Regarding Fourier transform

Question

Let $f\in L^1(\mathbb{T})$. Where $\mathbb{T}$ is the unite cirle in the complex plane. Define $$g(e^{it})=f(e^{2it}).$$ Show that $\hat{g}(n)= \hat{f}(n/2)$ if 2 divides $n$ and $\hat{g}(n)= 0$ if 2 does not divide $n$. I understand how the first part comes clearly, can anyone help me with the second part?

Answer 1

You have to show that $\int_{-\pi }^{\pi} e^{-int} f(e^{2it}) \, dt =0$ if $n$ is odd. Just make the substitution $s=t+\pi $ ad see what you get.

Answer 2

This should be intuitively clear: If $g$ is only made up of even frequencies, then it should be orthogonal to the odd frequencies.

Suggestion: What happens if $f$ is a polynomial? What do you know about the subset of all polynomials inside $L^1(\mathbb{T})$?