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| {{merge from|Mercer's condition|date=July 2012}}
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| In [[mathematics]], specifically [[functional analysis]], '''Mercer's theorem''' is a representation of a symmetric [[Definite bilinear form|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 (mathematician)|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.<ref>http://www.cs.berkeley.edu/~bartlett/courses/281b-sp08/7.pdf</ref>
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| == Introduction ==
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| To explain Mercer's [[theorem]], we first consider an important special case; see [[#Generalizations|below]] for a more general formulation.
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| A ''kernel'', in this context, is a symmetric continuous function that maps
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| :<math> K: [a,b] \times [a,b] \rightarrow \mathbb{R}</math>
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| where symmetric means that ''K''(''x'', ''s'') = ''K''(''s'', ''x'').
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| ''K'' is said to be ''non-negative definite'' (or [[positive semidefinite matrix|positive semidefinite]]) if and only if
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| :<math> \sum_{i=1}^n\sum_{j=1}^n K(x_i, x_j) c_i c_j \geq 0</math>
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| for all finite sequences of points ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> of [''a'', ''b''] and all choices of real numbers ''c''<sub>1</sub>, ..., ''c''<sub>''n''</sub> (cf. [[positive definite kernel]]).
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| Associated to ''K'' is a linear operator on functions defined by the integral
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| :<math> [T_K \varphi](x) =\int_a^b K(x,s) \varphi(s)\, ds. </math>
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| For technical considerations we assume φ can range through the space
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| ''L''<sup>2</sup>[''a'', ''b''] (see [[Lp space]]) of square-integrable real-valued functions.
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| Since ''T'' is a linear operator, we can talk about [[eigenvalues]] and [[eigenfunction]]s of ''T''.
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| '''Theorem'''. Suppose ''K'' is a continuous symmetric non-negative definite kernel. Then there is an [[orthonormal basis]]
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| {''e''<sub>i</sub>}<sub>i</sub> of ''L''<sup>2</sup>[''a'', ''b''] consisting of eigenfunctions of ''T''<sub>''K''</sub> such that the corresponding
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| sequence of eigenvalues {λ<sub>''i''</sub>}<sub>''i''</sub> is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on [''a'', ''b''] and ''K'' has the representation
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| :<math> K(s,t) = \sum_{j=1}^\infty \lambda_j \, e_j(s) \, e_j(t) </math>
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| where the convergence is absolute and uniform.
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| == Details ==
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| We now explain in greater detail the structure of the proof of
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| Mercer's theorem, particularly how it relates to [[spectral theory of compact operators]].
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| * The map ''K'' → ''T''<sub>''K''</sub> is injective.
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| * ''T''<sub>''K''</sub> is a non-negative symmetric compact operator on ''L''<sup>2</sup>[''a'',''b'']; moreover ''K''(''x'', ''x'') ≥ 0.
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| To show compactness, show that the image of the [[unit ball]] of ''L''<sup>2</sup>[''a'',''b''] under ''T''<sub>''K''</sub> [[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 ''L''<sup>2</sup>[''a'',''b''].
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| Now apply the [[spectral theorem]] for compact operators on Hilbert
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| spaces to ''T''<sub>''K''</sub> to show the existence of the
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| orthonormal basis {''e''<sub>i</sub>}<sub>i</sub> of
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| ''L''<sup>2</sup>[''a'',''b'']
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| :<math> \lambda_i e_i(t)= [T_K e_i](t) = \int_a^b K(t,s) e_i(s)\, ds. </math>
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| If λ<sub>i</sub> ≠ 0, the eigenvector ''e''<sub>i</sub> is seen to be continuous on [''a'',''b'']. Now
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| :<math> \sum_{i=1}^\infty \lambda_i |e_i(t) e_i(s)| \leq \sup_{x \in [a,b]} |K(x,x)|^2, </math> | |
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| which shows that the sequence
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| :<math> \sum_{i=1}^\infty \lambda_i e_i(t) e_i(s) </math>
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| converges absolutely and uniformly to a kernel ''K''<sub>0</sub> which is easily seen to define the same operator as the kernel ''K''. Hence ''K''=''K''<sub>0</sub> from which Mercer's theorem follows.
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| == Trace ==
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| The following is immediate:
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| '''Theorem'''. Suppose ''K'' is a continuous symmetric non-negative definite kernel; ''T''<sub>''K''</sub> has a sequence of nonnegative
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| eigenvalues {λ<sub>i</sub>}<sub>i</sub>. Then
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| :<math> \int_a^b K(t,t)\, dt = \sum_i \lambda_i. </math>
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| This shows that the operator ''T''<sub>''K''</sub> is a [[trace class]] operator and
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| :<math> \operatorname{trace}(T_K) = \int_a^b K(t,t)\, dt. </math>
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| == Generalizations ==
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| Mercer's theorem itself is a generalization of the result that any [[positive semidefinite matrix]] is the [[Gramian matrix]] of a set of vectors.
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| 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 sets|Borel algebra]] of ''X'' whose support is ''X''. This means that μ(''U'') > 0 for any nonempty open subset ''U'' of ''X''.
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| 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:
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| '''Theorem'''. Suppose ''K'' is a continuous symmetric non-negative definite kernel on ''X''. If the function κ is ''L''<sup>1</sup><sub>μ</sub>(''X''), where κ(x)=K(x,x), for all ''x'' in ''X'', then there is an [[orthonormal set]]
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| {''e''<sub>i</sub>}<sub>i</sub> of ''L''<sup>2</sup><sub>μ</sub>(''X'') consisting of eigenfunctions of ''T''<sub>''K''</sub> such that corresponding
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| sequence of eigenvalues {λ<sub>i</sub>}<sub>i</sub> is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on ''X'' and ''K'' has the representation
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| :<math> K(s,t) = \sum_{j=1}^\infty \lambda_j \, e_j(s) \, e_j(t) </math>
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| where the convergence is absolute and uniform on compact subsets of ''X''.
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| The next generalization deals with representations of ''measurable'' kernels.
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| Let (''X'', ''M'', μ) be a σ-finite measure space. An ''L''<sup>2</sup> (or square integrable) kernel on ''X'' is a function
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| :<math> K \in L^2_{\mu \otimes \mu}(X \times X). </math>
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| ''L''<sup>2</sup> kernels define a bounded operator ''T''<sub>''K''</sub> by the formula
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| :<math> \langle T_K \varphi, \psi \rangle = \int_{X \times X} K(y,x) \varphi(y) \psi(x) \,d[\mu \otimes \mu](y,x). </math>
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| ''T''<sub>''K''</sub> is a compact operator (actually it is even a [[Hilbert-Schmidt operator]]). If the kernel ''K'' is symmetric, by the [[compact operator on Hilbert space#Compact self adjoint operator|spectral theorem]], ''T''<sub>''K''</sub> has an orthonormal basis of eigenvectors. Those eigenvectors that correspond to non-zero eigenvalues can be arranged in a sequence {''e''<sub>''i''</sub>}<sub>''i''</sub> (regardless of separability).
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| '''Theorem'''. If ''K'' is a symmetric non-negative definite kernel on(''X'', ''M'', μ), then
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| :<math> K(y,x) = \sum_{i \in \mathbb{N}} \lambda_i e_i(y) e_i(x) </math> | |
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| where the convergence in the ''L''<sup>2</sup> norm. Note that when continuity of the kernel is not assumed, the expansion no longer converges uniformly.
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| == References ==
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| * Adriaan Zaanen, ''Linear Analysis'', North Holland Publishing Co., 1960,
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| * 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)
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| * [[Konrad Jörgens]], ''Linear integral operators'', Pitman, Boston, 1982,
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| * [[Richard Courant]] and [[David Hilbert]], ''[[Methods of Mathematical Physics]]'', vol 1, Interscience 1953,
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| * Robert Ash, ''Information Theory'', Dover Publications, 1990,
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| * {{citation
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| |first=J. |last=Mercer
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| |title=Functions of positive and negative type and their connection with the theory of integral equations
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| |journal=[[Philosophical Transactions of the Royal Society]] A
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| |year=1909 |volume=209 |pages=415–446
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| |doi=10.1098/rsta.1909.0016
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| |issue=441–458
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| }},
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| * {{springer|title=Mercer theorem|id=p/m063440}}
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| * H. König, ''Eigenvalue distribution of compact operators'', Birkhäuser Verlag, 1986. (Gives the generalization of Mercer's theorem for finite measures μ.)
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| == See also ==
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| * [[Kernel trick]]
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| * [[Representer theorem]]
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| * [[Spectral theory]]
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| * [[Mercer's condition]]
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| ==Notes==
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| {{reflist}}
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| [[Category:Functional analysis]]
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| [[Category:Theorems in functional analysis]]
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