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

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.