Mercer's theorem


In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in, is one of the most notable results of the work of James Mercer. It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel.

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 = K.
K is said to be non-negative definite if and only if
for all finite sequences of points x1, ..., xn of and all choices of real numbers c1, ..., cn.
Associated to K is a linear operator on functions defined by the integral
For technical considerations we assume can range through the space
L2 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
i of L2 consisting of eigenfunctions of TK such that the corresponding
sequence of eigenvalues 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.

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 under TK equicontinuous and apply Ascoli's theorem, to show that the image of the unit ball is relatively compact in C with the uniform norm and a fortiori in L2.
Now apply the spectral theorem for compact operators on Hilbert
spaces to TK to show the existence of the
orthonormal basis i of
L2
If λi ≠ 0, the eigenvector ei is seen to be continuous on . 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 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, implying.

Trace

The following is immediate:
Theorem. Suppose K is a continuous symmetric non-negative definite kernel; TK has a sequence of nonnegative
eigenvalues i. Then
This shows that the operator TK is a trace class operator and

Generalizations

Mercer's theorem itself is a generalization of the result that any symmetric positive-semidefinite matrix is the Gramian matrix of a set of vectors.
The first generalization replaces the interval with any compact Hausdorff space and Lebesgue measure on is replaced by a finite countably additive measure μ on the Borel algebra of X whose support is X. This means that μ > 0 for any nonempty open subset U of X.
A recent generalization replaces these conditions by the following: the set X is a first-countable topological space endowed with a Borel 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 L1μ, where κ=K, for all x in X, then there is an orthonormal set
i of L2μ consisting of eigenfunctions of TK such that corresponding
sequence of eigenvalues 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 be a σ-finite measure space. An L2 kernel on X is a function
L2 kernels define a bounded operator TK by the formula
TK is a compact operator. If the kernel K is symmetric, by the spectral theorem, TK has an orthonormal basis of eigenvectors. Those eigenvectors that correspond to non-zero eigenvalues can be arranged in a sequence i.
Theorem. If K is a symmetric positive-definite kernel on, 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 is said to fulfill Mercer's condition if for all square-integrable functions g one has

Discrete analog

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