Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space~$L_2$ and Widths of Classes of Functions

S. B. Vakarchuk ; V. I. Zabutnaya

Matematičeskie zametki, Tome 99 (2016) no. 2, pp. 215-238.

Résumé

We obtain exact constants in Jackson-type inequalities for smoothness characteristics $\Lambda_k(f)$, $k\in \mathbb{N}$, defined by averaging the $k$th-order finite differences of functions $f \in L_2$. On the basis of this, for differentiable functions in the classes $L^r_2$, $r\in \mathbb{N}$, we refine the constants in Jackson-type inequalities containing the $k$th-order modulus of continuity $\omega_k$. For classes of functions defined by their smoothness characteristics $\Lambda_k(f)$ and majorants $\Phi$ satisfying a number of conditions, we calculate the exact values of certain $n$-widths.

Détail

Keywords: best polynomial approximation, smoothness characteristics, Jackson-type inequality, modulus of continuity, Bernstein $n$-width of a function class, Rolle's theorem.

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