Shallow neural network approximation in mixed Sobolev spaces
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 $ρ$ 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 $\mathrm{ReLU}^k$, a matching algebraic lower bou...
| Authors | Yuwen Li, Guozhi Zhang |
|---|---|
| Primary category | math.NA |
| Published | 4 September 2026 |
| Last updated at source | 7 October 2026 |
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