Mercer theorem for RKHS on noncompact sets

Reproducing kernel Hilbert spaces are an important family of function spaces and play useful roles in various branches of analysis and applications including the kernel machine learning. When the domain of definition is compact, they can be characterized as the image of the square root of an integral operator, by means of the Mercer theorem. The purpose of this paper is to extend the Mercer theorem to noncompact domains, and to establish a functional analysis characterization of the reproducing kernel Hilbert spaces on general domains.