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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.
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 for all .
K is said to be positive-definite 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.
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 can range through the space
L2[a, b] (see Lp space) of square-integrable real-valued functions.
Since TK is a linear operator, we can talk about eigenvalues and eigenfunctions of TK.
Theorem. Suppose K is a continuous symmetric positive-definite kernel. 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.
The map K ↦ TK is injective.
TK is a non-negative symmetric compact operator on L2[a,b]; moreover K(x, x) ≥ 0.
Official source
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