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

Nikolay Kyurkchiev, Svetoslav Markov

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.

Keywords


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

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ISSN 1314-7897 - Online
ISSN 1312-6555 - Print