Shallow neural network approximation in mixed Sobolev spaces

We investigate the best L2 approximation of mixed Sobolev spaces by shallow neural networks with n neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order ρ in the sense of the Fourier-block property, then the global approximation rate has algebraic order minα,ρ for target functions of mixed smoothness α, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For ReLUk, a matching algebraic lower bound identifies minα,k+1 as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent minα,k+1 for cardinal B-splines and soft-ReLUk, and the full mixed-smoothness exponent α for ELU and cosine activations, again up to logarithmic~factors.