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 sr,n(f) that has degree r, defect 1, nodes at the points xi=i/n, i=0,1,...,n-1 and that interpolates f at these points for r odd and at the mid-points of the intervals [xi,xi+1] for r even. For the corresponding Lebesgue constants Lr,n, that is the norms of the operators f(x)→sr,n(f) from C to C, the asymptotic formula Lr,n=2/π log min(r,n)+O(l) is established, which holds uniformly in r and n