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⩽1 and ‖f(n)‖2⩽σ, where t∈I and σ>0 are fixed. For n=1 and n=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 supt∈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 higher-order Sturm–Liouville equation (−1)nu(2n)+λu=−λf with boundary conditions u(s)(−1)=u(s)(1)=0, s=0,1,…,n−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(n)‖2 with the smallest possible constant A>0 and the smallest possible constant B=B(A) for k∈{n−2,n−1}.