Про один клас ніде не монотонних функцій з фрактальними властивостями, який містить підклас сингулярних функцій
We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality$$f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j (x)j}\right]\equiv\Delta^{G^*_3}_{\alpha_1\alpha_2\ldots\alpha_k\ldots},$$where $||q^*_{ik}||$ is given infinit...
Saved in:
| Date: | 2017 |
|---|---|
| Author Affiliations: |
|
| Keywords: | keywords |
| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Інститут математики НАН України
2017
|
| Online Access: | https://trim.imath.kiev.ua/index.php/trim/article/view/402 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Transactions of Institute of Mathematics of NAS of Ukraine |
| Download file: |
|
Institution
Transactions of Institute of Mathematics of NAS of Ukraine| Summary: | We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality$$f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j (x)j}\right]\equiv\Delta^{G^*_3}_{\alpha_1\alpha_2\ldots\alpha_k\ldots},$$where $||q^*_{ik}||$ is given infinite stochastic positive matrix ($i=0,1,2$; $k \in N$); $\beta_{0k}=0$, $\beta_{1k}=q_{0k}$, $\beta_{2k}=q_{0k}+q_{1k}$;$(\varepsilon_k)$ is given sequence of numbers such that $0\leqslant \varepsilon_k \leqslant 1 $; $g_{0k}=\dfrac{1+\varepsilon_k}{3}=g_{2k}$, $g_ {1k}=\dfrac{1-2\varepsilon_k}{3}$, $\delta_{0k}=0$, $\delta_{1k}=g_{0k}$, $\delta_{2k}=g_{0k}+g_{1k}$, $k\in N$.
We found criteria of strict monotonicity, non monotonicity and nowhere monotonicity, non-differentiability and singularity of the functions. We pay attention to properties of level sets of the functions. |
|---|