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

Abstract

We solve the pointwise Landau-Kolmogorov problem on the interval I = [−1, 1] on finding f (k) (t) → sup under constraints f 2 δ and f (r) 2 1, where t ∈ I and δ > 0 are fixed. For r = 1 and r = 2, we solve the uniform version of the Landau-Kolmogorov problem on the interval I in the Taikov case by proving the Karlin-type conjecture sup t∈I f (k) (t) = f (k) (−1) under above constraints. The proof relies on the analysis of the dependence of the norm of the solution to higherorder Sturm-Liouville equation (−1) r u (2r) + λu = −λf with boundary conditions u (s) (−1) = u (s) (1) = 0, s = 0, 1, . . . , r − 1, on non-negative parameter λ, where f is some piece-wise polynomial function. Furthermore, we find sharp inequality f (k) ∞ A f 2 + B f (r) 2 with the smallest possible constant A > 0 and the smallest possible constant B = B(A) for k ∈ {r − 2, r − 1}.