Kolmogorov inequalities for functions in classes $W^rH^\omega$ with bounded $\mathbb L_p$-norm

S. K. Bagdasarov

Izvestiya. Mathematics , Tome 74 (2010) no. 2, pp. 219-279

Résumé

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.

Détail

Keywords: Kolmogorov–Landau inequalities, moduli of continuity.