Strengthening the Comparison Theorem and Kolmogorov Inequality in the Asymmetric Case

• Published in: Researches in mathematics (Online) 2022-07, Vol.30 (1), p.30-38
• Main Authors: Kofanov, V.A., Sydorovych, K.D.
• Format: Article
• Language: English
• Subjects: asymmetric case; kolmogorov comparison theorem; kolmogorov inequality; strengthening
• ISSN: 2664-4991; 2664-5009
• DOI: 10.15421/242204

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$$|||x|||_\infty:=\frac12 \sup_{\alpha ,\beta}\{ |x(\beta)-x(\alpha)|:x'(t)\neq 0 \;\;\forallt\in (\alpha ,\beta) \}$$$k,r \in \mathbb{N}$, $k 0$, $\varphi_r( \cdot \;;\alpha ,\beta )_r$ 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.