Convolution Theorem - Proof

Proof

The proof here is shown for a particular normalisation of the Fourier transform. As mentioned above, if the transform is normalised differently, then constant scaling factors will appear in the derivation.

Let f, g belong to L1(Rn). Let be the Fourier transform of and be the Fourier transform of :

where the dot between x and ν indicates the inner product of Rn. Let be the convolution of and

Now notice that

Hence by Fubini's theorem we have that so its Fourier transform is defined by the integral formula

\begin{align} H(\nu) = \mathcal{F}\{h\} &= \int_{\mathbb{R}^n} h(z) e^{-2 \pi i z\cdot\nu}\, dz \\ &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(x) g(z-x)\, dx\, e^{-2 \pi i z\cdot \nu}\, dz. \end{align}

Observe that and hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration):

Substitute ; then, so:

These two integrals are the definitions of and, so:

QED.

Read more about this topic:  Convolution Theorem

Famous quotes containing the word proof:

    “O, popular applause! what heart of man
    Is proof against thy sweet, seducing charms?”
    —William Cowper (1731–1800)

    “The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
    —Andrew Michael Ramsay (1686–1743)

    “He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
    —Herman Melville (1819–1891)