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 min{α,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 min{α,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 min{α,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.