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 $2\pi$-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