On one class of functions with complicated local structure that the solutions of infinite systems of functional equations (On one application of infinite systems of functional equations in the functions theory)

TL;DR

This paper investigates infinite systems of functional equations related to functions with complex local structures defined via nega-tilde Q-representations, proving existence, uniqueness, and properties of solutions, and exploring their applications in modeling distribution functions.

Contribution

It introduces new conditions for solutions of infinite functional systems to be distribution functions of certain random variables, with detailed analysis of their properties.

Findings

Unique bounded and continuous solutions are established.

Conditions for monotonicity and nonmonotonicity are identified.

Solutions can represent distribution functions of independent-tilde Q-symbol random variables.

Abstract

The article is devoted to investigation of applications of infinite systems of functional equations for modeling of functions with complicated local structure, that are defined in terms of the nega-$\tilde Q$-representation. The infinite system of functional equations $$ f\left(\hat \varphi^k(x)\right)=\tilde \beta_{i_{k+1},k+1}+\tilde p_{i_{k+1},k+1}f\left(\hat \varphi^{k+1}(x)\right), $$ where $k=0,1,...$, $\hat \varphi$ is a shift operator of the $\tilde Q$-expansion, $x=\Delta^{-\tilde Q} _{i_1(x)i_2(x)...i_n(x)...}$, are investigated. It is proved, that the system has the unique solution in the class of determined and bounded on $[0;1]$ functions and continuity of the solution. His analytical presentation is founded. Conditions of its monotonicity and nonmonotonicity, differential, integral properties are studied. Conditions under which the solution of the functional equations system is a distribution function of random variable $\eta=\Delta^{\tilde Q} _{\xi_1\xi_2...\xi_n...}$ with independent $\tilde Q$-symbols are discovered. The results of the article was represented in Fourth all-Ukrainian Scientific Conference of Young Scientists on Mathematics and Physics, Kyiv, April 23-25, 2015 (https://www.researchgate.net/publication/301765100). The investigation was represented at the seminar on fractal analysis of Institute of Mathematics of the National Academy of Sciences of Ukraine on, October 30, 2014 (http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1).