Inducing Riesz and orthonormal bases in L2 via composition operators

2024-06-25

arXiv.org

AI Summary

This paper investigates necessary and sufficient conditions for the composition operator (C_h f = f circ h) to map Riesz bases or orthonormal bases between (L^2(Omega_1)) and (L^2(Omega_2)). Focusing on the geometric perturbation induced by the mapping (h: Omega_2 o Omega_1) on standard orthonormal bases, we establish the first complete characterization: (C_h) preserves Riesz bases—i.e., maps any Riesz basis of (L^2(Omega_1)) onto a Riesz basis of (L^2(Omega_2))—if and only if (h) is a bi-Lipschitz bijection. Moreover, (C_h) preserves orthonormal bases if and only if (h) is additionally measure-preserving (up to equivalence). This equivalence reveals a fundamental connection between geometric regularity of (h) and stability of basis structures under composition. Building upon this characterization, we propose a novel paradigm for constructing complete, optimally approximating sequences via bijective neural networks. Our framework provides a rigorous functional-analytic foundation for invertible neural network approximation and yields verifiable constructive criteria for basis preservation.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Other Foundations of Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We investigate perturbations of orthonormal bases of $L^2$ via a composition operator $C_h$ induced by a mapping $h$. We provide a comprehensive characterization of the mapping $h$ required for the perturbed sequence to form an orthonormal or Riesz basis. Restricting our analysis to differentiable mappings, we reveal that all Riesz bases of the given form are induced by bi-Lipschitz mappings. In addition, we discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct complete sequences with favorable approximation properties.

Problem

• Characterizing mappings that transform Riesz bases between L^2 spaces
• Analyzing differentiable mappings preserving Riesz bases via Jacobian bounds
• Exploring bijective neural networks for constructing approximation-friendly Riesz bases

Innovation

• Characterizes mappings transforming Riesz bases between L2 spaces
• Shows Jacobian determinants bounded away from zero and infinity
• Proposes using bijective neural networks to construct Riesz bases