# Pointwise convergence of Fourier series and deep neural network for the indicator function of d-dimensional ball

arXiv: https://arxiv.org/abs/2304.08172 · HTML: https://mlsanity.com/paper/2304.08172/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 17 Apr 2023.

## Builds on (most cited first, 6 of 6)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [On the minimax optimality and superiority of deep neural network learning over sparse parameter spaces](https://mlsanity.com/paper/1905.09195/) (2019)
- [Estimation error analysis of deep learning on the regression problem on the variable exponent Besov space](https://mlsanity.com/paper/2009.11285/) (2020)
- [Optimal Function Approximation with Relu Neural Networks](https://mlsanity.com/paper/1909.03731/) (2019)

## Abstract (arXiv)

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 ℝd, 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.