Strengthening the Comparison Theorem and Kolmogorov Inequality in the Asymmetric Case
We obtain the strengthened Kolmogorov comparison theorem in asymmetric case. In particular, it gives us the opportunity to obtain the following strengthened Kolmogorov inequality in the asymmetric case: $$ \|x^{(k)}_{\pm }\|_{\infty}\le \frac {\|\varphi _{r-k}( \cdot \;;\alpha ,\beta )_\pm \|_{\...
| Published in: | Researches in Mathematics |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Oles Honchar Dnipro National University
2022-07-01
|
| Subjects: | |
| Online Access: | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/385/385 |
| Summary: | We obtain the strengthened Kolmogorov comparison theorem in asymmetric case.
In particular, it gives us the opportunity to obtain the following strengthened Kolmogorov inequality in the asymmetric case:
$$
\|x^{(k)}_{\pm }\|_{\infty}\le \frac
{\|\varphi _{r-k}( \cdot \;;\alpha ,\beta )_\pm \|_{\infty }}
{E_0(\varphi _r( \cdot \;;\alpha ,\beta ))^{1-k/r}_{\infty }}
|||x|||^{1-k/r}_{\infty}
\|\alpha^{-1}x_+^{(r)}+\beta^{-1}x_-^{(r)}\|_\infty^{k/r}
$$
for functions $x \in L^r_{\infty }(\mathbb{R})$, where
$$
|||x|||_\infty:=\frac12 \sup_{\alpha ,\beta}\{ |x(\beta)-x(\alpha)|:x'(t)\neq 0 \;\;\forall
t\in (\alpha ,\beta) \}
$$
$k,r \in \mathbb{N}$, $k<r$, $\alpha, \beta > 0$, $\varphi_r( \cdot \;;\alpha ,\beta )_r$ is the asymmetric perfect spline of Euler of order $r$ and $E_0(x)_\infty $ is the best uniform approximation of the function $x$ by constants. |
|---|---|
| ISSN: | 2664-4991 2664-5009 |