Let $f(x)$ be a real-valued function on $\mathbb{R}$ such that $x^nf(x), n=0,1,2,\ldots$ are Lebesgue integrable.
Suppose $$\int_{-\infty}^\infty x^n f(x) dx=0$$ for all $n=0,1,2,\ldots. $
Does it follow that $f(x)=0$ almost everywhere?
Let $f(x)$ be a real-valued function on $\mathbb{R}$ such that $x^nf(x), n=0,1,2,\ldots$ are Lebesgue integrable.
Suppose $$\int_{-\infty}^\infty x^n f(x) dx=0$$ for all $n=0,1,2,\ldots. $
Does it follow that $f(x)=0$ almost everywhere?