Originally Posted by

**Hartlw** That wraps it up. Thanks, very nice.

I note belatedly that f is given by you in post #3, which also invokes Parseval's theorem.

The theorem that if ($\displaystyle \widehat{f}$, $\displaystyle \widehat{\phi}_n$) = 0 for all n and some $\displaystyle \widehat{f}$ not zero implies $\displaystyle \widehat{\phi}_n$ is incomplete, might help the novice.

Finally, I found many versions of Parsevals theorem, with various (formula, theorem, identity) interchanged names, including for Fourier Series and one that states the Fourier transform is bijective, but none with your version. Could you please give a source for your version, preferably internet?