Editing Mercer's theorem (section)

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

:<math> \iint g(x)K(x,y)g(y)\,dx\,dy \geq 0. </math>

===Discrete analog===
This is analogous to the definition of a [[positive-semidefinite matrix]]. This is a matrix <math>K</math> of dimension <math>N</math>, which satisfies, for all vectors <math>g</math>, the property
:<math>(g,Kg)=g^{T}{\cdot}Kg=\sum_{i=1}^N\sum_{j=1}^N\,g_i\,K_{ij}\,g_j\geq0</math>.

===Examples===
A positive constant function
:<math>K(x, y)=c\,</math>
satisfies Mercer's condition, as then the integral becomes by [[Fubini's theorem]] 
:<math> \iint g(x)\,c\,g(y)\,dx \, dy = c\int\! g(x) \,dx \int\! g(y) \,dy = c\left(\int\! g(x) \,dx\right)^2</math>
which is indeed [[non-negative]].