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{{merge to|Mercer's theorem|date=July 2012}}
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 K(x,y)g(x)g(y)\,dx dy \geq 0. </math>
 
==Examples==
The constant function
:<math>K(x, y)=1\,</math>
satisfies Mercer's condition, as then the integral becomes by [[Fubini's theorem]]
:<math> \iint g(x)g(y)\,dx dy = \int\! g(x) \,dx \int\! g(y) \,dy = \left(\int\! g(x) \,dx\right)^2</math>
which is indeed [[non-negative]].
 
==See also==
* [[Definite bilinear form]]
{{Mathanalysis-stub}}
[[Category:Real analysis]]

Latest revision as of 12:35, 20 April 2013



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K(x,y)g(x)g(y)dxdy0.

Examples

The constant function

K(x,y)=1

satisfies Mercer's condition, as then the integral becomes by Fubini's theorem

g(x)g(y)dxdy=g(x)dxg(y)dy=(g(x)dx)2

which is indeed non-negative.

See also

Template:Mathanalysis-stub