A Constructive Proof and An Extension of Cybenko’s Approximation Theorem

Abstract

In this paper, we present a constructive proof of approximation by superposition of sigmoidal functions. We point out a sufficient condition that the set of finite linear combinations of the form $\sum \alpha _j\sigma (y_jx+\theta _j)$ is dense in $C(\mathbb{I}^n)$, is the boundedness of the sigmoidal function σ(x). Moreover, we show that if the set of finite linear combinations of the form $\sum c_j\omega (\xi _j+\eta _j)$, where ω is a univariate function, is dense in $L^p[a,b] (1\leq p< \infty )$ (or C[a,b]) for any finite a,b, then the set of finite linear combinations of the form $\sum c_j\omega (y_j.x+\theta _j)$ is dense in $L^p(\mathbb{I}^n)(or C(\mathbb{I}^n))$. An extension in another direction is also presented in Theorem 4 of this paper.