Suppose we are given an arbitrary two-sided sequence of complex numbers, say ${a_n}$. After forming the formal Fourier series $\sum_{n=-\infty} ^{\infty} a_n e^{inx}=a_0 + \sum_{n=1} ^{\infty} (a_n e^{inx} + a_{-n} e^{-inx} )$, suppose the series converges to zero for all x in $\mathbb{R}$. Then can we say that every ${a_n}$ is actually zero?
Here, the convergence is pointwise.
If there is a counterexample for the above question, can we impose some conditions on the prescribed sequence ${a_n}$ (e.g. it is in $l^2$) so that the abovementioned property holds?
Best Answer
This a uniqueness result due to Cantor. You can find a proof in this article by J. Marshal Ash.