I have a function with an irrational period and I know its values at integers. How do I calculate the Fourier series?

Question

As per the title, I have a function $f$. I know $f$ is periodic, and I can calculate $f(n)$ for any integer $n$. I also happen to know the period, but only because someone else calculated it, so I would also like to know how to calculate the period. And as previously stated, I would like to know how to calculate the Fourier series.

Answer 1

Let's say the period is $L$. When $L$ is irrational, the integer sequence $0,1,2\ldots$ will be equidistributed modulo $L$. For any Riemann integrable periodic function $g(x)$ whose period divides $L$, one will have

$$\frac{1}{L}\int_0^L g(x) dx = \lim_{N\to\infty} \frac1N \sum_{n=0}^{N-1} g(n)$$

In particular, you can choose $g(x)$ to be those of the form

$$f(x)\cos\left(\frac{2\pi k x}{L}\right)\quad\text{ or }\quad f(x)\sin\left(\frac{2\pi k x}{L}\right)\quad\text{ for } k \in \mathbb{N}$$ and compute the Fourier coefficients of $f(x)$ over $[0,L]$ by taking "average" of this sort of $g(x)$ over $\mathbb{N}$.