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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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"Why does my computer keep freezing up?" I was asked by a great deal of folks the cause of their pc freeze problems. And I am fed up with spending much time inside answering the query time and time again. This article is to tell you the real cause of the PC Freezes.
If it happens to be not because big of the problem as we think it's, it may probably be solved conveniently by running a Startup Repair or by System Restore Utility. Again it can be because effortless as running an anti-virus check or cleaning the registry.
Naturally, the upcoming logical step is to receive these false entries cleaned out. Fortunately, this really is not a difficult task. It is the 2nd thing you need to do when you noticed the computer has lost speed. The first will be to make certain there are no viruses or serious spyware present.
If you feel you don't have enough money at the time to upgrade, then the number one option is to free up some space by deleting a few of the unwelcome files plus folders.
There are numerous tuneup utilities s found on the market now. How do you understand which you to choose? Well, whenever we purchased a automobile we did some research on it, didn't you? You didn't just go out plus purchase the first red convertible you saw. The same thing works with registry products. On any look engine, kind inside "registry cleaner reviews" plus they might receive posted for you to read about.
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Perfect Optimizer is a wise Registry Product, updates consistently and has many qualities. Despite its cost, there are which the update are truly valuable. They provide plenty of support through telephone, send and forums. You might like to check out the free trial to check it out for yourself.
A system and registry cleaner may be downloaded from the internet. It's simple to use plus the procedure refuses to take lengthy. All it does is scan plus then when it finds errors, it may fix plus clean those errors. An error free registry can protect the computer from errors plus give you a slow PC fix.