Fractal functions of exponential type that is generated by the $\mathbf{Q_2^*}$-representation of argument

M.V. Pratsovytyi; Ya. V. Goncharenko; I. M.  Lysenko; S.P. Ratushniak

doi:10.30970/ms.56.2.133-143

M.V. Pratsovytyi Kyiv Drahomanov Pedagogical University
Ya. V. Goncharenko National Pedagogical Dragomanov University, Kyiv, Ukraine
I. M. Lysenko National Pedagogical Dragomanov University, Kyiv, Ukraine
S.P. Ratushniak Institute of Mathematics, National Academy of Science of Ukraine

DOI: https://doi.org/10.30970/ms.56.2.133-143

Keywords: two-symbol Q2 -representation of numbers of unit segment; function with fractal properties; si- ngular function; Hausdorff-Besicovith dimension; a set of incomplete sums of row.

Abstract

We consider function $f$ which is depended on the parameters $0<a\in R$, $q_{0n}\in (0;1)$, $n\in N$ and convergent positive series $v_1+v_2+...+v_n+...$, defined by equality $f(x=\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=a^{\varphi(x)}$, where $\alpha_n\in \{0,1\}$, $\varphi(x=\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=\alpha_1v_1+...+\alpha_nv_n+...$, $q_{1n}=1-q_{0n}$, $\Delta^{Q_2^*}_{\alpha_1...\alpha_n...}=\alpha_1q_{1-\alpha_1,1}+
\sum\limits_{n=2}^{\infty}\big(\alpha_nq_{1-\alpha_n,n}\prod\limits_{i=1}^{n-1}q_{\alpha_i,i}\big)$.
In the paper we study structural, variational, integral, differential and fractal properties of the function $f$.

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