Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that
$$ \int\_\Omega \left(f(x\_1,x\_2)- \frac{m}{(n+1)}g(x\_1,x\_2)\right) x\_1^n \,x\_2^m \,dx\_1\,dx\_2 = 0 \quad \text{for all $n,m=0,1,\ldots$}.$$


Does it follow that $f \equiv g \equiv 0$ on $\Omega$?



A density question
Certainly not. Assume $\partial h/\partial x\_1=g/x\_1$, and assume that $h$ and $g$ have compact support. Then integrate by parts to obtain
$$\int\int (f-x\_2{\partial h\over\partial x\_2})x\_1^nx\_2^m=0.$$
This implies $f=x\_2(\partial h/\partial x\_2)$ and nothing more.


