Approximation capabilities of certain nonlinear neural network operators

Abstract

In this paper, a new family of nonlinear neural network operators with a single hidden layer has been introduced for a class of continuous functions and Lebesgue integrable functions on a compact interval. We used specific activation functions constructed using a class of sigmoidal functions. The convergence of these operators is examined, and we establish both pointwise and uniform approximation theorems for the functions in the class of continuous functions $C\left(I\right)$, $I=[c,d]$. We also extend the study for the class of Lebesgue integrable functions $L^p\left(I\right)$ for p ≥ 1. Investigation of these nonlinear operators provides valuable insights into their generalization ability and practical applications in signal processing. We also discuss some examples, quantitative estimates, and graphical representations to highlight the applications.