On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions

Abstract

We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the logistic and the Gompertz functions. The limiting case of the interval-valued step function is discussed using Hausdorff metric. Various expressions for the error estimates of the corresponding uniform and Hausdorff approximations are obtained. Numerical examples are presented using CAS MATHEMATICA.

Keywords

• cut function
• step function
• sigmoid function
• logistic function
• Gompertz function
• squashing function
• Hausdorff approximation.