Positive-definite function

Summary

A positive-definite function on $\R^n$ is a complex function $f$ such that every matrix $\big(f(x_i - x_j)\big)_{i,j}$ built from finitely many points is positive semidefinite. It is the case $G = \R^n$ of a positive-definite function on a group. By Bochner's theorem, the continuous positive-definite functions on $\R^n$ are exactly the Fourier transforms of finite positive measures. They appear as characteristic functions in probability, as covariance functions of stationary processes, and as kernels in interpolation and machine learning.