Convolution power

Convolution power

In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if x is a function on Euclidean space Rd and n is a natural number, then the convolution power is defined by

 x^{*n} = \underbrace{x * x * x * \cdots * x * x}_n,\quad x^{*0}=\delta_0

where * denotes the convolution operation of functions on Rd and δ0 is the Dirac delta distribution. This definition makes sense if x is an integrable function (in L1), a compactly supported distribution, or is a finite Borel measure.

If x is the distribution function of a random variable on the real line, then the nth 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 L1 and L2 with mean zero and variance σ2, then

P\left(\frac{x^{*n}}{\sigma\sqrt{n}} < \beta\right) \to \Phi(\beta)\quad\rm{as}\ n\to\infty

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

In some cases, it is possible to define powers x*t for arbitrary real t > 0. If μ is a probability measure, then μ is infinitely divisible provided there exists, for each positive integer n, a probability measure μ1/n such that

\mu_{1/n}^{* n} = \mu.

That is, a measure is infinitely divisible if it is possible to define all nth 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 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

\pi_{\alpha,\mu} = e^{-\alpha}\sum_{n=0}^\infty \frac{\alpha^n}{n!}\mu^{*n}.

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 (Stroock 1993, §3.2).

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

F^*(x) = a_0\delta_0 + \sum_{n=1}^\infty a_n x^{*n}.

If x ∈ L1(Rd) or more generally is a finite Borel measure on Rd, 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

\exp^*(x) = \delta_0 + \sum_{n=1}^\infty \frac{x^{*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 by Ben Chrouda, El Oued & Ouerdiane (2002).

As convolution algebras are special cases of Hopf algebras, 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 (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

x = log  * (exp  * x) = exp  * (log  * 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.

Properties

If x is itself suitably differentiable, then the properties of convolution, one has

\mathcal{D}\big\{x^{*n}\big\} = (\mathcal{D}x)  * x^{*(n-1)} = x  * \mathcal{D}\big\{x^{*(n-1)}\big\}

where \mathcal{D} denotes the derivative operator. Specifically, this holds if x is a compactly supported distribution or lies in the Sobolev space W1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined.

See also

References

  • 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, MR1953702 .
  • Brouder, Christian; Frabetti, Alessandra; Patras, Frédéric (2008). "Decomposition into one-particle irreducible Green functions in many-body physics". arXiv:0803.3747. .
  • Feller, William (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: John Wiley & Sons, MR0270403 .
  • Stroock, Daniel W. (1993), Probability theory, an analytic view, Cambridge University Press, ISBN 978-0-521-43123-1, MR1267569 .

Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Convolution — For the usage in formal language theory, see Convolution (computer science). Convolution of two square pulses: the resulting waveform is a triangular pulse. One of the functions (in this case g) is first reflected about τ = 0 and then offset by t …   Wikipedia

  • Convolution for optical broad-beam responses in scattering media — Photon transport theories, such as the Monte Carlo method, are commonly used to model light propagation in tissue. The responses to a pencil beam incident on a scattering medium are referred to as Green’s functions or impulse responses. Photon… …   Wikipedia

  • Power series — In mathematics, a power series (in one variable) is an infinite series of the form:f(x) = sum {n=0}^infty a n left( x c ight)^n = a 0 + a 1 (x c)^1 + a 2 (x c)^2 + a 3 (x c)^3 + cdotswhere an represents the coefficient of the n th term, c is a… …   Wikipedia

  • Formal power series — In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful, especially in combinatorics, for… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Arithmetic function — In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers (i.e. positive integers) that expresses some arithmetical property of n. [1] An example of an arithmetic… …   Wikipedia

  • Schönhage-Strassen algorithm — The Schönhage Strassen algorithm is an asymptotically fast multiplication algorithm for large integers. It was developed by Arnold Schönhage and Volker Strassen in 1971. [A. Schönhage and V. Strassen, Schnelle Multiplikation großer Zahlen ,… …   Wikipedia

  • Fast Fourier transform — A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex number arithmetic to group …   Wikipedia

  • Overlap–add method — The overlap–add method (OA, OLA) is an efficient way to evaluate the discrete convolution of a very long signal x[n] with a finite impulse response (FIR) filter h[n]: where h[m]=0 for m outside the region [1, M]. The concept is to divide the… …   Wikipedia

  • Discrete Fourier transform — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms In mathematics, the discrete Fourier transform (DFT) is a specific kind of discrete transform, used in… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”