For the inner product space $V=\mathcal{R}([-\pi,\pi];\mathbb{C})$, if the inner product is defined by $$ \langle f,g\rangle = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\overline{g(x)}dx, $$ and the $n$-th Fourier coefficient of $f$ is $$ \hat{f}(n) = \langle f,e^{int}\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int}dt. $$
Now suppose that $f:\mathbb{R}\rightarrow\mathbb{C}$ is a $2\pi$-periodic function. My question is, what is the relation between the inner product $\langle f,e^{-int}\rangle$ and $\hat{f}(n)$?
Any of your help will be highly appreciated :)