Exact Constants in Jackson Inequalities for Periodic Differentiable Functions in the Space $L_\infty$

It is proved that, in the space ${L }_\infty[0,2\pi]$, the following equalities hold for all $k=0,1,2,\dots$, $n\in\mathbb N$, $r=1,3,5,\dots$, $\mu\ge r$:
$$ \sup_{\substack{f\in {L }_\infty^r\\ f\ne\operatorname{const}}} \frac{{E}_{n-1}(f)}{\omega(f^{(r)},\pi/(n(2k+1)))}= \sup_{\substack{f\in {L }_\infty^r\\ f\ne\operatorname{const}}} \frac{{E}_{n,\mu}(f)}{\omega(f^{(r)},\pi/(n(2k+1)))}= \frac{\|\psi_{r,2k+1}\|}{2n^r}\mspace{2mu}, $$
where ${E}_{n-1}(f)$ and ${E}_{n,\mu}(f)$ are the best approximations of $f$ by, respectively, trigonometric polynomials of degree $n-1$ and $2\pi$-periodic splines of minimal deficiency of order $\mu$ with $2n$ equidistant nodes, $\omega(f^{(r)},h)$ is the modulus of continuity of $f^{(r)}$, $\psi_{r,2k+1}$ is the $r$th periodic integral of the special function $\psi_{0,2k+1}$, which is odd and piecewise constant on the partition $j\pi/ (2k+1)$, $j\in\mathbb Z$. For $k=0$, this result was obtained earlier by Ligun.