Suppose that I am told that f(x) is some elementary function and that f(x) has the fourier series $\Sigma_{k=-\infty}^{\infty}c_ke^{ikx}$.
By "elementary function" I mean: https://en.wikipedia.org/wiki/Elementary_function.
I don't know the closed form of f, by which I mean that although I know that f can be written as a finite composition of of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations, I don't know how to actually do it.
Is there any way (maybe given certain extra conditions) that I can find the closed form of f from knowing its fourier series?
In other words: Given that I know all the complex numbers $c_k$, $k\in\mathbb{Z}$, and that I know that $\forall k \in \mathbb{Z}: c_k = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) e^{-ikx}dx$, can I then find the closed form f?
Thank you for your time.
The way elementary functions work, they essentially come with an enumeration. So one way would be to iterate through the elementary functions in order of increasing complexity, compute their Fourier series, and check if it matches yours. If your function really is elementary, this will eventually terminate, at least assuming that you have an effective way of comparing two sequences of Fourier coefficients.
I'd doubt there's anything that works better than this in general, since Fourier series aren't particularly well-behaved under arbitrary function composition. There may be a method that works better for a class of elementary functions you're more likely to encounter "in nature," though.