Inequality of the Bernshtein type for polynomial splines in the space $L_2$

Abstract

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

$\Delta_h^lf(x)=\sum_{k=0}^l(-1)^kC_1^kf(x+(l-2k)h)$.