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.

翻译：本文为空间$C[0,1]$构造了四类Schauder基：第一类基于ReLU函数，第二类基于Softplus函数，另外两类分别基于ReLU和Softplus的S型函数变体。该研究首次证明了使用这些函数构造基的可能性，并改进了其相关的通用逼近性质。基于所提出的ReLU基，我们证明了$O(\\frac{1}{n})$量级的逼近误差界，同时通过有限个ReLU函数组合构造多元函数时存在理论局限性。