Consider the following Fourier series
$ f\left(t\right)=\sum_{k=1}^{N}c_{k}e^{i\frac{2\pi}{T}kt} $
Can we prove that the function which its Fourier series is $ f $ is a complex function due to the fact that its Fourier series only contains positive indices?
Also, are there other cool properties of a function that we can tell just from looking at its Fourier series? I know that for real functions we can tell if the function is odd/even by looking at the Fourier series in the form of the $ \sin + \cos $ orthogonal system. Are there any other properties?
Thanks in advance.
If $f(t)$ is real valued then $$\sum_{k=1}^{N}c_{k}e^{i\frac{2\pi}{T}kt} =\sum_{k=1}^{N}c_{k}e^{-i\frac{2\pi}{T}kt} $$ and orthogonality of the functiosn $e^{i\frac{2\pi}{T}kt}$, $k \in \mathbb Z$ on $[0,T]$ show that each $c_k$ must be $0$.