Mercer's theorem

In mathematics and 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 James Mercer (1909). "Functions of positive and negative type and their connection with the theory of integral equations". Philos. Trans. Roy. Soc. London. 209.) 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 also used in the Hilbert space theory of stochastic processes, for example the Karhunen-Loève theorem (cf. Karhunen-Loève transform).

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 continuous function that maps

$K:[a,b]\times [a,b]\rightarrow \mathbb {R}$

such that K(x, s) = K(s, x).

K is said to be non-negative definite if and only if

$\sum _{i=1}^{n}\sum _{j=1}^{n}K(x_{i},x_{j})c_{i}c_{j}\geq 0$

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 on functions defined by the integral

$[T_{K}\varphi ](x)=\int _{a}^{b}K(x,s)\varphi (s)\,ds.$

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

$K(s,t)=\sum _{j=1}^{\infty }\lambda _{j}\,e_{j}(s)\,e_{j}(t)$

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 L²[a,b]; moreover K(x, x) ≥ 0.

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]

$\lambda _{i}e_{i}(t)=[T_{K}e_{i}](t)=\int _{a}^{b}K(t,s)e_{i}(s)\,ds.$

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

$\sum _{i=1}^{\infty }\lambda _{i}|e_{i}(t)e_{i}(s)|\leq \sup _{x\in [a,b]}|K(x,x)|^{2},$

which shows that the sequence

$\sum _{i=1}^{\infty }\lambda _{i}e_{i}(t)e_{i}(s)$

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.

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

$\int _{a}^{b}K(t,t)\,dt=\sum _{i}\lambda _{i}.$

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

$\operatorname {trace} (T_{K})=\int _{a}^{b}K(t,t)\,dt.$

Generalizations

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 open subset U of X. Then essentially the same result holds:

Theorem. Suppose K is a continuous symmetric non-negative definite kernel on X. Then there is an orthonormal basis {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

$K(s,t)=\sum _{j=1}^{\infty }\lambda _{j}\,e_{j}(s)\,e_{j}(t)$

where the convergence is absolute and uniform on 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

$K\in L_{\mu \otimes \mu }^{2}(X\times X).$

L² kernels define a bounded operator T_K by the formula

$\langle T_{K}\varphi ,\psi \rangle =\int _{X\times X}K(y,x)\varphi (y)\psi (x)\,d[\mu \otimes \mu ](y,x).$

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 non-negative definite kernel on(X, M, μ), then

$K(y,x)=\sum _{i\in \mathbb {N} }\lambda _{i}e_{i}(y)e_{i}(x)$

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

References

• Adriaan Zaanen, Linear Analysis, North Holland Publishing Co., 1960

• Jörgens Konrad, Linear integral operators, Pitman, Boston, 1982

• Richard Courant and David Hilbert, Methods of Mathematical Physics, vol 1, Interscience 1953,

• Robert Ash, Information Theory, Dover Publications, 1990

• J. Mercer, Functions of positive and negative type and their connection with the theory of integral equations, Philos. Trans. Roy. Soc. London 1909,

• H. König, Eigenvalue distribution of compact operators, Birkhäuser Verlag, 1986 (gives the generalization of Mercer's theorem for finite measures μ)