“Mercer's theorem”的意思、由来-开放百科全书

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Mercer's theorem

释义

Introduction
Details
Trace
Generalizations
Mercer's condition
Discrete analog Examples
See also
Notes
References

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 {{harv|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 stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel.^[1]

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

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 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.

Theorem. Suppose K is a continuous symmetric non-negative definite kernel. Then there is an orthonormal basis{e_i}_i of L²[a, b] consisting of eigenfunctions of T_K such that the corresponding

sequence of eigenvalues {λ_i}_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 of

Mercer's theorem, particularly how it relates to spectral theory of compact operators.

The map K → T_K is injective.
T_K 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 T_K 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 T_K to show the existence of the

orthonormal basis {e_i}_i of

L²[a,b]

If λ_i ≠ 0, the eigenvector (eigenfunction) e_i 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.

Finally, to show non-negativity of the eigenvalues one can write 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 , implying .

Trace

The following is immediate:

Theorem. Suppose K is a continuous symmetric non-negative definite kernel; T_K has a sequence of nonnegative

eigenvalues {λ_i}_i. Then

This shows that the operator T_K is a trace class 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 {{Citation needed|date=January 2019}} 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 nonempty open subset U of X.

A recent generalization {{Citation needed|date=January 2019}} 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. Then essentially the same result holds:

Theorem. Suppose K is a continuous symmetric positive-definite kernel on X. If the function κ is L¹_μ(X), where κ(x)=K(x,x), for all x in X, then there is an orthonormal set{e_i}_i of L²_μ(X) consisting of eigenfunctions of T_K such that corresponding

sequence of eigenvalues {λ_i}_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 {{Citation needed|date=January 2019}} 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 T_K by the formula

T_K is a compact operator (actually it is even a Hilbert–Schmidt operator). If the kernel K is symmetric, by the spectral theorem, T_K has an orthonormal basis of eigenvectors. Those eigenvectors that correspond to non-zero eigenvalues can be arranged in a sequence {e_i}_i (regardless of separability).

Theorem. If K is a symmetric positive-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.

Mercer's condition

In mathematics, a real-valued function K(x,y) is said to fulfill Mercer's condition if for all square-integrable functions g(x) one has

Discrete analog

This is analogous to the definition of a positive-semidefinite matrix. This is a matrix of dimension , which satisfies, for all vectors , the property

.

Examples

A positive constant function

satisfies Mercer's condition, as then the integral becomes by Fubini's theorem

which is indeed non-negative.

Notes

1. ^http://www.cs.berkeley.edu/~bartlett/courses/281b-sp08/7.pdf

References

Adriaan Zaanen, Linear Analysis, North Holland Publishing Co., 1960,
Ferreira, J. C., Menegatto, V. A., Eigenvalues of integral operators defined by smooth positive definite kernels, Integral equation and Operator Theory, 64 (2009), no. 1, 61–81. (Gives the generalization of Mercer's theorem for metric spaces. The result is easily adapted to first countable topological spaces)
Konrad Jörgens, Linear integral operators, Pitman, Boston, 1982,
Richard Courant and David Hilbert, Methods of Mathematical Physics, vol 1, Interscience 1953,
Robert Ash, Information Theory, Dover Publications, 1990,
{{citation

}},

{{springer|title=Mercer theorem|id=p/m063440}}
H. König, Eigenvalue distribution of compact operators, Birkhäuser Verlag, 1986. (Gives the generalization of Mercer's theorem for finite measures μ.)

1 : Theorems in functional analysis

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