Fourier series with positive indices defines a complex function?

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?

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 $$0$$.
• Take inner product on both sides with $e^{i\frac {2pi} T mt}$ for any particuar $m$ (with $|m| \leq N$) . All terms except one will vanish and you get $c_m=0$. @FreeZe Jul 12, 2021 at 23:14