Series of $\sin(nx)$ terms that sum up to $0$.

Question

Working in $\mathbb{R}$, what sequences $a_n$ satisfy $\sum_{n=0}^{\infty} a_n \sin(nx)=0$ for all $x$, pointwise ?

I've never thought about this and I'm not sure whether I'm not missing something trivial - I would have expected that $a_n = 0$ for all $n$ is the only solution, but I'm not sure I've seen this stated anywhere.

Answer 1

Riemann's uniqueness theorem says that if a trigonometric series converges pointwise to $0$, then its coefficients are all equal to $0$. This was later generalized by Cantor, leading him to the development of set theory.

Answer 2

It may depend on the meaning of the summation (and, as a consequence, the speed of the decay of $a$).

To solve all convergence problems, we assume that the series converges normally on $\mathbb{R}$, ie $a \in \ell^1$.

Let then $f(x)=\sum_n{a_n\sin(nx)}$. $f$ is continuous.

I suggest that you compute $\int_0^{2\pi}{f(x)\sin(nx)\,dx}$ as a function of $a_n$ (using the normal convergence of the series).