Mercer's condition

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

$\iint g(x)K(x,y)g(y)\,dx\,dy \geq 0.$

Discrete analog

This is analogous to the definition of a positive-semidefinite matrix. This is a matrix $K$ of dimension $N$, which satisfies, for all vectors $g$, the property

$(g,Kg)=g^{T}{\cdot}Kg=\sum_{i=1}^N\sum_{j=1}^N\,g_i\,K_{ij}\,g_j\geq0$.

Examples

A positive constant function

$K(x, y)=c\,$

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

$\iint g(x)\,c\,g(y)\,dx dy = c\int\! g(x) \,dx \int\! g(y) \,dy = c\left(\int\! g(x) \,dx\right)^2$

which is indeed non-negative.

See also

• Definite bilinear form