Mercer's theorem
Encyclopedia
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
. It is an important theoretical tool in the theory of integral equation
s; it is used in the Hilbert space
theory of stochastic process
es, for example the Karhunen-Loève theorem
; and it is also used to characterize a symmetric positive semi-definite kernel.
, we first consider an important special case; see below for a more general formulation.
A kernel, in this context, is a symmetric continuous function that maps
symmetric meaning 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, ..., xn of [a, b] and all choices of real numbers c1, ..., cn (cf. positive definite kernel).
Associated to K is a linear operator on functions defined by the integral
For technical considerations we assume φ 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 eigenfunction
s of T.
Theorem. Suppose K is a continuous symmetric non-negative 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.
Mercer's theorem, particularly how it relates to spectral theory of compact operators
.
To show compactness, show that the image of the unit ball of L2[a,b] under TK equicontinuous and apply Ascoli's theorem, to show that the image of the unit ball is relatively compact in C([a,b]) with the uniform norm and a fortiori in L2[a,b].
Now apply the spectral theorem
for compact operators on Hilbert
spaces to TK to show the existence of the
orthonormal basis {ei}i of
L2[a,b]
If λi ≠ 0, the eigenvector ei is seen to be continuous on [a,b]. Now
which shows that the sequence
converges absolutely and uniformly to a kernel K0 which is easily seen to define the same operator as the kernel K. Hence K=K0 from which Mercer's theorem follows.
Theorem. Suppose K is a continuous symmetric non-negative definite kernel; TK has a sequence of nonnegative
eigenvalues {λi}i. Then
This shows that the operator TK is a trace class
operator and
The first generalization replaces the interval [a, b] with any compact Hausdorff space and Lebesgue measure on [a, b] 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 L1μ(X), where κ(x)=K(x,x), for all x in X, then there is an orthonormal set
{ei}i of L2μ(X) consisting of eigenfunctions of TK such that corresponding
sequence of eigenvalues {λi}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 TK by the formula
TK is a compact operator (actually it is even a Hilbert-Schmidt operator). 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).
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, Mercer's theorem is a representation of a symmetric positive-definite
Definite bilinear form
In mathematics, a definite bilinear form is a bilinear form B over some vector space V such that the associated quadratic formQ=B \,...
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
James Mercer (mathematician)
James Mercer FRS was a mathematician, born in Bootle, close to Liverpool, England. He was educated at University of Manchester, and then University of Cambridge...
. It is an important theoretical tool in the theory of integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
s; it is used in the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
theory of stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es, for example the Karhunen-Loève theorem
Karhunen-Loève theorem
In the theory of stochastic processes, the Karhunen–Loève theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval...
; and it is also used to characterize a symmetric positive semi-definite kernel.
Introduction
To explain Mercer's theoremTheorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
, we first consider an important special case; see below for a more general formulation.
A kernel, in this context, is a symmetric continuous function that maps
symmetric meaning 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, ..., xn of [a, b] and all choices of real numbers c1, ..., cn (cf. positive definite kernel).
Associated to K is a linear operator on functions defined by the integral
For technical considerations we assume φ can range through the space
L2[a, b] (see Lp space
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
) of square-integrable real-valued functions.
Since T is a linear operator, we can talk about eigenvalues and eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of T.
Theorem. Suppose K is a continuous symmetric non-negative definite kernel. Then there is an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
{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.
Details
We now explain in greater detail the structure of the proof ofMercer's theorem, particularly how it relates to spectral theory of compact operators
Spectral theory of compact operators
In functional analysis, compact operators are linear operators that map bounded sets to precompact sets. The set of compact operators acting on a Hilbert space H is the closure of the set of finite rank operators in the uniform operator topology. In general, operators on infinite dimensional spaces...
.
- The map K → TK is injective.
- TK is a non-negative symmetric compact operator on L2[a,b]; moreover K(x, x) ≥ 0.
To show compactness, show that the image of the unit ball of L2[a,b] under TK equicontinuous and apply Ascoli's theorem, to show that the image of the unit ball is relatively compact in C([a,b]) with the uniform norm and a fortiori in L2[a,b].
Now apply the spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
for compact operators on Hilbert
spaces to TK to show the existence of the
orthonormal basis {ei}i of
L2[a,b]
If λi ≠ 0, the eigenvector ei is seen to be continuous on [a,b]. Now
which shows that the sequence
converges absolutely and uniformly to a kernel K0 which is easily seen to define the same operator as the kernel K. Hence K=K0 from which Mercer's theorem follows.
Trace
The following is immediate:Theorem. Suppose K is a continuous symmetric non-negative definite kernel; TK has a sequence of nonnegative
eigenvalues {λi}i. Then
This shows that the operator TK is a trace class
Trace class
In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis....
operator and
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 [a, b] with any compact Hausdorff space and Lebesgue measure on [a, b] 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 L1μ(X), where κ(x)=K(x,x), for all x in X, then there is an orthonormal set
{ei}i of L2μ(X) consisting of eigenfunctions of TK such that corresponding
sequence of eigenvalues {λi}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 TK by the formula
TK is a compact operator (actually it is even a Hilbert-Schmidt operator). 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).
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.