Approximation order to a function in $\text{C}$($\text{R}$) by superposition of a sigmoidal function

Abstract

We investigate the approximation error to a continuous function defined on the whole real line by superposition of a sigmoidal function. With the minimal constraints on a continuous function, we show that the approximation order by 3n superposition of a sigmoidal function is $\text{O}$(1/n). Furthermore, we show that our result is “almost best possible” for a certain continuous function and a certain sigmoidal function.