Mercer's theorem

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 (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 semi-definite kernel.[1] Contents 1 Introduction 2 Details 3 Trace 4 Generalizations 5 Mercer's condition 5.1 Discrete analog 5.2 Examples 6 See also 7 Notes 8 References 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 {displaystyle K:[a,b]times [a,b]rightarrow mathbb {R} } where symmetric means that {displaystyle K(x,y)=K(y,x)} for all {displaystyle x,yin [a,b]} .

K is said to be non-negative definite (or positive semidefinite) if and only if {displaystyle sum _{i=1}^{n}sum _{j=1}^{n}K(x_{i},x_{j})c_{i}c_{j}geq 0} 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 (more specifically a Hilbert–Schmidt integral operator) on functions defined by the integral {displaystyle [T_{K}varphi ](x)=int _{a}^{b}K(x,s)varphi (s),ds.} 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 TK is a linear operator, we can talk about eigenvalues and eigenfunctions of TK.

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 {displaystyle K(s,t)=sum _{j=1}^{infty }lambda _{j},e_{j}(s),e_{j}(t)} 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 → 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 for compact operators on Hilbert spaces to TK to show the existence of the orthonormal basis {ei}i of L2[a,b] {displaystyle lambda _{i}e_{i}(t)=[T_{K}e_{i}](t)=int _{a}^{b}K(t,s)e_{i}(s),ds.} If λi ≠ 0, the eigenvector (eigenfunction) ei is seen to be continuous on [a,b]. Now {displaystyle sum _{i=1}^{infty }lambda _{i}|e_{i}(t)e_{i}(s)|leq sup _{xin [a,b]}|K(x,x)|,} which shows that the sequence {displaystyle sum _{i=1}^{infty }lambda _{i}e_{i}(t)e_{i}(s)} 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 {displaystyle lambda langle f,frangle =langle f,T_{K}frangle } 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 {displaystyle lambda langle f,frangle geq 0} , implying {displaystyle lambda geq 0} .

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 {displaystyle int _{a}^{b}K(t,t),dt=sum _{i}lambda _{i}.} This shows that the operator TK is a trace class operator and {displaystyle operatorname {trace} (T_{K})=int _{a}^{b}K(t,t),dt.} Generalizations Mercer's theorem itself is a generalization of the result that any symmetric positive-semidefinite matrix is the Gramian matrix of a set of vectors.

The first generalization[citation needed] 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] 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 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 {displaystyle K(s,t)=sum _{j=1}^{infty }lambda _{j},e_{j}(s),e_{j}(t)} where the convergence is absolute and uniform on compact subsets of X.

The next generalization[citation needed] 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 {displaystyle Kin L_{mu otimes mu }^{2}(Xtimes X).} L2 kernels define a bounded operator TK by the formula {displaystyle langle T_{K}varphi ,psi rangle =int _{Xtimes X}K(y,x)varphi (y)psi (x),d[mu otimes mu ](y,x).} 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 positive-definite kernel on (X, M, μ), then {displaystyle K(y,x)=sum _{iin mathbb {N} }lambda _{i}e_{i}(y)e_{i}(x)} 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 {displaystyle iint g(x)K(x,y)g(y),dx,dygeq 0.} Discrete analog This is analogous to the definition of a positive-semidefinite matrix. This is a matrix {displaystyle K} of dimension {displaystyle N} , which satisfies, for all vectors {displaystyle g} , the property {displaystyle (g,Kg)=g^{T}{cdot }Kg=sum _{i=1}^{N}sum _{j=1}^{N},g_{i},K_{ij},g_{j}geq 0} . Examples A positive constant function {displaystyle K(x,y)=c,} satisfies Mercer's condition, as then the integral becomes by Fubini's theorem {displaystyle iint g(x),c,g(y),dxdy=cint !g(x),dxint !g(y),dy=cleft(int !g(x),dxright)^{2}} which is indeed non-negative.

See also Kernel trick Representer theorem Spectral theory Notes ^[bare URL 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, Mercer, J. (1909), "Functions of positive and negative type and their connection with the theory of integral equations", Philosophical Transactions of the Royal Society A, 209 (441–458): 415–446, Bibcode:1909RSPTA.209..415M, doi:10.1098/rsta.1909.0016, "Mercer theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] H. König, Eigenvalue distribution of compact operators, Birkhäuser Verlag, 1986. (Gives the generalization of Mercer's theorem for finite measures μ.) hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Theorems in functional analysis

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