So here we are. We have finally made it to a place where we can transition with confidence from the classical continuous Fourier transform to the discrete version, which is the foundation for applications of Fourier analysis to programming. Indeed, we are quite close to unfurling the might of the Fast Fourier Transform algorithm, which efficiently computes the discrete Fourier transform. But because of its focus on algorithmic techniques, we will save it for a main content post and instead focus here on the intuitive connections between the discrete and continuous realms.

The goal has roughly three parts:

- Find a reasonable discrete approximation to a continuous function.
- Find a reasonable discrete approximation to the Fourier transform of a continuous function.
- Find a way to transition between the two discrete representations.

We should also note that there will be some notational clashes in the sequel. Rigorously, item 3 will result in…

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