Approximation by local $\mathcal L$-splines that are exact on subspaces of the kernel of a differential operator

We construct local $\mathcal L$-splines with uniform nodes that preserve subsets from the kernel of a linear differential operator $\mathcal L$ of order $r$ with constant real coefficients and pairwise distinct roots of the characteristic polynomial. Pointwise estimates are found for the error of approximation by the constructed $\mathcal L$-splines on classes of functions defined by differential operators of orders smaller than $r$.