Fourier series - Understanding an equality

Question

Why is this equality true: $$\left\langle {f,g} \right\rangle = \sum\limits_{n = - N}^N {\hat{f}(n)\hat{g}(n)}$$

where $$f = \sum_{n=-N}^N c_n e^{int}, g=\sum_{n=-N}^N d_n e^{int} $$

and $\hat{f},\hat{g}$ are the Fourier coefficients

Answer 1

Assuming that $\langle f,g\rangle$ is defined as $(2\pi)^{-1}\int_0^{2\pi}f(x)\overline{g(x)}\mathrm dx$, the wanted formula follows from the fact that $\langle e_n,e_n\rangle=1$ and $\langle e_n,e_m\rangle=0$ if $n\neq m$ (where $e_n(x):=e^{inx}$).