Functional analysis and partial differential equations in spectral Barron spaces

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.