A convolution is an integral that expresses the amount of overlap of one function 
as it is shifted over another function 
. It therefore "blends" one
function with another. For example, in synthesis imaging,
the measured dirty map is a convolution of the "true" CLEAN map with the
dirty beam (the Fourier transform of the sampling
distribution). The convolution is sometimes also known by its German name, faltung
("folding").
Convolution is implemented in the Wolfram Language as Convolve[f, g, x, y] and DiscreteConvolve[f, g, n, m].
Convolution defines a product on the algebra of Schwartz functions on 
.
For functions 
and 
that vanish for negative arguments, their convolution for 
reduces to the finite interval ![[0,t]](/images/equations/Convolution/Inline7.svg)
and is given by
=int_0^tf(tau)g(t-tau)dtau,](/images/equations/Convolution/NumberedEquation1.svg) |
(1)
|
where the symbol ](/images/equations/Convolution/Inline8.svg)
denotes convolution of 
and 
, provided the integral
exists.
More generally, convolution is taken over the entire real line,
 |  |  |
(2)
|
 |  |  |
(3)
|
(Bracewell 1965, p. 25), when the integral exists, with the variable (in this case 
) implied, and also occasionally written as 
.
|
|
The animations above graphically illustrate the convolution of two boxcar functions (left) and two Gaussians (right).
In the plots, the green curve shows the convolution of the blue and red curves as
a function of 
, the position indicated by the vertical green line. The gray
region indicates the product 
as a function of
for fixed 
, so its area as a function
of 
is precisely the convolution. One feature
to emphasize and which is not conveyed by these illustrations (since they both exclusively
involve symmetric functions) is that the function 
must be mirrored before lagging it across
and integrating.
The convolution of two boxcar functions 
and 
has the particularly simple form, for 
and 
,
![f*g=[(t-t_1-u_1)H(t-t_1-u_1)-(t-t_2-u_1)H(t-t_2-u_1)
-(t-t_1-u_2)H(t-t_1-u_2)+(t-t_2-u_2)H(t-t_2-u_2)],](/images/equations/Convolution/NumberedEquation2.svg) |
(4)
|
where 
is the Heaviside step function. The convolution
of two Gaussians, with 
and 
,
 |  |  |
(5)
|
 |  |  |
(6)
|
is another Gaussian
![f*g=1/(sqrt(2pi(sigma_1^2+sigma_2^2)))e^(-[t-(mu_1+mu_2)]^2/[2(sigma_1^2+sigma_2^2)]).](/images/equations/Convolution/NumberedEquation3.svg) |
(7)
|
For the following algebraic and differentiation identities, let 
, 
,
and 
be Schwartz
functions on 
and 
a constant.
Convolution satisfies the properties
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
(Bracewell 1965, p. 27), as well as
 |  |  |
(11)
|
 |  |  |
(12)
|
(Bracewell 1965, p. 49).
Taking the derivative of a convolution gives
 |  |  |
(13)
|
 |  |  |
(14)
|
(Bracewell 1965, p. 119).
For Lebesgue-integrable functions 
and 
, the integral of their convolution
is the product of their integrals,
 |  | ![int_(-infty)^infty[int_(-infty)^inftyf(u)g(t-u)du]dt](/images/equations/Convolution/Inline69.svg) |
(15)
|
 |  | ![int_(-infty)^inftyf(u)[int_(-infty)^inftyg(t-u)dt]du](/images/equations/Convolution/Inline72.svg) |
(16)
|
 |  | ![[int_(-infty)^inftyf(u)du][int_(-infty)^inftyg(t)dt].](/images/equations/Convolution/Inline75.svg) |
(17)
|
For nonnegative functions 
and 
with finite second
raw moments and positive finite integrals, the horizontal function
centroids of their convolution add
 |
(18)
|
and provided that either 
or 
has its function
centroid at the origin, the second raw moments
do as well
 |
(19)
|
(Bracewell 1965, p. 142), where
 |
(20)
|
For cumulative distribution functions 
and 
, their convolution in probability
theory is given by
 |
(21)
|
where the Stieltjes integral is taken over the real line.
