Functional analysis and partial differential equations in spectral Barron spaces

TL;DR

This paper explores the analytical foundations of spectral Barron spaces, focusing on their duality, embeddings, and applications to PDEs, thereby bridging mathematical analysis, PDEs, and machine learning.

Contribution

It provides a rigorous dual space characterization, embedding results, and PDE applications within spectral Barron spaces, advancing their theoretical understanding.

Findings

Dual space structure of spectral Barron spaces characterized.

Continuous embedding into Hölder spaces established.

Applications to Schrödinger equation boundary value problems analyzed.

Abstract

Spectral Barron spaces, constituting a specialized class of function spaces that serve as an interdisciplinary bridge between mathematical analysis, partial differential equations (PDEs), and machine learning, are distinguished by the decay profiles of their Fourier transform. In this work, we shift from conventional numerical approximation frameworks to explore advanced functional analysis and PDE theoretic perspectives within these spaces. Specifically, we present a rigorous characterization of the dual space structure of spectral Barron spaces, alongside continuous embedding in H\"older spaces established through real interpolation theory. Furthermore, we investigate applications to boundary value problems governed by the Schr\"odinger equation, including spectral analysis of associated linear operators. These contributions elucidate the analytical foundations of spectral Barron spaces while underscoring their potential to unify approximation theory, functional analysis, and machine learning.