Let $f(x,y) \in \mathcal{L}^2(\mathbb{R^2})$ be a probability density function (PDF).
Is there any way to expand $f(x,y)$ in terms of its marginals $\int f(x,y) dx, \int f(x,y)dy$ and its moments $\int x^n f(x,y) dx, \int y^n f(x,y) dy, \int x^n y^m f(x,y)$?
I know that knowledge of the marginals is not enough to recover the joint PDF but I don't know whether the additional knowledge of the moments can help. Thus, I was wondering if there is some analog of the Taylor expansion for joint PDFs in terms of its marginals and moments.