# 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 (Mercer 1909), is one of the most notable results of the work of James Mercer (1883–1932). 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 in the reproducing kernel Hilbert space theory where it characterizes a symmetric positive-definite kernel as a reproducing 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 for all .

*K* is said to be a positive-definite kernel if and only if

for all finite sequences of points *x*_{1}, ..., *x*_{n} of [*a*, *b*] and all choices of real numbers *c*_{1}, ..., *c*_{n}. Note that the term "positive-definite" is well-established in literature despite the weak inequality in the definition.

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 can range through the space
*L*^{2}[*a*, *b*] (see Lp space) of square-integrable real-valued functions.
Since *T _{K}* is a linear operator, we can talk about eigenvalues and eigenfunctions of

*T*.

_{K}**Theorem**. Suppose *K* is a continuous symmetric positive-definite kernel. Then there is an orthonormal basis
{*e*_{i}}_{i} of *L*^{2}[*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.

- The map
*K*↦*T*_{K}is injective. *T*_{K}is a non-negative symmetric compact operator on*L*^{2}[*a*,*b*]; moreover*K*(*x*,*x*) ≥ 0.

To show compactness, show that the image of the unit ball of *L*^{2}[*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 *L*^{2}[*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
*L*^{2}[*a*,*b*]

If λ_{i} ≠ 0, the eigenvector (eigenfunction) *e*_{i} is seen to be continuous on [*a*,*b*]. Now

which shows that the sequence

converges absolutely and uniformly to a kernel *K*_{0} which is easily seen to define the same operator as the kernel *K*. Hence *K*=*K*_{0} 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 positive-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 symmetric positive-semidefinite matrix is the Gramian matrix of a set of vectors.

The first generalization^{[citation needed]} 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^{[citation needed]} replaces these conditions by the following: 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 *L*^{1}_{μ}(*X*), where κ(x)=K(x,x), for all *x* in *X*, then there is an orthonormal set
{*e*_{i}}_{i} of *L*^{2}_{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^{[citation needed]} deals with representations of *measurable* kernels.

Let (*X*, *M*, μ) be a σ-finite measure space. An *L*^{2} (or square-integrable) kernel on *X* is a function

*L*^{2} 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 *L*^{2} 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 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.