Sharp estimates of linear approximations by nonperiodic splines in terms of linear combinations of moduli of continuity

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
$$ \|f-\mathcal X_{\sigma,r,\mu}(f)\|_p\leqslant\frac{\mathcal K_r}{2\sigma^r}\omega_1\left(f^{(r)},\frac\pi\sigma\right)_p. $$