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In [[mathematics]], the '''convolution power''' is the ''n''-fold iteration of the [[convolution]] with itself. Thus if <math>x</math> is a [[Function (mathematics)|function]] on [[Euclidean space]] '''R'''<sup>''d''</sup> and <math>n</math> is a [[natural number]], then the convolution power is defined by

:<math> x^{*n} = \underbrace{x * x * x * \cdots * x * x}_n,\quad x^{*0}=\delta_0 </math>

where '''*''' denotes the convolution operation of functions on '''R'''<sup>''d''</sup> and δ<sub>0</sub> is the [[Dirac delta distribution]].  This definition makes sense if ''x'' is an [[integrable]] function (in [[Lp space|L<sup>1</sup>]]), a compactly supported [[distribution (mathematics)|distribution]], or is a finite [[Borel measure]].

If ''x'' is the distribution function of a [[random variable]] on the real line, then the ''n''<sup>th</sup> convolution power of ''x'' gives the distribution function of the sum of ''n'' independent random variables with identical distribution ''x''.  The [[central limit theorem]] states that if ''x'' is in L<sup>1</sup> and L<sup>2</sup> with mean zero and variance σ<sup>2</sup>, then
:<math>P\left(\frac{x^{*n}}{\sigma\sqrt{n}} < \beta\right) \to \Phi(\beta)\quad\rm{as}\ n\to\infty</math>
where Φ is the cumulative [[standard normal distribution]] on the real line.  Equivalently, <math>x^{*n}/\sigma\sqrt{n}</math> tends weakly to the standard normal distribution.

In some cases, it is possible to define powers ''x''<sup>*''t''</sup> for arbitrary real ''t''&nbsp;>&nbsp;0.  If μ is a [[probability measure]], then μ is [[Infinite divisibility (probability)|infinitely divisible]] provided there exists, for each positive integer ''n'', a probability measure μ<sub>1/''n''</sub> such that

:<math>\mu_{1/n}^{* n} = \mu.</math>

That is, a measure is infinitely divisible if it is possible to define all ''n''th roots.  Not every probability measure is infinitely divisible, and a characterization of infinitely divisible measures is of central importance in the abstract theory of [[stochastic process]]es.  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 process|Poisson]] type, given in the form

:<math>\pi_{\alpha,\mu} = e^{-\alpha}\sum_{n=0}^\infty \frac{\alpha^n}{n!}\mu^{*n}.</math>

In fact, the [[Lévy–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 the [[vague topology]], of the class of Poisson measures {{harv|Stroock|1993|loc=§3.2}}.

Many applications of the convolution power rely on being able to define the analog of [[analytic function]]s as [[formal power series]] with powers replaced instead by the convolution power.  Thus if <math>\textstyle{F(z) = \sum_{n=0}^\infty a_n z^n}</math> is an analytic function, then one would like to be able to define

:<math>F^*(x) = a_0\delta_0 + \sum_{n=1}^\infty a_n x^{*n}.</math>

If ''x''&nbsp;∈&nbsp;''L''<sup>1</sup>('''R'''<sup>''d''</sup>) or more generally is a finite Borel measure on '''R'''<sup>''d''</sup>, then the latter series converges absolutely in norm provided that the norm of ''x'' is less than the radius of convergence of the original series defining ''F''(''z'').  In particular, it is possible for such measures to define the '''complex exponential'''

:<math>\exp^*(x) = \delta_0 + \sum_{n=1}^\infty \frac{x^{*n}}{n!}.</math>

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 by {{harvtxt|Ben Chrouda|El Oued|Ouerdiane|2002}}.

As [[convolution algebra]]s are special cases of [[Hopf algebra]]s, the convolution power is a special case of the (ordinary) power in a Hopf algebra.  In applications to [[quantum 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 {{harv|Brouder|Frabetti|Patras|2008}}.  If, in addition, the algebra is a [[Banach algebra]], then convergence of the series can be determined as above.  In the formal setting, familiar identities such as
:<math>x = \log^*(\exp^*x) = \exp^*(\log^*x)</math>
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.

==Properties==
If ''x'' is itself suitably differentiable, then the [[Convolution#Differentiation|properties]] of convolution, one has

:<math>\mathcal{D}\big\{x^{*n}\big\} = (\mathcal{D}x)  * x^{*(n-1)} = x  * \mathcal{D}\big\{x^{*(n-1)}\big\}</math>

where <math>\mathcal{D}</math> denotes the [[derivative]] operator.  Specifically, this holds if ''x'' is a compactly supported distribution or lies in the [[Sobolev space]] ''W''<sup>1,1</sup> to ensure that the derivative is sufficiently regular for the convolution to be well-defined.

==See also==
*[[Convolution]]
*[[Convolution theorem]]
*[[Fourier transform]]
*[[Taylor series]]

==References==
* {{Citation | last1=Ben Chrouda | first1=Mohamed | last2=El Oued | first2=Mohamed | last3=Ouerdiane | first3=Habib | title=Convolution calculus and applications to stochastic differential equations | mr=1953702 | year=2002 | journal=Soochow Journal of Mathematics | issn=0250-3255 | volume=28 | issue=4 | pages=375–388}}.
* {{cite arxiv | last1=Brouder|first1=Christian|last2=Frabetti|first2=Alessandra|last3=Patras|first3=Frédéric|title=Decomposition into one-particle irreducible Green functions in many-body physics|year=2008|eprint=0803.3747 }}.
* {{Citation | last1=Feller | first1=William | author1-link=William Feller | title=An introduction to probability theory and its applications. Vol. II. | publisher=[[John Wiley & Sons]] | location=New York | series=Second edition | mr=0270403 | year=1971}}.
* {{Citation | last1=Stroock | first1=Daniel W. | title=Probability theory, an analytic view | publisher=[[Cambridge University Press]] | isbn=978-0-521-43123-1 | mr=1267569 | year=1993}}.

[[Category:Functional analysis]]
[[Category:Fourier analysis]]