I'm reading Stein&Shakarchi's Fourier Analysis: An Introduction. Section 2 of Chapter 2 has the title of "Uniqueness of Fourier series". However, after thinking about the results given there, it seems to me this title is misleading. Because the aim of this section is to establish, in a rough sense, the following
If $f$ and $g$ have the same Fourier coefficients, then $f$ and $g$ are necessarily equal.
So the title should be "Uniqueness of limit functions of Fourier series", rather than the original.
I also did some searching, and it turns out, by "uniqueness of trigonometric series", the correct question to ask is
Can a function $f$ be represented by more than one trigonometric series $\sum_{n=-\infty}^{\infty}c_n e^{int}$, where $c_n$ are complex numbers?
Or equivalently,
Can a trigonometric series $\sum_{n=-\infty}^{\infty}c_n e^{int}$, where $c_n\neq 0$ for some $n\in \mathbb{Z}$, converge to the zero function?
The two problems seem to be exactly opposite to me. Is my understanding correct? Thanks in advance.