A Green Function Approach to (Conditionally) Positive Definite Functions and Reproducing Kernels of Generalized Sobolev Spaces
Department of Applied Mathematics
Illinois Institute of Technology
In this talk we introduce a generalization of the classical L2(Rd)-based Sobolev spaces with the help of a vector operator P which consists of finitely or countably many operators Pn that may be of a differential or even more general type. We find that certain proper full-space Green functions G with respect to L = PTP are (conditionally) 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 use G to construct multivariate minimum-norm interpolants s f,X to data sampled from a generalized Sobolev function f on X. Among other examples we show how the Gaussian kernel K(x, y) := e−2kx−yk22 is the reproducing kernel of a generalized Sobolev space.