Let be the space of all functions with absolutely continuous derivatives and a square integrable th derivative. The choice of an inner product involving a linear differential operator of th order as well as linear functionals yields a reproducing kernel Hilbert space. Then the subspaces and , defined by the linear functionals, represent orthogonal complements. Each subspace exhibits a reproducing kernel of its own.
The construction of the reproducing kernel for the subspace is complicated and thus considered in the paper. The authors restrict to the case of a linear differential operator with constant coefficients and pairwise different roots of its characteristic polynomial. For the linear functionals, two choices are investigated: functionals determining an initial value condition (of derivatives up to order ) at one point and functionals specifying an interpolation at nodes. In each case, the authors derive an explicit formula for the calculation of the reproducing kernel. The specific situation of arithmetic roots is analysed in detail. Three examples are presented.