On the Approximation of the Generalized Cut Function of Degree p+1 By Smooth Sigmoid Functions
DOI:
https://doi.org/10.55630/sjc.2015.9.93-104Keywords:
Sigmoid Functions, Cut Function, Generalized Cut Function of Degree P 1, Step Function, Logistic Function, Shifted Logistic Function, Uniform and Hausdorff ApproximationAbstract
We introduce a modification of the familiar cut function byreplacing the linear part in its definition by a polynomial of degree p + 1
obtaining thus a sigmoid function called generalized cut function of degree
p + 1 (GCFP). We then study the uniform approximation of the (GCFP)
by smooth sigmoid functions such as the logistic and the shifted logistic
functions. The limiting case of the interval-valued Heaviside step function
is also discussed which imposes the use of Hausdorff metric. Numerical
examples are presented using CAS MATHEMATICA.
