Point of order, jp -

Edited quote ----place a source at one focal point, the (spacial) Fourier transform of the[/quote]

spacial arrangement of the source will show up at the other focal plane.

People may be confused, if they are thinking about frequencies and time functions of light waves etc. The time / frequency thing has a parallel in geometry but all the sums are the same. Spacial frequency refers to ' lines per inch' or, more accurately, 'sinewave stripes per inch'.

Put a point at one focus and you get a 'zone plate' image at the other - the well known sin x over x pattern.

The Fourier transform, in fact, should be quoted in its complex form, to be accurate - this means that, rather than just building up an arbitrary waveform with sinewaves, you also need a corresponding set of cosine waves, too, or phase information about your sinewaves. If you don't have the cosines, too or the phases , you cannot get the shape right. It doesn't matter much for audio waveforms, which get so dispersed that the ear more or less ignores phase structure information in order to make sense of what it hears but edges and lines in pictures get distorted, in particular.

Have just read your post s-s.

NO - Fourier Analysis is really what you are describing. The Fourier Transform, in general, is defined f or ANY F(t). When the waveform repeats, the transform gives you discrete harmonics and also allows you to do DST and FFT. But it is a sub-set of the Fourier thing.

Wiki is full of it but it is a bit over the top for non-maths people. But the whole thing is a bit of a mouthful and relies on integral calculus.

To relate to the 'real world' , you have to be careful with FTs and resort to 'windowing' and other 'fudges'. but this is just like a lot of the Maths which is used in Science.

You even have to be careful when solving a quadratic equation, after all. . . .