Norms on $L$ of Periodic Interpolation Splines with Equidistant Nodes

We consider the set $S_{r,n}$ of periodic (with period 1) splines of degree $r$ with deficiency 1 whose nodes are at $n$ equidistant points $x_i=i / n$. For $n$-tuples $\mathbf y=(y_0, y_1, \dots,y_{n-1})$, we take splines $s_{r,n}(\mathbf y, x)$ from $S_{r,n}$ solving the interpolation problem
$$s_{r, n} (\mathbf y, t_i)=y_i,$$
where $t_i = x_i$ if $r$ is odd and $t_i$ is the middle of the closed interval $[x_i , x_{i+1}]$ if $r$ is even. For the norms $L_{r,n}^*$ of the operator $\mathbf y\to s_{r,n} (\mathbf y, x)$ treated as an operator from $l^1$ to $L^1 [0,1]$ we establish the estimate
$$L_{r, n}^*=\frac{4}{\pi^2 n} \log \min (r, n)+O\biggl(\frac{1}{n} \biggr)$$
with an absolute constant in the remainder. We study the relationship between the norms $L_{r,n}^*$ and the norms of similar operators for nonperiodic splines.