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The convolution of and is written , denoting the operator with the symbol . [B] It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
Convolution is the process of adding each element of the image to its local neighbors, weighted by the kernel. This is related to a form of mathematical convolution . The matrix operation being performed—convolution—is not traditional matrix multiplication, despite being similarly denoted by *.
If the operator is translation invariant, that is, when has constant coefficients with respect to x, then the Green's function can be taken to be a convolution kernel, that is, (,) = (). In this case, Green's function is the same as the impulse response of linear time-invariant system theory .
In mathematics, deconvolution is the inverse of convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy. [1]
Then, from the perspective of operator theory, a circulant matrix is the kernel of a discrete integral transform, namely the convolution operator for the function (,, …,); this is a discrete circular convolution.
If x is itself suitably differentiable, then from the properties of convolution, one has {} = () = {()}where denotes the derivative operator. Specifically, this holds if x is a compactly supported distribution or lies in the Sobolev space W 1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined.