Inequality of the Bernshtein type for polynomial splines in the space L_2 - Babenko , V. F.; Pichugov , S. A. (Ukrains’kyi Matematychnyi Zhurnal, 1991)
For 2π-periodic polynomial splines of order r, of minimal defect, with nodes at the points kπ /n, nN, there are established the sharp inequalities \[|| s^(l) || _ 2 ≤ || φ _n,r^(l) || _ 2 || Δ _h^l φ _n, r || _ 2 || Δ _h^l s|| _ 2 ≤ || φ _n,r^(l) || _ 2 || φ _n, r || _ 2 || s || _ 2 ,l = 1, …, r - 1,\] valid for 0<h<π/2ln and 0<h<π/4ln respectively, where φ_n,r is the r -th periodic integral of the function φ_n,0(x)=sign sin nx, and Δ_h^lf(x)=_k=0^l(-1)^kC_1^kf(x+(l-2k)h).