Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operators
نویسنده
چکیده
In this paper we introduce a generalization of the classical L2(R)-based Sobolev spaces with the help of a vector differential operator P which consists of finitely or countably many differential operators Pn which themselves are linear combinations of distributional derivatives. We find that certain proper full-space Green functions G with respect to L = P∗TP are positive definite functions. Here we ensure that the vector distributional adjoint operator P∗ of P is well-defined in the distributional sense. We then provide sufficient conditions under which our generalized Sobolev space will become a reproducing-kernel Hilbert space whose reproducing kernel can be computed via the associated Green function G. As an application of this theoretical framework we useG to construct multivariate minimumnorm interpolants s f ,X to data sampled from a generalized Sobolev function f on X . Among other examples we show the reproducing-kernel Hilbert space of the Gaussian function is equivalent to a generalized Sobolev space. Mathematics Subject Classification (2000): Primary 41A30, 65D05; Secondary 34B27, 41A63, 46E22, 46E35
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