Inducing Riesz and orthonormal bases in $L^2$ via composition operators
This provides theoretical foundations for constructing approximation bases using neural networks, though it appears to be an incremental theoretical extension of existing composition operator theory.
The paper characterizes which composition operators transform Riesz bases between L² spaces, showing that differentiable mappings preserving Riesz bases must have Jacobian determinants bounded away from zero and infinity. It discusses applications to approximation theory, particularly using bijective neural networks to construct Riesz bases with good approximation properties.
Let $C_h$ be a composition operator mapping $L^2(Ω_1)$ into $L^2(Ω_2)$ for some open sets $Ω_1, Ω_2 \subseteq \mathbb{R}^n$. We characterize the mappings $h$ that transform Riesz bases of $L^2(Ω_1)$ into Riesz bases of $L^2(Ω_2)$. 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.