ReLU networks approximate spectral Barron functions of smoothness as low as 1/2 at the N^{-1/2} rate, and L-layer networks achieve sharp N^{-sL} rates for 0<sL<=1/2.
Artin, The Gamma function , Athena Series: Selected Topics in Mathematics, Holt, Rine - hart and Winston, 1964
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Sharp uniform approximation for spectral Barron functions by deep neural networks
ReLU networks approximate spectral Barron functions of smoothness as low as 1/2 at the N^{-1/2} rate, and L-layer networks achieve sharp N^{-sL} rates for 0<sL<=1/2.