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

Abstract

The embedding constants of the Sobolev spaces $\mathring{W}^n_2[0;1] \hookrightarrow \mathring{W}^k_\infty[0; 1]$ ($0\leqslant k \leqslant n-1$) are studied. A relation of the embedding constants with 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 to be extremal for the embedding constants. A relation of the embedding constants to the Legendre polynomials is put forward. A detailed study is made of the embedding constants with k = 3 and k = 5: we found explicit formulas for extreme points, calculate global maximum points, and give the values of the sharp embedding constants. A link between the embedding constants and some class of spectral problems with distribution coefficients is discovered.