On~Jackson's inequality for a~generalized modulus of continuity in~$L_2$

A. I. Kozko ; A. V. Rozhdestvenskii

Sbornik. Mathematics, Tome 195 (2004) no. 8, pp. 1073-1115

Résumé

The value of the sharp constant $\varkappa$ in the Jackson type inequality in the space $L_2(\mathbb T^d)$ \begin{equation} E_{n-1}(f)\leqslant\varkappa\overline\omega_\psi(f,T) \end{equation} is studied for the generalized modulus of continuity $$ \overline\omega_\psi(f,T)=\max_{t\in T}\biggl(\sum_{s}\psi(st)|\widehat f_s|^2\biggr)^{1/2}. $$ The value $\overset{*}{\varkappa}$ of the minimum sharp constant in inequality (1) is found. A class of generalized moduli of continuity is introduced which contains the moduli $\widetilde\omega_{a,r}(f,\delta):=\sup_{0\leqslant t\leqslant\delta}\|\Delta_{a^{r-1}t}\dotsb \Delta_{at}\Delta_{t}f\|_2$, with even $a$. The relation $\varkappa=\overset{*}\varkappa$ is proved in this class for all $\delta\geqslant\pi/n$.