Asymptotic behaviour of the Lebesgue constants of periodic interpolation splines with equidistant nodes

Associated with each continuous function $f$ of period 1 is the periodic spline $s_{r,n}(f)$ that has degree $r$, defect 1, nodes at the points $x_i=i/n$, $i=0,1,\dots,n-1$ and that interpolates $f$ at these points for $r$ odd and at the mid-points of the intervals $[x_i,x_{i+1}]$ for $r$ even.
For the corresponding Lebesgue constants $L_{r,n}$, that is the norms of the operators $f(x)\to s_{r,n}(f)$ from $C$ to $C$, the asymptotic formula
$$ L_{r,n}=\frac2\pi\log\min(r,n)+O(1), $$
is established, which holds uniformly in $r$ and $n$.