Mercer's Theorem

Mercer's Theorem, proved by the British mathematician James Mercer, is a result of functional analysis representing a continuous, symmetric, positive-definite function on a square as a convergent sum of simpler product functions built from the eigenvalues and eigenfunctions of an associated integral operator. It is a key theoretical tool in the theory of integral equations and in the Hilbert space treatment of stochastic processes, underlying the Karhunen-Loeve theorem, and it also characterizes when a symmetric positive-definite kernel is a reproducing kernel, a fact used throughout reproducing kernel Hilbert space theory and kernel methods in statistics.