Let $\vec{w}, \vec{a} , \vec{x} \in \mathbb{R}^n$. Now we look at say the integral,
$\int e^{2 \pi i \vec{w}.\vec{x} } (\vec{a}.\vec{x})^p d \vec{x}$ for some $p \in \mathbb{R}$. Feel free to assume that $p$ is a positive (or a negative) integer if that helps!
I want to know if there is any meaningful sense that can be given to this integral.
For example if $\vec{x} \perp \vec{a}$ then this integral is $0$. I am wondering if there is a succinct way this property is seen by this Fourier transform.