Strengthening the Comparison Theorem and Kolmogorov Inequality in the Asymmetric Case

V.A. Kofanov (Oles Honchar Dnipro National University), https://orcid.org/0000-0003-0392-2257
K.D. Sydorovych (Oles Honchar Dnipro National University)

Abstract


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.


Keywords


Kolmogorov comparison theorem; Kolmogorov inequality; asymmetric case; strengthening

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References


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Kofanov V.A. "Strengthening the comparison theorem and Kolmogorov's inequality and their applications", Ukrainian Math. J., 2002; 54(10): pp. 1348-1355. doi:10.1023/A:1023728202727

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DOI: https://doi.org/10.15421/242204

  

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Copyright (c) 2022 V.A. Kofanov, K.D. Sydorovych

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ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
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