Form preservation under approximation by local exponential splines of an arbitrary order

We continue the study of the properties of local $\mathcal L$-splines with uniform knots (such splines were constructed in the authors' earlier papers) corresponding to a linear differential operator $\mathcal L$ of order $r$ with constant coefficients and real pairwise distinct roots of the characteristic polynomial. Sufficient conditions (which are also necessary) are established under which the $\mathcal L$-spline locally inherits the property of the generalized $k$-monotonicity of $(k\le r-1)$ input data, which are the values of the approximated function at the nodes of a uniform grid shifted with respect to the grid of knots of the $\mathcal L$-spline. The parameters of an $\mathcal L$-spline that is exact on the kernel of the operator $\mathcal L$ are written explicitly.