Mercer's condition

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

\iint K(x,y)g(x)g(y)\,dxdy\geq 0.

Examples

The constant function

K(x,y)=1\,

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

\iint g(x)g(y)\,dxdy=\int \!g(x)\,dx\int \!g(y)\,dy=\left(\int \!g(x)\,dx\right)^{2}

which is indeed non-negative.

See also

  • Definite bilinear form
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