---
tag: B5AB451TB7
kind: concept
name: Difference kernel
msc: ["45E10", 45P05]
specializes: [GSMBXT7FPK]
sources:
  - url: https://en.wikipedia.org/wiki/Convolution
---

## Summary

A difference kernel is a [[kernel|GSMBXT7FPK]] of the form $K(x, s) = k(x - s)$, which depends only on the difference of its two variables.
Its integral operator is a [[convolution|2Z29JN3VES]] with $k$, so it commutes with translations.
Integral equations with difference kernels are called convolution equations.
Transforms that turn convolutions into products, such as the [[Laplace transform|0V49JEZW06]] and the Fourier transform, reduce them to algebraic equations.

## Explanations

### L1 human

The influence of the point $s$ on the point $x$ depends only on how far apart they are, and in which direction.
Shifting both points by the same amount does not change the kernel.
For example, $K(x, s) = e^{-\abs{x - s}}$ is a difference kernel with $k(t) = e^{-\abs{t}}$.
The Volterra equation $\varphi(x) = f(x) + \int_0^x (x - s) \varphi(s) \, ds$ has the difference kernel with $k(t) = t$.