In this paper, we clarify the crucial difference between a deep neural network and the Fourier series. For the multiple Fourier series of 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。针对$\mathbb{R}^d$上某类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现象之外的第三种现象。特别地，第三种现象表现为对逐点收敛的阻碍。与之形成对比的是，我们构建了一个特定的深度神经网络，并证明了其逐点收敛性。