---
tag: JGR1E7Q9KQ
kind: concept
name: Convolution power
msc: [44A35, 42A85]
sources:
  - url: https://en.wikipedia.org/wiki/Convolution_power
same_as:
  wikidata: Q5166605
  wikipedia: Convolution power
---

## Summary

The $k$-th convolution power $f^{*k}$ of a function or measure $f$ is the [[convolution|2Z29JN3VES]] of $k$ copies of $f$.
It is the analogue of the ordinary power $x^k$ when the product is convolution.
If $f$ is the density of a random variable $X$, then $f^{*k}$ is the density of the sum of $k$ independent copies of $X$.
Under the Fourier transform, $f^{*k}$ becomes the ordinary power $\hat f^{\,k}$.

## Explanations

### L1 human

Convolving a function with itself again and again gives its convolution powers.
For the uniform density on $[0, 1]$, the second power is a tent on $[0, 2]$, the third is a rounded hump on $[0, 3]$ made of quadratic pieces, and so on.
The higher powers look more and more like a bell curve, which is the central limit theorem in action.