I have two functions $y(\theta)$ and $x(\theta)$ that are related by the following non-linear relationship:
\begin{equation} y(\theta) = \begin{cases} \left(x(\theta)\right)^n & \text{for $x(\theta) \ge 0$} \\ 0 & \text{for $x(\theta) < 0$} \end{cases} \end{equation}
where $n$ is any positive real number. (The particular context I am looking at is based on Hertzian contact mechanics of rolling element bearings, where $n = 3/2$ for a ball bearing, and $n=10/9$ for a roller bearing) If it facilitates discussion, I am only really interested in the case of $1 < n < 2$.
$x(\theta)$ is a periodic function of period $2\pi$. Therefore, we can represent it as a Fourier series:
\begin{equation} x(\theta) = \sum_{-J \le j \le J} x_j \,\mathrm{e}^{\mathrm{i}j\theta} \end{equation}
where $J$ is some integer, assuming that the Fourier series of this finite number of terms accurately/exactly represents $x(\theta)$.
Since $x(\theta + 2\pi) = x(\theta)$, it can be seen that $y(\theta + 2\pi) = y(\theta)$, and therefore $y(\theta)$ must also be $2\pi$-periodic. This then implies that $y(\theta)$ has its own Fourier series:
\begin{equation} y(\theta) = \sum_{-K \le k \le K} y_k \,\mathrm{e}^{\mathrm{i}k\theta} \end{equation}
(I get the impression that $K$ might need to be infinite for non-integer $n$, but I am happy to settle with an approximation. i.e. choose an integer $J=K$ that is large enough to do the job.)
This then naturally gives rise the following question: does there exist a practical relationship between the sets of Fourier coefficients $\{x_j\}$ and $\{y_k\}$? That is, if I'm given the Fourier coefficients for $x(\theta)$, can I directly calculate the Fourier coefficients for $y(\theta)$?
I am attemping to implement this in MATLAB, so it is ideal for me to represent the input function $x(\theta)$ using a finite set of Fourier coefficients. So far, my approach is to discretely sample $x(\theta)$ over a uniformly distributed set of values $\theta_p$ within $0 \le \theta < 2\pi$, getting $\{x(\theta_p)\}$ (I think this could be done using FFT), and then using the non-linear relationship at each value $\theta_p$ to determine $\{y(\theta_p)\}$, and then using the FFT to obtain the Fourier coefficients.
Does there exist a more direct/computationally faster approach that this? I get the feeling that there probably isn't, given the severe non-linearity due to the relationship being piecewise, but perhaps this sort of problem has been analysed before, especially if $n$ is considered to be only slightly larger than 1?