Approximation by local parabolic splines constructed on the basis of interpolationin the mean

The paper deals with approximative and form–retaining properties of the local parabolic splines of the form $S(x)=\sum\limits_j y_j B_2 (x-jh), \ (h>0),$ where $B_2$ is a normalized parabolic spline with the uniform nodes and functionals $y_j=y_j(f)$ are given for an arbitrary function $f$ defined on $\mathbb{R}$ by means of the equalities
$$y_j=\frac{1}{h_1}\int\limits_{\frac{-h_1}{2}}^{\frac{h_1}{2}} f(jh+t)dt \quad (j\in\mathbb{Z}). $$
On the class $W^2_\infty$ of functions under $0<h_1\leq 2h$, the approximation error value is calculated exactly for the case of approximation by such splines in the uniform metrics.