I found this as an exercise in Kolmogorov-Fomin, let me state the problem precisely:
Let $S_{\infty}$ be the space of functions on the real line for which there exists a colelction of constants $C_{pq}$ for all $p, q$ such that $$ |x^pf^{(q)}(x)|\leq C_{pq}. $$ Is it true that if for all $p\geq 0$ $\int_{-\infty}^{\infty} x^pf(x)=0$, then $f=0$?
The exercise follows a section on the Fourier transform of functions of $S_{\infty}$. Now, I guess one should prove this exercise (I am fairly sure it is true) using some properties of the Fourier transform, but I cannot see how.
So my question is, how is this to be proved with the fundamental properties of the Fourier transform in $S_{\infty}$?