Positive-definite function

A complex-valued function on a group is positive-definite when the displayed inequality holds for every finite choice of points and coefficients. This includes positive semidefinite, rather than necessarily strictly positive definite, Gram matrices. In particular $p(e)\ge 0$ and $|p(x)|\le p(e)$. On an Abelian group, every convolution square $u\ast u^{\ast }$ with $u\in C_c(G)$ is continuous and positive-definite: its quadratic form is the integral of $|{\sum }_i{z}_{i}u(t+x_i){|}^2$.