An explicit form for extremal functions in the embedding constant problem for Sobolev spaces

The embedding constants for the Sobolev spaces $\mathring W^n_2[0;1]\hookrightarrow \mathring W^k_2[0;1]$ ($0\le k\le n-1$) are studied. A relationship between the embedding constants and the norms of the functionals $f\mapsto f^{(k)}(a)$ in the space $\mathring W^n_2[0;1]$ is given. An explicit form of the functions $g_{n,k}\in \mathring W^n_2[0;1]$ on which these functionals attain their norm is found. These functions are also extremals for the embedding constants. A connection between the embedding constants and the Legendre polynomials is put forward. A detailed study is made of the embedding constants for $k=3$ and $k=5$: explicit formulas for extreme points are obtained, global maximum points calculated, and the values of the sharp embedding constants is given. A link between the embedding constants and some class of spectral problems with distribution coefficients is established.