On intertwining operators via reproducing kernels

Reproducing kernels of Sobolev spaces via a green kernel approach with differential operators and boundary operators

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L := P∗T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional ad...

Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operators

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. H...

Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to la...

Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators

In this paper we extend the definition of generalized Sobolev space and subsequent theoretical results established recently for positive definite kernels and differential operators in the article [21]. In the present paper the semi-inner product of the generalized Sobolev space is set up by a vector distributional operator P consisting of finitely or countably many distributional operators Pn, ...

Reproducing kernels , Bergman kernels , Poisson kernels