NOTICE: Citizendium is still being set up on its newer server, treat as a beta for now; please see here for more.
Citizendium - a community developing a quality comprehensive compendium of knowledge, online and free. Click here to join and contribute—free
CZ thanks our previous donors. Donate here. Treasurer's Financial Report -- Thanks to our content contributors. --

Convolution (mathematics)

From Citizendium, the Citizens' Compendium
(Redirected from Convolution)
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

In mathematics, convolution is a process which combines two functions on a set to produce another function on the set. The value of the product function depends on a range of values of the argument.

Convolution of real functions by means of an integral are found in probability, signal processing and control theory. Algebraic convolutions are found in the discrete analogues of those applications, and in the foundations of algebraic structures.

Integral convolutions

The convolution of integrable real functions f and g may be defined as the real function

Conditions need to be imposed on f and g for this to make sense, such as having compact support or rapid decay at infinity. Other ranges of integration, that is, other domains of definition for the functions involved, may also be used.


Algebraic convolutions

Let M be a set with a binary operation and R a ring. Let f and g be functions from M to R. The convolution of f and g is a function from M to R

where the addition and multiplication are those of R. For this sum to make sense it must be finite. If M has the "locally finite" property that for any given x there are only finitely many pairs (t,u) such that , this definition makes sense for any functions f,g. Alternatively, if we restrict to functions of finite support, then no condition on M is needed.



The Fourier transform translates convolution into pointwise multiplication of functions.