arXiv:2406.18613v3 Announce Type: replace-cross
Abstract: Let $C_h$ be a composition operator mapping $L^2(Omega_1)$ into $L^2(Omega_2)$ for some open sets $Omega_1, Omega_2 subseteq mathbb{R}^n$. We characterize the mappings $h$ that transform Riesz bases of $L^2(Omega_1)$ into Riesz bases of $L^2(Omega_2)$. Restricting our analysis to differentiable mappings, we demonstrate that mappings $h$ that preserve Riesz bases have Jacobian determinants that are bounded away from zero and infinity. We discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct Riesz bases with favorable approximation properties.