Exact estimates for higher order derivatives in Sobolev spaces

T. A. Garmanova ; I. A. Sheipak

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2024), pp. 3-10

Résumé

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