Exact Constants in Generalized Inequalities for Intermediate Derivatives

A. A. Lunev ; L. L. Oridoroga

Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 737-744

Résumé

Consider the Sobolev space $W_2^n(\mathbb R_+)$ on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients. We study the problem of exact constants $A_{n,k}$ in inequalities of Kolmogorov type for the values of intermediate derivatives $|f^{(k)}(0)|\le A_{n,k}\|f\|$. In the general case, the expression for the constants $A_{n,k}$ is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants $A_{n,k}$ in the case of the following norms: $$ \|f\|_1^2=\|f\|_{L_2}^2+\|f^{(n)}\|_{L_2}^2\qquad\text{and}\qquad \|f\|_2^2=\sum_{l=0}^n\|f^{(l)}\|_{L_2}^2. $$ In the case of the norm $\|\cdot\|_1$, formulas for the constants $A_{n,k}$ were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants $A_{n,k}$ is also studied in the case of the norm $\|\cdot\|_2$. In addition, we prove a symmetry property of the constants $A_{n,k}$ in the general case.

Détail

Keywords: Sobolev space, Kolmogorov-type inequalities, intermediate derivative, linear functional in Hilbert space, Cramer's rule.
Mots-clés : Vandermonde matrix