I was studying Fourier analysis recently and I came to the following question:
Given a trigonometric series (Here I express it in the form of Fourier series, but it's not necessarily the Fourier series of some functions, don't be mesled!) $$\frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\cos > nx+b_n\sin nx),x\in \mathbb{R}$$ where $a_0,a_n,b_n\in\mathbb{R},\forall n\geq 1.$ If such a series converges to $0$ for any $x\in\mathbb{R}$ (just pointwise convergence), namely $$\frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\cos nx+b_n\sin nx)=0,\forall x\in\mathbb{R}\tag{1}$$ can we derive that $a_0=0$ and $a_n=b_n=0,\forall n\geq 1$ ?
I find it hard to handle this question because the convergence in $(1)$ is only pointwise, so neither the tool of integral nor differential can be applied. Moreover, the theorems I have studied in Fourier analysis only deal with the properties of the coefficients of Fourier series, so I cannot apply them, too. I also take some values of $x$ in $(1)$ ($x=k\pi,2k\pi,k\pi+\frac{1}{2}\pi, e.t.c$) but I only get some equations of power series and cannot solve it. Is there anyone who can help me or give me some hints?