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].
Abstractly, a convolution is defined as a product of functions 
and 
that are objects
in the algebra of Schwartz functions in 
. Convolution of two functions 
and 
over a finite
range ![[0,t]](/images/equations/Convolution/Inline8.svg)
is given by
=int_0^tf(tau)g(t-tau)dtau,](/images/equations/Convolution/NumberedEquation1.svg) |
(1)
|
where the symbol ](/images/equations/Convolution/Inline9.svg)
denotes convolution of 
and 
.
Convolution is more often taken over an infinite range,
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(2)
|
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(3)
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(Bracewell 1965, p. 25) with the variable (in this case 
) implied, and also
occasionally written as 
.
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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 
, 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
![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)
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where 
is the Heaviside
step function. Even more amazingly, the convolution of two Gaussians
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(5)
|
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(6)
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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)
|
Let 
, 
, and 
be arbitrary functions
and 
a constant. Convolution satisfies the properties
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(8)
|
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(9)
|
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(10)
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(Bracewell 1965, p. 27), as well as
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(11)
|
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(12)
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(Bracewell 1965, p. 49).
Taking the derivative of a convolution gives
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(13)
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(14)
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(Bracewell 1965, p. 119).
The area under a convolution is the product of areas under the factors,
 |  | ![int_(-infty)^infty[int_(-infty)^inftyf(u)g(t-u)du]dt](/images/equations/Convolution/Inline62.svg) |
(15)
|
 |  | ![int_(-infty)^inftyf(u)[int_(-infty)^inftyg(t-u)dt]du](/images/equations/Convolution/Inline65.svg) |
(16)
|
 |  | ![[int_(-infty)^inftyf(u)du][int_(-infty)^inftyg(t)dt].](/images/equations/Convolution/Inline68.svg) |
(17)
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The horizontal function centroids of a convolution add
 |
(18)
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and provided that either 
or 
has its function
centroids at its origin, the variances do as well
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(19)
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(Bracewell 1965, p. 142), where
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(20)
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There is also a definition of the convolution which arises in probability theory and is given by
 |
(21)
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where 
is a Stieltjes
integral.
