On Tue, 22 Dec 2015, Fred Lunnon wrote:
It would be helpful to the non-analysts amongst us to indicate where this argument might be explained in greater detail --- though I have a nasty feeling that considerable expansion may be required! WFL
For the modern part of it, Paul Garrett's web site, especially the writeups from his courses. The Friedrichs extension idea is incredible, I think. The Fejer kernel argument that the oscillations are a basis of L^2(\R/\Z) should be in any respectable undergraduate text on Fourier analysis. For the complex analytic arguments being made in this thread, a \Z-periodic function on the complex upper half plane is analytic on the punctured disk under the change of variables q=e^{2\pi iz}, and so it has a representation \sum_{n\in\Z}a_n q^n, though the representation needn't be a power series or even a Laurent series. But this is a much smaller setting than L^2.