An Operator Theoretical Approach to Mercer's Theorem

TL;DR

This survey explores Mercer's Theorem through an operator-theoretical lens, connecting it with reproducing kernel Hilbert spaces and spectral theory, offering a modern and unified perspective.

Contribution

It introduces a novel operator-theoretical approach to Mercer's Theorem, linking it with spectral theory and reproducing kernel Hilbert spaces in a reverse manner.

Findings

Provides a modern introduction to reproducing kernel Hilbert spaces

Connects Mercer's Theorem with spectral theory of compact operators

Offers new insights into the relations between analysis domains

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.