Exact constants of approximation for differentiable periodic functions

For all odd $r$ we construct a linear operator $B_{n,r}(f)$ which maps the set of $2\pi$-periodic functions $f(t)\in X^{(r)}$ ($X^{(r)}=X^{(r)}$ or $L_1^{(r)}$) into a set of trigonometric polynomials of order not higher than $n-1$ such that
$$ \sup_{f\in X^{(r)}}\frac{n^rE_n(f)_X}{\omega(f^{(r)},\pi/n)_X}=\sup_{f\in X^{(r)}}\frac{n^r\|f-B_{n,r}(f)\|_X}{\omega(f^{(r)},\pi/n)_X}=\frac{K_r}2, $$
where $X$ is the $C$ or $L_1$ metric, $E_n(f)_X$ and $\omega(f,\delta)_X$ are the best approximation by means of trigonometric polynomials of order not higher than $n-1$ and the modulus of continuity of the function $f$ in the $X$ metric, respectively; $K_r$ are the known Favard constants.