Mercer's theorem

The set of Mercer is a mathematical statement from the branch of functional analysis. It is named after the mathematician James Mercer and states that the integral kernel of a positive, self-adjoint integral operator can be represented as a convergent series of its eigenvalues ​​and eigenvectors.

Statement

Let be a compact subset of. Furthermore, let a continuous function, for which the condition holds for all, so that the integral operator defined by

Is self-adjoint. Be also the counted according to their geometric multiplicity of the eigenvalues ​​of the integral operator with corresponding eigenfunctions. Is the operator In addition, positive, that is

Then applies

Where the convergence is absolute and uniform.