Effective Formulas for Constants in the Stechkin--Gabushin Problem

G. A. Kalyabin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 124-129

Résumé

Explicit and transparent expressions are found for the numbers $S_{n,k}$ involved in the formula $E(N,n,k)= S_{n,k} N^{-\beta /\alpha }$, where $\alpha :=(2k+1)/2n$, $\beta := 1-\alpha $, and $k\in \{0,1,\dots ,n-1\}$, for the best approximation of the operators $d^k/dx^k$ in the $C(\mathbb R_+)$ metric on the class of functions $f$ such that $\|f\|_{L_2(\mathbb R_+)} \infty$ and $\|f^{(n)}\|_{L_2(\mathbb R_+)}\le 1$ by means of linear operators $V$ whose norms satisfy the inequality $\|V\|_{L_2(\mathbb R_+)\to C(\mathbb R_+)}\le N$. Simultaneously, the values of the sharp constants $K_{n,k}$ in the Kolmogorov inequality $\|f^{(k)}\|_{C(\mathbb R_+)}\le K_{n,k}\|f^{(n)}\|^{\alpha }_{L_2(\mathbb R_+)} \|f\|^{\beta }_{L_2 (\mathbb R_+)}$ are determined. The symmetry and regularity properties of the constants, as well as their asymptotic behavior as $n\to \infty$, are studied.