Interpolation and inverse problems in spectral Barron spaces
This work addresses theoretical aspects of function approximation and inverse problems for researchers in mathematical analysis and machine learning, but it appears incremental as it builds on existing spectral Barron space concepts.
The paper investigates interpolation and scaling relationships among spectral Barron spaces by connecting them to a positive linear operator, and links these spaces to inverse problems with examples, validating an error bound for Tikhonov regularization penalized by the spectral Barron norm.
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.