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

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 $\delta\geqslant\pi/n$.