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 (1883âÂÂ1932). 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 KarhunenâÂÂLoève 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.
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 for all .
K is said to be a positive-definite kernel if and only if
for all finite sequences of points x<sub>1</sub>, ..., x<sub>n</sub> of [a, b] and all choices of real numbers c<sub>1</sub>, ..., c<sub>n</sub>. Note that the term "positive-definite" is well-established in literature despite the weak inequality in the definition.
The fundamental characterization of stationary positive-definite kernels (where ) is given by Bochner's theorem. It states that a continuous function is positive-definite if and only if it can be expressed as the Fourier transform of a finite non-negative measure :
This spectral representation reveals the connection between positive definiteness and harmonic analysis, providing a stronger and more direct characterization of positive definiteness than the abstract definition in terms of inequalities when the kernel is stationary, e.g, when it can be expressed as a 1-variable function of the distance between points rather than the 2-variable function of the positions of pairs of points.
Associated to K is a linear operator (more specifically a HilbertâÂÂSchmidt integral operator when the interval is compact) on functions defined by the integral
We assume can range through the space of real-valued square-integrable functions L<sup>2</sup>[a, b]; however, in many cases the associated reproducing kernel Hilbert space can be strictly larger than L<sup>2</sup>[a, b]. Since T<sub>K</sub> is a linear operator, the eigenvalues and eigenfunctions of T<sub>K</sub> exist.
Theorem. Suppose K is a continuous symmetric positive-definite kernel. Then there is an orthonormal basis {e<sub>i</sub>}<sub>i</sub> of L<sup>2</sup>[a, b] consisting of eigenfunctions of T<sub>K</sub> such that the corresponding sequence of eigenvalues {λ<sub>i</sub>}<sub>i</sub> 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.
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 L<sup>2</sup>[a,b] under T<sub>K</sub> is 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<sup>2</sup>[a,b].
Now apply the spectral theorem for compact operators on Hilbert spaces to T<sub>K</sub> to show the existence of the orthonormal basis {e<sub>i</sub>}<sub>i</sub> of L<sup>2</sup>[a,b]
If λ<sub>i</sub> ≠ 0, the eigenvector (eigenfunction) e<sub>i</sub> is seen to be continuous on [a,b]. Now
which shows that the sequence
converges absolutely and uniformly to a kernel K<sub>0</sub> which is easily seen to define the same operator as the kernel K. Hence K=K<sub>0</sub> 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 .
The following is immediate:
Theorem. Suppose K is a continuous symmetric positive-definite kernel; T<sub>K</sub> has a sequence of nonnegative eigenvalues {λ<sub>i</sub>}<sub>i</sub>. Then
This shows that the operator T<sub>K</sub> is a trace class operator and
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 [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 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<sup>1</sup><sub>μ</sub>(X), where κ(x) := K(x,x) for all x in X, then there is an orthonormal set {e<sub>i</sub>}<sub>i</sub> of L<sup>2</sup><sub>μ</sub>(X) consisting of eigenfunctions of T<sub>K</sub> such that corresponding sequence of eigenvalues {λ<sub>i</sub>}<sub>i</sub> 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 L<sup>2</sup> (or square-integrable) kernel on X is a function
L<sup>2</sup> kernels define a bounded operator T<sub>K</sub> by the formula
T<sub>K</sub> is a compact operator (actually it is even a HilbertâÂÂSchmidt operator). If the kernel K is symmetric, by the spectral theorem, T<sub>K</sub> has an orthonormal basis of eigenvectors. Those eigenvectors that correspond to non-zero eigenvalues can be arranged in a sequence {e<sub>i</sub>}<sub>i</sub> (regardless of separability).
Theorem. If K is a symmetric positive-definite kernel on (X, M, μ), then
where the convergence in the L<sup>2</sup> norm. Note that when continuity of the kernel is not assumed, the expansion no longer converges uniformly.
A real-valued function K(x,y) is said to fulfill Mercer's condition if for all square-integrable functions g(x) one has
This is analogous to the definition of a positive-semidefinite matrix. This is a matrix of dimension , which satisfies, for all vectors , the property
A positive constant function
satisfies Mercer's condition, as then the integral becomes by Fubini's theorem
which is indeed non-negative.