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:

    “When children feel good about themselves, it’s like a snowball rolling downhill. They are continually able to recognize and integrate new proof of their value as they grow and mature.”
    —Stephanie Martson (20th century)

    “a meek humble Man of modest sense,
    Who preaching peace does practice continence;
    Whose pious life’s a proof he does believe,
    Mysterious truths, which no Man can conceive.”
    —John Wilmot, 2d Earl Of Rochester (1647–1680)

    “The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
    —Lenny Bruce (1925–1966)