Spectra of averages of unitary representations of LCA groups

What the math gives to ML

The paper provides a concrete spectral criterion for stability of repeated averages of group actions: a probability measure whose Fourier transform satisfies the bounded angular ratio condition produces a Ritt operator when averaged over positive isometries. This gives controlled transients and almost-everywhere convergence for iterates, rather than merely nonexpansiveness. A transferable neural-network construction is an equivariant averaging layer over a finite group of feature permutations or spatial transformations, with mixing weights constrained or selected to satisfy the Fourier BAR criterion. The strongest initial use is as a repeated stabilization or message-passing operator, where the criterion can be checked exactly by a finite Fourier transform.