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This article is cited in 1 scientific paper (total in 1 paper)
Kolmogorov inequalities for functions in classes $W^rH^\omega$ with bounded $\mathbb L_p$-norm
S. K. Bagdasarov Parametric Technology Corporation, Needham, MA, USA
Abstract:
We find the general solution and describe the structural properties
of extremal functions of the Kolmogorov problem
$\|f^{(m)}\|_{\mathbb L_\infty(\mathbb I)}\to\sup$,
$f\in W^r\!H^\omega\!(\mathbb I)$, $\|f\|_{\mathbb L_p(\mathbb I)}\le B$,
for all $r,m\in\mathbb Z$, $0\le m\le r$,
all $p$, $1\le p<\infty$, concave moduli of continuity $\omega$,
all positive $B$ and $\mathbb I=\mathbb R$ or
$\mathbb{I}=\mathbb R_+$, where $W^rH^\omega(\mathbb I)$ is the
class of functions whose $r$th derivatives have modulus of continuity
majorized by $\omega$. We also obtain sharp constants in the additive
(and multiplicative in the case of Hölder classes) inequalities
for the norms $\|f^{(m)}\|_{\mathbb L_\infty(\mathbb I)}$ of the
derivatives of functions $f\in W^rH^\omega(\mathbb I)$ with
finite norm $\|f^{(r)}\|_{\mathbb L_p(\mathbb I)}$.
We also investigate some properties of extremal functions
in the special case $r=1$ (such as the property of being
compactly supported) and obtain inequalities between the
knots of the corresponding $\omega$-splines.
In the case of the Hölder moduli of continuity
$\omega(t)=t^\alpha$, we find that the lengths of the
intervals between the knots of extremal $\omega$-splines
decrease in geometric progression while the graphs
of the solutions exhibit the fractal property of self-similarity.
Keywords:
Kolmogorov–Landau inequalities, moduli of continuity.
Received: 07.05.2007 Revised: 14.05.2008
Citation:
S. K. Bagdasarov, “Kolmogorov inequalities for functions in classes $W^rH^\omega$ with bounded $\mathbb L_p$-norm”, Izv. RAN. Ser. Mat., 74:2 (2010), 5–64; Izv. Math., 74:2 (2010), 219–279
Linking options:
https://www.mathnet.ru/eng/im2659https://doi.org/10.1070/IM2010v074n02ABEH002486 https://www.mathnet.ru/eng/im/v74/i2/p5
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Abstract page: | 684 | Russian version PDF: | 251 | English version PDF: | 25 | References: | 87 | First page: | 30 |
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