---
tag: EY75YD988Z
kind: concept
name: Convolution integral equation
msc: ["45E10", 44A35]
specializes: [R3SG1SN7JQ]
sources:
  - url: https://en.wikipedia.org/wiki/Convolution
---

## Summary

A convolution integral equation is an [[integral equation|R3SG1SN7JQ]] whose kernel depends only on the difference $x - t$ of its two variables.
The integral in it is then a [[convolution|2Z29JN3VES]] of the kernel with the unknown function.
Because integral transforms turn convolutions into products, such equations can often be solved explicitly with the [[Fourier transform|DPVM34ADZV]] or the [[Laplace transform|0V49JEZW06]].
They model systems that are invariant under translation, such as linear filters, renewal processes and homogeneous media.

## Explanations

### L1 human

Think of a signal that passes through a blurring device that treats every position in the same way.
The output at a point is a weighted average of the input around that point, with weights that depend only on the distance.
Asking for the input that produced a given output is a convolution integral equation.
For example, $\varphi(x) - \frac{1}{2} \int_{-\infty}^{\infty} e^{-\abs{x - t}} \varphi(t) \, dt = f(x)$ asks for $\varphi$ when $f$ is known.
Since the weight $e^{-\abs{x - t}}$ depends only on $x - t$, the equation is of convolution type.