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. It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of , for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite 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 symmetric means that K(x, s) = K(s, x).

K is said to be non-negative definite (or positive semidefinite) if and only if

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 (more specifically a Hilbert–Schmidt integral operator) on functions defined by the integral

For technical considerations we assume ${\displaystyle \varphi }$ 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.

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