Interpolation and inverse problems in spectral Barron spaces

TL;DR

This paper explores the properties of spectral Barron spaces, establishing their relationships, and applies these insights to inverse problems and regularization techniques, advancing understanding of function approximation and inverse problem solutions.

Contribution

It introduces a new connection between spectral Barron spaces and positive linear operators, and relates these spaces to inverse problems and regularization methods.

Findings

Established interpolation and scaling relations among spectral Barron spaces.

Linked spectral Barron spaces to inverse problems via a new link condition.

Validated an error bound for Tikhonov regularization using spectral Barron norms.

Abstract

Spectral Barron spaces, which quantify the absolute value of weighted Fourier coefficients of a function, have gained considerable attention due to their capability for universal approximation across certain function classes. By establishing a connection between these spaces and a specific positive linear operator, we investigate the interpolation and scaling relationships among diverse spectral Barron spaces. Furthermore, we introduce a link condition by relating the spectral Barron space to inverse problems, illustrating this with three exemplary cases. We revisit the notion of universal approximation within the context of spectral Barron spaces and validate an error bound for Tikhonov regularization, penalized by the spectral Barron norm.