[1] To explain Mercer's theorem, we first consider an important special case; see below for a more general formulation.
K is said to be a positive-definite kernel if and only if for all finite sequences of points x1, ..., xn of [a, b] and all choices of real numbers c1, ..., cn.
Note that the term "positive-definite" is well-established in literature despite the weak inequality in the definition.
can range through the space of real-valued square-integrable functions L2[a, b]; however, in many cases the associated RKHS can be strictly larger than L2[a, b].
Then there is an orthonormal basis {ei}i of L2[a, b] consisting of eigenfunctions of TK such that the corresponding sequence of eigenvalues {λi}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.
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.
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
Suppose K is a continuous symmetric positive-definite kernel; TK has a sequence of nonnegative eigenvalues {λi}i.
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.
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.
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 {ei}i (regardless of separability).
Note that when continuity of the kernel is not assumed, the expansion no longer converges uniformly.
, the property A positive constant function satisfies Mercer's condition, as then the integral becomes by Fubini's theorem which is indeed non-negative.