Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)

We solve Tikhomirov's problem on the explicit computation of sharp constants in the Kolmogorov type inequalities
$$|f^{(k)}(0)|\le A_{n,k}\bigg(\int_0^{+\infty}(|f(x)|^2+|f^{(n)}(x)|^2)\,dx\bigg)^{1/2}.$$
Specifically, we prove that
$$A_{n,k}=\bigg(\sin\frac{\pi(2k+1)}{2n}\bigg)^{-1/2} \prod_{s=1}^k\operatorname{cot}\frac{\pi s}{2n}\,$$
for all $n\in\{1,2,\dots\}$ and $k\in\{0,\dots,n-1\}$. We establish symmetry and regularity properties of the numbers $A_{n,k}$ and study their asymptotic behavior as $n\to\infty$ for the cases $k=O(n^{2/3})$ and $k/n\to\alpha\in(0,1)$.
Similar problems were previously studied by Gabushin and Taikov.