[hpcjoe]

I gotta say I disagree. I got the space change bit quickly (piece-wise linear space into frequency space). Complex plane's aren't a crutch, really they are the firm theoretical basis in which to express the detailed discussion properly.

You could always just switch to the Fourier sine and cosine forms, and avoid all of the other theoretical basis baggage. Sort of like physics for poets (I'm a computational theoretical physicist by training), leaving out the more detailed derivations and background, for a more straightforward approach.

Moreover, the DFT is not the FT. In the limit as your sampled points get very large, it will approach FT. There's a great book covering lots of these things in depth[1], with a pragmatic as well as theoretical approach. I think I gave my daughter this one (math phd student) last year.

FTs aren't merely a change of basis, there is quite a bit more to them than that. For DFT you can look at the process as a sequence of operator applications, but in the FT case this becomes a continuous sequence. Hence the space bits.

[1] https://epubs.siam.org/doi/book/10.1137/1.9781611971514 I highly recommend this book.