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

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

This is analogous to the idea of a positive-semidefinite matrix, which is a matrix $M$ satisfying $z^{T}Mz\geq0$. The quantity $z^{T}Mz$ can be thought of as a double sum, whereas the above integral is also analogous to a double sum.

Examples

The constant function

$K(x, y)=1\,$

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

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

which is indeed non-negative.

See also

• Definite bilinear form