Mercer's Theorem - Introduction

Introduction

To explain Mercer's theorem, we first consider an important special case; see below for a more general formulation. A kernel, in this context, is a symmetric continuous function that maps

symmetric meaning that K(x, s) = K(s, x).

K is said to be non-negative definite (or positive semidefinite) if and only if

for all finite sequences of points x₁, ..., x_n of and all choices of real numbers c₁, ..., c_n (cf. positive definite kernel).

Associated to K is a linear operator on functions defined by the integral

For technical considerations we assume φ can range through the space L2 (see Lp space) of square-integrable real-valued functions. Since T is a linear operator, we can talk about eigenvalues and eigenfunctions of T.

Theorem. Suppose K is a continuous symmetric non-negative definite kernel. Then there is an orthonormal basis {e_i}_i of L2 consisting of eigenfunctions of T_K such that the corresponding sequence of eigenvalues {λ_i}_i is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on and K has the representation

where the convergence is absolute and uniform.