# On one application of infinite systems of functional equations in function theory

In: Tatra Mountains Mathematical Publications, vol. 74, no. 2
Symon Serbenyuk

## Details:

Year, pages: 2019, 117 - 144
Language: eng
Keywords:
Function with complicated local structure, systems of functional equations, singular function, nowhere differentiable function, distribution function, nowhere monotone function, $\tilde Q$-representation, nega-$\tilde Q$-representation, Lebesg
Article type: mathematics
Document type: Scientific article *.pdf
The paper presents the investigation of applications of infinite systems of functional equations for modeling functions with complicated local structure that are defined in terms of the nega-$\tilde Q$-representation. The infinite systems of functional equations

$$f(\hat φk(x)) = \widetilde βik+1,k+1+\widetilde pik+1,k+1f(\hat φk+1(x)),$$

where $k=0,1,…$, $x=Δ-\widetilde Q i1(x)i2(x)… in(x)…$, and $\hat φ$ is the shift operator of the $\widetilde Q$-expansion, are investigated. It is proved that the system has a unique solution in the class of determined and bounded on $[0,1]$ functions. Its analytical presentation is founded. The continuity of the solution is studied. Conditions of its monotonicity and nonmonotonicity, differential, and integral properties are studied. Conditions under which the solution of the system of functional equations is a distribution function of the random variable $η=Δ\widetilde Q ξ1ξ2… ξn$ with independent $\widetilde Q$-symbols are founded.
How to cite:
ISO 690:
Serbenyuk, S. 2019. On one application of infinite systems of functional equations in function theory. In Tatra Mountains Mathematical Publications, vol. 74, no.2, pp. 117-144. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2019-0024

APA:
Serbenyuk, S. (2019). On one application of infinite systems of functional equations in function theory. Tatra Mountains Mathematical Publications, 74(2), 117-144. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2019-0024