Configuration-Dependent Lower Bounds for Approximation by Shallow ReLUk Networks on the Sphere

We establish two related but logically distinct results for shallow ReLUk neural networks on the unit sphere SSd. First, for an arbitrary set of inner neural-network parameters, the best L2(SSd) approximation of a fixed target function with smoothness r>tfracd+2k+12 admits an asymptotic lower bound given by a constant multiple of n-1/2underlinehk+1/2, where underlineh denotes the antipodal separation distance of the normalized inner-parameter set. This lower bound depends explicitly on the parameter configuration through underlineh and applies without additional assumptions on the parameters. Second, for antipodally quasi-uniform parameters, underlinehsimeq n-1/d, and the lower bound establishes the exact saturation order n^-fracd+2k+12d for such parameter families: a target function with regularity greater than fracd+2k+12 and satisfying the required parity condition can be approximated at this rate, whereas approximation at any strictly faster rate forces the target function to be zero. Our results therefore place linearized neural-network approximation within the classical saturation framework and show that, although ReLUk network spaces can outperform finite elements of the same degree, this advantage is intrinsically limited.