Suppose that I have a nice differentiable (or maybe even analytic) monotonous function $\theta(x)\colon \lbrack -\pi, +\pi \rbrack \mapsto \lbrack -\pi, +\pi \rbrack$, $\theta(-\pi) = -\pi, \theta(+\pi) = +\pi$. I am calling function $\cos \theta(x)$ a "distorted" cosine: it basically has all the properties of a regular cosine, but the shape is distorted depending on function $\theta(x)$. My question is the following: is anything known about Fourier series of $\cos \theta(x)$? I'm not sure if this problem has specific name and I understand that it is a very broad question, but any reference or keywords would be useful. I'm mostly interested in figuring out whether such "distorted" cosine can be represented as a finite sum of Fourier harmonics. It feels like that it couldn't be represented this way, but I'm struggling to find a general explanation why so. I've got an example where it's pretty easy to show why Fourier series of $\cos{\theta(x)}$ is infinite, but I don't think that it generalizes well.
Thanks in advance for your help!