Completeness of $\{\sin nx \}$ over $(0, \pi)$

Question

This is stated as a miscellaneous theorem in the book 'Trigonometric series' by Zygmund.

Can anyone prove it? It is of use in quantum mechanics.

Answer 1

Start with $f \in L^2[0,\pi]$, and extend $f$ to an odd function $f_o$ on $L^2[-\pi,\pi]$. Now expand $f_o$ in a Fourier series on $L^2[-\pi,\pi]$. The resulting Fourier sin series converges in $L^2[-\pi,\pi]$ to $f_o$ and, hence, also converges in $L^2[0,\pi]$ to $f$.