In this work, we consider the approximation capabilities of shallow neural networks in weighted Sobolev spaces for functions in the spectral Barron space. The existing literature already covers several cases, in which the spectral Barron space can be approximated well, i.e., without curse of dimensionality, by shallow networks and several different classes of activation function. The limitations of the existing results are mostly on the error measures that were considered, in which the results are restricted to Sobolev spaces over a bounded domain. We will here treat two cases that extend upon the existing results. Namely, we treat the case with bounded domain and Muckenhoupt weights and the case, where the domain is allowed to be unbounded and the weights are required to decay. We first present embedding results for the more general weighted Fourier-Lebesgue spaces in the weighted Sobolev spaces and then we establish asymptotic approximation rates for shallow neural networks that come without curse of dimensionality.

翻译：本文研究了浅层神经网络在加权Sobolev空间中对谱Barron空间函数的逼近能力。现有文献已涵盖多种情形，表明谱Barron空间可通过浅层网络及多类激活函数实现高效逼近（即规避维数灾难）。现有成果的主要局限在于所考虑误差度量仅限于有界域上的Sobolev空间。本研究将拓展两种情形：其一处理具有Muckenhoupt权重的有界域情形，其二处理允许无界域且要求权重衰减的情形。我们首先建立更一般的加权Fourier-Lebesgue空间到加权Sobolev空间的嵌入结果，继而证明浅层神经网络可获得无维数灾难的渐近逼近率。