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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.
UDC:
517.988
MSC: 41A17,
41A44,
26A16,
26D10,
58C30,
90C30 Received: 07.05.2007
Revised: 14.05.2008
DOI:
10.4213/im2659
Fulltext:
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