möbius instability of sampling for small weighted spaces of analytic functions

Abstract

in this paper, the space $\mathcal{a}_\psi(\mathbb{d})$ is considered, consisting of those holomorphic functions $f$ on the unit disk $\mathbb{d}$ such that $\|f\|_\psi=\sup_{z\in\mathbb{d}}|f(z)|\psi(|z|)<+\infty$, with $\psi(1)=0$. the sampling problem is studied for weights satisfying $\ln\psi(r)/\ln(1-r)\to0$. möbius stability of sampling is shown to fail in this space.