On one application of infinite systems of functional equations in the functions theory

Abstract

The article is devoted to investigation of applications
of infinite systems of functional equations for modeling of functions
with complicated local structure, that defined in terms of
nega-$\tilde{Q}$-representation. The infinite system of functional equations
$$
f\bigl(\^\varphi^{k}(x)\bigr) = \tilde{\beta}_{i_{k+1,}k+1} + p_{i_{k+1}, k+1} f\bigl(\^\varphi^{k+1}(x)\bigr) ;
$$
where
$ k = 0, 1, \ldots,\^{\varphi} $
is a shift operator of $\tilde{Q}$-expansion,
$ x = \Delta_{i_{1}(x)i_{2}(x) \ldots i_{n}(x) \ldots} $,
are investigated. It is proved, that the system has an 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_{\xi_{1}\xi_{2}\ldots\xi_{n}\ldots}^{\tilde{Q}}  $  
with independent $\tilde{Q}$-symbols are discovered.