Метрична та ймовірнісна теорії $G_2$-зображення чисел
For numbers of interval $[0,g_0]$, $g_0<1$, we consider asystem of encoding with two bases having different signs $g_0$ and$g_1\equiv g_0-1$ by the means of alphabet $A=\{0,1\}$:\begin{equation}\label{ex:ab:1}x=\alpha_1 g_{1-\alpha_1}+\sum\limits^\infty_{k=2}(\alpha_kg_{1-\alpha_k}\prod\l...
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| Date: | 2019 |
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| Keywords: | keywords |
| Main Authors: | , , |
| Format: | Article |
| Language: | Ukrainian |
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Інститут математики НАН України
2019
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| Online Access: | https://trim.imath.kiev.ua/index.php/trim/article/view/521 |
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| Journal Title: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Transactions of Institute of Mathematics of NAS of Ukraine| Summary: | For numbers of interval $[0,g_0]$, $g_0<1$, we consider asystem of encoding with two bases having different signs $g_0$ and$g_1\equiv g_0-1$ by the means of alphabet $A=\{0,1\}$:\begin{equation}\label{ex:ab:1}x=\alpha_1 g_{1-\alpha_1}+\sum\limits^\infty_{k=2}(\alpha_kg_{1-\alpha_k}\prod\limits^{k-1}_{j=1}g_{\alpha_j})\equiv\Delta^{G_2}_{\alpha_1\alpha_2\ldots\alpha_k\ldots},\end{equation}where $\alpha_n\in A$.We develop a probabilistic theory for this representation, i.e., weconsider distributions of digits of random variable $X$ with a givendistribution as well as distribution of random variable $\xi$determined by distributions of digits of its $G_2$-representation(\ref{ex:ab:1}) if the digits are independent.Lebesgue structure of the probability distribution is described.Differential properties of the probability distribution function aswell as properties of its spectrum and support are also studied. |
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