Inducing Riesz Bases in $L^2$ via Composition Operators

Let Ch be a composition operator mapping L2(1) into L2(2) for some open sets1, 2 ⊆ Rn. We characterize the mappings h that transform Riesz bases of L2(1)into Riesz bases of L2(2). Restricting our analysis to differentiable mappings, wedemonstrate that mappings h that preserve Riesz bases have Jacobian determinantsthat are bounded away from zero and infinity. We discuss implications of these resultsfor approximation theory, highlighting the potential of using bijective neural networksto construct Riesz bases with favorable approximation properties.