Direct and inverse inequalities for $\varphi$-Fejér mean-square approximations

We consider approximation of a function $f\in W_2^l(R_1)$, $l\ge0$, by linear operators of the form
$$ K_\sigma^\varphi(f;x)=\frac1{\sqrt{2\pi}}\int_{R_1}\varphi\Bigl(\frac u\sigma\Bigr)\widetilde f(u)e^{iux}\,du,\quad \sigma>0. $$
We elucidate the conditions for the existence of direct and inverse inequalities between the quantities $\|f-K_\sigma^\varphi(f)\|_{L_2}$ and $\omega_k(f;\tau/\sigma)_{L_2}$, viz., the $k$-th integral modulus of continuity of the function $f(x)$, $k=1,2,\dots,$. Under some restrictions on $\varphi(u)$, $u\in R_1$ the exact constants in these inequalities are found.