Interpolation and inverse problems in spectral Barron spaces

2025/02/06 by Shuai Lu, Lu, Shuai, Peter Mathé +1 · 3 citations
Mathematics · Computer Science · #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2502.03819

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.

Cited by

• A mixed residual method for biharmonic equations in spectral Barron spaces
• Statistical Learning Guarantees for Group-Invariant Barron Functions
• Functional analysis and partial differential equations in spectral Barron spaces

Related

• Functional analysis and partial differential equations in spectral Barron spaces
• A Regularity Theory for Static Schrödinger Equations on ℝ^d in Spectral Barron Spaces
• Spectral Barron space for deep neural network approximation
• Spectral Barron spaces arising from quantum harmonic analysis
• Spectral representations of interpolation spaces of reproducing kernel Hilbert spaces