Sharp Estimates of High-Order Derivatives in Sobolev Spaces

Abstract

The paper describes the splines \(Q_{n,k}(x,a)\), which define the relations \(y^{(k)}(a)=\int\limits_{0}^{1}y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx\) 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 \(\|Q^{(n)}_{n,k}\|_{L_{p^{\prime}}[0;1]}\) (\(1/p+1/p^{\prime}=1\)) 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 \(p=\infty\).