Fractal functions related to the $Δ^μ$ -representation of numbers

Pratsiovytyi Mykola; Isayeva  T. M.

Pratsiovytyi Mykola ^1,2 , Isayeva T. M. ³

¹ Department of dynamic systems and fractal analysis, Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, 01001, Ukraine

² Department of Higher Mathematics, National Pedagogical Dragomanov University, Kyiv, 01001, Ukraine

³ National Pedagogical Dragomanov University, Kyiv, 01001, Ukraine

Keywords: fractal functions, $Δ^μ$ -representation of numbers

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Abstract

In the paper, we consider functions with fractal properties defined in terms of $Δ^μ$ -representation of a number with infinite alphabet $A = \mathbb{N}$:

$(0;1] ∋ x = \sum_n (B_n - B'_n) ≡ Δ_{α_1α_2...α_n....}^μ,$

where $B_n = (1 - μ)^{α_1+α_3+...+α_{2n-1}-1} μ^{α_2+α_4+...+α_{2n-2}}, B'_n = (1 - μ)^{α_1+α_3+...+α_{2n-1}-1} μ^{α_2+α_4+...+α_{2n}}, (0;1) ∋ μ$ is a fixed parameter, and $(α_n)$ is a sequence of positive integers depending on a number $x$. Functions

$f(Δ_{α_1α_2...α_n....}^μ) = Δ_{α_2α_1α_4α_3....}^μ, φ(Δ_{α_1α_2...α_n....}^μ) = Δ_{[α_1α_2][α_3α_4]....}^μ, γ(Δ_{α_1α_2...α_n....}^μ) = Δ_{[α_1α_2][α_2α_3]....}^μ.$

are among them. Structural properties of the function and fractal (self-similar, self-affine, etc.) properties of essential sets for this function are studied.

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Cite

ACS Style: Pratsiovytyi, M.; Isayeva , T.M. Fractal functions related to the $Δ^μ$ -representation of numbers. Bukovinian Mathematical Journal. 2016, 3
AMA Style: Pratsiovytyi M, Isayeva TM. Fractal functions related to the $Δ^μ$ -representation of numbers. Bukovinian Mathematical Journal. 2016; 3(3-4).
Chicago/Turabian Style: Mykola Pratsiovytyi, T. M. Isayeva . 2016. "Fractal functions related to the $Δ^μ$ -representation of numbers". Bukovinian Mathematical Journal. 3 no. 3-4.

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