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arXiv · 2506.07884

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

Anand Ganesh·Babhrubahan Bose·Anand Rajagopalan
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Abstract

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 $O(\frac{1}{n})$ approximation bound based on our ReLU basis, and a negative result on constructing multivariate functions using finite combinations of ReLU functions.

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Anand Ganesh, Babhrubahan Bose, Anand Rajagopalan. 2025-06-09. Schauder Bases for $C[0, 1]$ Using ReLU, Softplus and Two Sigmoidal Functions. https://arxiv.org/abs/2506.07884

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