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

Q3 Mathematics
M. Pratsovytyi, O. Baranovskyi, O. Bondarenko, S. Ratushniak
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引用次数: 1

Abstract

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 $\tau=\frac{\sqrt{5}-1}{2}$: $\displaystyle x=b_{\alpha_1}+\sum\limits_{k=2}^{m}(b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i})\equiv\Delta^{\Phi}_{\alpha_1\alpha_2...\alpha_m(\emptyset)},\quadx=b_{\alpha_1}+\sum\limits_{k=2}^{\infty}(b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i})\equiv\Delta^{\Phi}_{\alpha_1\alpha_2...\alpha_n...},$ 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 functionswithout monotonicity intervals except for constancy intervals, Cantor-type andquasi-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 thefunction to be of unbounded variation are established.
一类与无穷符号$\Phi$有关的连续局部复函数——数字表示
在本文中,我们引入并研究了一类在区间$(0;1)$上定义的连续函数,使用字母表$\mathbb{Z}=\{0,\pm1,\pm2,…\}$和基$\tau=\frac{\sqrt的自变量的特殊编码(表示){5}-1}{2} $:$\displaystyle x=b_{k-1}\Theta_{\alpha_i})\equiv\Delta^{\Phi}_{\alpha_1\alpha_2…\alpha_n…},$其中$\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{casses}$函数$f$是研究的主要对象,由等式$\displaystyle\boot定义{cases}f(x=\Δ^{\Phi}_{i_1…i_k…}^{k-1}p_{i_jj}\equiv\Delta_{i_1…i_k…},\\f(x=\Delta^{\Phi}_{i1…i_m(\pemptyset)}^{k-1}p_{i_jj}\equiv\Delta_{i_1…i_m(\emptyset)},\end{cases}$其中一个无限矩阵$|p_{ik}|$($i\in\mathbb{Z}$,$k\in\math bb N$)满足条件1)$|p_{ik}|<1$\ for all i\in\athbb{Z}$,$\ for ll k\in\ath bb N;\quad$2)$\sum\limits_{i\in\mathbb{Z}}p_{ik}=1$$\对于所有k\in\math bb N$;3) $0^{k-1}p_{i_jj}<\infty~~\forall(i_j)\在L中;\quad$4)$0^{i-1}p_{jk}<1$$\for all i\in\mathbb Z,\for all k\in\math bb N$这类函数包含单调、非单调、无单调函数和除恒定区间、Cantor型和准Cantor型函数以及有界和无界变差函数外没有单调区间的函数。证明了函数$f$是单调的、是Cantor型函数的判据以及无单调性的判据。得到了函数的非恒定集的Lebesgue测度和函数的变分的表达式。建立了函数具有无界变分的充要条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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