Mercer's theorem

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

where symmetric means 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 [a, b] and all choices of real numbers c₁, ..., c_n (cf. positive definite kernel).

Associated to K is a linear operator (more specifically a Hilbert–Schmidt integral operator) on functions defined by the integral

For technical considerations we assume φ can range through the space L²[a, b] (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[a, b] 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 [a, b] and K has the representation

where the convergence is absolute and uniform.

Details

We now explain in greater detail the structure of the proof of Mercer's theorem, particularly how it relates to spectral theory of compact operators.

To show compactness, show that the image of the unit ball of L²[a,b] under T_K equicontinuous and apply Ascoli's theorem, to show that the image of the unit ball is relatively compact in C([a,b]) with the uniform norm and a fortiori in L²[a,b].

Now apply the spectral theorem for compact operators on Hilbert spaces to T_K to show the existence of the orthonormal basis {e_i}_i of L²[a,b]

If λ_i ≠ 0, the eigenvector e_i is seen to be continuous on [a,b]. Now

which shows that the sequence

converges absolutely and uniformly to a kernel K₀ which is easily seen to define the same operator as the kernel K. Hence K=K₀ from which Mercer's theorem follows.

Finally, to show non-negativity of the eigenvalues one can write λ ⟨ f , f ⟩ = ⟨ f , T K f ⟩ and expressing the right hand side as an integral well approximated by its Riemann sums, which are non-negative by positive definiteness of K, implying λ ⟨ f , f ⟩ ≥ 0 , implying λ ≥ 0 .

Trace

The following is immediate:

Theorem. Suppose K is a continuous symmetric non-negative definite kernel; T_K has a sequence of nonnegative eigenvalues {λ_i}_i. Then

This shows that the operator T_K is a trace class operator and

Generalizations

Mercer's theorem itself is a generalization of the result that any positive semidefinite matrix is the Gramian matrix of a set of vectors.

The first generalization replaces the interval [a, b] with any compact Hausdorff space and Lebesgue measure on [a, b] is replaced by a finite countably additive measure μ on the Borel algebra of X whose support is X. This means that μ(U) > 0 for any nonempty open subset U of X.

A recent generalization replaces this conditions by that follows: the set X is a first-countable topological space endowed with a Borel (complete) measure μ. X is the support of μ and, for all x in X, there is an open set U containing x and having finite measure. Then essentially the same result holds:

Theorem. Suppose K is a continuous symmetric positive definite kernel on X. If the function κ is L¹_μ(X), where κ(x)=K(x,x), for all x in X, then there is an orthonormal set {e_i}_i of L²_μ(X) consisting of eigenfunctions of T_K such that corresponding sequence of eigenvalues {λ_i}_i is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on X and K has the representation

where the convergence is absolute and uniform on compact subsets of X.

The next generalization deals with representations of measurable kernels.

Let (X, M, μ) be a σ-finite measure space. An L² (or square integrable) kernel on X is a function

L² kernels define a bounded operator T_K by the formula

T_K is a compact operator (actually it is even a Hilbert–Schmidt operator). If the kernel K is symmetric, by the spectral theorem, T_K has an orthonormal basis of eigenvectors. Those eigenvectors that correspond to non-zero eigenvalues can be arranged in a sequence {e_i}_i (regardless of separability).

Theorem. If K is a symmetric positive definite kernel on(X, M, μ), then

where the convergence in the L² norm. Note that when continuity of the kernel is not assumed, the expansion no longer converges uniformly.

Mercer's condition

In mathematics, a real-valued function K(x,y) is said to fulfill Mercer's condition if for all square integrable functions g(x) one has

Discrete analog

This is analogous to the definition of a positive-semidefinite matrix. This is a matrix K of dimension N , which satisfies, for all vectors g , the property

Examples

A positive constant function

satisfies Mercer's condition, as then the integral becomes by Fubini's theorem

which is indeed non-negative.