On One Application of Infinite Systems of Functional Equations in Function Theory

Symon Serbenyuk

doi:10.2478/tmmp-2019-0024

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On One Application of Infinite Systems of Functional Equations in Function Theory

Tatra Mountains Mathematical Publications

Volume 74 (2019): Issue 1 (December 2019)

By: Symon Serbenyuk

Open Access

|Nov 2019

Abstract

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-˜Q-representation. The infinite systems of functional equations

$f ({\hat{φ}}^{k} (x)) = \tilde{β} i_{k + 1}, k + 1 + \tilde{p} i_{k + 1}, k + 1 f ({\hat{φ}}^{k + 1} (x)),$ f\left( {{{\hat \varphi }^k}(x)} \right) = \tilde \beta {i_{k + 1}},k + 1 + \tilde p{i_{k + 1}},k + 1f\left( {{{\hat \varphi }^{k + 1}}(x)} \right),

where $x = Δ_{i_{1} (x) i_{2} (x) \dots i_{n} (x) \dots}^{- \tilde{Q}}$ x = \Delta _{{i_1}(x){i_2}(x) \ldots {i_n}(x) \ldots }^{ - \tilde Q}, and φ ̑ is the shift operator of the Q̃-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 $η = Δ_{ξ_{1} ξ 2 \dots ξ_{n} \dots}^{\tilde{Q}}$ \eta = \Delta _{{\xi _1}\,\xi 2 \ldots {\xi _n} \ldots }^{\tilde Q} with independent Q̃-symbols are founded.

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Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2019-0024 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 117 - 144

Submitted on: Jul 27, 2016

Published on: Nov 15, 2019

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Function with complicated local structure,

systems of functional equations,

singular function,

nowhere differentiable function,

distribution function,

nowhere monotone function,

Q̃-representation,

nega-Q̃-representation,

Lebesgue integral

Related subjects:

Mathematics,

General mathematics

© 2019 Symon Serbenyuk, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 74 (2019): Issue 1 (December 2019)