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

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 $\Vert f{\Vert }_2⩽\delta$ and ${\Vert f^{(r)}\Vert }_2⩽1$, 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 $\underset{t\in \mathbb{I}}{\sup }\left|f^{(k)}(t)\right|=\left|f^{(k)}(-1)\right|$ under above constraints. The proof relies on the analysis of the dependence of the norm of the solution to higher-order Sturm-Liouville equation $(-1{)}^r{u}^{(2r)}+\lambda u=-\lambda f$ with boundary conditions $u^{(s)}(-1)=u^{(s)}(1)=0$, $s=0,1,\dots ,r-1$, on non-negative parameter $\lambda$, where $f$ is some piece-wise polynomial function. Furthermore, we find sharp inequality ${\Vert f^{(k)}\Vert }_{\infty }⩽A\Vert f{\Vert }_2+B{\Vert f^{(r)}\Vert }_2$ with the smallest possible constant $A>0$ and the smallest possible constant $B=B(A)$ for $k\in \{r-2,r-1\}$.