In this paper we clarify the crucial difference between a deep neural network and the Fourier series. For the multiple Fourier series of the periodization of some radial functions on $\mathbb{R}^d$, Kuratsubo (2010) investigated the behavior of the spherical partial sum, and discovered the third phenomenon other than the well-known Gibbs-Wilbraham and Pinsky phenomena. In particular, the third one exhibits prevention of pointwise convergence. In contrast to it, we give a specific deep neural network and prove pointwise convergence.

翻译：本文阐明了深度神经网络与傅里叶级数之间的关键差异。对于$\mathbb{R}^d$上某些径向函数周期化的多重傅里叶级数，Kuratsubo（2010）研究了球面部分和的行为，并发现了除众所周知的Gibbs-Wilbraham现象和Pinsky现象之外的第三类现象。第三类现象尤其表现出对点态收敛的阻碍。与此相反，我们给出了一种特定的深度神经网络，并证明了其点态收敛性。