The Landau-Kolmogorov Problem on a Finite Interval in the Taikov Case

TL;DR

This paper solves the Landau-Kolmogorov problem on a finite interval for specific derivatives, proving a conjecture and deriving sharp inequalities for functions under certain norm constraints.

Contribution

It provides solutions to the Landau-Kolmogorov problem for r=1,2, proves the Karlin-type conjecture, and finds optimal constants in related inequalities.

Findings

Confirmed the Karlin-type conjecture for r=1,2

Derived sharp inequalities with minimal constants

Solved the pointwise extremal problem on a finite interval

Abstract

We solve the pointwise Landau-Kolmogorov problem on the interval $\mathbb{I} = [-1,1]$ on finding $\left|f^{(k)}(t)\right|\to\sup$ under constraints $\|f\|_2 \leqslant \delta$ and $\left\|f^{(r)}\right\|_2\leqslant 1$, where $t\in\mathbb{I}$ and $\delta > 0$ are fixed. For $r = 1$ and $r = 2$, we solve the uniform version of the Landau-Kolmogorov problem on the interval $\mathbb{I}$ in the Taikov case by proving the Karlin-type conjecture $\sup\limits_{t\in \mathbb{I}}\left|f^{(k)}(t)\right| = \left|f^{(k)}(-1)\right|$ under above constraints. The proof relies on the analysis of the dependence of the norm of the solution to higher-order Sturm-Liouville equation $(-1)^ru^{(2r)} + \lambda u = -\lambda f$ with boundary conditions $u^{(s)}(-1) = u^{(s)}(1) = 0$, $s = 0,1,\ldots,r-1$, on non-negative parameter $\lambda$, where $f$ is some piece-wise polynomial function. Furthermore, we find sharp inequality $\left\|f^{(k)}\right\|_\infty \leqslant A\|f\|_2 + B\left\|f^{(r)}\right\|_2$ with the smallest possible constant $A > 0$ and the smallest possible constant $B = B(A)$ for $k \in \{r-2, r-1\}$.