Pointwise convergence of Fourier series and deep neural network for the indicator function of d-dimensional ball
This addresses a theoretical convergence issue in function approximation for researchers in mathematical analysis and machine learning, but it is incremental as it builds on prior work on Fourier series phenomena.
The paper tackles the problem of pointwise convergence for the indicator function of a d-dimensional ball, comparing Fourier series and deep neural networks. It demonstrates that while Fourier series exhibit a third phenomenon preventing pointwise convergence, a specific deep neural network achieves pointwise convergence.
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.