Take the 2-minute tour ×  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 only
on its values at the positive integers?   Note: I am not asking
about the moment problem and the