Extrapolations with the Least Norms in the Sobolev Spaces $W_2^n$ on the Half-Axis and the Whole Axis

The spaces $W_2^n(\mathbb R_+)$ of functions with finite norms $\| f| W_2^n(\mathbb R_+)\|_{\sigma} := (\|f|L_2(\mathbb R_+)\|^2 +{\sigma}^{-2n} \|f^{(n)}|L_2(\mathbb R_+)\|^2)^{1/2}$, $\sigma>0$, are studied. Let $\Omega _{n,\sigma }$ and $\omega _{n,\sigma }$ be the maximum and minimum of $\|f|W_2^n(\mathbb R_+ )\|_{\sigma}$ under the condition $\sum _0^{n-1} |f^{(s)}(0)|^2 = 1$. It is proved that, as $n\to\infty$, the quantities $n^{-1}\ln \Omega _{n,\sigma}$ and $n^{-1} \ln \omega _{n,\sigma}$ tend to explicitly calculated limits that depend on the number $\sigma$. The behavior of similar quantities $\Omega ^*_{n,\sigma}$ and $\omega ^*_{n,\sigma}$ for the functions defined on the whole axis $\mathbb R$ instead of the half-axis $\mathbb R_+$ is analyzed. The results obtained can be applied to inequalities between the $l_2$-norm of the set of coefficients of an algebraic polynomial of degree $<n$ and the norm of this polynomial in the space $L_2$ with the weight $(1+(x/\sigma )^{2n})^{-1}$.