Sharp Bernstein type inequalities for splines in the mean square metrics

O. L. Vinogradov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 82-90

Résumé

We give an elementary proof of the sharp Bernstein type inequality $$ \|f^{(s)}\|_2\le\frac{n^s}{2^s}\left(\frac{\mathcal K_{2r+1-2s}}{\mathcal K_{2r+1}}\right)^{1/2}\|\delta^s_\frac\pi n f\|_2. $$ Here $n,r,s\in\mathbb N$, $f$ is a $2\pi$-periodic spline of order $r$ and of minimal defect with nodes $\frac{j\pi}n$ ($j\in\mathbb Z$), $\delta^s_h$ is the difference operator of order $s$ with step $h$, and the $\mathcal K_m$ are the Favard constants. A similar inequality for the space $L_2(\mathbb R)$ is also established.