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

TL;DR

This paper characterizes when composition operators induced by certain mappings transform Riesz bases in $L^2$ spaces, with implications for neural network-based basis construction in approximation theory.

Contribution

It provides a complete characterization of mappings that preserve Riesz bases under composition operators, especially for differentiable functions with bounded Jacobian determinants.

Findings

Mappings with bounded Jacobian determinants preserve Riesz bases.

Characterization of composition operators that transform Riesz bases.

Implications for neural network-based basis construction.

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.