---
tag: XGQT4WEVGG
kind: concept
name: Colombeau algebra
msc: [46F30]
sources:
  - url: https://en.wikipedia.org/wiki/Colombeau_algebra
same_as:
  wikidata: Q5147873
  wikipedia: Colombeau algebra
---

## Summary

A Colombeau algebra is an algebra of generalized functions that contains the distributions and in which any two elements can be multiplied.
Its elements are classes of families of smooth functions indexed by a small parameter, with bounds that are polynomial in the inverse of the parameter.
It keeps the product of smooth functions and has a derivative obeying the Leibniz rule.
It is used for nonlinear equations with singular data.

## Explanations

### L1 human

The delta function cannot be squared as a distribution.
A Colombeau algebra replaces the delta by the whole family of narrower and narrower spikes that approximate it.
Families can be multiplied term by term, so products such as the square of the delta make sense.
Families that become small faster than any power of the parameter are treated as zero.