Kolmogorov-type inequalities for functions with asymmetric restrictions on the highest derivative
For $k, r\in {\rm \bf N}$, $k<r$; $q\ge 1$, $p>0$; $\alpha, \beta>0$ and for functions $x\in L_{\infty}^r({\rm\bf R})$ inequalities that estimate the norm $\|x_{\pm }^{(k)}\|_{L_q[a,b]}$ on an arbitrary segment $[a,b] \subset {\rm\bf R}$ such that $\;x^{(k)}(a)=x^{(k)}(b)=0$ via a local no...
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| Format: | Article |
| Language: | English |
| Published: |
Oles Honchar Dnipro National University
2024-12-01
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| Series: | Researches in Mathematics |
| Subjects: | |
| Online Access: | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/434/434 |
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| Summary: | For $k, r\in {\rm \bf N}$, $k<r$; $q\ge 1$, $p>0$; $\alpha, \beta>0$ and for functions $x\in L_{\infty}^r({\rm\bf R})$ inequalities that estimate the norm $\|x_{\pm }^{(k)}\|_{L_q[a,b]}$ on an arbitrary segment $[a,b] \subset {\rm\bf R}$ such that $\;x^{(k)}(a)=x^{(k)}(b)=0$ via a local norm of the function $|||x^{\uparrow \downarrow}|||_p :=\sup \left\{ E_0(x)_{L_p[a,b]}: \; \pm x'(t) > 0 \; \forall t\in (a,b), \;\; a,b\in \rm \bf R \right\},$ and the asymmetric norm $\|\alpha^{-1}x_+^{(r)}+\beta ^{-1}x_-^{(r)}\| _{\infty}$ of its highest derivative are proved, where $E_0(x)_{L_p([a,b])}:= \inf \{\|x - c\|_{L_p([a,b])}: c \in {\rm \bf R }\}$.
As a consequence, generalizations of a number of well-known Kolmogorov-type inequalities are obtained. |
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| ISSN: | 2664-4991 2664-5009 |