## Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?

PROBLEM. Let $$\theta(t)$$ and $$\phi(t)$$ be two real analytic non-constant functions $$[0,2\pi]\rightarrow \mathbb{R}$$. I am trying to prove the following claim

If the integral $$\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$$ for all $$n\in\mathbb{N}_0$$ than the first derivative $$\theta’$$ and $$\phi$$ are periodic of common period $$2\pi/l$$ with $$1\neq l\in\mathbb{N}$$.

Note that this is equivalent to $$F(\lambda):=\int_0^{2\pi} e^{i(\theta(t)+\lambda\phi(t))} dt=0$$ for all $$\lambda \in \mathbb{R}$$. In fact, $$F(\lambda)$$ is analytic in $$\lambda$$ and its being constantly equal to 0 is equivalent to the vanishing of all its derivatives $$F^{(n)}(0)=\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt$$. Geometrically this means that the curve obtained by integrating the (tangent) vector function $$(\cos(\theta+\lambda\phi),\sin(\theta+\lambda\phi))$$ over $$[0,2\pi]$$ is closed $$\forall \lambda$$.

Just in case, a back-up less general claim for which I would like to see a clean solution is

If, in the hypotesis above, $$\phi$$ is a polynomial, then $$\phi$$ is constantly $$0$$.

OBSERVATION. If $$\theta’$$ and $$\phi$$ are periodic of common period $$\frac{2\pi}{l}$$ with $$1\neq l \in \mathbb{N}$$ and $$\int_0^{\frac{2\pi}{l}} e^{i\theta}\neq 0$$ then the converse implication is true. In fact, in this setting $$\theta=c\cdot t+\theta_p(t)$$ with $$c=\frac{2\pi}{l}(\theta(\frac{2\pi}{l})-\theta(0))$$ and $$\theta_p$$ periodic of period $$\frac{2\pi}{l}$$. Then \begin{align} \int_0^{2\pi} e^{i(\theta(t)+\lambda\phi(t))} dt &=& \sum_{j=0}^{l-1} \int_{j \frac{2\pi}{l}}^{(j+1) \frac{2\pi}{l}} e^{i(c\cdot t+\theta_p(t)+\lambda\phi(t))} dt \ &=& \sum_{j=0}^{l-1} e^{i\cdot j \cdot \frac{2\pi}{l}} \int_{0}^{\frac{2\pi}{l}} e^{i(c\cdot t+\theta_p(t)+\lambda\phi(t))} dt, \end{align} where the last equality is obtained by repetedly applying the substitution $$t’=t-\frac{2\pi}{l}$$. Since we know $$\sum_{j=0}^{l-1} e^{i\cdot j \cdot \frac{2\pi}{l}} \int_{0}^{\frac{2\pi}{l}} e^{i\theta(t)}dt=\int_0^{2\pi} e^{i\theta(t)} dt=0$$ then also the integral above must be $$0$$. In the following picture the curve associated to $$\theta(t)=t + \cos( 12 t)$$ deformated in the direction $$\cos(3 t)$$. In this case $$l=3$$ and the curve is closed $$\forall \lambda$$.

$\theta(t)=t + \cos( 12 t)$ deformated in the direction $$\cos( 3 t)$$. In this case $$l=3$$ and the curve is closed $$\forall \lambda$$.”>

IDEA. If $$\theta$$ monotone one can substitute $$s=\theta(t)$$ in the integral and get $$\int_{\theta(0)}^{\theta(2\pi)} e^{i s} \frac{(\phi(\theta^{-1}(s)))^n}{\theta'(\theta^{-1}(s))} ds=0.$$ In this case the idea behind the hypotesis becomes apperent: $$\phi(\theta^{-1}(s))$$ is periodic of non-trivial period iff $$\phi$$ and $$\theta’$$ have the common period property. It seems here that looking at the Fourier expansion of our functions on $$[\theta(0),\theta(2\pi)]$$ could be a good idea: the condition we have means indeed that, $$\forall n$$, the first harmonic of the function $$\frac{(\phi(\theta^{-1}(s)))^n}{\theta'(\theta^{-1}(s))}$$ is $$0$$. Fourier coefficients of a product are obtained by convolutions and therefore the condition above becomes, $$\forall n$$: $$\sum_{k_n=-\infty}^{+\infty} \sum_{k_{n-1}} … \sum_{k_{2}}\sum_{k_{1}} \widehat{\frac{1}{\theta’}}(1-\sum_{i=1}^{n} k_i) \prod_{i=1}^{n} \widehat{\phi}(k_i)=0.$$ Is this approach viable? Can one from here exploit the fact that a function is periodic of non-trivial period iff there exists $$k$$ such that only harmonics multiple of $$k$$ are different from 0? Other way round, do non-zero harmonics of coprime orders imply a contradiction with our constraints? As for a toy example, if $$\theta(t)=t$$,$$\theta'(s)=1$$ and $$\phi(s)=\cos(2s)+\cos(3s)$$ already $$\widehat{f^2}(1)= 2 \widehat{f}(3)\widehat{f}(-2) \neq 0$$; in the general setting interaction of coefficients is not straightforward.

NOTE: This question originated from Orthogonality relation in $L^2$ implying periodicity. As suggested in the comments to the previous post, since the target of the question changed over time and edits were major, here I hope I gave a clearer and more consistent presentation of my problem.