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In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (and ) that produces a third function (). The term convolution refers to both the resulting function and to the process of computing it.
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).
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.
As seen above, the discrete Fourier transform has the fundamental property of carrying convolution into componentwise product. A natural question is whether it is the only one with this ability. It has been shown [9] [10] that any linear transform that turns convolution into pointwise product is the DFT up to a permutation of coefficients ...
the solution of the initial-value problem = is the convolution (). Through the superposition principle , given a linear ordinary differential equation (ODE), L y = f {\displaystyle Ly=f} , one can first solve L G = δ s {\displaystyle LG=\delta _{s}} , for each s , and realizing that, since the source is a sum of delta functions , the solution ...
We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above. A sequence of convolution polynomials defined in the notation above has the following properties: The sequence n! · f n (x) is of binomial type
In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if x {\displaystyle x} is a function on Euclidean space R d and n {\displaystyle n} is a natural number , then the convolution power is defined by
Visual comparison of convolution, cross-correlation and autocorrelation.For the operations involving function f, and assuming the height of f is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point.