Let $F\colon \mathbb R \to \mathbb C$ be a complex-valued smooth function of period $2 \pi$ that we consider as a function on the unit circle.I have the feel that the following statement is true but my proof may have some glitches.
Claim: If a diffeo $\alpha \colon \mathbb R \to \mathbb R$ makes $F(\alpha(x))$ a trigonometric polynomial then it must be a linear transformation.
Proof:If $P(x) = F(\alpha(x)) = \sum_{|n| \leq N} p_n \exp(inx)$ is a trigonometric polynomial of degree $N$, then also $P'(x) = \alpha'(x) F(\alpha(x))$ is, and still of degree $N$.
Hence by comparison $\alpha'(x)$ must be constant.
Is there a glitch somewhere in the degree comparison?
And a follow-up question:What can one say about $F(\alpha)$ being a trigonometric polynomial in the new variable$\alpha = \alpha(x)$?
