Exact Jackson–Stechkin inequality in the space $L^2(\mathbb R^m)$

Let $\mathcal K=\mathcal K_{\sigma}(\tau,r,m)$ be the exact constant in the Jackson–Stechkin inequality
$$ E_{\sigma}(f)\leq\mathcal K\omega_{\tau}\biggl(f,\frac{\tau}{\sigma}\biggr),\quad f\in L^2(\mathbb R^m),\quad\sigma>0,\quad\tau>0,\quad r>0,\quad m=1,2,3,\dots, $$
where $E_{\sigma}(f)$ is the best $L^2$ approximation of a function $f$ by entire functions of exponential spherical type $\sigma$ and $\omega_r(f,t)$ is the $r$th spherical modulus of continuity of $f$. For $r\geq 1$, the following relations are proved:
$$ \min_{t>0}\mathcal K_{\sigma}(t,r,m)=1;\quad\tau_{(m-2)/2}\leq\rm{int}\biggl\{\tau>0\colon\mathcal K_{\sigma}(\tau,r,m)=1\biggr\}\leq 2\tau_{(m-2)/2}, $$
where $\tau_{\nu}$ is the first positive zero of the Bessel function $J_{\nu}$.