---
tag: ERP5737VZ1
kind: concept
name: Convolution on the line
msc: [44A35, 42A85]
specializes: [2Z29JN3VES]
sources:
  - url: https://en.wikipedia.org/wiki/Convolution
---

## Summary

The convolution of two functions $f$ and $g$ on the real line is the function $(f * g)(x) = \int_\R f(y) g(x - y) \, dy$.
It is the case $G = \R$ of the general [[convolution|2Z29JN3VES]] on a group.
It models the output of a linear time-invariant system and the density of a sum of independent random variables.
It also describes the smoothing of a function by a kernel.
The [[Fourier transform|DPVM34ADZV]] turns it into a pointwise product.

## Explanations

### L1 human

To compute $(f * g)(x)$, flip $g$, slide it so that it sits at $x$, multiply by $f$, and take the area under the product.
Doing this for every $x$ gives a new function, usually smoother than either of the two.
For example, the indicator function of $[0, 1]$ convolved with itself is a tent of height $1$ over $[0, 2]$.
Convolving a signal with a narrow bump of area $1$ replaces each value by a local weighted average.