Geodesic
Sharp estimates of linear approximations by nonperiodic splines in terms of linear combinations of moduli of continuity
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 55-76 
O. L. Vinogradov; A. V. Gladkaya. Sharp estimates of linear approximations by nonperiodic splines in terms of linear combinations of moduli of continuity. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 55-76. http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a4/
@article{ZNSL_2017_456_a4,
     author = {O. L. Vinogradov and A. V. Gladkaya},
     title = {Sharp estimates of linear approximations by nonperiodic splines in terms of linear combinations of moduli of continuity},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {55--76},
     year = {2017},
     volume = {456},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a4/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Suppose that $\sigma>0$, $r,\mu\in\mathbb N$, $\mu\geqslant r+1$, $r$ is odd, $p\in[1,+\infty]$, $f\in W^{(r)}_p(\mathbb R)$. We construct linear operators $\mathcal X_{\sigma,r,\mu}$ whose values are splines of degree $\mu$ and of minimal defect with knots $\frac{k\pi}\sigma$ ($k\in\mathbb Z$) such that \begin{gather*} \|f-\mathcal X_{\sigma,r,\mu}(f)\|_p\\ \leqslant\left(\frac\pi\sigma\right)^r\left\{\frac{A_{r,0}}2\omega_1\left(f^{(r)},\frac\pi\sigma\right)_p+\sum_{\nu=1}^{\mu-r-1}A_{r,\nu}\omega_\nu\left(f^{(r)},\frac\pi\sigma\right)_p\right\}\\ +\left(\frac\pi\sigma\right)^r\biggl( \frac{\mathcal K_r}{\pi^r}-\sum_{\nu=0}^{\mu-r-1}2^\nu A_{r,\nu}\biggr)2^{r-\mu}\omega_{\mu-r}\left(f^{(r)},\frac\pi\sigma\right)_p, \end{gather*} where for ${p=1,+\infty}$ the constants cannot be reduced on the class $W^{(r)}_p(\mathbb R)$. Here $\mathcal K_r=\frac4\pi\sum_{l=0}^\infty\frac{(-1)^{l(r+1)}}{(2l+1)^{r+1}}$ are the Favard constants, the constants $A_{r,\nu}$ are constructed explicitly, $\omega_\nu$ is a modulus of continuity of order $\nu$. As a corollary, we get the sharp Jackson type inequality $$ \|f-\mathcal X_{\sigma,r,\mu}(f)\|_p\leqslant\frac{\mathcal K_r}{2\sigma^r}\omega_1\left(f^{(r)},\frac\pi\sigma\right)_p. $$

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