Let $f$ be an integrable real valued function defined on $[0,\infty)$. Let $$m_n=\int_0^\infty f(x)x^n \mathrm dx$$ be the $n^{th}$ moment, and suppose that all of these integrals converge absolutely. Are there conditions that can we impose on $f$ that would allow us to write $f$ **explicitly** in terms of its moments and certain simple functions.

This idea is similar to the fact that we can reconstruct sufficiently nice functions on $[0,1]$ from a sum of their Fourier coefficients and $e^{inx}$. Also, on $(-\infty,\infty)$ we can reconstruct $f$ from an integral over the real line.

The analogy for $[0,\infty)$ and $x^s$ is of course the Mellin transform, which has an inversion formula as a line integral in the complex plane.

My question is then: Can we impose nice enough (non-trivial) conditions on a class of functions so that we can invert the Mellin transform on the real line based

onlyon its values at the positive integers?

Thanks!

**Note:** I am **not** asking about the moment problem and the requirements for uniqueness. (Such as Carleman's Condition etc..)