Positive-definite function

A positive-definite function is a function whose values generate non-negative quadratic forms for every finite collection of points and weights.

For a translation-invariant function, this means the matrix with entries $f(x_i-x_j)$ is positive semidefinite for every finite choice of the $x_i$. Pointwise negativity does not automatically rule out positive definiteness.

See also: positive definiteness, positive semidefinite matrix, quadratic form, translation invariance..