Sharp jackson - Chernykh type inequality for spline approximations on the line

An analog of the Jackson - Chernykh inequality for spline approximations in the space $L_2(\mathbb{R})$ is established. For $r \in \mathbb{N}$, $\sigma > 0$, we denote by $A_{\sigma r}(f)_2$ the best approximation of a function $f \in L_2(\mathbb{R})$ by the space of splines of degree $r$ and of minimal defect with knots $\frac{j \pi}{\sigma}$, $j \in \mathbb{Z}$, and by $\omega(f, \delta)$ its first order modulus of continuity in $L_2(\mathbb{R})$. The main result of the paper is the following. For every $f \in L_2(\mathbb{R})$
$$A_{\sigma r}(f)_2 \leqslant \frac{1}{\sqrt{2}}\omega(f,\frac{\theta_r \pi}{\sigma})_2$$
, where $\varepsilon_r$ is the positive root of the equation
$$\frac{4 \varepsilon^2(ch \frac{\pi \varepsilon}{\tau}-1)}{ch \frac{\pi \varepsilon}{\tau}+\cos \frac{\pi}{\tau}}= \frac{1}{3^{2r-2}}, \tau = \sqrt{1-\varepsilon^2}$$
, $\theta_r = \frac{1}{\sqrt{1-\varepsilon_r^2}}$. The constant $\frac{1}{\sqrt{2}}$ cannot be reduced on the whole class $L_2(\mathbb{R})$, even if one insreases the step of the modulus of continuity.