---
tag: F73VWGF637
kind: concept
name: Positive-definite function
msc: [42A82, 43A35]
specializes: [3PRZ02APFA]
sources:
  - url: https://en.wikipedia.org/wiki/Positive-definite_function
same_as:
  wikidata: Q2293170
  wikipedia: 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|3PRZ02APFA]].
By [[Bochner's theorem|ZRHAG4NXHC]], 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.

## Explanations

### L1 human

Pick some points on the line and form the table of values $f(x_i - x_j)$.
For a positive-definite function, this table always behaves like a table of inner products, so it never produces a negative "squared length".
The cosine is the simplest example, and so is the Gaussian $e^{-x^2}$.
Being positive-definite forces $f(0)$ to be the largest value of $\abs{f}$.