A nonperiodic analogue of the Akhiezer–Krein–Favard operators

In what follows, $\sigma>0$, $m,r\in\mathbb N$, $m\geqslant r$, $\mathbf S_{\sigma,m}$ is the space of splines of order $m$ and minimal defect with nodes $\frac{j\pi}\sigma$ ($j\in\mathbb Z$), $A_{\sigma,m}(f)_p$ is the best approximation of a function $f$ by the set $\mathbf S_{\sigma,m}$ in the space $L_p(\mathbb R)$. It is known that for $p=1,+\infty$
\begin{equation} \sup_{f\in W^{(r)}_p(\mathbb R)}\frac{A_{\sigma,m}(f)_p}{\|f^{(r)}\|_p}=\frac{\mathcal K_r}{\sigma^r}.\end{equation}
In this paper we construct linear operators $\mathcal X_{\sigma,r,m}$ with their values in $\mathbf S_{\sigma,m}$, such that for all $p\in[1,+\infty]$ and $f\in W_p^{(r)}(\mathbb R)$
$$ \|f-\mathcal X_{\sigma,r,m}(f)\|_p\leqslant\frac{\mathcal K_r}{\sigma^r}\|f^{(r)}\|_p. $$
So we establish the possibility to achieve the upper bounds in (1) by linear methods of approximation, which was unknown before.