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It is easy to show that in L^2 space if $f(x)$ is written as:

$$f(x)=\sum_{n=-\infty}^{\infty} c_{n} e^{i n x}$$

Then the basis exponential functions are orthogonal and thus the co-efficients can be expressed as:

$$c_{n}=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(x) e^{-i n x} d x$$

In the textbook being read this is shown as the proof that the functions are orthogonal and span the space. However the above does not prove that these set of functions spans the L^2 space, it just shows they are an orthonormal set. How can one prove that they will also space the space?

David C. Ullrich
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Rahul Deora
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  • What book says that? – David C. Ullrich Apr 24 '21 at 17:05
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    In general the proof of completeness is significantly more involved than the proof of orthonormality. A decent elementary reference should either prove it or make it clear that they're skipping it for brevity (the latter is the treatment I got in undergrad). Anyway, you can find one proof here: https://math.stackexchange.com/questions/316235/proving-that-the-fourier-basis-is-complete-for-cr-2-pi-c-with-l2-norm – Ian Apr 24 '21 at 17:08
  • Do you know much about complex analysis? – Disintegrating By Parts May 05 '21 at 15:34
  • Not that much @DisintegratingByParts – Rahul Deora May 06 '21 at 06:12
  • There is a conservation law at work, which equates a quantity at infinity with something along the real axis. You can approach this through Complex Analysis, or through Harmonic functions, which are essentially the same argument. Any other approach requires a lot of extra detail that is not especially general. – Disintegrating By Parts May 06 '21 at 07:02

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