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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 $m$th derivative. The choice of an inner product involving a linear differential operator $L$ of $m$th order as well as $m$ linear functionals yields a reproducing kernel Hilbert space. Then the subspaces ${H}_{1}=\text{ker}\left(L\right)$ 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:
 46E22 Hilbert spaces with reproducing kernels 47E05 Ordinary differential operators 34L99 Ordinary differential operators
##### References:
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