Sorry Tony, but krw is correct on this one.

Fourier postulated that every periodic wave could be represented by the sum of a series of sinusoidal waves.

So far as I know, the vice versa isn't true. You can't represent a sinusoidal wave as the sum of a series of square waves or saw tooth waves or any other kind of wave (spherical wave) for that matter..

A Pictorial Introduction to Fourier Analysis/Synthesis

If that were the case, then anyone explaining Fourier Synthesis would make the point loud and clear that you can represent any periodic wave as the sum of a series of ANY KIND of wave, square, sinusoidal, sawtooth, take your pick. But they don't say that. They say that you can represent any periodic wave as the sum of a series of sinusoidal waves, and they stop there. So, the idea that a sine wave can be represented by the sum of a series of square waves just doesn't work for me.

That's because in Fourier synthesis, there is a mathematical relationship between the waves comprising the series; you just can't hobble together various sine waves of varying amplitudes and frequencies to get something that fits. There has to be a mathematical expression by which one can determine what each wave in the series to be summed should be.