On error estimates of local approximation by splines

We consider the so-called simplest formula for local approximation by polynomial splines of order $n$ (Schoenberg splines). The spline itself and all derivatives except that of the highest order, approximate a given function and its corresponding derivatives with the second order. We show that the jump of the highest derivative of order $n-1$; i. e., the value of discontinuity, divided by the meshsize, approximates the $n$th derivative of the original function. We found an asymptotic expansion of the jump.