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

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.

Direct and Converse Theorems for Generalized Bernstein-Type Operators

2000 Mathematics Subject Classification: 41A25, 41A27, 41A36.

Best Approximation and Moduli of Smoothness

2010 Mathematics Subject Classification: 41A25, 41A10.