Editing Convolution (section)

=== Translational equivariance ===
The convolution commutes with translations, meaning that
: <math>\tau_x (f * g) = (\tau_x f) * g = f * (\tau_x g)</math>

where τ<sub>''x''</sub>f is the translation of the function ''f'' by ''x'' defined by
: <math>(\tau_x f)(y) = f(y - x).</math>

If ''f'' is a [[Schwartz function]], then ''τ<sub>x</sub>f'' is the convolution with a translated Dirac delta function ''τ''<sub>''x''</sub>''f'' = ''f'' ∗ ''τ''<sub>''x''</sub> ''δ''. So translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution.

Furthermore, under certain conditions, convolution is the most general translation invariant operation. Informally speaking, the following holds
: Suppose that ''S'' is a bounded [[linear operator]] acting on functions which commutes with translations: ''S''(''τ<sub>x</sub>f'') = ''τ<sub>x</sub>''(''Sf'') for all ''x''. Then ''S'' is given as convolution with a function (or distribution) ''g''<sub>''S''</sub>; that is ''Sf'' = ''g''<sub>''S''</sub> ∗ ''f''.

Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study of [[time-invariant system]]s, and especially [[LTI system theory]]. The representing function ''g''<sub>''S''</sub> is the [[impulse response]] of the transformation ''S''.

A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that ''S'' must be a [[continuous linear operator]] with respect to the appropriate [[topology]]. It is known, for instance, that every continuous translation invariant continuous linear operator on ''L''<sup>1</sup> is the convolution with a finite [[Borel measure]]. More generally, every continuous translation invariant continuous linear operator on ''L''<sup>''p''</sup> for 1 ≤ ''p'' < ∞ is the convolution with a [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]] whose [[Fourier transform]] is bounded. To wit, they are all given by bounded [[Fourier multiplier]]s.