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Construction and calculation of reproducing kernel determined by various linear differential operators. (English) Zbl 1181.46019

Let H m be the space of all functions with m-1 absolutely continuous derivatives and a square integrable mth derivative. The choice of an inner product involving a linear differential operator L of mth order as well as m linear functionals yields a reproducing kernel Hilbert space. Then the subspaces H 1 =ker(L) and H 2 , 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 H 2 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 m-1) at one point and functionals specifying an interpolation at m 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.

MSC:
46E22Hilbert spaces with reproducing kernels
47E05Ordinary differential operators
34L99Ordinary differential operators
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