English: Fourier transform of the rect function and sinc functions.

The continuous Fourier transform takes an input function f(x) in the time domain and turns it into a new function, ƒ̂(x) in the frequency domain.

In the first part of the animation, the Fourier transform (as usually defined in signal processing) is applied to the rectangular function, returning the normalized sinc function.

In the second part, the transform is reapplied to the normalized sinc function, and we get our original rect function back.

It takes four iterations of the Fourier transform to get back to the original function.

However, in this particular example, and with this particular definition of the Fourier transform, the rect function and the sinc function are exact inverses of each other. Using other definitions would require four applications, as we would get a distorted rect and sinc function in the intermediate steps.

For simplicity, I opted for this so I don't have very tall and very wide intermediate functions, or the need for a very long animation. It doesn't really work visually, and the details can be easily extrapolated once the main idea gets across.

In this example, it also happens that there are no imaginary/sine components, so only the real/cosine components are displayed.

Shown at left, overlaid on the red time domain curve, there's a changing yellow curve. This is the approximation using the components extracted from the frequency domain "found" so far (the blue cosines sweeping the surface). The approximation is calculated by adding all the components, integrating along the entire surface, with the appropriate amplitude correction due to the specific Fourier transform and ranges used.