An Operator Theoretical Approach to Mercer's Theorem

Abstract

This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of the theory of reproducing kernel Hilbert spaces, an overview on Weyl's kernel and the Gaussian kernels, and finally an approach to Mercer's Theorem within the theory of reproducing kernel Hilbert spaces and the spectral theory of integral operators. This approach is reverse to the known approaches to Mercer's Theorem and sheds some light on the intricate relations between different domains in analysis.