I'm looking to find the inverse of
$$g(n)= \int_{-\infty}^\infty f(x)x^n dx$$
I'm not sure if an inverse exists but i suspect the kernel is well behaved enough to have an inverse.If an inverse exists I'd like to know how I'd go about inverting this, I am only concerned with real functions here, $f(x)$ is real. $g(n)$ is also known for all real $n$.
I looked up fredholm integral equation but I'm not sure how I'd go about applying that here since i need a kernel something like.
$$(1-x)^n$$