Configuration-Dependent Lower Bounds for Approximation by Shallow ReLU$^k$ Networks on the Sphere

We establish two related but logically distinct results for shallow ReLU$^k$ neural networks on the unit sphere $\SS^d$. First, for an arbitrary set of inner neural-network parameters, the best $\mathcal{L}^2(\SS^d)$ approximation of a fixed target function with smoothness $r>\tfrac{d+2k+1}{2}$ admi