Inmathematics, theconvolution power is then-fold iteration of theconvolution with itself. Thus if${\displaystyle x}$ is afunction onEuclidean spaceR^d and${\displaystyle n}$ is anatural number, then the convolution power is defined by

${\displaystyle x^{\ast n}=\underset{n}{\underbrace{x\ast x\ast x\ast \cdots \ast x\ast x}},x^{\ast 0}={\delta }_0}$

where∗ denotes the convolution operation of functions onR^d and δ₀ is theDirac delta distribution. This definition makes sense ifx is anintegrable function (inL¹), a rapidly decreasingdistribution (in particular, a compactly supported distribution) or is a finiteBorel measure.

Ifx is the distribution function of arandom variable on the real line, then then^th convolution power ofx gives the distribution function of the sum ofn independent random variables with identical distributionx. Thecentral limit theorem states that ifx is in L¹ and L² with mean zero and variance σ², then

where Φ is the cumulativestandard normal distribution on the real line. Equivalently,${\displaystyle x^{\ast n}/\sigma \sqrt{n}}$ tends weakly to the standard normal distribution.

In some cases, it is possible to define powersx^*t for arbitrary realt > 0. If μ is aprobability measure, then μ isinfinitely divisible provided there exists, for each positive integern, a probability measure μ_1/n such that

${\displaystyle {\mu }_{1/n}^{\ast n}=\mu .}$

That is, a measure is infinitely divisible if it is possible to define allnth roots. Not every probability measure is infinitely divisible, and a characterization of infinitely divisible measures is of central importance in the abstract theory ofstochastic processes. Intuitively, a measure should be infinitely divisible provided it has a well-defined "convolution logarithm." The natural candidate for measures having such a logarithm are those of (generalized)Poisson type, given in the form

${\displaystyle {\pi }_{\alpha ,\mu }=e^{-\alpha }\sum _{n=0}^{\mathrm{\infty }}\frac{{\alpha }^n}{n!}{\mu }^{\ast n}.}$

In fact, theLévy–Khinchin theorem states that a necessary and sufficient condition for a measure to be infinitely divisible is that it must lie in the closure, with respect to thevague topology, of the class of Poisson measures (Stroock 1993, §3.2).

Many applications of the convolution power rely on being able to define the analog ofanalytic functions asformal power series with powers replaced instead by the convolution power. Thus if${\displaystyle F(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n}$ is an analytic function, then one would like to be able to define

${\displaystyle F^{\ast }(x)=a_0{\delta }_0+\sum _{n=1}^{\mathrm{\infty }}{a}_n{x}^{\ast n}.}$

Ifx ∈ L¹(R^d) or more generally is a finite Borel measure onR^d, then the latter series converges absolutely in norm provided that the norm ofx is less than the radius of convergence of the original series definingF(z). In particular, it is possible for such measures to define theconvolutional exponential

${\displaystyle {\exp }^{\ast }(x)={\delta }_0+\sum _{n=1}^{\mathrm{\infty }}\frac{x^{\ast n}}{n!}.}$

It is not generally possible to extend this definition to arbitrary distributions, although a class of distributions on which this series still converges in an appropriate weak sense is identified byBen Chrouda, El Oued & Ouerdiane (2002).

Ifx is itself suitably differentiable, then from theproperties of convolution, one has

${\displaystyle \mathcal{D}\{x^{\ast n}\}=(\mathcal{D}x)\ast x^{\ast (n-1)}=x\ast \mathcal{D}\{x^{\ast (n-1)}\}}$

where${\displaystyle \mathcal{D}}$ denotes thederivative operator. Specifically, this holds ifx is a compactly supported distribution or lies in theSobolev spaceW^1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined.

In the configuration random graph, the size distribution ofconnected components can be expressed via the convolution power of the excessdegree distribution (Kryven (2017)):

Here,${\displaystyle w(n)}$ is the size distribution for connected components,${\displaystyle u_1(k)=\frac{k+1}{{\mu }_1}u(k+1),}$ is the excess degree distribution, and${\displaystyle u(k)}$ denotes thedegree distribution.

Asconvolution algebras are special cases ofHopf algebras, the convolution power is a special case of the (ordinary) power in a Hopf algebra. In applications toquantum field theory, the convolution exponential, convolution logarithm, and other analytic functions based on the convolution are constructed as formal power series in the elements of the algebra (Brouder, Frabetti & Patras 2008). If, in addition, the algebra is aBanach algebra, then convergence of the series can be determined as above. In the formal setting, familiar identities such as

${\displaystyle x={\log }^{\ast }({\exp }^{\ast }x)={\exp }^{\ast }({\log }^{\ast }x)}$

continue to hold. Moreover, by the permanence of functional relations, they hold at the level of functions, provided all expressions are well-defined in an open set by convergent series.

• Convolution
• Convolution theorem
• Fourier transform
• Taylor series

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• Horváth, John (1966),Topological Vector Spaces and Distributions, Addison-Wesley Publishing Company: Reading, MA, USA.
• Ben Chrouda, Mohamed; El Oued, Mohamed; Ouerdiane, Habib (2002), "Convolution calculus and applications to stochastic differential equations",Soochow Journal of Mathematics,28 (4):375–388,ISSN 0250-3255,MR 1953702.
• Brouder, Christian; Frabetti, Alessandra; Patras, Frédéric (2008). "Decomposition into one-particle irreducible Green functions in many-body physics".arXiv:0803.3747 [cond-mat.str-el]..
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• Stroock, Daniel W. (1993),Probability theory, an analytic view,Cambridge University Press,ISBN 978-0-521-43123-1,MR 1267569.
• Kryven, I (2017), "General expression for component-size distribution in infinite configuration networks",Physical Review E,95 (5): 052303,arXiv:1703.05413,Bibcode:2017PhRvE..95e2303K,doi:10.1103/physreve.95.052303,PMID 28618550,S2CID 8421307.

• Functional analysis
• Fourier analysis

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