Editing Mercer's theorem (section)

== Introduction ==
To explain Mercer's theorem, we first consider an important special case; see [[#Generalizations|below]] for a more general formulation.
A ''kernel'', in this context, is a [[Symmetric_function|symmetric]] continuous function

:<math> K: [a,b] \times [a,b] \rightarrow \mathbb{R}</math>

where <math>K(x,y) = K(y,x)</math> for all <math>x,y \in [a,b]</math>.

''K'' is said to be a [[positive-definite kernel]] if and only if

:<math> \sum_{i=1}^n\sum_{j=1}^n K(x_i, x_j) c_i c_j \geq 0</math>

for all finite sequences of points ''x''<sub>1</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub> of [''a'',&nbsp;''b''] and all choices of real numbers ''c''<sub>1</sub>,&nbsp;...,&nbsp;''c''<sub>''n''</sub>.  Note that the term "positive-definite" is well-established in literature despite the weak inequality in the definition.<ref>{{Cite book |last=Mohri |first=Mehryar |url=https://www.worldcat.org/oclc/1041560990 |title=Foundations of machine learning |date=2018 |others=Afshin Rostamizadeh, Ameet Talwalkar |isbn=978-0-262-03940-6 |edition=Second |location=Cambridge, Massachusetts |oclc=1041560990}}</ref><ref>{{Cite book |last=Berlinet |first=A. |url=https://www.worldcat.org/oclc/844346520 |title=Reproducing kernel Hilbert spaces in probability and statistics |date=2004 |publisher=Springer Science+Business Media |others=Christine Thomas-Agnan |isbn=1-4419-9096-8 |location=New York |oclc=844346520}}</ref> 

The fundamental characterization of stationary positive-definite kernels (where <math>K(x,y) = K(x-y)</math>) is given by '''[[Bochner's theorem]]'''. It states that a continuous function <math>K(x-y)</math> is positive-definite if and only if it can be expressed as the [[Fourier transform]] of a finite non-negative measure <math>\mu</math>:

:<math>K(x-y) = \int_{-\infty}^{\infty} e^{i(x-y)\omega} \, d\mu(\omega)</math>

This spectral representation reveals the connection between positive definiteness and harmonic analysis, providing a stronger and more direct characterization of positive definiteness than the abstract definition in terms of inequalities when the kernel is stationary, e.g, when it can be expressed as a 1-variable function of the distance between points rather than the 2-variable function of the positions of pairs of points.

Associated to ''K'' is a [[linear operator]] (more specifically a [[Hilbert–Schmidt integral operator]] when the interval is compact) on functions defined by the integral

:<math> [T_K \varphi](x) =\int_a^b  K(x,s) \varphi(s)\, ds. </math>

We assume <math>\varphi</math> can range through the space
of real-valued [[square-integrable functions]] ''L''<sup>2</sup>[''a'',&nbsp;''b'']; however, in many cases the associated RKHS can be strictly larger than ''L''<sup>2</sup>[''a'',&nbsp;''b'']. Since ''T<sub>K</sub>'' is a linear operator, the [[eigenvalues]] and [[eigenfunction]]s of ''T<sub>K</sub>'' exist.

'''Theorem'''. Suppose ''K'' is a continuous symmetric positive-definite kernel.  Then there is an [[orthonormal basis]]
{''e''<sub>i</sub>}<sub>i</sub> of ''L''<sup>2</sup>[''a'',&nbsp;''b''] consisting of eigenfunctions of ''T''<sub>''K''</sub> such that the corresponding
sequence of eigenvalues {&lambda;<sub>''i''</sub>}<sub>''i''</sub> is nonnegative.  The eigenfunctions corresponding to non-zero eigenvalues are continuous on [''a'',&nbsp;''b''] and ''K'' has the representation

:<math> K(s,t) = \sum_{j=1}^\infty \lambda_j \, e_j(s) \, e_j(t) </math>

where the convergence is absolute and uniform.