Singular monotone functions, which are determined by convergent series and double stochastic matrix
The paper considers functions, which are defined by $$F(x)=\overline{\Delta}_{a_1(x)a_2(x)\ldots a_n(x)\ldots}^2={\Delta}_{a_1(x)a_2(x)\ldots a_n(x)\ldots}=y,$$where $\overline{\Delta}_{a_1a_2\ldots a_n\ldots}^2=\frac{2}{3}+\frac{\alpha_1}{(-2)^1}+\frac{\alpha_2}{(-2)^2}+\frac{\alpha_3}{(-2)^3}+\ldo...
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| Datum: | 2020 |
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| Автори та афіліації: |
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| Ключові слова: | keywords |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2020
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| Online Zugang: | https://trim.imath.kiev.ua/index.php/trim/article/view/398 |
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| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Transactions of Institute of Mathematics of NAS of Ukraine| Zusammenfassung: | The paper considers functions, which are defined by $$F(x)=\overline{\Delta}_{a_1(x)a_2(x)\ldots a_n(x)\ldots}^2={\Delta}_{a_1(x)a_2(x)\ldots a_n(x)\ldots}=y,$$where $\overline{\Delta}_{a_1a_2\ldots a_n\ldots}^2=\frac{2}{3}+\frac{\alpha_1}{(-2)^1}+\frac{\alpha_2}{(-2)^2}+\frac{\alpha_3}{(-2)^3}+\ldots$ is the binary nega-positional representation of the number on the interval $[0; 1]$,${\Delta}_{a_1a_2\ldots a_n\ldots}$ is the Markov representation determined by the positive doubly-stochastic matrix$$\|p_{ik}\|=\begin{pmatrix}p_{00} & p_{01}\\p_{10} & p_{11}\end{pmatrix}.$$The singularity and self-similar properties of the functions are established. Functional relations between them are found. |
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