Shallow neural network approximation in mixed Sobolev spaces

Summary

We investigate the best $L_2$ 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 $\rho$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{\alpha,\rho\}$ for target functions of mixed smoothness $\alpha$, 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 $\mathrm{ReLU}^k$, a matching algebraic lower bound identifies $\min\{\alpha,k+1\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\min\{\alpha,k+1\}$ for cardinal B-splines and soft-$\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $\alpha$ for ELU and cosine activations, again up to logarithmic~factors.