Notes on the Fourier Transform

The Fourier series is a great tool for analyzing periodic functions. But what about functions that don’t repeat? We’ve seen that we can compute Fourier series for a non-periodic function defined on a finite interval, as long as we don’t care about its behavior beyond that interval. Let’s extend this idea to functions that never repeat; that is, non-periodic functions defined on the interval . Visualizing Fourier series for non-repeating functions To motivate the subject ahead, let’s look back at the example used in the earlier post about Fourier series: With an odd extension into . In that post, to make the Fourier series work, we assumed keeps repeating with a period on the entire axis. Here, let’s face the reality that it does not - in fact - repeat, and observe how our Fourier series work out. Recall that the Fourier series approximating are the sine series (since it’s an odd function): The following visualization is interactive. By default, it shows (with its odd extension) and no…