Stability to perturbations of Stieltjes continued fractions on domain in the space R and applications

This paper focuses on the study and practical application of Stieltjes continued fractions as a stability to perturbations instrument for function approximation. A framework for the stability analysis of S-fractions is developed. We establish conditions under which the continued fraction is stable to perturbations of its coefficients and variable. The derived results connect the fraction’s stability with the properties of its coefficients and the variable’s domain. To illustrate the theoretical findings, we present an experiment modeling an asymptotic process: the reduction of consumer demand after an advertising campaign has ended. A comparative analysis is performed between a second-order S-fractional model and a polynomial model of the same order. The results show that the S-fraction model not only achieves a lower approximation error and a higher coefficient of determination but also, in contrast to the polynomial model, provides an accurate long-term forecast. The key finding is that the S-fractional model maintains its accuracy to coefficient perturbations, while similar perturbations cause the polynomial model to deviate significantly.