Local approximation by parabolic splines in the mean with large averaging intervals

In the paper, local parabolic splines on the whole real line $\mathbb R$ with equidistant nodes are considered. These splines realize the simplest local approximation scheme, but instead of the function values at the nodes, their average values in symmetric neighborhoods of the nodes are approximated. For an arbitrary averaging step $H$, which more than twice is more than the spline grid step $h$, the approximation errors in the uniform metric for functions and their derivatives are precisely calculated on the class $W_{\infty}^2$. For small steps of averaging $H\leq 2h$, these values were found by E.V.Strelkova in 2007.