Spectral Barron spaces of vector-valued functions on compact groups
In this article, we study spectral Barron spaces whose elements are Banach space-valued functions on a compact group whose Fourier transforms admit a certain summability property. We investigate the functional properties of these spaces and establish continuous embeddings with respect to other function spaces, among which are Sobolev spaces of vector-valued functions and the space of bounded vecto-valued functions on compact groups. Beyond these structural results, we prove a quantitative approximation theorem: when the target space is a separable Hilbert space, every function in the spectral Barron space admits an approximation by matrix coefficients of unitary representations of the group with $L^2$-error decaying at the rate $O(n^{-1/2})$ and a constant depending only on the spectral Barron norm of the function. This extends, via Maurey's empirical method, the classical dimension-independent approximation rate of Barron's theorem beyond the Euclidean setting, and we further indicate how the argument persists, with an inflated constant, when target space is only assumed to have Rademacher type 2.