Inequality of the Bernshtein type for polynomial splines in the space $L_2$ - Babenko , V. F.; Pichugov , S. A.; Бабенко , В. Ф.; Пичугов , С. А. (Institute of Mathematics, NAS of Ukraine, 1991)
For $2\pi$-periodic polynomial splines of order $r$, of minimal defect, with nodes at the points $k\pi /n, n\in\mathbb{N}$, there are established the sharp inequalities \[|| s^{(l)} || _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \Delta _h^l \varphi _{n, r} || _{ 2} }|| \Delta _h^l s|| _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \varphi _{n, r} || _{ 2} }|| s || _{ 2} ,\quad l = 1, \dots, r - 1,\] valid for $0<h<\pi/2ln$ and $0<h<\pi/4ln$ respectively, where $\varphi_{n,r}$ is the $r$ -th periodic integral of the function $\varphi_{n,0}(x)={\rm sign} \, {\rm sin} nx$, and $\Delta_h^lf(x)=\sum_{k=0}^l(-1)^kC_1^kf(x+(l-2k)h)$.