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
Keywords: $\Phi$-representation; cylindrical sets; Cantor-type function; quasi-Cantor type function; nowhere monotonic function; continuous function of unbounded variation; Lebesgue structure of a function; Lebesgue structure of a probability measure


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

where $\alpha_n\in \mathbb{Z}$, $\Theta_n=\Theta_{-n}=\tau^{3+|n|}$,
\tau^{2-n}, & \mbox{if } n\leq0, \\
1-\tau^{n+1}, & \mbox{if } n\geq 0.

The function $f$, which is the main object of the study, is defined by equalities$\displaystyle\begin{cases}
\end{cases}$ 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.


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How to Cite
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. Matematychni Studii, 59(2), 123-131.