Explicit Approximation Rates for Shallow Networks on Additive Besov Classes

We study the approximation of functions by localized shallow neural networks on bounded Lipschitz domains. For a non-polynomial activation, we establish a direct, Jackson-type estimate on the Besov scale Bp,qs(Ω), the approximation error decaying as N−s/d in the width N with an explicit constant tracked through the activation and the domain. In one spatial variable we prove the corresponding Bernstein-type converse on the nonlinear free-knot Besov scale Bτ,τα. For coordinate-subset additive classes we refine the direct estimate to the rate N−s/d0, in which the effective dimension d0 governs the exponent while the ambient dimension enters only through the constant. A deterministic construction corroborates the predicted decay rates, and an effective-dimension reading of educational tabular data is consistent with the additive setting.