Consider the sampled sine function, $f(n)=\sin(\omega n)$, where $n$ is an integer.
If $\omega_2 = 3\pi/2$, does there exist an $0 \leq \omega_1 \leq \pi$ such that $\sin(\omega_1 n)=\sin(\omega_2 n)$ for all $n$?
Note the answer is yes for the cosine function - that is, $\cos(3\pi/2n)=\cos(\pi/2n)$ identically.
I think for sine, there does not exist $\omega_1$, but have not been able to prove it or find a counterexample.
Not. If it were possible, taking $n=1$ we have $\sin(w_1)=\sin(\frac{3 \pi}{2})$ for some $w_1\in[0,\pi]$. But $\sin(x)\ge0$ for all $x\in[0,\pi]$, and $\sin(\frac{3 \pi}{2})=-1$.