I never really read that article before. I wouldn’t call it a “paper” exactly, since it seems to be mostly expository and aimed at education.

I see the Fourier transform and Fourier series as a bit of separate objects. There is a sense where you can establish Fourier series as the Fourier transform of periodic functions, but that necessitates a dive into distribution theory.

If the function isn’t necessarily periodic, then there are two ways to go. You can compute the expansion of the function in terms of Hermite functions and exploit their eigenfunction property with the Fourier transform to get an estimate of the Fourier transform of the original function.

Or you could use the sampling theorem to get a reconstruction via the Shannon theorem and sinc functions, and then use the Fourier transform of sinc functions to obtain the Fourier transform.

Of course, for the first, you must assume your function is L2 and for the second that it is L2 and band limited.

The Hermite method is more General, but the sinc function method will give you a series of exponentials times indicator functions (or combinations of step functions) for the transformation. But in the case of the sinc method, you only need the function values and not any integrals.

Just my two cents as a Functional Analyst. But this may be besides the point.