Mercer's condition

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.

Examples[edit]

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[edit]