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DOI: https://doi.org/10.35634/2226-3594-2023-61-10

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.

Keywords: piecewise continuous monotone functions, uniform approximation, Laplace integral, Gaussian function, quadratic exponential.

Received: 10.03.2023
Accepted: 25.04.2023

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Citation: A. V. Chernov, “On monotone approximation of piecewise continuous monotone functions with the help of translations and dilations of the Laplace integral”, Izv. IMI UdGU, 61 (2023), 187–205

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Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles