Editing Convolution (section)

=== Algebraic properties ===
{{See also|Convolution algebra}}
The convolution defines a product on the [[linear space]] of integrable functions. This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative [[associative algebra]] without [[identity element|identity]] {{harv|Strichartz|1994|loc=§3.3}}. Other linear spaces of functions, such as the space of continuous functions of compact support, are [[closure (mathematics)|closed]] under the convolution, and so also form commutative associative algebras.

; [[Commutativity]]: <math display="block">f * g = g * f </math> Proof: By definition: <math display="block">(f * g)(t) = \int^\infty_{-\infty} f(\tau)g(t - \tau)\, d\tau</math> Changing the variable of integration to <math>u = t - \tau</math> the result follows.

; [[Associativity]]: <math display="block">f * (g * h) = (f * g) * h</math> Proof: This follows from using [[Fubini's theorem]] (i.e., double integrals can be evaluated as iterated integrals in either order).

; [[Distributivity]]: <math display="block">f * (g + h) = (f * g) + (f * h)</math> Proof: This follows from linearity of the integral.

; Associativity with scalar multiplication: <math display="block">a (f * g) = (a f) * g</math> for any real (or complex) number <math>a</math>.

; [[Multiplicative identity]]: No algebra of functions possesses an identity for the convolution. The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a [[Dirac delta|delta distribution]] (a unitary impulse, centered at zero) or, at the very least (as is the case of ''L''<sup>1</sup>) admit [[Nascent delta function|approximations to the identity]]. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically, <math display="block">f * \delta = f</math> where ''δ'' is the delta distribution.

; Inverse element: Some distributions ''S'' have an [[inverse element]] ''S''<sup>−1</sup> for the convolution which then must satisfy <math display="block">S^{-1} * S = \delta</math> from which an explicit formula for ''S''<sup>−1</sup> may be obtained.{{paragraph}}The set of invertible distributions forms an [[abelian group]] under the convolution.

; Complex conjugation: <math display="block">\overline{f * g} = \overline{f} * \overline{g}</math>

; Time reversal:  If &nbsp;<math>q(t) = r(t)*s(t),</math>&nbsp; then &nbsp;<math>q(-t) = r(-t)*s(-t).</math>

<blockquote>
Proof (using [[convolution theorem]]):

<math>q(t) \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ Q(f) = R(f)S(f)</math>

<math>q(-t) \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ Q(-f) = R(-f)S(-f)</math>

<math>
\begin{align}
q(-t) &= \mathcal{F}^{-1}\bigg\{R(-f)S(-f)\bigg\}\\
&= \mathcal{F}^{-1}\bigg\{R (-f)\bigg\} * \mathcal{F}^{-1}\bigg\{S(-f)\bigg\}\\
&= r(-t) * s(-t)
\end{align}
</math>
</blockquote>
; Relationship with differentiation: <math display="block">(f * g)' = f' * g = f * g'</math> Proof:
:<math>
\begin{align}
(f * g)' & = \frac{d}{dt} \int^\infty_{-\infty} f(\tau) g(t - \tau) \, d\tau \\
& =\int^\infty_{-\infty} f(\tau) \frac{\partial}{\partial t} g(t - \tau) \, d\tau \\
& =\int^\infty_{-\infty} f(\tau) g'(t - \tau) \, d\tau = f* g'.
\end{align}
</math>

; Relationship with integration: If <math display="inline">F(t) = \int^t_{-\infty} f(\tau) d\tau,</math> and <math display="inline">G(t) = \int^t_{-\infty} g(\tau) \, d\tau,</math> then <math display="block">(F * g)(t) = (f * G)(t) = \int^t_{-\infty}(f * g)(\tau)\,d\tau.</math>