Inducing Riesz and orthonormal bases in $L^2$ via composition operators

Let $C_h$ be a composition operator mapping $L^2({\mathrm{\Omega }}_1)$ into $L^2({\mathrm{\Omega }}_2)$ for some open sets ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2\subseteq {\mathbb{R}}^n$. We characterize the mappings $h$ that transform Riesz bases of $L^2({\mathrm{\Omega }}_1)$ into Riesz bases of $L^2({\mathrm{\Omega }}_2)$. 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.