On monotone approximation of piecewise continuous monotone functions with the help of translations and dilations of the Laplace integral

Abstract

For piecewise continuous monotone functions defined on a bounded interval $[-b;b]$, a monotone smooth approximation $Q(x)$ of any prescribed accuracy in the metric of the space $\mathbf{C}(\Pi)$ with as small as desired measure of the difference $[-b;b]\setminus\Pi$, $\Pi\subset[-b;b]$, is constructed using translations and dilations of the Laplace function (integral). In fact, this extends to the case of piecewise continuous monotone functions the result (obtained by the author formerly) on arbitrarily exact in the metric of the space $\mathbf{C}[-b;b]$ monotone approximation of continuous monotone functions with the help of translations and dilations of the Laplace integral. Besides, we suggest a new way of approximation in the form of linear combination of translations and dilations of the Laplace integral. Finally, we give and discuss concrete numerical examples of using approximation ways under study for a piecewise constant (stepwise) monotone function and for a piecewise continuous monotone function. Here, we also compare the results obtained for two discussed ways of approximation.