The Best Extension Operators for Sobolev Spaces on the Half-Line

We describe the construction of extension operators with minimal possible norm $\tau_m$ from the half-line to the entire real line for the spaces $W_2^m$ and derive the asymptotic estimate $\ln\tau_m\approx K_0m$ (as $m\to\infty$), where
$$ K_0:=\frac4\pi\int_0^{\pi/4}\ln(\operatorname{\cot}x)\,dx=1.166243\ldots=\ln3.209912\dots. $$

The proof is based on the investigation of the maximum and minimum eigenvalues and the corresponding eigenvectors of some special matrices related to Vandermonde matrices and their inverses, which can be of interest in themselves.