Im proving the following statement:
Let $\alpha>0$, the gaussian weight $\omega(z)= \frac{\alpha}{\pi}e^{-\alpha \left | z \right |^2}$ is the unique continuous, radial weight verifying $\int_\mathbb{C} \omega(z) dA(z)=1$ such that for polynomials p and q under the inner product $$ \left \langle p,q \right \rangle = \int_\mathbb{C} p(z)\overline{q(z)}\omega(z) dA(z) $$ the operations of multiplication for $\alpha z$ and differentiation are adjoint.
I supposed $\omega$ a weight verifying the conditions and got to te final equality, $\forall n \in \mathbb{N} \cup \{ 0 \}$ $$ \int_0^\infty r^{2n + 1} \omega (r) dr = \frac{\alpha}{\pi}\int_0^\infty r^{2n + 1} e^{-\alpha r^2} dr $$ I need to prove the following property in order to end the proof. Let $f : [0,+ \infty ) \rightarrow \mathbb{R}$ a continuous function that verifies $\forall n\in \mathbb{N} \cup \{0 \}$ the following equality: $$ \int_0^\infty r^{2n + 1} f (r) dr = 0 $$ Prove that $f$ is the zero function.