Geodesic
Sharp Bernstein type inequalities for splines in the mean square metrics
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 82-90 
O. L. Vinogradov. Sharp Bernstein type inequalities for splines in the mean square metrics. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 82-90. http://geodesic.mathdoc.fr/item/ZNSL_2015_434_a6/
@article{ZNSL_2015_434_a6,
     author = {O. L. Vinogradov},
     title = {Sharp {Bernstein} type inequalities for splines in the mean square metrics},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {82--90},
     year = {2015},
     volume = {434},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_434_a6/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

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