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

Serbenyuk, Symon

doi:10.2478/tmmp-2019-0024

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

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