We define a function $T:[0,1)\rightarrow [0,1)$ s.t $T(x)=2x$ if $x\in [0,\frac{1}{2})$ and $T(x)=2x-1$ if $x\in [\frac{1}{2},1)$. $T$ obviously has a period of $\frac{1}{2}$. Now we look at $f:\mathbb{R}\rightarrow\mathbb{C}$ 1-periodic and integrable function, and we want to find the relation between its Fourier coefficients and the Fourier coefficients of $f\circ T$.
My go was: $2\hat f\circ T(k)=2\int^{\frac{1}{2}}_0f(2x)e^{-4\pi ikx}dx+2\int^{\frac{1}{2}}_0f(2x-1)e^{-4\pi ikx}dx$. By change of variables $u=2x,t=2x-1$, I then get:
$$2\hat f\circ T(k)=\int^{1}_0f(u)e^{-2\pi iku}du+\int^{1}_0f(t)e^{-2\pi ik(t+1)}dt=\hat f(k)+e^{-2\pi it}\hat f(k)=2\hat f(k).$$
Is it possible for two different functions to have the same Fourier coefficients? (The goal is to prove that such $T$ is ergodic).