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The convolution of two functions
is the function
In a sense,
is the sum of all the terms where . Such sums occur when investigating sums of independent random variables, and discrete versions appear in the coefficients of products of polynomials and power series. Convolution is an important tool in data processing, in particular in digital signal and image processing. We will first define the concept in various general settings, discuss its properties and then list several convolutions of probability distributions.
If is a locally compact abelian topological group with Haar measure and and are measurable functions on , we define the convolution
whenever the right hand side integral exists (this is for instance the case if
,
and
).
The case
is the most important one, but is also useful, since it recovers the convolution of sequences which occurs when computing the coefficients of a product of polynomials or power series. The case
yields the so-called cyclic convolution which is often discussed in connection with the discrete Fourier transform.
The (Dirichlet) convolution of multiplicative functions considered in number theory does not quite fit the above definition, since there the functions are defined on a commutative monoid (the natural numbers under multiplication) rather than on an abelian group.
If and are independent random variables with probability densities and respectively, and if has a probability density, then this density is given by the convolution
. This motivates the following definition: for probability distributions and on , the convolution is the probability distribution on given by
for every Borel set . The last equation is the result of Fubini's theorem.
The convolution of two distributions and on is defined by
for any test function for , assuming that
is a suitable test function for .
The convolution operation, when defined, is commutative, associative and distributive with respect to addition. For any we have
where is the Dirac delta distribution. The Fourier transform translates between convolution and pointwise multiplication:
Because of the availability of the Fast Fourier Transform and its inverse, this latter relation is often used to quickly compute discrete convolutions, and in fact the fastest known algorithms for the multiplication of numbers and polynomials are based on this idea.
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Cross-references: order, width, finite, distribution function, vanishes, quasi-uniform, slope, lines, straight, parallel, exponential distribution, exponential, parameters, Poisson distributions, degrees of freedom, variances, mean, normal distributions, algorithms, pointwise, translates, Fourier transform, delta distribution, addition, distributive, associative, commutative, operation, Fubini's theorem, equation, Borel set, densities, abelian group, multiplication, natural numbers, commutative monoid, number theory, multiplicative functions, discrete Fourier transform, cyclic, sequences, integral, right hand side, measurable functions, Haar measure, topological group, abelian, locally compact, distributions, properties, power series, polynomials, products, coefficients, discrete, random variables, independent, terms, sum, functions
There are 22 references to this entry.
This is version 21 of convolution, born on 2002-03-13, modified 2007-06-26.
Object id is 2790, canonical name is Convolution.
Accessed 45872 times total.
Classification:
| AMS MSC: | 44A35 (Integral transforms, operational calculus :: Convolution) | | | 94A12 (Information and communication, circuits :: Communication, information :: Signal theory ) |
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Pending Errata and Addenda
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