Fourier tranform of the Euclidean norm

Question

where can I find the Fourier transform of the power of the Euclidean norm?, that is:

$$\mathcal{F}[\|x\|^{p}](\omega) = \int_{\mathbb{R}^{d}}\exp(-2\pi i \langle\omega, x\rangle) \|x\|^{p} dx$$

Thank you!

Answer 1

In vol. 1, Chap.II, Sec.3.3 of the book Generalized functions by Gelfand-Shilov you will find the formula

$$ F\bigl[\Vert x\Vert^\lambda\bigr](\omega)=2^{\lambda+d}\pi^{\frac{n}{2}}\frac{\Gamma\left(\frac{\lambda+d}{2}\right)}{\Gamma\left(-\frac{\lambda}{2}\right)} \Vert\omega\Vert^{-\lambda-d}, $$

or all $\lambda\neq -d,-d-2,\dotsc $, where, for $\lambda>0$ you should think of $\Vert\omega\Vert^{-\lambda-d}$ as a generalized function in the precise sense described in Chap.I. Sec. 3.9 op.cit.