One class of continuous locally complicated functions related to infinite-symbol $\Phi$-representation of numbers

  • M. V. Pratsovytyi Mykhailo Drahomanov Ukrainian State University
  • O. M. Baranovskyi Mykhailo Drahomanov Ukrainian State University
  • O.I. Bondarenko Mykhailo Drahomanov Ukrainian State University
  • S.P. Ratushniak Mykhailo Drahomanov Ukrainian State University

Анотація

In the paper, we introduce and study a massive class of continuous functions defined on the interval $(0;1)$ using a special encoding (representation) of the argument with an alphabet $ \mathbb{Z}=\{0,\pm 1, \pm 2,...\}$ and base $\tau=\frac{\sqrt{5}-1}{2}$: $\displaystyle x=b_{\alpha_1}+\sum\limits_{k=2}^{m}(b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i})\equiv
\Delta^{\Phi}_{\alpha_1\alpha_2...\alpha_m(\emptyset)},\quad
x=b_{\alpha_1}+\sum\limits_{k=2}^{\infty}(b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i})\equiv
\Delta^{\Phi}_{\alpha_1\alpha_2...\alpha_n...},$

where $\alpha_n\in \mathbb{Z}$, $\Theta_n=\Theta_{-n}=\tau^{3+|n|}$,
$b_n=\sum\limits_{i=-\infty}^{n-1}\Theta_i=\begin{cases}
\tau^{2-n}, & \mbox{if } n\leq0, \\
1-\tau^{n+1}, & \mbox{if } n\geq 0.
\end{cases}$

The function $f$, which is the main object of the study, is defined by equalities$\displaystyle\begin{cases}
f(x=\Delta^{\Phi}_{i_1...i_k...})=\sigma_{i_11}+\sum\limits_{k=2}^{\infty}\sigma_{i_kk}\prod\limits_{j=1}^{k-1}p_{i_jj}\equiv
\Delta_{i_1...i_k...},\\
f(x=\Delta^{\Phi}_{i_1...i_m(\emptyset)})=\sigma_{i_11}+\sum\limits_{k=2}^{m}\sigma_{i_kk}\prod\limits_{j=1}^{k-1}p_{i_jj}\equiv
\Delta_{i_1...i_m(\emptyset)},
\end{cases}$ where an infinite matrix $||p_{ik}||$ ($i\in \mathbb{Z}$, $k\in \mathbb N$) satisfies the conditions

1) $|p_{ik}|<1$ $\forall i\in \mathbb{Z}$, $\forall k\in \mathbb N;\quad$
2) $\sum\limits_{i\in \mathbb{Z}}p_{ik}=1$ $\forall k\in\mathbb N$;

3) $0<\sum\limits_{k=2}^{\infty}\prod\limits_{j=1}^{k-1}p_{i_jj}<\infty~~\forall (i_j)\in L;\quad$
4) $0<\sigma_{ik}\equiv\sum\limits_{j=-\infty}^{i-1}p_{jk}<1$ $\forall i\in \mathbb Z, \forall k\in \mathbb N.$

This class of functions contains monotonic, non-monotonic, nowhere monotonic functions and functions
without monotonicity intervals except for constancy intervals, Cantor-type and
quasi-Cantor-type functions as well as functions of bounded and unbounded variation. The criteria for the function $f$ to be monotonic and to be a function of the Cantor type as well as the criterion of nowhere monotonicity are proved. Expressions for the Lebesgue measure of the set of non-constancy of the function and for the variation of the function are found. Necessary and sufficient conditions for the
function to be of unbounded variation are established.

Посилання

1. M. Pratsiovytyi, I. Lysenko, O. Voitovska, Distribution of values of classic singular Cantor function of random argument, Random Operators and Stochastic Equations, 26 (2018), No4, 193–200.
2. M.V. Pratsovytyi, O.L. Leshchinskii, Properties of random variable defined by the distributions of elements of their Qe∞-representation, Theor. Probability and Math. Statist. 57 (1998), 143–148.
3. O.M. Baranovskyi, M.V. Pratsiovytyi, G.M. Torbin, Ostrogradsky-Sierpiński-Pierce series and their applications, Kyiv: Nauk. Dumka, 2013, 288 p. (in Ukrainian)
4. O.Yu. Feschenko, Properties of distributons of random variables with independent symbol of their G2 ∞codes, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Dragomanova, Ser 1. Fiz.-Mat. Nauky, 6 (2005), 225–234. (in Ukrainian)
5. M.V. Prats’ovytyi, O.V. Svynchuk, Spread of values of a Cantor-type fractal continuous nonmonotone function, Journal Math. Sc., 240 (2019), No3, 342–357.
6. M.V. Pratsiovytyi, A.V. Kalashnikov, Self-affine singular and nowhere monotone functions related to the Q-representation of real numbers, Ukrainian Math. J., 65 (2013), No3, 405–417. (in Ukrainian)
7. M.V. Pratsiovytyi, O.Yu. Feshchenko, Topological-metric and fractal properties of the distributions on the set of the incomplete sums of series of positive terms, Theory of Stochastic Processes, 13 (29) (2007), No1-2, 205–224.
8. M.V. Pratsiovytyi, Fractal approach to the study of singular distributions, Kyiv: Nats. Pedagog. Mykhailo Dragomanov Univ., 1998. (in Ukrainian)
9. M.V. Pratsiovytyi, Ya.V. Goncharenko, N.V. Dyvliash, S.P. Ratushniak, Inversor of digits of Q∗ 2representation of numbers, Mat. Stud., 55 (2021), No1, 37–43. https://doi.org/10.30970/ms.55.1.37-43
10. M.V. Pratsiovytyi, Ya.V. Goncharenko, I.M. Lysenko, S.P. Ratushniak, Fractal functions of exponential type that is generated by the Q∗ 2-representation of argument, Mat. Stud., 56 (2021), No2, 133–143. https://doi.org/10.30970/ms.56.2.133-143
11. M.V. Pratsiovytyi, S.P. Ratushniak, Independent digits of Q2 -representation of random variable withgiven distribution, Proc. Inst. Math. Nat. Acad. Sc. Ukraine, 16 (2019), No3, 79–91. (in Ukrainian)
12. M.V. Pratsiovytyi, S.P. Ratushniak, Continuous nowhere differentiable function with fractal properties defined in terms of Q2-representation, J. Math. Sc., 258 (2021), No5, 670–697. https://doi.org/10.1007/s10958-021-05573-2
13. M. Pratsiovytyi, N. Vasylenko, Fractal properties of functions defined in terms of Q-representation, International Journal of Math. Analysis, 7 (2013), No61–67, 3155–3169.
14. M. Jarnicki, P. Pflug, Continuous nowhere differentiable functions. The monsterrs of analysis, Springer Monographs in Math. https://doi.org/10.1007/978-3-319-12670-8
Опубліковано
2023-06-24
Як цитувати
Pratsovytyi, M. V., Baranovskyi, O. M., Bondarenko, O., & Ratushniak, S. (2023). One class of continuous locally complicated functions related to infinite-symbol $\Phi$-representation of numbers. Математичні студії, 59(2), 123-131. https://doi.org/10.30970/ms.59.2.123-131
Розділ
Статті