Orders of approximation by local exponential splines

We continue the study of approximation properties of local exponential splines on a uniform grid with step $h>0$ corresponding to a linear differential operator $\mathcal L$ with constant coefficients and real pairwise different roots of the characteristic polynomial (such splines were constructed by E. V. Strelkova and V. T. Shevaldin). We find order estimates as $h\to0$ for the error of approximation of certain Sobolev classes of functions by the mentioned splines, which are exact on the kernel of the operator $\mathcal L$.