Sampling Theorems In Reproducing Kernel Hilbert Spaces

Sampling theorem is a basic problem of computational mathematics, and it is also a problem of classic mathematics. Now it has been widely used in communication theory, image processing, geological exploration, measurement and etc field. The research and application of the sampling theorem is very important and has been widely used in technology of engineering. In this paper, the problem of Reproducing Kernel Hilbert Spaces has been discussed.First, some equivalent problems about Riesz basis in the Reproducing Kernel Hilbert Spaces have been researched and proved by using some special skills of Reproducing Kerne function Spaces. The corresponding Sampling theorems in the Reproducing Kernel Hilbert space have been given when the Riesz basis in the Reproducing Kernel Hilbert Spaces has been produced by function of the Reproducing Kernel.Second, some Sampling formula have been established through translation invariant Reproducing Kernel Hilbert Spaces for practical application in translation invariant Reproducing Kernel Hilbert Spaces . In this paper, weight function in the space has been set up through Fourier transform. The Reproducing Kernel function has been given on the basis of them, then the Sampling theorem in translation invariant Reproducing Kernel Hilbert Spaces, at last the convergence and estimation of it has been discussed.At last, in the Reproducing Kernel Hilbert Spaces which have been established by basic theorem of linear transform, Sampling theorem which more general than before has been given through some properties of Reproducing Kernel function in the Reproducing Kernel Hilbert Spaces, and the convergence and estimation of the theorem has been researched. The Sampling formula which been given make the numerical computation easier than before, and it also provides theoretical base.This paper focused on using some different and special properties of the Reproducing Kernel Hilbert function in the Reproducing Kernel Hilbert Spaces, different Sampling problems of function have been discussed in corresponding Reproducing Kernel Hilbert Spaces. These supply basic theorem for solving the general Sampling problems in the Reproducing Kernel Hilbert Spaces.