Transform graphs to memorize:

boxcar <=> sinc

Gaussian <=> Gaussian

sine <=> positive and negative spikes

cosine <=> pair of spikes with same sign

squarewave <=> decaying series of odd order harmonic spikes

closely spaced spikes <=> widely spaced spikes

triangle <=> sinc**2

Hilbert operator

These are the things you have to deal with the most.. Other transforms popup from time to time, but generally you can get a pretty close approximation from some combination of these. So if you're discussing it in the cafeteria or a bar after work you can convey what is going on.

I have The Fourier Transform and Its Application" 2nd ed. I'm pretty sure they all have the dictionary of transforms. It's one of the reasons it was so popular. It's mathematically rigorous, though not at the level of "Operational Mathematics" by Churchill. The latter consumed much of my life for two semesters.

Book prices have gotten crazy. Publishers like to push out new editions even though they are not justified in order to obsolete older editions. But it makes the old edition cheap. If you're building a personal library that's the way to go.

As 720 is not prime, there are FFTs which exist. However, the most common transforms are the radix 2 transforms popularized by Cooley and Tukey and those are restricted to powers of 2 lengths. Prime factor routines will handle 720. My favorite is an algorithm attributed to Glassmann. The FORTAN version is only two pages with lots of white space. Probably the best is Dave Hale's prime factor algorithm which uses a series of tests to select the fastest transform larger than the input series.

I don't mean to beat up on anyone, but the proper way to address the coarse sampling in the frequency domain is to pad the end of the series with zeros. Adding 19,456 zeros will give sub Hertz resolution in the frequency domain. That will also make clear the implicit sinc function imposed by truncating the series to 1024 values.

I did 3/4ths of the problem because Simon was being misled. both by the person who prepared the spreadsheet and forum members who don't quite know as much as they think they do.

I ran an "orphan home for lost problems" at large oil companies. My tools of the trade for a lot of it were Octave, gnuplot, awk and CWP/SU. The latter is a seismic processing package. I supported it for many years but got weary of John's renaming my contributions. I thought authors were entitled to choose the names of their own programs. It's somewhat buggy, but it's the best of the lot. Madagascar has potential if Sergey ever gets after documenting and debugging it. The function fit routine in gnuplot is the best Marquardt-Levenberg L2 solver I've ever used. If you know what you're doing it will fit things nothing else will short of a sparse L1 pursuit. The latter being the current sate of the art method for a vast array of problems. However, it makes the traditional Fourier-Wiener-Shannon math look simple.