# Schauder Bases for C[0, 1] Using ReLU, Softplus and Two Sigmoidal Functions

arXiv: https://arxiv.org/abs/2506.07884 · HTML: https://mlsanity.com/paper/2506.07884/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 10 references that are in the map.
- First posted on arXiv 9 Jun 2025.

## Builds on (most cited first, 1 of 1)

- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)

## Abstract (arXiv)

We construct four Schauder bases for the space C[0,1], one using ReLU functions, another using Softplus functions, and two more using sigmoidal versions of the ReLU and Softplus functions. This establishes the existence of a basis using these functions for the first time, and improves on the universal approximation property associated with them. We also show an O(frac1n) approximation bound based on our ReLU basis, and a negative result on constructing multivariate functions using finite combinations of ReLU functions.