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Concept# Mercer's theorem

Summary

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 (1883–1932). 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 Karhunen–Loève theorem; and it is also used to characterize a symmetric positive-definite kernel.
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 symmetric continuous function
: K: [a,b] \times [a,b] \rightarrow \mathbb{R}
where symmetric means that K(x,y) = K(y,x) for all x,y \in [a,b].
K is said to be positive-definite if and only if
: \sum_{i=1}^n\sum_{j=1}^n K(x_i

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