On the Approximation of the Generalized Cut Function of Degree p+1 By Smooth Sigmoid Functions

Authors

  • Nikolay Kyurkchiev Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev Str., Bl. 8 1113 Sofia, Bulgaria
  • Svetoslav Markov

Keywords:

Sigmoid Functions, Cut Function, Generalized Cut Function of Degree P 1, Step Function, Logistic Function, Shifted Logistic Function, Uniform and Hausdorff Approximation

Abstract

We introduce a modification of the familiar cut function by
replacing 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.

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Published

2022-04-13

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Section

Articles