Mercer's Theorem - Generalizations

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 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(U) > 0 for any 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 non-negative 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 non-negative 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.