Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere

We prove a saturation theorem for linearized shallow ReLU${}^k$ neural networks on the unit sphere ${\mathbb{S}}^d$. For any antipodally quasi-uniform set of centers, if the target function has smoothness $r>\frac{d+2k+1}{2}$, then the best ${\mathcal{L}}^2({\mathbb{S}}^d)$ approximation cannot converge faster than order $n^{-\frac{d+2k+1}{2d}}$. This lower bound matches existing upper bounds, thereby establishing the exact saturation order $\frac{d+2k+1}{2d}$ for such networks. Our results place linearized neural-network approximation firmly within the classical saturation framework and show that, although ReLU${}^k$ networks outperform finite elements under equal degrees $k$, this advantage is intrinsically limited.