  The paper describes the splines $Q_{n,k}(x,a)$, which for an arbitrary point
$a\in(0;1)$ and an arbitrary function $y\in\mathring{W}^n_p[0;1]$ set the
relations $y^{(k)}(a)=\int_0^1 y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx$. The relation of
the $L_{p'}[0;1]$ norm minimization for $Q^{(n)}_{n,k}$ ($1/ p+1/p'=1$) with
the problem of the best estimates of derivatives of $y^{(k)}(a)\leqslant
A_{n,k,p}(a)\|y^{(n)}\|_{L_p[0;1]}$, and also with the problem of finding the
exact embedding constants of the Sobolev space $\mathring{W}^n_p[0;1]$ into the
space $\mathring{W}^k_\infty[0;1]$, $n\in\mathbb{N}$, $k=0,1,\ldots, n-1$.
Exact embedding constants are found for $k=n-1$ and $p=\infty$, as well as for
all $n\in\mathbb{N}$, $k=0,1,\ldots, n-1$ and $p=1$.
