sets, extending previous results.

Findings

Derived a sharp upper bound for the interpolation constant c(\sigma, X, H^{\infty})

Connected the bound to the size and location of the interpolation set \sigma

Provided a formula involving the evaluation functional norm \\phi_{X} and set parameters

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.