Reconstruction of a function from its moments

Question

The n-th moment of a real valued function $f$ is defined as: $m_n(f)=\int_{-\infty}^{+\infty}x^nf(x)dx$. I heard that a function $f$ is uniquely determined by its moments. I would be quite surprised if this worked for any $f$. Is it true? And in that case, does anyone know how to re-build $f$ from its moments?

Thanks

Answer 1

One example of nonuniqueness was found (I think) by Stieltjes in 1894: The function $$ f(x) = \cases{x^{-\ln(x)} \sin(2\pi \ln(x))& $x > 0$\cr 0 & otherwise} $$
has all its moments $0$.

EDIT: The proof of this is a nice exercise in integration and symmetry. Start with the change of variables $x = \exp(t)$...