Sharp Bernstein type inequalities for splines in the mean square metrics

We give an elementary proof of the sharp Bernstein type inequality
$$ \|f^{(s)}\|_2\le\frac{n^s}{2^s}\left(\frac{\mathcal K_{2r+1-2s}}{\mathcal K_{2r+1}}\right)^{1/2}\|\delta^s_\frac\pi n f\|_2. $$
Here $n,r,s\in\mathbb N$, $f$ is a $2\pi$-periodic spline of order $r$ and of minimal defect with nodes $\frac{j\pi}n$ ($j\in\mathbb Z$), $\delta^s_h$ is the difference operator of order $s$ with step $h$, and the $\mathcal K_m$ are the Favard constants. A similar inequality for the space $L_2(\mathbb R)$ is also established.