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

For continuous monotone functions defined on a bounded interval $[-b;b]$, a monotone approximation $Q(x)$ of any prescribed accuracy in the metric of the space $\mathbf{C}[-b;b]$ is constructed using translations and dilations of the Laplace function (integral). In turn, a highly accurate approximation of the Laplace function in the same metric is constructed using a sum of a linear function and a linear combination of quadratic exponentials (also known as Gaussian functions). The stability of the monotonicity of $Q(x)$ when the Laplace integral is replaced by its approximation is analyzed. The problem of approximating a continuous monotone function arises, for example, when continuous multivariable functions are approximated using Kolmogorov’s theorem, according to which they are represented by single-variable functions (specifically, by several external functions and one monotone internal one), which are then approximated instead of the original multivariable functions. A corresponding approach in which the external and internal functions were approximated by linear combinations of Gaussian functions was earlier investigated by the author. Since an internal function in Kolmogorov’s representation is always the same monotone continuous function $\Psi$ of one variable, the question arises as to how it can be efficiently approximated with the preservation of monotonicity. The present paper answers this question.