# A density question

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 $$\Omega$$?

• Smoothness of $f$ and $g$ does not guarantee that the "moments" $\int f(x_1,x_2)x_1^nx_2^mdx_1dx_2$ exist. Apr 17, 2020 at 15:06
• The set is bounded here, Jochen Wengenroth Apr 17, 2020 at 16:19

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.